# List of space groups

There are 230 space groups in three dimensions, given by a number index, and a full name in Hermann–Mauguin notation, and a short name (international short symbol). The long names are given with spaces for readability. The groups each have a point groups of the unit cell.

## Symbols

In Hermann–Mauguin notation, space groups are named by a symbol combining the point group identifier with the uppercase letters describing the lattice type. Translations within the lattice in the form of screw axes and glide planes are also noted, giving a complete crystallographic space group.

These are the Bravais lattices in three dimensions:

• P primitive
• I body centered (from the German "Innenzentriert")
• F face centered (from the German "Flächenzentriert")
• A centered on A faces only
• B centered on B faces only
• C centered on C faces only
• R rhombohedral

A reflection plane m within the point groups can be replaced by a glide plane, labeled as a, b, or c depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, which is along a quarter of either a face or space diagonal of the unit cell. The d glide is often called the diamond glide plane as it features in the diamond structure.

• , , or glide translation along half the lattice vector of this face
• glide translation along with half a face diagonal
• glide planes with translation along a quarter of a face diagonal.
• two glides with the same glide plane and translation along two (different) half-lattice vectors.

A gyration point can be replaced by a screw axis is noted by a number, n, where the angle of rotation is . The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. For example, 21 is a 180° (twofold) rotation followed by a translation of ½ of the lattice vector. 31 is a 120° (threefold) rotation followed by a translation of ⅓ of the lattice vector.

The possible screw axis are: 21, 31, 32, 41, 42, 43, 61, 62, 63, 64, and 65.

In Schoenflies notation, the symbol of a space group is represented by the symbol of corresponding point group with additional superscript. The superscript doesn't give any additional information about symmetry elements of the space group. It is related to the order in which Shoenflies derived space groups.

In Fedorov symbol, the type of space group is denoted as s (symmorphic ), h (hemisymmorphic), or a (asymmorphic). The number is related to the order in which Fedorov derived space groups. There are 73 symmorphic, 54 hemisymmorphic, and 103 asymmorphic space groups. Symmorphic space groups can be obtained as combination of Bravais lattices with corresponding point group. These groups contain the same symmetry elements as the corresponding point groups. Hemisymmorphic space groups contain only axial combination of symmetry elements from the corresponding point groups. All the other space groups are asymmorphic. Example for point group 4/mmm (): the symmorphic space groups are P4/mmm (, 36s) and I4/mmm (, 37s); hemisymmorphic space groups should contain axial combination 422, these are P4/mcc (, 35h), P4/nbm (, 36h), P4/nnc (, 37h), and I4/mcm (, 38h).

## List of Triclinic

Triclinic crystal system
Number Point group Short name Full name Schoenflies Fedorov Shubnikov
11P1 P 1 1s
21P1 P 1 2s

## List of Monoclinic

Monoclinic Bravais lattice
Simple
(P)
Base
(C)
Monoclinic crystal system
Number Point group Short name Full name(s) Schoenflies Fedorov Shubnikov
32P2 P 1 2 1P 1 1 2 3s
42P21P 1 21 1P 1 1 21 1a
52C2 C 1 2 1B 1 1 2 4s
6mPm P 1 m 1P 1 1 m 5s
7mPc P 1 c 1P 1 1 b 1h
8mCm C 1 m 1B 1 1 m 6s
9mCc C 1 c 1B 1 1 b 2h
102/mP2/mP 1 2/m 1P 1 1 2/m 7s
112/mP21/mP 1 21/m 1P 1 1 21/m 2a
122/mC2/mC 1 2/m 1B 1 1 2/m 8s
132/mP2/cP 1 2/c 1P 1 1 2/b 3h
142/mP21/cP 1 21/c 1P 1 1 21/b 3a
152/mC2/cC 1 2/c 1B 1 1 2/b 4h

## List of Orthorhombic

Orthorhombic crystal system
Number Point group Short name Full name Schoenflies Fedorov Shubnikov
16222P222P 2 2 2 9s
17222P2221P 2 2 21 4a
18222P21212P 21 21 2 7a
19222P212121P 21 21 21 8a
20222C2221C 2 2 21 5a
21222C222C 2 2 2 10s
22222F222F 2 2 2 12s
23222I222I 2 2 2 11s
24222I212121I 21 21 21 6a
25mm2Pmm2P m m 2 13s
26mm2Pmc21P m c 21 9a
27mm2Pcc2P c c 2 5h
28mm2Pma2P m a 2 6h
29mm2Pca21P c a 21 11a
30mm2Pnc2P n c 2 7h
31mm2Pmn21P m n 21 10a
32mm2Pba2P b a 2 9h
33mm2Pna21P n a 21 12a
34mm2Pnn2P n n 2 8h
35mm2Cmm2C m m 2 14s
36mm2Cmc21C m c 21 13a
37mm2Ccc2C c c 2 10h
38mm2Amm2A m m 2 15s
39mm2Aem2A b m 2 11h
40mm2Ama2A m a 2 12h
41mm2Aea2A b a 2 13h
42mm2Fmm2F m m 2 17s
43mm2Fdd2F dd2 16h
44mm2Imm2I m m 2 16s
45mm2Iba2I b a 2 15h
46mm2Ima2I m a 2 14h
47PmmmP 2/m 2/m 2/m 18s
48PnnnP 2/n 2/n 2/n 19h
49PccmP 2/c 2/c 2/m 17h
50PbanP 2/b 2/a 2/n 18h
51PmmaP 21/m 2/m 2/a 14a
52PnnaP 2/n 21/n 2/a 17a
53PmnaP 2/m 2/n 21/a 15a
54PccaP 21/c 2/c 2/a 16a
55PbamP 21/b 21/a 2/m 22a
56PccnP 21/c 21/c 2/n 27a
57PbcmP 2/b 21/c 21/m 23a
58PnnmP 21/n 21/n 2/m 25a
59PmmnP 21/m 21/m 2/n 24a
60PbcnP 21/b 2/c 21/n 26a
61PbcaP 21/b 21/c 21/a 29a
62PnmaP 21/n 21/m 21/a 28a
63CmcmC 2/m 2/c 21/m 18a
64CmcaC 2/m 2/c 21/a 19a
65CmmmC 2/m 2/m 2/m 19s
66CccmC 2/c 2/c 2/m 20h
67CmmeC 2/m 2/m 2/e 21h
68CcceC 2/c 2/c 2/e 22h
69FmmmF 2/m 2/m 2/m 21s
70FdddF 2/d 2/d 2/d 24h
71ImmmI 2/m 2/m 2/m 20s
72IbamI 2/b 2/a 2/m 23h
73IbcaI 2/b 2/c 2/a 21a
74ImmaI 2/m 2/m 2/a 20a

## List of Tetragonal

Tetragonal crystal system
Number Point group Short name Full name Schoenflies Fedorov Shubnikov
754P4P 4 22s
764P41P 41 30a
774P42P 42 33a
784P43P 43 31a
794I4I 4 23s
804I41I 41 32a
814P4P 4 26s
824I4I 4 27s
834/mP4/mP 4/m 28s
844/mP42/mP 42/m 41a
854/mP4/nP 4/n 29h
864/mP42/nP 42/n 42a
874/mI4/mI 4/m 29s
884/mI41/aI 41/a 40a
89422P422P 4 2 2 30s
90422P4212P4212 43a
91422P4122P 41 2 2 44a
92422P41212P 41 21 2 48a
93422P4222P 42 2 2 47a
94422P42212P 42 21 2 50a
95422P4322P 43 2 2 45a
96422P43212P 43 21 2 49a
97422I422I 4 2 2 31s
98422I4122I 41 2 2 46a
994mmP4mmP 4 m m 24s
1004mmP4bm P 4 b m 26h
1014mmP42cm P 42 c m 37a
1024mmP42nm P 42 n m 38a
1034mmP4cc P 4 c c 25h
1044mmP4nc P 4 n c 27h
1054mmP42mc P 42 m c 36a
1064mmP42bc P 42 b c 39a
1074mmI4mm I 4 m m 25s
1084mmI4cm I 4 c m 28h
1094mmI41md I 41 m d 34a
1104mmI41cd I 41 c d 35a
11142mP42m P 4 2 m 32s
11242mP42c P 4 2 c 30h
11342mP421m P 4 21 m 52a
11442mP421c P 4 21 c 53a
11542mP4m2 P 4 m 2 33s
11642mP4c2 P 4 c 2 31h
11742mP4b2 P 4 b 2 32h
11842mP4n2 P 4 n 2 33h
11942mI4m2 I 4 m 2 35s
12042mI4c2 I 4 c 2 34h
12142mI42m I 4 2 m 34s
12242mI42d I 4 2 d 51a
1234/m 2/m 2/mP4/mmm P 4/m 2/m 2/m 36s
1244/m 2/m 2/mP4/mcc P 4/m 2/c 2/c 35h
1254/m 2/m 2/mP4/nbm P 4/n 2/b 2/m 36h
1264/m 2/m 2/mP4/nnc P 4/n 2/n 2/c 37h
1274/m 2/m 2/mP4/mbm P 4/m 21/b 2/m 54a
1284/m 2/m 2/mP4/mnc P 4/m 21/n 2/c 56a
1294/m 2/m 2/mP4/nmm P 4/n 21/m 2/m 55a
1304/m 2/m 2/mP4/ncc P 4/n 21/c 2/c 57a
1314/m 2/m 2/mP42/mmc P 42/m 2/m 2/c 60a
1324/m 2/m 2/mP42/mcm P 42/m 2/c 2/m 61a
1334/m 2/m 2/mP42/nbc P 42/n 2/b 2/c 63a
1344/m 2/m 2/mP42/nnm P 42/n 2/n 2/m 62a
1354/m 2/m 2/mP42/mbc P 42/m 21/b 2/c 66a
1364/m 2/m 2/mP42/mnm P 42/m 21/n 2/m 65a
1374/m 2/m 2/mP42/nmc P 42/n 21/m 2/c 67a
1384/m 2/m 2/mP42/ncm P 42/n 21/c 2/m 65a
1394/m 2/m 2/mI4/mmm I 4/m 2/m 2/m 37s
1404/m 2/m 2/mI4/mcm I 4/m 2/c 2/m 38h
1414/m 2/m 2/mI41/amd I 41/a 2/m 2/d 59a
1424/m 2/m 2/mI41/acd I 41/a 2/c 2/d 58a

## List of Trigonal

Unit cells for trigonal crystal system
Rhombohedral
(R)
Hexagonal
(P)
Trigonal crystal system
Number Point group Short name Full name Schoenflies Fedorov Shubnikov
1433P3 P 3 38s
1443P31 P 31 68a
1453P32 P 32 69a
1463R3 R 3 39s
1473P3 P 3 51s
1483R3 R 3 52s
14932P312 P 3 1 2 45s
15032P321 P 3 2 1 44s
15132P3112 P 31 1 2 72a
15232P3121 P 31 2 1 70a
15332P3212 P 32 1 2 73a
15432P3221 P 32 2 1 71a
15532R32 R 3 2 46s
1563mP3m1 P 3 m 1 40s
1573mP31m P 3 1 m 41s
1583mP3c1 P 3 c 1 39h
1593mP31c P 3 1 c 40h
1603mR3m R 3 m 42s
1613mR3c R 3 c 41h
1623 2/mP31m P 3 1 2/m 56s
1633 2/mP31c P 3 1 2/c 46h
1643 2/mP3m1 P 3 2/m 1 55s
1653 2/mP3c1 P 3 2/c 1 45h
1663 2/mR3m R 3 2/m 57s
1673 2/mR3c R 3 2/c 47h

## List of Hexagonal

Hexagonal lattice cell
(P)
Hexagonal crystal system
Number Point group Short name Full name Schoenflies Fedorov Shubnikov
1686P6 P 6 49s
1696P61 P 61 74a
1706P65 P 65 75a
1716P62 P 62 76a
1726P64 P 64 77a
1736P63 P 63 78a
1746P6 P 6 43s
1756/mP6/m P 6/m 53s
1766/mP63/m P 63/m 81a
177622P622 P 6 2 2 54s
178622P6122 P 61 2 2 82a
179622P6522 P 65 2 2 83a
180622P6222 P 62 2 2 84a
181622P6422 P 64 2 2 85a
182622P6322 P 63 2 2 86a
1836mmP6mm P 6 m m 50s
1846mmP6cc P 6 c c 44h
1856mmP63cm P 63 c m 80a
1866mmP63mc P 63 m c 79a
1876m2P6m2 P 6 m 2 48s
1886m2P6c2 P 6 c 2 43h
1896m2P62m P 6 2 m 47s
1906m2P62c P 6 2 c 42h
1916/m 2/m 2/mP6/mmm P 6/m 2/m 2/m 58s
1926/m 2/m 2/mP6/mcc P 6/m 2/c 2/c 48h
1936/m 2/m 2/mP63/mcm P 63/m 2/c 2/m 87a
1946/m 2/m 2/mP63/mmc P 63/m 2/m 2/c 88a

## List of Cubic

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(221) Caesium chloride. Different colors for the two atom types.
Cubic crystal system
Number Point group Short name Full name Schoenflies Fedorov Shubnikov Fibrifold
19523P23 P 2 3 59s 2o
19623F23 F 2 3 61s 1o
19723I23 I 2 3 60s 4oo
19823P213 P 21 3 89a 1o/4
19923I213 I 21 3 90a 2o/4
2002/m 3Pm3 P 2/m 3 62s 4
2012/m 3Pn3 P 2/n 3 49h 4+o
2022/m 3Fm3 F 2/m 3 64s 2
2032/m 3Fd3 F 2/d 3 50h 2+o
2042/m 3Im3 I 2/m 3 63s 8−o
2052/m 3Pa3 P 21/a 3 91a 2/4
2062/m 3Ia3 I 21/a 3 92a 4/4
207432P432 P 4 3 2 68s 4−o
208432P4232 P 42 3 2 98a 4+
209432F432 F 4 3 2 70s 2−o
210432F4132 F 41 3 2 97a 2+
211432I432 I 4 3 2 69s 8+o
212432P4332 P 43 3 2 94a 2+/4
213432P4132 P 41 3 2 95a 2+/4
214432I4132 I 41 3 2 96a 4+/4
21543mP43m P 4 3 m 65s 2o:2
21643mF43m F 4 3 m 67s 1o:2
21743mI43m I 4 3 m 66s 4o:2
21843mP43n P 4 3 n 51h 4o
21943mF43c F 4 3 c 52h 2oo
22043mI43d I 4 3 d 93a 4o/4
2214/m 3 2/mPm3m P 4/m 3 2/m 71s 4:2
2224/m 3 2/mPn3n P 4/n 3 2/n 53h 8oo
2234/m 3 2/mPm3n P 42/m 3 2/n 102a 8o
2244/m 3 2/mPn3m P 42/n 3 2/m 103a 4+:2
2254/m 3 2/mFm3m F 4/m 3 2/m 73s 2:2
2264/m 3 2/mFm3c F 4/m 3 2/c 54h 4−−
2274/m 3 2/mFd3m F 41/d 3 2/m 100a 2+:2
2284/m 3 2/mFd3c F 41/d 3 2/c 101a 4++
2294/m 3 2/mIm3m I 4/m 3 2/m 72s 8o:2
2304/m 3 2/mIa3d I 41/a 3 2/d 99a 8o/4
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