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 denoted 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 axes 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, but is instead related to the order in which Schoenflies derived the 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 Bravais lattice
Triclinic crystal system
Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold
11 P1P 1 1s
21 P1P 1 2s

List of Monoclinic

Monoclinic Bravais lattice
Simple
(P)
Base
(C)
Monoclinic crystal system
Number Point group Orbifold Short name Full name(s) Schoenflies Fedorov Shubnikov Fibrifold (primary) Fibrifold (secondary)
32 P2P 1 2 1P 1 1 2 3s
4P21P 1 21 1P 1 1 21 1a
5C2C 1 2 1B 1 1 2 4s ,
6m PmP 1 m 1P 1 1 m 5s
7PcP 1 c 1P 1 1 b 1h ,
8CmC 1 m 1B 1 1 m 6s ,
9CcC 1 c 1B 1 1 b 2h ,
102/m P2/mP 1 2/m 1P 1 1 2/m 7s
11P21/mP 1 21/m 1P 1 1 21/m 2a
12C2/mC 1 2/m 1B 1 1 2/m 8s ,
13P2/cP 1 2/c 1P 1 1 2/b 3h ,
14P21/cP 1 21/c 1P 1 1 21/b 3a ,
15C2/cC 1 2/c 1B 1 1 2/b 4h ,

List of Orthorhombic

Orthorhombic crystal system
Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold (primary) Fibrifold (secondary)
16222 P222P 2 2 2 9s
17P2221P 2 2 21 4a
18P21212P 21 21 2 7a
19P212121P 21 21 21 8a
20C2221C 2 2 21 5a
21C222C 2 2 2 10s
22F222F 2 2 2 12s
23I222I 2 2 2 11s
24I212121I 21 21 21 6a
25mm2 Pmm2P m m 2 13s
26Pmc21P m c 21 9a ,
27Pcc2P c c 2 5h
28Pma2P m a 2 6h ,
29Pca21P c a 21 11a
30Pnc2P n c 2 7h ,
31Pmn21P m n 21 10a ,
32Pba2P b a 2 9h
33Pna21P n a 21 12a ,
34Pnn2P n n 2 8h
35Cmm2C m m 2 14s
36Cmc21C m c 21 13a ,
37Ccc2C c c 2 10h
38Amm2A m m 2 15s ,
39Aem2A b m 2 11h ,
40Ama2A m a 2 12h ,
41Aea2A b a 2 13h ,
42Fmm2F m m 2 17s
43Fdd2F dd2 16h
44Imm2I m m 2 16s
45Iba2I b a 2 15h
46Ima2I m a 2 14h ,
47 PmmmP 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 Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold
754 P4P 4 22s
76P41P 41 30a
77P42P 42 33a
78P43P 43 31a
79I4I 4 23s
80I41I 41 32a
814 P4P 4 26s
82I4I 4 27s
834/m P4/mP 4/m 28s
84P42/mP 42/m 41a
85P4/nP 4/n 29h
86P42/nP 42/n 42a
87I4/mI 4/m 29s
88I41/aI 41/a 40a
89422 P422P 4 2 2 30s
90P4212P4212 43a
91P4122P 41 2 2 44a
92P41212P 41 21 2 48a
93P4222P 42 2 2 47a
94P42212P 42 21 2 50a
95P4322P 43 2 2 45a
96P43212P 43 21 2 49a
97I422I 4 2 2 31s
98I4122I 41 2 2 46a
994mm P4mmP 4 m m 24s
100P4bmP 4 b m 26h
101P42cmP 42 c m 37a
102P42nmP 42 n m 38a
103P4ccP 4 c c 25h
104P4ncP 4 n c 27h
105P42mcP 42 m c 36a
106P42bcP 42 b c 39a
107I4mmI 4 m m 25s
108I4cmI 4 c m 28h
109I41mdI 41 m d 34a
110I41cdI 41 c d 35a
11142m P42mP 4 2 m 32s
112P42cP 4 2 c 30h
113P421mP 4 21 m 52a
114P421cP 4 21 c 53a
115P4m2P 4 m 2 33s
116P4c2P 4 c 2 31h
117P4b2P 4 b 2 32h
118P4n2P 4 n 2 33h
119I4m2I 4 m 2 35s
120I4c2I 4 c 2 34h
121I42mI 4 2 m 34s
122I42dI 4 2 d 51a
1234/m 2/m 2/m P4/mmmP 4/m 2/m 2/m 36s
124P4/mccP 4/m 2/c 2/c 35h
125P4/nbmP 4/n 2/b 2/m 36h
126P4/nncP 4/n 2/n 2/c 37h
127P4/mbmP 4/m 21/b 2/m 54a
128P4/mncP 4/m 21/n 2/c 56a
129P4/nmmP 4/n 21/m 2/m 55a
130P4/nccP 4/n 21/c 2/c 57a
131P42/mmcP 42/m 2/m 2/c 60a
132P42/mcmP 42/m 2/c 2/m 61a
133P42/nbcP 42/n 2/b 2/c 63a
134P42/nnmP 42/n 2/n 2/m 62a
135P42/mbcP 42/m 21/b 2/c 66a
136P42/mnmP 42/m 21/n 2/m 65a
137P42/nmcP 42/n 21/m 2/c 67a
138P42/ncmP 42/n 21/c 2/m 65a
139I4/mmmI 4/m 2/m 2/m 37s
140I4/mcmI 4/m 2/c 2/m 38h
141I41/amdI 41/a 2/m 2/d 59a
142I41/acdI 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 Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold
1433 P3P 3 38s
144P31P 31 68a
145P32P 32 69a
146R3R 3 39s
1473 P3P 3 51s
148R3R 3 52s
14932 P312P 3 1 2 45s
150P321P 3 2 1 44s
151P3112P 31 1 2 72a
152P3121P 31 2 1 70a
153P3212P 32 1 2 73a
154P3221P 32 2 1 71a
155R32R 3 2 46s
1563m P3m1P 3 m 1 40s
157P31mP 3 1 m 41s
158P3c1P 3 c 1 39h
159P31cP 3 1 c 40h
160R3mR 3 m 42s
161R3cR 3 c 41h
1623 2/m P31mP 3 1 2/m 56s
163P31cP 3 1 2/c 46h
164P3m1P 3 2/m 1 55s
165P3c1P 3 2/c 1 45h
166R3mR 3 2/m 57s
167R3cR 3 2/c 47h

List of Hexagonal

Hexagonal lattice cell
(P)
Hexagonal crystal system
Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold
1686 P6P 6 49s
169P61P 61 74a
170P65P 65 75a
171P62P 62 76a
172P64P 64 77a
173P63P 63 78a
1746 P6P 6 43s
1756/m P6/mP 6/m 53s
176P63/mP 63/m 81a
177622 P622P 6 2 2 54s
178P6122P 61 2 2 82a
179P6522P 65 2 2 83a
180P6222P 62 2 2 84a
181P6422P 64 2 2 85a
182P6322P 63 2 2 86a
1836mm P6mmP 6 m m 50s
184P6ccP 6 c c 44h
185P63cmP 63 c m 80a
186P63mcP 63 m c 79a
1876m2 P6m2P 6 m 2 48s
188P6c2P 6 c 2 43h
189P62mP 6 2 m 47s
190P62cP 6 2 c 42h
1916/m 2/m 2/m P6/mmmP 6/m 2/m 2/m 58s
192P6/mccP 6/m 2/c 2/c 48h
193P63/mcmP 63/m 2/c 2/m 87a
194P63/mmcP 63/m 2/m 2/c 88a

List of Cubic

Cubic Bravais lattice
Simple
(P)
Body centered
(I)
Face centered
(F)
(221) Caesium chloride. Different colors for the two atom types.
Cubic crystal system
Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Conway Fibrifold (preserving ) Fibrifold (preserving , , )
19523 P23P 2 3 59s
196F23F 2 3 61s
197I23I 2 3 60s
198P213P 21 3 89a
199I213I 21 3 90a
2002/m 3 Pm3P 2/m 3 62s
201Pn3P 2/n 3 49h
202Fm3F 2/m 3 64s
203Fd3F 2/d 3 50h
204Im3I 2/m 3 63s
205Pa3P 21/a 3 91a
206Ia3I 21/a 3 92a
207432 P432P 4 3 2 68s
208P4232P 42 3 2 98a
209F432F 4 3 2 70s
210F4132F 41 3 2 97a
211I432I 4 3 2 69s
212P4332P 43 3 2 94a
213P4132P 41 3 2 95a
214I4132I 41 3 2 96a
21543m P43mP 4 3 m 65s
216F43mF 4 3 m 67s
217I43mI 4 3 m 66s
218P43nP 4 3 n 51h
219F43cF 4 3 c 52h
220I43dI 4 3 d 93a
2214/m 3 2/m Pm3mP 4/m 3 2/m 71s
222Pn3nP 4/n 3 2/n 53h
223Pm3nP 42/m 3 2/n 102a
224Pn3mP 42/n 3 2/m 103a
225Fm3mF 4/m 3 2/m 73s
226Fm3cF 4/m 3 2/c 54h
227Fd3mF 41/d 3 2/m 100a
228Fd3cF 41/d 3 2/c 101a
229Im3mI 4/m 3 2/m 72s
230Ia3dI 41/a 3 2/d 99a
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