List of Point Groups 2D and 3D


This article lists all the 3D and 2D point groups, including alternate notations, related space groups, and allowed/forbidden properties in tabular format.

If you want help understanding concepts, check out my article which explains the basics about point groups.

This article is meant as a helpful reference for you to print out or bookmark.

List of 32 3D Point Groups with Alternative Notation

This table shows the relationship between point groups, crystal systems, and space groups, with alternate notations including full and short Hermann-Mauguin (international), Schoenflies, and Orbifold notation.

Crystal SystemHermann-Mauguin
(full name)
Hermann-Mauguin (short name)SchoenfliesOrbifoldAssociated Space Groups
Cubic2323 C1332195-199
Cubic\frac{2}{m}\bar{3}m\bar{3} Th3*2200-206
Cubic432432 O432207-214
Cubic\bar{4}3m\bar{4}3m Td*332215-220
Cubic\frac{4}{m}\bar{3}mm\bar{3}m Oh*432221-230
Hexagonal66 C666168-173
Hexagonal\bar{6}\bar{6} C3h3*174
Hexagonal\frac{6}{m}6/m C6h6*175-176
Hexagonal622622 D6622177-182
Hexagonal6mm6mm C6v*66183-186
Hexagonal\bar{6}2m\bar{6}2m D3h*322187-190
Hexagonal\frac{6}{m} \frac{2}{m} \frac{2}{m}6/mmm D6h*622191-194
Trigonal33 C333143-146
Trigonal\bar{3}\bar{3} C3i = S6147-148
Trigonal3232 D3322149-155
Trigonal3m3m C3v*33156-161
Trigonal\bar{3}\frac{2}{m}\bar{3}m D3d2*3162-167
Tetragonal44 C44475-80
Tetragonal\bar{4}\bar{4} S481-82
Tetragonal\frac{4}{m}4/m C4h4*83-88
Tetragonal422422 D442289-98
Tetragonal4mm4mm C4v*4499-110
Tetragonal\bar{4}2m\bar{4}2m D2d = Vd2*2111-122
Tetragonal\frac{4}{m} \frac{2}{m} \frac{2}{m}4/mmm D4h*422123-142
Orthorhombic222222 D2 = V22216-24
Orthorhombicmm2mm2 C2v*2224-46
Orthorhombic\frac{2}{m} \frac{2}{m} \frac{2}{m}mmm D2h = Vh*22247-74
Monoclinic22 C2223-5
Monoclinicmm Cs = S1h*6-9
Monoclinic\frac{2}{m}2/m C2h2*10-15
Triclinic11 C1111
Triclinic\bar{1}\bar{1} Ci = S2×2

Forbidden/Allowed Properties of 3D Point Groups

Certain properties are forbidden due to symmetry relationships, which point groups concisely capture. 

This table shows whether each point group has a center of inversion, enantiomorphic pairs, piezoelectric properties, pyroelectric properties, or the potential to be ferroelectric.

Note that ferroelectric properties are not determined solely by the point group, so certain point groups can only forbid the existence of ferroelectric properties, but not guarantee them.

Point Group
(Hermann-Mauguin)
Centrosymmetric (center of inversion)Enantiomorphic Pairs (only rotation axes)PiezoelectricPyroelectric (polar)Ferroelectric
23 
m\bar{3} 
432 
\bar{4}3m 
m\bar{3}m 
6 Has the potential to be ferroelectric
\bar{6} 
6/m 
622 
6mm Has the potential to be ferroelectric
\bar{6}2m 
6/mmm 
3 Has the potential to be ferroelectric
\bar{3} 
32 
3m Has the potential to be ferroelectric
\bar{3}m 
4 Has the potential to be ferroelectric
\bar{4} 
4/m 
422 
4mm Has the potential to be ferroelectric
\bar{4}2m 
4/mmm 
222 
mm2 Has the potential to be ferroelectric
mmm 
2 Has the potential to be ferroelectric
m Has the potential to be ferroelectric
2/m 
1 Has the potential to be ferroelectric
\bar{1} 

List of 10 2D Point Groups

Here is a list of the 10 2D point groups.

Crystal SystemPoint Group
Oblique (Rhomboidal)1
Oblique (Rhomboidal)2
Rectangularm
Rectangular2mm
Trigonal3m
Trigonal3
Square4mm
Square4
Hexagonal6mm
Hexagonal6

References and Further Reading

If you want a simple explanation of crystallographic point groups, make sure to check out this article.

Our list of 2D point groups comes from: Anthony Kelly, Kevin M. Knowles, Crystallography and Crystal Defects, Second Edition,  John Wiley & Sons, Ltd, (2012)

If you’re reading this article because you’re taking a class on structures, you may be interested in my other crystallography articles. Here is this list, in recommended reading order:

What are Crystals and Grains
Introduction to Bravais Lattices
What is the Difference Between “Crystal Structure” and “Bravais Lattice”
Atomic Packing Factor
How to Read Miller Indices
How to Read Hexagonal Miller-Bravais Indices
Close-Packed Crystals and Stacking Order
Interstitial Sites
Primitive Cells
How to Read Crystallography Notation
What are Point Groups
List of Point Groups

If you are interested in more details about any specific crystal structure, I have written individual articles about simple crystal structures which correspond to each of the 14 Bravais lattices:

1. Simple Cubic
2. Face-Centered Cubic
2a. Diamond Cubic
3. Body-Centered Cubic
4. Simple Hexagonal
4a. Hexagonal Close-Packed
4b. Double Hexagonal Close-Packed (La-type)
5. Rhombohedral
5a. Rhombohedral Close-Packed (Sm-type)
6. Simple Tetragonal
7. Body-Centered Tetragonal
7a. Diamond Tetragonal (White Tin)
8. Simple Orthorhombic
9. Base-Centered Orthorhombic
10. Face-Centered Orthorhombic
11. Body-Centered Orthorhombic 
12. Simple Monoclinic 
13. Base-Centered Monoclinic 
14. Triclinic

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