List of Point Groups 2D and 3D – Materials Science & Engineering

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.

Outline

List of 32 3D Point Groups with Alternative Notation
Forbidden/Allowed Properties of 3D Point Groups
List of 10 2D Point Groups
References and Further Reading

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 System	Hermann-Mauguin (full name)	Hermann-Mauguin (short name)	Schoenflies	Orbifold	Associated Space Groups
Cubic	$23$	$23$	C₁	332	195-199
Cubic	$\frac{2}{m}\bar{3}$	$m\bar{3}$	T_h	3*2	200-206
Cubic	$432$	$432$	O	432	207-214
Cubic	$\bar{4}3m$	$\bar{4}3m$	T_d	*332	215-220
Cubic	$\frac{4}{m}\bar{3}m$	$m\bar{3}m$	O_h	*432	221-230
Hexagonal	$6$	$6$	C₆	66	168-173
Hexagonal	$\bar{6}$	$\bar{6}$	C_3h	3*	174
Hexagonal	$\frac{6}{m}$	$6/m$	C_6h	6*	175-176
Hexagonal	$622$	$622$	D₆	622	177-182
Hexagonal	$6mm$	$6mm$	C_6v	*66	183-186
Hexagonal	$\bar{6}2m$	$\bar{6}2m$	D_3h	*322	187-190
Hexagonal	$\frac{6}{m} \frac{2}{m} \frac{2}{m}$	$6/mmm$	D_6h	*622	191-194
Trigonal	$3$	$3$	C₃	33	143-146
Trigonal	$\bar{3}$	$\bar{3}$	C_3i= S₆	3×	147-148
Trigonal	$32$	$32$	D₃	322	149-155
Trigonal	$3m$	$3m$	C_3v	*33	156-161
Trigonal	$\bar{3}\frac{2}{m}$	$\bar{3}m$	D_3d	2*3	162-167
Tetragonal	$4$	$4$	C₄	44	75-80
Tetragonal	$\bar{4}$	$\bar{4}$	S₄	2×	81-82
Tetragonal	$\frac{4}{m}$	$4/m$	C_4h	4*	83-88
Tetragonal	$422$	$422$	D₄	422	89-98
Tetragonal	$4mm$	$4mm$	C_4v	*44	99-110
Tetragonal	$\bar{4}2m$	$\bar{4}2m$	D_2d= V_d	2*2	111-122
Tetragonal	$\frac{4}{m} \frac{2}{m} \frac{2}{m}$	$4/mmm$	D_4h	*422	123-142
Orthorhombic	$222$	$222$	D₂= V	222	16-24
Orthorhombic	$mm2$	$mm2$	C_2v	*22	24-46
Orthorhombic	$\frac{2}{m} \frac{2}{m} \frac{2}{m}$	$mmm$	D_2h= V_h	*222	47-74
Monoclinic	$2$	$2$	C₂	22	3-5
Monoclinic	$m$	$m$	C_s= S_1h	*	6-9
Monoclinic	$\frac{2}{m}$	$2/m$	C_2h	2*	10-15
Triclinic	$1$	$1$	C₁	11	1
Triclinic	$\bar{1}$	$\bar{1}$	C_i= 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)	Piezoelectric	Pyroelectric (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 System	Point Group
Oblique (Rhomboidal)	1
Oblique (Rhomboidal)	2
Rectangular	m
Rectangular	2mm
Trigonal	3m
Trigonal	3
Square	4mm
Square	4
Hexagonal	6mm
Hexagonal	6

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
What are Space Groups
List of Space Groups
The 7 Crystal Systems

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

Outline

List of 32 3D Point Groups with Alternative Notation

Forbidden/Allowed Properties of 3D Point Groups

List of 10 2D Point Groups

References and Further Reading

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