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

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 | C_{1} | 332 | 195-199 | ||

Cubic | T_{h} | 3*2 | 200-206 | ||

Cubic | O | 432 | 207-214 | ||

Cubic | T_{d} | *332 | 215-220 | ||

Cubic | O_{h} | *432 | 221-230 | ||

Hexagonal | C_{6} | 66 | 168-173 | ||

Hexagonal | C_{3h} | 3* | 174 | ||

Hexagonal | C_{6h} | 6* | 175-176 | ||

Hexagonal | D_{6} | 622 | 177-182 | ||

Hexagonal | C_{6v} | *66 | 183-186 | ||

Hexagonal | D_{3h} | *322 | 187-190 | ||

Hexagonal | D_{6h} | *622 | 191-194 | ||

Trigonal | C_{3} | 33 | 143-146 | ||

Trigonal | C_{3i }= S_{6} | 3× | 147-148 | ||

Trigonal | D_{3} | 322 | 149-155 | ||

Trigonal | C_{3v} | *33 | 156-161 | ||

Trigonal | D_{3d} | 2*3 | 162-167 | ||

Tetragonal | C_{4} | 44 | 75-80 | ||

Tetragonal | S_{4} | 2× | 81-82 | ||

Tetragonal | C_{4h} | 4* | 83-88 | ||

Tetragonal | D_{4} | 422 | 89-98 | ||

Tetragonal | C_{4v} | *44 | 99-110 | ||

Tetragonal | D_{2d }= V_{d} | 2*2 | 111-122 | ||

Tetragonal | D_{4h} | *422 | 123-142 | ||

Orthorhombic | D_{2 }= V | 222 | 16-24 | ||

Orthorhombic | C_{2v} | *22 | 24-46 | ||

Orthorhombic | D_{2h }= V_{h} | *222 | 47-74 | ||

Monoclinic | C_{2} | 22 | 3-5 | ||

Monoclinic | C_{s }= S_{1h} | * | 6-9 | ||

Monoclinic | C_{2h} | 2* | 10-15 | ||

Triclinic | C_{1} | 11 | 1 | ||

Triclinic | C_{i }= S_{2} | × | 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 |

✘ | ✔ | ✔ | ✘ | ✘ | |

✔ | ✘ | ✘ | ✘ | ✘ | |

✘ | ✔ | ✘ | ✘ | ✘ | |

✘ | ✘ | ✘ | ✘ | ✘ | |

✔ | ✘ | ✘ | ✘ | ✘ | |

✘ | ✘ | ✘ | ✔ | Has the potential to be ferroelectric | |

✘ | ✘ | ✘ | ✘ | ✘ | |

✔ | ✘ | ✘ | ✘ | ✘ | |

✘ | ✔ | ✔ | ✘ | ✘ | |

✘ | ✘ | ✘ | ✔ | Has the potential to be ferroelectric | |

✘ | ✘ | ✘ | ✘ | ✘ | |

✔ | ✘ | ✘ | ✘ | ✘ | |

✘ | ✘ | ✘ | ✔ | Has the potential to be ferroelectric | |

✔ | ✘ | ✘ | ✘ | ✘ | |

✘ | ✔ | ✔ | ✘ | ✘ | |

✘ | ✘ | ✘ | ✔ | Has the potential to be ferroelectric | |

✔ | ✘ | ✘ | ✘ | ✘ | |

✘ | ✘ | ✘ | ✔ | Has the potential to be ferroelectric | |

✘ | ✘ | ✘ | ✘ | ✘ | |

✔ | ✘ | ✘ | ✘ | ✘ | |

✘ | ✔ | ✔ | ✘ | ✘ | |

✘ | ✘ | ✘ | ✔ | Has the potential to be ferroelectric | |

✘ | ✘ | ✘ | ✘ | ✘ | |

✔ | ✘ | ✘ | ✘ | ✘ | |

✘ | ✔ | ✔ | ✘ | ✘ | |

✘ | ✘ | ✘ | ✔ | Has the potential to be ferroelectric | |

✔ | ✘ | ✘ | ✘ | ✘ | |

✘ | ✘ | ✘ | ✔ | Has the potential to be ferroelectric | |

✘ | ✘ | ✘ | ✔ | Has the potential to be ferroelectric | |

✔ | ✘ | ✘ | ✘ | ✘ | |

✘ | ✘ | ✘ | ✔ | Has the potential to be ferroelectric | |

✔ | ✘ | ✘ | ✘ | ✘ |

### 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

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