Chapter 1.6 Relating crystal structures by group–subgroup relations

Symmetry relations between space groups

Second Online Edition (2011)

Part 1. Space groups and their subgroups

  1. Ulrich Müller

Published Online: 15 DEC 2011

DOI: 10.1107/97809553602060000795

International Tables for Crystallography

International Tables for Crystallography

How to Cite

Müller, U. 2011. Relating crystal structures by group–subgroup relations. International Tables for Crystallography. A1:1:1.6:44–56.

Author Information

  1. Fachbereich Chemie, Philipps‐Universität, D‐35032 Marburg, Germany

Publication History

  1. Published Online: 15 DEC 2011



The relations between crystal structures that are related by symmetry can be set forth in a concise manner with a tree of group–subgroup relations of their space groups (called a Bär­nig­hausen tree). At its top, the tree starts from the space‐group symbol of an aristotype, i.e. a simple, highly symmetrical crystal structure. Arrows pointing downwards depict symmetry reductions that result from structural distortions or partial substitutions of atoms; each arrow represents the relation from a space group to a maximal subgroup. In the middle of each arrow the kind of the subgroup is marked by a t (translationengleiche), k (klassengleiche) or i (isomorphic) followed by the index of the symmetry reduction. In addition, changes of the basis vectors and origin shifts are marked. Each step of the symmetry reduction may involve moderate changes of the atomic coordinates that have to be monitored carefully. An aristotype can be at the head of a large family of structures. From the kinds of subgroups it can be deduced what and how many kinds of domains can result at a phase transition or topotactic reaction involving a symmetry reduction.


  • group–subgroup relations;
  • Bärnighausen trees