The International Tables Symmetry Database (https://symmdb.iucr.org/), part of International Tables for Crystallography, is a collection of individual databases of crystallographic space-group and point-group information. Programs are provided to access and interactively visualize the data, and also allow new data to be calculated `on the fly', facilitating the in-depth study of group–subgroup relations, domain structures and twinning, families of related crystal structures and phase transitions, and the prediction of new crystal structures.

Symmetry relations between space groups for an arrangement before and after a transition in the thortveitite structure were used to build structural models for compositionally induced phase transitions.

Sub- and supergroups of space groups are important tools for the prediction and investigation of continuous phase transitions. A new method is presented which facilitates the search for sub- and particularly supergroups of space groups. A new kind of phase transition is discussed.

A periodic-like table of the 230 space groups, a sweeping overview in the spirit of the periodic table of the chemical elements, is presented. It organizes the 73 symmorphic space groups along two non-orthogonal axes of point group symmetry and general position multiplicity that separates the crystal systems in discrete color fields. The remaining non-symmorphic groups, for reasons of graphical economy, are enumerated as `isotopes' of their parent symmorphic groups.

Subgroups of 3D crystallographic space groups that are also normal subgroups of 3D space groups have long been of interest. For each arithmetic crystal class of 3D space groups, the sublattices capable of supporting such subgroups are tabulated.

Alternative settings of space groups are explored: what they are, why they can be useful and how to obtain them.

The program LSITESYM of the Bilbao Crystallographic Server (http://www.cryst.ehu.es), which establishes symmetry relations between localized and extended states in crystals with layer symmetry, is presented. The utility of the program is discussed and illustrated with several examples.

The article reports on the computer databases and group-theoretical tools related to representations of crystallographic point groups and space groups that are available on the Bilbao Crystallographic Server (http://www.cryst.ehu.es).

The main aim of SUBGROUPGRAPH is a detailed study of a group–subgroup relation between two space groups, G > H, including the construction of chains of maximal subgroups for G > H for a specified/non-specified index i = |G:H| (the so-called G–H graph), the determination of the number of the different H_k < G, H_k ≅ H, and their distribution into classes of conjugate subgroups relative to G, etc.

A general scheme is proposed to classify and determine the crystallographic properties of twisted bilayers of any homophase 2D structures using complex numbers.