Section 13.2.1. Introduction

The three-dimensional subgroups of space group P1 and the two-dimensional subgroups of plane group p1 are all isomorphic subgroups; i.e. these subgroups are pure translation groups and correspond to lattices. In the past, these lattices have often been called `superlattices' (the term `sublattice' perhaps would be more precise). To avoid confusion, the lattices that correspond to the isomorphic subgroups of P1 and p1 are designated here as derivative lattices.

The number of derivative lattices (both maximal and nonmaximal) of a lattice is infinite and always several derivative lattices of index exist. Only for prime indices are maximal derivative lattices obtained; for any prime p, there are three-dimensional derivative lattices of P1, whereas there are two-dimensional derivative lattices of p1. The number of nonmaximal derivative lattices is given by more complicated formulae (cf. Billiet & Rolley Le Coz, 1980).