International
Tables for Crystallography Volume A Spacegroup symmetry Edited by Th. Hahn © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. A, ch. 13.2, pp. 843844
https://doi.org/10.1107/97809553602060000529 Chapter 13.2. Derivative latticesThe lattices that correspond to the isomorphic subgroups of space group P1 and plane group p1 are termed derivative lattices. Formulae for the construction of three and twodimensional derivative lattices are given, and the three and twodimensional lattices of indices 2 to 7 are tabulated. 
The threedimensional subgroups of space group P1 and the twodimensional 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 threedimensional derivative lattices of P1, whereas there are twodimensional derivative lattices of p1. The number of nonmaximal derivative lattices is given by more complicated formulae (cf. Billiet & Rolley Le Coz, 1980).
It is possible to construct in a simple way all threedimensional derivative lattices of a lattice (Table 13.2.2.1). Starting from a primitive unit cell defined by a, b, c, each derivative lattice possesses exactly one primitive unit cell defined by , , by means of the following relation

Note that the vector has the same direction as the vector a and the plane is parallel to the plane (a, b), i.e. the matrix of the transformation is triangular. Equivalent formulae can be derived by permutations of the vectors a, b, c which keep the directions of or and which preserve the parallelism of the planes with (b, c) or with (a, c).
Another primitive cell of a given derivative lattice is obtained if one of the following three elementary transformations is performed on the vectors of a primitive cell of this derivative lattice:
(i) and (ii) are lefthanded transformations, (iii) is righthanded.
Example
The primitive cell , , (, , ) belongs to the derivative lattice of index 10 given by the primitive cell because these two cells are related by the following sequence of elementary transformations: (, , ) and (, , ) have the same handedness.
All previous considerations are valid also for twodimensional lattices and their derivative lattices (Table 13.2.3.1). The relevant formula for any index is A similar formula is obtained by interchange of a and b.
