Lattice of subgroups

In addition, there are two subgroups of the form Z2 × Z2, generated by pairs of order-two elements.

The lattice formed by these ten subgroups is shown in the illustration.

The Zassenhaus lemma gives an isomorphism between certain combinations of quotients and products in the lattice of subgroups.

The fact that normal subgroups form a modular lattice is a particular case of a more general result, namely that in any Maltsev variety (of which groups are an example), the lattice of congruences is modular (Kearnes & Kiss 2013).

Lattice-theoretic information about the lattice of subgroups can sometimes be used to infer information about the original group, an idea that goes back to the work of Øystein Ore (1937, 1938).

For instance, as Ore proved, a group is locally cyclic if and only if its lattice of subgroups is distributive.

If additionally the lattice satisfies the ascending chain condition, then the group is cyclic.