Let be a group.

A subgroup of is said to be a maximal subgroup of if and there is no subgroup of such that . Note that a maximal subgroup of is not maximal among all subgroups of , but only among all proper subgroups of . For this reason, maximal subgroups are sometimes called maximal proper subgroups.

Similarly, a normal subgroup of is said to be a maximal normal subgroup (or maximal proper normal subgroup) of if and there is no normal subgroup of such that . A normal subgroup of is a maximal normal subgroup if and only if the quotient is a simple group.