The subgroup lattice of a group is the Hasse diagram of the subgroups under the partial ordering of set inclusion. This Demonstration displays the subgroup lattice for each of the groups (up to isomorphism) of orders 2 through 12. You can highlight the cyclic subgroups, the normal subgroups, or the center of the group. Moving the cursor over a subgroup displays a description of the subgroup.

A subgroup of a group is normal in if for all elements in . The center of a group is the normal subgroup consisting of those elements in that commute with all elements in . A group is Abelian iff .