I am interested in families of finite groups arising from direct products of other groups. For instance, abelian groups are a simple example of such families (direct products of cyclic groups), but there are more. The family of hamiltonian groups is another such family; a hamiltonian group is a group where all the subgroups are normal, and it can be expressed as the direct product of the quaternion group of order 8, an abelian group of odd order and a group formed only by involutions. Coxeter groups are other examples; they are represented as direct products of irreducible Coxeter groups.

Could you point out to any other such families??

Thanks in advance, and regards, Guillermo