The Klein-four group is a [[characteristically simple group]], since it is a direct power of a simple group. Hence, the only characteristic subgroups are the trivial subgroup and the whole group.
Groups containing it as a subgroup==
* [[Symmetric group:S4]] which is the holomorph of the Klein-four group, and in which the Klein-four group is a [[characteristic subgroup]]
* [[Dihedral group:D8]] which is the dihedral group of order 8, acting on a set of four elements. It sits between the Klein-four group and the symmetric group on 4 elements