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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==

These include:

* [[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

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