There are three important normal subgroups of prime power index, each being the smallest normal subgroup in a certain class:

As these are weaker conditions on the groups K, one obtains the containments

These groups have important connections to the Sylow subgroups and the transfer homomorphism, as discussed there.

and further, G does not act on this geometry, nor does it reflect any of the non-abelian structure (in both cases because the quotient is abelian).