# Characteristic maximal subgroups may be isomorphic and distinct in group of prime power order

From Groupprops

## Statement

It is possible to have a group of prime power order and two subgroups and of such that:

- and are both Characteristic subgroup (?)s of .
- and are both maximal subgroups of .
- and are isomorphic groups: They are hence also Series-equivalent subgroups (?) of , because the quotients and are both cyclic of the same prime order.
- and are
*distinct*subgroups, i.e., they are not equal as subgroups of .

## Related facts

See series-equivalent not implies automorphic#Related facts.

## Proof

The smallest known example is with a group of order 64. There are many examples with this order, one of which has isomorphic to SmallGroup(64,13) and and both isomorphic to nontrivial semidirect product of Z4 and Z8 (group ID: (32,12)).