Group of prime power order in which all maximal subgroups are isomorphic

Definition
A group of prime power order in which all maximal subgroups are isomorphic is a group of prime power order in which all maximal subgroups are isomorphic to each other, i.e., there is (at most) one isomorphism class of maximal subgroups.

Examples

 * Any homocyclic group of prime power order, i.e., a direct product of cyclic groups of the same prime power order, has this property. Examples are elementary abelian groups and cyclic groups.
 * The quaternion group is a non-abelian group with this property: all its maximal subgroups are cyclic of order four.
 * For odd $$p$$, the prime-cube order group:U(3,p), which is the unique non-abelian group of order $$p^3$$ and exponent $$p$$. All its maximal subgroups are elementary abelian of order $$p^2$$.
 * SmallGroup(16,4) is a group of order $$16$$, where all the maximal subgroups are isomorphic to direct product of Z4 and Z2.

Stronger properties

 * Weaker than::Cyclic group of prime power order
 * Weaker than::Finite elementary abelian group
 * Weaker than::Homocyclic group of prime power order
 * Weaker than::Group of prime power order in which all maximal subgroups are automorphic