Abelian subgroups of maximum order need not be isomorphic

Statement

It is possible to have a group of prime power order $P$ with Abelian subgroups $A,B$ of maximum order such that $A$ is not isomorphic to $B$.

Proof

Example of the dihedral group

Further information: dihedral group:D8

Let $P$ be the dihedral group of order eight, specifically:

$P=⟨a,x\mid a^4=x^2=e,xa{x}^{-1}=a^{-1}⟩$.

$P$ has three Abelian subgroups of maximum order (i.e., order four): the cyclic subgroup $A$ generated by $a$, and the following two Klein-four groups: the group $B=⟨a^2,x⟩$, and the group $C=⟨a^2,ax⟩$. $A$ is not isomorphic to either $B$ or $C$.