Abelian not implies contained in abelian subgroup of maximum order

Statement

It is possible to have a group of prime power order $P$ with an abelian subgroup $B$ such that $B$ is not contained in any Abelian subgroup of maximum order (?) in $P$.

Proof

Example of a dihedral group

Further information: dihedral group:D16

Consider the dihedral group of order $16$, specifically:

$P=⟨a,x\mid a^8=x^2=e,xa{x}^{-1}=a^{-1}⟩$.

This has an Abelian subgroup of order eight: the cyclic subgroup $A$ generated by $a$. Consider now the cyclic subgroup $B$ given as:

$B:=⟨a^4,x⟩$.

$B$ is a subgroup of order four.

Clearly, $B$ is self-centralizing in $P$: $C_P(B)=B$. Thus, $B$ is not contained in a bigger Abelian subgroup, and hence $B$ is not contained in an Abelian subgroup of maximum order.