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 fact about::abelian subgroup of maximum order in $$P$$.

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

$$P = \langle a,x \mid a^8 = x^2 = e, xax^{-1} = a^{-1} \rangle$$.

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

$$B := \langle a^4, x \rangle$$.

$$B$$ 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.