Let $G$ be an extraspecial group of order $p^{2n+1}$. Then
$$
cp(G)=\frac{p^{2n}+p-1}{p^{2n+1}}=\frac1p+\frac{p-1}{p^{2n+1}}.
$$
All maximal abelian subgroups have index $p^n$ in $G$. Hence, the answer is `not'.