Let $G$ be a $p$-group for some prime $p$. Let $n$ be the positive integer so that $|G:Z(G)|=p^n$. Suppose $A$ is a maximal abelian subgroup of $G$. Let $p^l=max\{|Z(C_G(g)):Z(G)| : g \in G\setminus Z(G)\}$, let $p^b=max\{|cl(g)| : g\in G\setminus Z(G)\}$, and let $p^a=|A:Z(G)|$. Then we show that $a\ge n/(b+l)$.