# Every elementary abelian p-group occurs as the Frattini quotient of a p-group in which every maximal subgroup is characteristic

Suppose $p$ is a prime number and $V$ is an elementary Abelian $p$-group (equivalently, $V$ is a vector space over the prime field $\mathbb{F}_p$. Then, there exists a finite $p$-group $P$ such that $P/\Phi(P) \cong V$ (where $\Phi(P)$ denotes the Frattini subgroup of $P$) and such that every maximal subgroup of $P$ is a characteristic subgroup.
For the case that $V$ is one-dimensional, we can take $P = V$. For the case that $\operatorname{dim}(V) > 1$, apply fact (1) with the subgroup $G$ as the trivial group.