Maximal elementary abelian subgroup of order four implies subgroup rank at most four

Statement

Suppose $P$ is a group whose order is a power of two, having an elementary abelian subgroup of order four (namely a Klein four-group) that is not contained in any bigger elementary abelian subgroup. In other words, $P$ is a Group of prime power order having a maximal elementary abelian subgroup of prime-square order (?).

Then, the subgroup rank of $P$ is at most four. In other words, every subgroup of $P$ is generated by a subset of size at most four.

Related facts

• Maximal elementary abelian subgroup of prime-square order implies normal rank at most the prime