  In this paper we investigate the structure of finite $p$-groups with the
property that every subgroup of index $p^i$ is powerful for some $i$. For odd
primes $p$, we show that under certain conditions these groups must be potent.
Then, motivated by a question of Mann, we investigate in detail the case when
all maximal subgroups are powerful. We show that for odd $p$ any finite
$p$-group $G$ with all maximal subgroups powerful has a regular power structure
- with precisely one exceptional case which is a $3$-group of maximal class and
order $81$. To show this counterexample is unique we use a computational
approach. We briefly discuss the case $p=2$ and some generalisations.
