  The maximal normal subgroup growth type of a finitely generated group is
$n^{\log n}$. Very little is known about groups with this type of growth. In
particular, the following is a long standing problem: Let $\Gamma$ be a group
and $\Delta$ a subgroup of finite index. Suppose $\Delta$ has normal subgroup
growth of type $n^{\log n}$, does $\Gamma$ has normal subgroup growth of type
$n^{\log n}$? We give a positive answer in some cases, generalizing a result of
M\"uller and the second author and a result of Gerdau. For instance, suppose
$G$ is a profinite group and $H$ an open subgroup of $G$. We show that if $H$
is a generalized Golod-Shafarevich group, then $G$ has normal subgroup growth
of type of $n^{\log n}$. We also use our methods to show that one can find a
group with characteristic subgroup growth of type $n^{\log n}$.

Group Theory