Let $s(n)$ denote the number of distinct crystallographic groups in $Isom(\mathbb{R}^n)$.
Apparently the best known upper bound so far is

$$
s(n)\le e^{e^{4n^2}},
$$
given by Peter Buser in $1985$. On the other hand, it is not difficult to see (using a fixed-point lemma
by Burnside-Cachy) that the number of $n$-dimensional crystallographic groups in the arithmetic class
of the group of diagonal matrices has the asymptotic value
$$
\frac{2^{n(n-1)}}{n!}.
$$
This should give a lower bound for $s(n)$. Moreover, this class seems to be quite large in general,
so that I have the feeling that the upper bound of Buser is not yet optimal.

What is known about lower and upper bounds for $s(n)$ ? Is it possible to use some other results on bounds of conjugacy classes of finite groups (since $s(n)$ is related to that by the number of conjugacy classes of finite subgroups of $GL(n,\mathbb{Z})$) ?

Update: Are there upper bounds known for the number of $n$-dimensional Bieberbach groups (torsionfree crystallographic groups), better than Buser's bound ?