A rough estimate can be obtained by comparison with the number of isomorphism classes of (symmorphic) space groups (see also Agol's comments): The map $Q \mapsto Q \ltimes\mathbb{Z}^n$ induces a bijection between the conjugacy classes $C_n$ of finite subgroups of $GL_n(\mathbb{Z})$ and the isomorphism classes of symmorphic space groups. A proof thereof can be found in my answer to this question:
Subgroups of the Euclidean group as semidirect products
In particular, $|C_n| = 73,\; 710,\; 6079,\; 85311$ for $n=3,4,5,6$ respectively:
http://www.math.ru.nl/~souvi/papers/acta03.pdf
The number of isomorphism classes of (all) space groups has been estimated (cf. Remark 5.5) in
http://www.unige.ch/math/folks/bucher/Affine/pdfAffine/BuserBieberbach.pdf
and yields the upper bound $|C_n| \le e^{\displaystyle e^{4n^2}}$.
Actually it is conjectured by Schwarzenberger in a 1974 paper that the number of isomorphism classes of space groups is asymptotically $O(2^{\displaystyle n^2})$. But I don't know if this has been proved in the meanwhile.