This is related to [this][1] MO question (and others as well).

Hoping that this will not turn out to be too broad, I would like to know about the 'state of the art' of:

> 1) The problem of classifying finite subgroups of $\mathrm{GL}(n,\mathbb{Z})$ up to conjugacy (within $\mathrm{GL}(n,\mathbb{Z})\;$). The conjugacy classes are called "arithmetic crystal classes in dimension $n$".

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> 2) The problem of classifying finite subgroups of $\mathrm{GL}(n,\mathbb{Z})$ up to conjugacy (by elements of $\mathrm{GL}(n,\mathbb{Q})\;$). The conjugacy classes are called "geometric crystal classes in dimention $n$".

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Also, a couple of more vague questions:

> 3) For given $n$, are the above problems 1) and 2) solvable by a (sensible) algorithm? Is the difficulty of the problems more of a computational or of a conceptual nature? 
 
It seems that, as for the classification of lattices in Euclidean space (which seems to be related), the problem presents some unexpected patterns as the dimension changes: for example, in dimension $24$ the Leech lattice appears which enjoys some uniqueness properties and an analogous is not found in other dimensions. 

> 4) Are the classification problems 1) and 2) more "tame" as $n$ varies?

 

(I set "community wiki" because questions 1) and 2) may include a reference request)


  [1]: http://mathoverflow.net/questions/91610/isomorphic-but-non-conjugate-subgroups-of-gln-mathbbz/91616#91616