This is related to this 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)

This is related to this 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 $n$".

--

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 $n$".

--

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)