Every discrete group of Euclidean isometries acts cocompactly on some subspace of $\mathbb{R}^n$ (a result of Bieberbach). If we are in the crystallographic case, then the index is bounded by the maximal order of a finite subgroup of $GL(n,\mathbb{Z})$. Here we have several results in the literature, e.g., the result of Friedland in $1997$, that
$$m\le 2^nn!
$$ 
for $n\ge n_0$.
There are exceptions for certain $n$, see the comment of Geoff Robinson.
There are conditions given in Friedland's paper "The maximal orders of finite subgroups of $GL(n,\mathbb{Q})$", when equality is attained.
[Rockmore](http://link.springer.com/article/10.1007%2FBF01198081) in $1995$ proved the following: for every $\epsilon>0$ there exists a constant $c(\epsilon)$ such that $m\le c(\epsilon)(n!)^{1+\epsilon}$.