I am not sure if this question is more suited for mathOverflow, I will give it a chance here first.
An n-dimensional crystallographic group is a discrete co-compact subgroup of the group of isometries of $\mathbb R^n$.
The cornerstone of n-dimensional crystallographic group theory is based on a series of three theorems by L. Bieberbach (1911 and 1912 ) together with the work of H. Zassenhaus (1948).
In addition to the algebraic understanding provided by these theorems, many works went to explicitly count crystallographic groups, at least in low dimensions, which is relevant not only to math but also to physics.
See for example
Plesken, W. and Schulz, T., (2000). "Counting crystallographic groups in low dimensions.",pdf
Fischer, M., Ratz, M., Torrado, J. and Vaudrevange, P.K., (2013). "Classification of symmetric toroidal orbifolds.", ArXiv
Contrastly, in my experience, it is difficult to find such theoretical and computational studies for non-cocompact discrete subgroups of Euclidean isometries in dimensions greater than $2$. In the case dimension $2$, I am aware of Rosette and Frieze groups. I want to know what we know about those non-crystallographic discrete subgroups in dimensions greater than $2$.
If $\Gamma$ is a discrete subgroup of $\mathrm{Isom}(E)$, $E$ a Euclidean space, then there exists a unique minimal $\Gamma$-invariant affine subspace $V$ in $E$. Then $\Gamma$ acts cocompactly and properly on $V$ (hence with finite kernel); it acts as a finite group on $E/V$; $\Gamma$ has a finite index subgroup isomorphic to $\mathbf{Z}^{\dim(V)}$, and is cocompact iff $V=E$.
This is essentially (but not completely) a reduction to the cocompact case.
It is possible that you get more complete information on MO (think of tagging your question, in addition to gr.group-theory, also gt.geometric-topology and lie-groups).