There is an algorithm due to Zassenhaus which, in principle, lists all conjugacy classes of finite subgroups of compact Lie groups. I believe that the algorithm was used for $\mathrm{SO}(n)$ for at least $n=6$ if not higher. I believe it is expensive to run, which means that in practice it is only useful for low dimension.
Added
Now that I'm in my office I have my orbifold folder with me and I can list some relevant links:
- Zassenhaus's original paper (in German) Über einen Algorithmus zur Bestimmung der Raumgruppen
- There is a book by RLE Schwarzenberger N-dimensional crystallography with lots of references
- There are a couple of papers in Acta Cryst. by Neubüser, Wondratschek and Bülow titled On crystallography in higher dimensions
- There is a sequence of papers in Math. Comp. by Plesken and Pohst titled On maximal finite irreducible subgroups of GL(n,Z) which I remember were relevant.
Independent of this algorithm, there is some work on $\mathrm{SU}(n)$ from the physics community motivated by elementary particle physics and more modern considerations of the use of orbifolds in the gauge/gravity correspondence.
The case of $\mathrm{SU}(3)$ was done in the mid 1960s and is contained in the paper Finite and Disconnected Subgroups of SU(3) and their Application to the Elementary-Particle Spectrum by Fairbairn, Fulton and Klink. For the case of $\mathrm{SU}(4)$ there is a more recent paper A Monograph on the Classification of the Discrete Subgroups of SU(4) by Hanany and He, and references therein.
Further edit
The paper Non-abelian finite gauge theories by Hanany and He have the correct list of finite subgroups of SU(3), based on Yau and Yu's paper Gorenstein quotient singularities in dimension three.