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.
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**Added**
Now that I'm in my office I have my orbifold folder with me and I can list some relevant links:
1. Zassenhaus's original paper (in German) [Über einen Algorithmus zur Bestimmung der Raumgruppen][1]
2. There is a book by RLE Schwarzenberger *N-dimensional crystallography* with lots of references
3. There are a couple of papers in *Acta Cryst.* by Neubüser, Wondratschek and Bülow titled *On crystallography in higher dimensions*
4. 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.
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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][2] 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)][3] by Hanany and He, and references therein.
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**Further edit**
The paper [Non-abelian finite gauge theories][4] 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][5].
[1]: https://doi.org/10.1007/BF02568029
[2]: http://link.aip.org/link/?JMAPAQ/5/1038/1
[3]: https://arxiv.org/abs/hep-th/9905212
[4]: https://arxiv.org/abs/hep-th/9811183
[5]: https://mathscinet.ams.org/mathscinet-getitem?mr=1169227