This question is inspired by the recent question ["The finite subgroups of SL(2,C)"](https://mathoverflow.net/questions/16026/the-finite-subgroups-of-sl2-c). While reading the answers there I remembered reading once that identifying the finite subgroups of SU(3) is still an open problem. I have tried to check this and it seems it was at least still open in the Eighties.


Can anyone confirm or deny that the finite subgroups of SU(3) are not all known? And if this is true, then what is the source of the difficulty? 


Secondly, what is known of the finite subgroups of SU(n) for n > 3?


**UPDATE:** Thanks to those below who have corrected my ignorance! It seems that I may have been tricked by some particularly sensationalised abstracts (or perhaps just misunderstood them.)



The finite subgroups of SU(n)
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](https://doi.org/10.1007/BF02568029)
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](http://link.aip.org/link/?JMAPAQ/5/1038/1) 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)](https://arxiv.org/abs/hep-th/9905212) by Hanany and He, and references therein.




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**Further edit**


The paper [Non-abelian finite gauge theories](https://arxiv.org/abs/hep-th/9811183) 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](https://mathscinet.ams.org/mathscinet-getitem?mr=1169227).


