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. It is expensive to run, which means that in practice it is only useful for low dimension.
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.