The finite subgroups of SU(3) have been known for a century. I think you can find it in these references (my Departmental library does not go back this far):
MR1500676 Blichfeldt, H. F. On the order of linear homogeneous groups. II. Trans. Amer. Math. Soc. 5 (1904), no. 3, 310--325. (doi:10.1090/S0002-9947-1904-1500676-6)
MR1511301 Blichfeldt, H. F. The finite, discontinuous primitive groups of collineations in four variables. Math. Ann. 60 (1905), no. 2, 204--231. (EuDML)
MR1560123 Blichfeldt, H. F. Blichfeldt's finite collineation groups. Bull. Amer. Math. Soc. 24 (1918), no. 10, 484--487. (Project Euclid, open access)
and also in this book
MR0123600 (23 #A925) Miller, G. A. ; Blichfeldt, H. F. ; Dickson, L. E. Theory and applications of finite groups. Dover Publications, Inc., New York 1961 xvii+390 pp.
