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 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 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.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 Dover Publications, Inc., New York 1961 xvii+390 pp.