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](https://doi.org/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](https://eudml.org/doc/158186))

MR1560123  Blichfeldt, H. F. _Blichfeldt's finite collineation groups._
 Bull. Amer. Math. Soc.  24  (1918),  no. 10, 484--487. ([Project Euclid](https://projecteuclid.org/euclid.bams/1183424613), 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.