There are exactly five types of such threefolds: the quintic in P^4, the complete intersections (3, 3) and (2, 4) in P^5, the complete intersection (2, 2, 3) in P^6 and finally (2, 2, 2, 2) in P^7. The counts are listed in that order.

Geir Ellingsrud and Stein Arild Stromme, Bott's formula and enumerative geometry, J. Amer. Math. Soc. 9 (1996), 175-193; arXiv:alg-geom/9411005, 1994. See Th. 1.1.