We investigate the quadratic homogeneous holomorphic vector fields on C n that are semicomplete, this is, those whose solutions are single-valued in their maximal definition domain. To a generic quadratic vector field we rationally associate some complex numbers that turn out to be integers in the semicomplete case, thus showing that the linear equivalence classes of semicomplete vector fields are contained in some sort of lattice in the space of linear equivalence classes of quadratic ones. We prove that the foliations of C P n - 1 induced by semicomplete quadratic vector fields are linearizable in a neighborhood of their singular points and give some new families of examples in C 3 . Finally, we classify the semicomplete isochoric vector fields in C 3 having an isolated singularity at the origin.