  In this paper we introduce a class of `parabolic' subgroups for the
generalized braid group associated to an arbitrary irreducible complex
reflection group, which maps onto the collection of parabolic subgroups of the
reflection group. Except for one case, which is proven separately elsewhere, we
prove that this collection forms a lattice, so that intersections of parabolic
subgroups are parabolic subgroups. In particular, every element admits a
parabolic closure, which is the smallest parabolic subgroup containing it. We
furthermore prove that it provides a simplicial complex which generalizes the
curve complex of the usual braid group. In the case of real reflection groups,
this complex generalizes the one previously introduced by Cumplido, Gebhardt,
Gonz\'alez-Meneses and Wiest for Artin groups of spherical type. We show that
it shares similar properties, and similarly conjecture its hyperbolicity.
