In this talk, we consider the following variant on the inverse Galois problem: Given a field F, which finite groups occur as Galois groups of maximal subfields of F-central division algebras? This question was first asked by Schacher when F is the field of rational numbers; he gave a necessary condition and conjectured it was also sufficient. . Using patching methods, we give a complete answer in the case when F is the function field of a curve over a complete discrete valuation field. This is joint work with David Harbater and Daniel Krashen.