Given a finite group G, let C(G) be the set of all cosets of all proper subgroups of G, ordered by inclusion. In joint work with Russ Woodroofe, we show that the order complex of C(G) is not acyclic in characteristic two, and therefore not contractible. This answers a question of K. S. Brown. Our proof uses P. A. Smith Theory and the Classification of Finite Simple Groups. From our proof, we are led to the following elementary problem on binomial coefficients, which remains open.