Usually, groups increase their productivity by the specialization of their group members. In these cases, group output is no longer simply a sum of individual outputs. We analyze contests with group-specific public goods that allow for different degrees of complementarity between group members' efforts. More specifically, we use a Tullock contest success function and a CES-impact function. We show that in equilibrium the degree of complementarity is irrelevant if groups do not differ in size and group members have an identical valuation of the public good. The equilibrium is discontinuous as the CES function converges to the Cobb-Douglas case. Except for the effects at the discontinuity, higher complementarity tends to favor larger groups. In groups with diverse valuations, higher complementarity also leads to higher similarity in group members' efforts, which however is not necessarily an advantage for a more diverse group.