We analyze a family of clustered excitatory-inhibitory neural networks and the underlying bifurcation structures that arise because of permutation symmetries in the network as the global coupling strength g is varied. We primarily consider two network topologies: an all-to-all connected network which excludes self-connections, and a network in which the excitatory cells are broken into clusters of equal size. Although in both cases the bifurcation structure is determined by symmetries in the system, the behavior of the two systems is qualitatively different. In the all-to-all connected network, the system undergoes Hopf bifurcations leading to periodic orbit solutions; notably, for large g, there is a single, stable periodic orbit solution and no stable fixed points. By contrast, in the clustered network, there are no Hopf bifurcations, and there is a family of stable fixed points for large g.