Abstract Minimal full-access (MFA) networks are the class of all interconnection networks for 2 N = 2 n+1 inputs and outputs that require a minimum number of switching elements and provide full access capability. In this paper, MFA networks with 2 × 2 switching elements are studied. Graph-theoretic ideas used in developing previous results concerning uniform MFA networks [M. A. Sridhar and C. S. Raghavendra, J. Parallel Distrib. Comput. 5, (1988), 383–403] are generalized to show how to enumerate a large class of MFA networks. An exponential lower bound on the number of such networks is derived, and routing algorithms are outlined. In the process, a characterization of the set of automorphisms of the Omega network is derived. A characterization of the permutations realizable by a certain subclass of these networks is also derived. Finally, it is shown that a simple self-routing algorithm exists for the class of networks introduced here.