We construct networks from simple robust heteroclinic cycles in R4. Under appropriate assumptions only very few ways exist by which cycles can be joined together in a network. Those of types B and C have been known and investigated in the last decades, while type A networks are mostly absent from the literature. Using the stability index defined by Podvigina and Ashwin (Nonlinearity 24, 887–929, 2011) we compare non-asymptotic stability properties of networks that are geometrically identical, but made up of cycles of different type, i.e. their corresponding vector fields commute with different symmetry groups.