The aim of this article is to develop a Bayesian random graph mixture model (RGMM) to detect the latent class network (LCN) structure of brain connectivity networks and estimate the parameters governing this structure. The use of conjugate priors for unknown parameters leads to efficient estimation, and a well-known nonidentifiability issue is avoided by a particular parameterization of the stochastic block model (SBM). Posterior computation proceeds via an efficient Markov Chain Monte Carlo algorithm. Simulations demonstrate that LCN outperforms several other competing methods for community detection in weighted networks, and we apply our RGMM to estimate the latent community structures in the functional resting brain networks of 185 subjects from the ADHD-200 sample. We find overlap in the estimated community structure across subjects, but also heterogeneity even within a given diagnosis group.