Abstract An undirected or a symmetric graph consists of a set of nodes and a set of nonoriented edges connecting between pairs of nodes. In widely differing disciplines of science and engineering, symmetric graphs find important uses. In many of these application areas, an often encountered problem is that of finding all the maximal complete subgraphs of a symmetric graph. In this paper, borrowing the concept of strong connectedness in a nonsymmetric graph, the idea of minimally strongly connected (MSC) and maximal minimally strongly connected (MMSC) subgraphs in a symmetric graph is introduced. The MMSC subgraphs play a kind of role identical to that played by maximal complete subgraphs in symmetric graphs. Many important properties of MMSC subgraphs are discussed in the paper, and an explicit, computer-oriented algorithm is developed for finding all the MMSC subgraphs, given an undirected graph.