A wealth of research on resting-state functional MRI (R-fMRI) data has revealed modularity as a fundamental characteristic of the human brain functional network. The modular structure has recently been suggested to be overlapping, meaning that a brain region may engage in multiple modules. However, not only the overlapping modular structure remains inconclusive, the topological features and functional roles of overlapping regions are also poorly understood. To address these issues, the present work utilized the maximal-clique based multiobjective evolutionary algorithm to explore the overlapping modular structure of the R-fMRI data obtained from 57 young healthy adults. Without prior knowledge, brain regions were optimally grouped into eight modules with wide overlap. Based on the topological features captured by graph theory analyses, overlapping regions were classified into an integrated club and a dominant minority club through clustering. Functional flexibility analysis found that overlapping regions in both clubs were significantly more flexible than non-overlapping ones. Lesion simulations revealed that targeted attack at overlapping regions were more damaging than random failure or even targeted attack at hub regions. In particular, overlapping regions in the dominant minority club were more flexible and more crucial for information communication than the others were. Together, our findings demonstrated the highly organized overlapping modular architecture and revealed the importance as well as complexity of overlapping regions from both topological and functional aspects, which provides important implications for their roles in executing multiple tasks and maintaining information communication.