In large-scale brain network dynamics, brain nodes switching between modules has been found to correlate with cognition. However, how the brain nodes engage in this kind of reorganization of modules is unclear. Based on a functional magnetic resonance imaging dataset, we construct dynamic brain functional networks and investigate nodal module temporal dynamic behavior by applying the multilayer network analysis approach. We reveal three cohesive communities that are groups of brain nodes linked in the same community during brain module dynamic reorganization. We show that the cohesive communities have higher clustering coefficients and lower characteristic path lengths than the controlled community, indicating cohesive communities are the parts of brain networks with high information processing efficiency. The smaller sample entropy of functional connectivity in cohesive communities also proves their property of being more "static" compared with the controlled community in brain dynamics. Specifically, compared with the controlled community, the functional connectivity of cohesive communities is restricted strictly by structure connectivity and shows more similarity to structure connectivity. More importantly, we find that the cohesive communities are stable not only in the resting state but also when processing cognitive tasks. Our results not only show that cohesive communities may be the fundamental community organization to support brain network dynamics but also provide insights into the intrinsic structural relationship between the resting state and task states of the brain.