We perform molecular simulations to study the self-assembly of tetratethered nanoparticles with a cubic geometry. We develop a minimal model of the tethered nanoscale building block (NBB) to represent a polyhedral oligomeric silsesquioxane (POSS) molecule with polymeric functionalities based on information about the molecular structure and interactions obtained from ab initio density functional theory calculations. Using this model, we explore the rich nanostructures formed from self-assembly of the NBBs and make analogies with the morphologies observed in block copolymer, surfactant, and liquid crystalline systems. On the basis of the assembled structures produced and determination of the location and nature of the order-disorder transitions in the system, we propose phase diagrams to describe the behavior of these molecules. We find that qualitative similarities exist between the phase diagrams for the tetratethered NBBs and those for block copolymer and surfactant systems.