The solution-based self-assembly of block copolymers (BCPs) into nanoparticulate or microparticulate inverse cubic mesophases, or polymer cubosomes, is of growing interest. This phenomenon could yield new polymeric mesoporous materials with three-dimensionally organized mazes of large water channels. In addition to the ratio of its hydrophilic and hydrophobic blocks, the architecture of a BCP critically influences self-assembly. BCP bilayers having triply periodic minimal surfaces with cubic lattice structures are formed. Here we report the synthesis and self-assembly of BCPs with T8 polyhedral oligomeric silsesquioxane (POSS), an inorganic cage molecule. POSS can be asymmetrically functionalized to host structural modules as branching units of the hydrophilic block that contain various spatial and chemical environments. BCPs, each containing seven poly(ethylene glycol) chains and the hydrophobic polystyrene block PEG350₇-POSS-b-PS, were prepared to investigate the effect of the highly branched hydrophilic block architecture on self-assembly. We found that the BCPs self-assembled into polymer cubosomes. Regardless of the block ratio, only the primitive cubic phase (Schwarz P surface, Imm space group) was observed in the cubosomes. This is in contrast to the self-assembly of conventional BCPs into inverse mesophases, in which the internal lattice preferentially transforms into double-diamond lattices to yield a Schwarz D surface as a consequence of block ratio changes. Our results suggest that BCP architecture may be related to the symmetry of the lattice formed via self-assembly in solution.