Discrete protein assemblies ranging from hundreds of kilodaltons to hundreds of megadaltons in size are a ubiquitous feature of biological systems and perform highly specialized functions. Despite remarkable recent progress in accurately designing new self-assembling proteins, the size and complexity of these assemblies has been limited by a reliance on strict symmetry. Inspired by the pseudosymmetry observed in bacterial microcompartments and viral capsids, we developed a hierarchical computational method for designing large pseudosymmetric self-assembling protein nanomaterials. We computationally designed pseudosymmetric heterooligomeric components and used them to create discrete, cage-like protein assemblies with icosahedral symmetry containing 240, 540, and 960 subunits. At 49, 71, and 96 nm diameter, these nanoparticles are the largest bounded computationally designed protein assemblies generated to date. More broadly, by moving beyond strict symmetry, our work represents an important step towards the accurate design of arbitrary self-assembling nanoscale protein objects.