Grand canonical Monte Carlo simulations on a cubic lattice are used to examine aggregation and phase separation of model amphiphiles with bulky head groups. The amphiphiles studied consist of a rigid, roughly spherical nanoparticle attached to one or more flexible chains. Overlapping distributions of energy and density are combined via histogram reweighting to obtain the free energy and osmotic pressure as a function of temperature and concentration. Finite size effects are used to distinguish between first order transitions to a disordered liquid or lamellar phase and continuous transitions to micelles. The transition type depends on the relative size of the solvophobic and neutral portions of the amphiphiles; none of the systems studied here exhibit both types of transition. The critical micellar concentration increases with temperature over the range of conditions examined. Solvophobic nanoparticles with neutral chains phase separate when the attached chain is short and form micelles for longer attached chains. For structures with neutral nanoparticles and solvophobic chains, amphiphile geometry plays a key role in determining whether the micelles that form are spheres or flat bilayers. Nanoparticles with many chains tend to form flat bilayers, while those with only one or two chains form nearly spherical aggregates. Particles with long chains undergo macroscopic phase separation instead of micellization, and the temperature range over which the first order transition occurs depends on the total volume occupied by the solvophobic segments.