The self-assembly of miktoarm star quaterpolymers, composed of one solvophilic arm and three solvophobic arms connected at a common junction point, in dilute solutions has been investigated using a simulated annealing technique. By tuning the strength of incompatibility between the different solvophobic arms (alpha) or the quaterpolymer composition, unique multicompartment aggregates are predicted and their formation mechanisms are elucidated. It is the competition between the interfacial energy and the surface energy that results in the morphological transitions. At large alpha values, interfacial energy dominates over the system, and hence, hexagonally shaped laterally patterned nanosheets with smaller interfacial areas always form to decrease the interfacial energy. The smaller interfacial area of this morphology comes from the high local order of the three-colored short-cylinders where any three centers of mass of the nearest neighboring A, B, and C domains constitute an approximately regular triangle. The paving of the nearly rigid triangles results in the hexagonal shape of the whole nanosheet. At relatively small alpha values, transitions from vesicles to nanosheets and further to micelles are observed with increasing the volume fraction of the solvophilic arm, f(P). The vesicles can be of lateral patterns with six protrusions which distributed symmetrically, and each corresponds to a packing defect of the three-colored short-cylinders. Micelles can be highly anisotropic and multiple-patched. The higher curvature of micelles favors the maximal contact for the solvophilic arm with solvent molecules. The morphological sequence obtained with increasing f(P) has a similar trend to that obtained with decreasing alpha. Our simulation results are compared with related experiments. The self-assembly of miktoarm star quaterpolymers could provide a powerful strategy for fabricating nanoscaled multicompartment aggregates with special shape and morphology, which may offer tremendous potential in nanotechnology.