Frank-Kasper phases, a family of ordered structures formed from particles with spherical motifs, are found in a host of materials, such as metal alloys, inorganic colloids and various types of soft matter. All the experimentally observed Frank-Kasper phases can be constructed from the basic units of three fundamental structures called the A15, C15 and Z phases. The Z phase, typically observed in metal alloys, is associated with a relatively large volume ratio between its constituents, and this constraint inhibits its formation in most self-assembled single-component soft-matter systems. We have assembled a series of nanosized shape amphiphiles that comprise a triphenylene core and six polyhedral oligomeric silsesquioxane cages grafted onto it through linkers to give a variety of unconventional structures, which include the Z phase. This structure was obtained through fine tuning of the linker lengths between the core and the peripheral polyhedral oligomeric silsesquioxane cages, and exhibits a relatively large volume asymmetry between its constituent polyhedral particle motifs.