Amphiphilic molecules such as cetyl trimethylammonium bromide (CTAB) and minerals such as graphene oxide (GO) self-assemble to form lyotropic liquid crystal (LLC) systems in water. Here we describe the preparation, structures and mechanical properties of the hybrid CATB and GO LLC system. We present a series of phase diagrams for the CTAB/GO/water ternary system with independently varying CTAB and GO loadings, covering various combinations of the isotropic and the different lyotropic phases of each that occur as a function of concentration and temperature. The corresponding LLC microstructures are identified through polarised optical microscopy (POM) in a confined environment. We find that GO either promotes suppresses the formation of the different LLC phases of CTAB depending on the concentration of GO and whether the GO itself is in its isotropic or lyotropic phase, resulting in the formation of a complex hybrid system. GO also significantly depresses the melting point of CTAB at moderate loadings, but the melting point recovers for higher GO loading. Rheology of the CTAB/GO/water system reveals a reinforcement effect of the GO sheets through the formation of CTAB/GO complexes that alter the hexagonal phase nanostructure. Our work reveals the diverse effects on molecular LC phases by an interpenetrating mineral LC phase, and the potential to design a wide range of novel nanostructured hybrid LLC materials.