This study investigates the dispersion relation of two dimensional photonic crystals conformed in a hybrid triangular-graphite configuration. This lattice includes, as limiting cases, two major well-known structures, the triangular and the graphite lattices. The analysis has been carried out by using preconditioned block-iterative algorithms for computing eigenstates of Maxwell's equations for periodic dielectric systems, using a plane-wave basis. We present the evolution of the so-called gap maps as a function of the radii of the structures. We conclude that a number of gaps exist for both TM and TE polarizations. We also predict the appearance of sizeable complete band gaps for structures the can be achieved using present fabrication capabilities.