Plane wave propagation in infinite two dimensional self-similar and non-self-similar square periodic lattices have been investigated using Floquet-Bloch Theory. Finite element analysis was applied to calculate dispersion properties for different hierarchical lattice structures along the irreducible Brillouin zone (IBZ) path. The structural symmetry of square hierarchical lattices were exploited. The mechanism for generation of bandgaps were analyzed by studying the mode shapes for different characteristic length ratios. The effects of material symmetry for symmetric hierarchical lattices were analyzed, the results of which demonstrated that material asymmetry generates bandgaps at low frequency ranges. A new technique was also introduced to classify P and S waves. By observing the mode shapes of P and S waves, polarized bandgaps were identified to selectively suppress individual bands. Changing the material properties of hierarchical squares helps engineer individual modes of P to generate directional bandgaps. An extension to the proposed concept applied to triangular and hexagonal lattices is also presented.