The band structures of shear horizontal (SH) waves in a periodically corrugated piezoelectric plate (PCPP) are studied by using the supercell plane wave expansion (SC-PWE) method. The effect of plate symmetry on the defect state caused by a defect in the plate is investigated in detail. The PCPPs with different types of symmetry give rise to different kinds of band gaps and the associated defect states. The increase of defect size lowers the frequency of defect bands, and it can be used to tune the narrow-passband frequencies in acoustic band gaps. Symmetry breaking is also introduced by reducing the lower corrugation depth of the PCPP. Results show that symmetry breaking leads to both the appearance and disappearance of new kinds of gaps and the corresponding defect bands in these gaps.