Optimization of a truncated, dielectric photonic crystal cavity leads to configurations that are far from truncated crystal cavities, and which have significantly better radiation confinement. Starting from a two-dimensional truncated photonic crystal cavity with optimal Q-factor, moving the rods from the lattice positions can increase the Q-factor by orders of magnitude, e.g., from 130 to 11 000 for a cavity constructed from 18 rods. In the process, parity symmetry breaking occurs. Achieving the same Q-factor with a regular lattice requires 60 rods. Therefore, using optimized irregular structures for photonic cavities can greatly reduce material requirements and device size.