  We develop a class of supercell photonic crystals supporting complete
photonic bandgaps based on breaking spatial symmetries of the underlying
primitive photonic crystal. One member of this class based on a two-dimensional
honeycomb structure supports a complete bandgap for an index-contrast ratio as
low as $n_{high}/n_{low} = 2.1$, making this the first such 2D photonic crystal
to support a complete bandgap in lossless materials at visible frequencies. The
complete bandgaps found in such supercell photonic crystals do not necessarily
monotonically increase as the index-contrast in the system is increased,
disproving a long-held conjecture of complete bandgaps in photonic crystals.
