We performed a systematic structural search of high-pressure carbon allotropes for unit cells containing from 6 to 24 atoms using the minima hopping method. We discovered a series of new structures that are consistently lower in enthalpy than the ones previously reported. Most of these include (5 + 7)- or (4 + 8)-membered rings and can therefore be placed in the families proposed by H. Niu et al. [Phys. Rev. Lett. 108, 135501 (2012)]. However, we also found three more families with competitive enthalpies that contain (5 + 5 + 8)-membered rings, sp(2) motives, or buckled hexagons. These structures are likely to play an important role in dislocation planes and structural defects of diamond and hexagonal diamond.