  We use a projective symmetry group analysis to determine all symmetric spin
liquids on the stuffed honeycomb lattice Heisenberg model. This lattice
interpolates between honeycomb, triangular and dice lattices, always preserving
hexagonal symmetry, and it already has one spin liquid candidate, TbInO$_3$,
albeit with strong spin-orbit coupling not considered here. In addition to the
stuffed honeycomb lattice itself, we gain valuable insight into potential spin
liquids on the honeycomb and triangular lattices, as well as how they might be
connected. For example, the sublattice pairing state proposed on the honeycomb
lattice connects to the uniform spinon Fermi surface that may be relevant for
the triangular lattice with ring exchange, while there are no spin liquids
competitive on both the $J_1-J_2$ honeycomb and triangular lattice limits. In
particular, we find three stuffed honeycomb descendants of the U(1) Dirac spin
liquid widely believed to be found on the $J_1-J_2$ triangular lattice. We also
discuss how spin liquids near the honeycomb limit can potentially explain the
physics of LiZn$_2$Mo$_3$O$_8$.
