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This paper reports hierarchical hybridization as a mode-mixing scheme to account for the unique optical properties of non-Bravais lattices of plasmonic nanoparticles (NPs). The formation of surface lattice resonances (SLRs) mediated by localized surface plasmons (LSPs) of different multipolar orders (dipole and quadrupole) can result in asymmetric electric near-field distributions surrounding the NPs. This asymmetry is because of LSP hybridization at the individual NP level from LSPs of different multipole order and at the unit cell level (NP dimer) from LSPs of the same multipole order. Fabricated honeycomb lattices of silver NPs exhibit ultrasharp SLRs at the Γ point that can also facilitate nanolasing. Modeling of the stimulated emission process revealed that the multipolar component of the lattice plasmon mode was responsible for feedback for lasing. By leveraging multipolar LSP responses in Al NP lattices, we achieved two distinct Γ point band-edge modes from a single honeycomb lattice. This work highlights how multipolar LSP coupling in plasmonic lattices with a non-Bravais symmetry has important implications for the design of SLRs and their associated plasmonic near-field distributions. These relatively unexplored degrees of freedom can decrease both ohmic and radiative losses in nanoscale systems and enable SLRs to build unanticipated connections among photonics and nanochemistry.