  We propose two mechanisms to realize the second order topological insulator
(SOTI) state in spinless hexagonal lattices, viz., chemical modification and
anti-Kekul\'e/Kekul\'e distortion of hexagonal lattice. Correspondingly, we
construct two models and demonstrate the nontrivial band topology of the SOTI
state characterized by the second Stiefel-Whitney class $w_2$ in the presence
of inversion symmetry ($\textit{P}$) and time-reversal symmetry ($\textit{T}$).
Based on the two mechanisms and using first-principles calculations and
symmetry analysis, we predict three categories of real light element material
candidates, i.e., hydrogenated and halogenated 2D hexagonal group IV materials
XY (X=C, Si, Ge, Sn, Y=H, F, Cl), 2D hexagonal group V materials (blue
phosphorene, blue arsenene, and black phosphorene, black arsenene), and the
recent experimentally synthesized anti-Kekul\'e/Kekul\'e order graphenes and
the counterparts of silicene/germanene/stanene. We explicitly demonstrate the
nontrivial topological invariants and existence of the protected corner states
with fractional charge for these candidates with giant bulk band gap (up to 3.5
eV), which could facilitate the experimental verification by STM. Our
approaches and proposed abundant real material candidates will greatly enrich
2D SOTIs and promote their intriguing physics research.
