  Notion of square-root topological insulators have been recently generalized
to higher-order topological insulators. In two-dimensional square-root
higher-order topological insulators, emergence of in-gap corner states are
inherited from the squared Hamiltonian which hosts higher-order topology. In
this paper, we propose that the martini lattice model serves as a concrete
example of higher-order topological insulators. Furthermore, we also propose a
suquare-root higher-order topological insulator based on the martini model.
Specifically, we propose that the honeycomb lattice model with two-site
decoration, whose squared Hamiltonian consists of two martini lattice models,
realizes square-root higher-order topological insulators. We show, for both of
these two models, that in-gap corner states appear at finite energies and that
they are portected by non-trivial bulk $\mathbb{Z}_3$ topological invariant.
