  We consider a trimer Su-Schrieffer-Heeger (SSH) tight-binding Hamiltonian
keeping up to next-nearest-neighbor (NNN) hopping terms and on-site potential
energy. The Bloch Hamiltonian can be expressed in terms of all the eight
generators (i.e. Gell-Mann matrices) of the SU(3) group. We provide exact
analytical expressions of three dispersive energy bands and the corresponding
eigenstates for any choices of the system parameters. The system lacks full
chiral symmetry since the energy spectrum is not symmetric around zero, except
at isolated Bloch wavevectors. We explore parity, time reversal, and certain
special chiral symmetries for various system parameters. We discuss the
bulk-boundary correspondence by numerically computing the Zak phase for all the
bands and the boundary modes in the open boundary condition. There are three
different kinds of topological phase transitions which are classified based on
the gap closing points in the Brillouin zone (BZ) while tuning the
nearest-neighbor (NN) and NNN hopping terms. We find that quantized changes (in
units of $\pi$) in two out of three Zak phases characterize these topological
phase transitions. We propose another bulk topological invariant, namely the
$\textit{sub-lattice winding number}$, which also characterizes the topological
phase transitions changing from $ \nu^{\alpha} = 0 \leftrightarrow 2 $ and $
\nu^{\alpha} = 0 \leftrightarrow 1 \leftrightarrow 2 $ ($\alpha $: sub-lattice
index). The sub-lattice winding number provides a relatively simple analytical
understanding of topological phases and may help in characterizing topological
phases of systems without chiral symmetry.
