  We have screened 71 two-dimensional (2D) materials with $C_3$ symmetry for
non-trivial second order topological order and find that 28 compounds exhibit
an obstructed atomic limit (OAL). In the case of $C_3$ symmetry, the second
order topology can be calculated from bulk symmetry indicator invariants, which
predict the value of fractional corner charges in symmetry conserving
nanoflakes. The procedure is exemplified by MoS$_2$ in the H-phase, which
constitutes a generic example of a 2D OAL material and the predicted fractional
corner charges is verified by direct calculations of nanoflakes with armchair
edges. We also determine the bulk topological polarization, which always lead
to gapless states at zigzag edges and thus deteriorates the concept of
fractional corner charges in nanoflakes with zigzag edges that are typically
more stable that armchair flakes. We then consider the case of TiCl$_2$, which
has vanishing polarization as well as an OAL and we verify that the edge states
of nanoflakes with zigzag edges may indeed by passivated such that the edges
remain insulating and the corner charges are well defined. For the 28 OAL
materials we find that 16 have vanishing polarization and these materials thus
constitute a promising starting point for experimental verification of second
order topology in a 2D material.
