We consider the discrete Allen-Cahn equation with cubic and quintic nonlinearity on the Lieb lattice. We study localized nonlinear solutions of the system that have linear multistability and hysteresis in their bifurcation diagram. In this work, we investigate the system's homoclinic snaking, i.e., snaking-like structure of the bifurcation diagram, particularly the effect of the lattice type. Numerical continuation using a pseudo-arclength method is used to obtain localized solutions along the bifurcation diagram. We then develop