A finite n-element semilattice is planar if it has at least 127 * 2^(n-8) subsemilattices, and this bound is sharp for n > 8 via an explicit non-planar counterexample with one fewer subsemilattice.
K.: Yet two additional large numbers of subuniverses of finite lattices, Discussiones Mathematicae—General Algebra and Applications, accepted for publication
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One hundred twenty-seven subsemilattices and planarity
A finite n-element semilattice is planar if it has at least 127 * 2^(n-8) subsemilattices, and this bound is sharp for n > 8 via an explicit non-planar counterexample with one fewer subsemilattice.