  Finite (upper) nearlattices are essentially the same mathematical entities as
finite semilattices, finite commutative idempotent semigroups, finite
join-enriched meet semilattices, and chopped lattices. We prove that if an
$n$-element nearlattice has at least $83\cdot 2^{n-8}$ subnearlattices, then it
has a planar Hasse diagram. For $n>8$, this result is sharp.
