  Finite monoids that generate monoid varieties with uncountably many
subvarieties seem rare, and surprisingly, no finite monoid is known to generate
a monoid variety with countably infinitely many subvarieties. In the present
article, it is shown that there are, nevertheless, many finite monoids that
generate monoid varieties with continuum many subvarieties; these include any
finite inherently non-finitely based monoid and any monoid for which $xyxy$ is
an isoterm. It follows that the join of two Cross monoid varieties can have a
continuum cardinality subvariety lattice that violates the ascending chain
condition.
  Regarding monoid varieties with countably infinitely many subvarieties, the
first example of a finite monoid that generates such a variety is exhibited. A
complete description of the subvariety lattice of this variety is given. This
lattice has width three and contains only finitely based varieties, all except
two of which are Cross.
