  We study the lattice of submonoids of the uniform block permutation monoid
containing the symmetric group (which is its group of units). We prove that
this lattice is distributive under union and intersection by relating the
submonoids containing the symmetric group to downsets in a new partial order on
integer partitions. Furthermore, we show that the sizes of the
$\mathscr{J}$-classes of the uniform block permutation monoid are sums of
squares of dimensions of irreducible modules of the monoid algebra.
