An occurrence of noncrossing partitions in type B/C

There is a map from the noncrossing partitions NC(n) on the set [n] to subgroups of an elementary abelian group of rank n-1 over the field with two elements. The image of this map is surjective for n=1,2,3, but then fails to be surjective for larger n (missing two subgroups for n=4). We discuss an occurrence of this map for the algebraic groups of types B and C, as well as a conjectural (related) occurrence involving Springer fibers. This is joint work with George Lusztig.