The characteristic subspace lattice of a linear transformation

Given a square matrix A∈Mn(F), the lattices of the hyperinvariant (Hinv(A)) and characteristic (Chinv(A)) subspaces coincide whenever F≠GF(2). If the characteristic polynomial of A splits over F, A can be considered nilpotent. In this paper we investigate the properties of the lattice Chinv(J) when F=GF(2) for a nilpotent matrix J. In particular, we prove it to be self-dual.