In response to Woodroofe, thank you but the question is about finite local algebras. Given a unital commutative ring R, by an R-algebra it is meant, as usual, a ring A with unity for which there exists a ring homomorphism f:R->Z(R) that maps the unity of R to the unity of A, where Z(R) is the center of A. So, R need not be a field. Moreover, by a subalgebra of A it is meant a unital subring of A whose unity is the same as the unity of A. Finally, as I believe Suarez-Alvarez was pointing out, the subfield lattice of a finite field is indeed distributive.

Perhaps, to apply said theorem of Ore, it would help to ask whether every such lattice is isomorphic to the subgroup lattice of a finite group.

Thank you, Ralph, for the reference. In response to Goldstern, yes, thank you, (a) and (b) is precisely what I meant if in (b) by "the family" it is meant the lattice L (a priori) and not necessarily the entire family of all subalgebras as in (a). In any case, these two axioms are basic parts of the definition of a lattice and I hope I have stated the problem clearly enough.

That is, the simplification given above is true for finite commutative algebras of prime characteristic.