If Fq is the nite eld of characteristic p and order q = p s , let F(Fq) be the category whose objects are functors from nite dimensional Fq{vector spaces to Fq{vector spaces, and with morphisms the natural trans- formations between such functors. We dene an innite lattice of thick subcategories of F(Fq). Our main re- sult then identies various subquotients as categories of modules over products of symmetric groups, via recollement diagrams. Our lattice of thick subcategories is a renemen t of the Eilenberg{MacLane polynomial degree ltration F 0 (Fq) F 1 (Fq) F 2 (Fq) : : : of F(Fq) which has been extensively studied and used in the algebraic K{theory lit- erature. Our main theorem implies a description of F d (Fq)=F d 1 (Fq) that renes and extends earlier results of Pirashvili and others. If q r, one of the subcategories is the Friedlander{Suslin category P r of 'strict polynomial functors of degree r', equivalent to the category of modules over the Schur algebra S(n; r) with n r. Our results can thus also be viewed as rening and extending the classic relationship between S(n; r){modules and r{modules via the Schur functor. In fact, (essentially) all our subcategories are equivalent to categories of modules over various nite dimensional algebras, and our lattice can be interpreted in terms of lattices of idempotent two sided ideals in these generalized Schur algebras. Applications include a simple proof, free of algebraic group theory, of a generalized Steinberg Tensor Product Theorem. This then implies the classic theorem for GLn(Fq), shedding some new light on this classic result. Our tensor product theorem is then used to study when various of our generalized Schur algebras are Morita equivalent. We use two technical tricks which may be of some independent interest. Firstly, we 'twist' by an action of the Galois group Gal(Fq; Fp) to be able to work entirely with vector spaces over the prime eld Fp, making discussions of 'base change' unnecessary. Secondly, we systematically use 'functors with product', a.k.a. lax symmetric monoidal functors, to dene our subcategories.