LetAbe a finite dimensional commutative semisimple algebra over a fieldkand letVbe a finitely generatedA—module. In previous work the author examined the action of the general linear group GLA(V) on the Grassmannians ofk—subspaces ofV. The present paper examines the structure of the orbits in greater detail, in particular by working out the structure of the stabilizers in each of the cases when dimkA≤3. From an algebraic point of view the most interesting situation occurs forAa cubic extension field ofk