Let A A be a finite-dimensional commutative semisimple algebra over a field k k and let V V be a finitely generated faithful A A -module. We study the action of the general linear group GL A ( V ) {\text {GL}}_A(V) on the set of all k k -subspaces of V V and show that, if the field k k is infinite, there are infinitely many orbits as soon as A A has dimension at least four. If A A has dimension two or three, the number of orbits is finite and independent of the field; in each such case we completely classify the orbits by means of a certain number of integer parameters and determine the structure of the quotient poset obtained from the action of GL A ( V ) {\text {GL}}_A(V) on the poset of k k -subspaces of V V .