Note first that is a vector space over . Moreover, if then it's also trivially a subspace of since it's closed under addition. Thus, the subgroups of index , which are precisely the subgroups of order , which are precisely the subspaces of dimension . Now, using the common formula that the number of subspaces of an -dimensional -space is (which is just a simple counting argument about choosing linearly independent subsets of size --it's on page 412 of Dummit and Foote for example) and pluggin in gives the desired result.