It's known that the Gaussian binomial coefficient $\binom{n}{k}_p$ counts $k$-dimensional subspaces of the vector space $\mathbb{F}_p^n$ ($p \in \mathbb{P}$). However, if we use another field, say $\mathbb{F}_{p^m}$ ($m \ge 2$), the coefficients don't work anymore.
For instance, when $n = 2$, we have that the actual number of subspaces of the vector space $\mathbb{F}_{p^m}^2$ is $$C_{p^m} = \dfrac{p^{m+2}+p^{m+1}-(2m+3)p+2m+1}{(p-1)^2}$$ (Ref: oeis). On the other hand, the coefficients give us another answer: $$G_{p^m} = \sum_{k=0}^2 \binom{2}{k}_{p^m} = p^m+3$$. When $m = 1$, two answers actually collide.
My questions are that why the coefficients don't work and how we can fix the counting process so that they work again.
I've googled quite a lot, and it seems that the coefficients do work. However, my "counter"-example has yet explained.