Hi mathoverflow community, may be some one may give me a hint on the following problem before I spend much time on brute force search.

For $q$ a prime number and $n=6$, let $\mathbb {F}_{q}^{n}$ be an $n-$dimensional vector space over $GF_{q}$.
Furthermore let $ U_i, U_j$ and $U_k$ be disjoint subspaces of $\mathbb{F}_{q}^n$ having each dimension $2$, with $i, j, k \in \{1,\dots, n\}$, $i\neq j \neq k$.
What is the *effective* maximum number of disjoint subspaces of dimension $2$ in $\mathbb{F}_{q}^{n}$ so that $\langle U_i, U_j \rangle \cap U_k = \{0\}$?

Please note the keyword effective because using the Gaussian binomial coefficients it is possible to calculate the maximum number of all the subsets possible in $\mathbb{F}_{q}^{n}$ but probably not all fulfill the condition that $\langle U_i, U_j \rangle \cap U_k = \{0\}$.

Thanks.