 Solution

The stabilizer of the indicated flag consists of block upper triangular matrices with diagonal block sizes $1,2,2$. The diagonal blocks can be any elements of $\mathrm{GL}(1,q)$, $\mathrm{GL}(2,q)$ and $\mathrm{GL}(2,q)$. The upper off-diagonal blocks have $2+2+4=8$ arbitrary entries. Thus the stabilizer order is $(q-1)|\mathrm{GL}(2,q){|}^2{q}^8$. The orbit-stabilizer theorem gives