Fano point-line stabilizer intersection ($|G_p\cap G_L|=8\text{ or }6$)

There are 21 incident point-line pairs and 28 nonincident pairs in the Fano plane, with $G{L}_3(2)$ transitive on each type. Orbit-stabilizer theorem gives the two orders as $168/21=8$ and $168/28=6$.