= Fano plane duality group
{c}
{title2=$GL_3(2)\rtimes\langle\tau\rangle,\quad|\overline G|=336$}
The <Fano plane duality group> includes both incidence-preserving permutations of <points> and <projective lines> and the dualities that exchange those two types. Its type-preserving <subgroup> is $GL_3(\mathbb F_2)$, of order $(8-1)(8-2)(8-4)=168$. A polarity supplies the <inverse-transpose automorphism> $A\mapsto(A^{-1})^{\mathsf T}$, giving the <semidirect product>
$$
\overline G=GL_3(2)\rtimes C_2,\qquad |\overline G|=336.
$$
There are $7\cdot3=21$ unordered incident point-line pairs and $7\cdot4=28$ unordered nonincident point-line pairs. The two transitive actions therefore have stabilizers of orders $336/21=16$ and $336/28=12$, respectively. These are <dihedral groups>. The normalizer of a <Sylow subgroup> of order seven has order42 and is $C_7\rtimes C_6$.
Every maximal proper <subgroup> is a <conjugate subgroup> of one of these three types or of the index-two collineation <subgroup>. The extension is also isomorphic to $PGL_2(7)$, identifying the geometric stabilizers with the standard finite-group classification.<ATLAS of Finite Group Representations>[L2(7)=L3(2) page, automorphisms and maximal subgroups of L2(7):2]{ref}
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