I’m reading a note on the subgroup structure of classical subgroups. According to Corollary 2.2 (d), since $8=2^3$ is a prime power of even number $2$, ${\rm PSL}(2,8)$ has a maximal subgroup ${\rm PSL}(2,2)=S_3$. But ${\rm PSL}(2,8)$ has exactly $3$ conjugacy classes of maximal subgroups. The three non-conjugate maximal subgroups of ${\rm PSL}(2,8)$ are $C_9\rtimes C_2$, $C_7\rtimes C_2$ and $C_2^3\rtimes C_7$, none of which is $S_3$.
I believe I misunderstood something when reading the reference, but I don’t know what that is. Could you give me some help? Any help would be appreciated.
The list you refer to is incorrect. For $q_0=2$, the subgroup $\mathrm{PSL}_2(q_0)$ is not a maximal subgroup of $\mathrm{PSL}_2(q)$, where $q=q_0^r$ for $r$ a prime. In this case the subgroup is contained in the subgroup $D_{2(q+1)}$ or $D_{2(q-1)}$, depending on whichever is divisible by $3$.