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The subgroups of ${\rm PSL}_3(q)$ for odd $q$ were first enumerated by H.H. Mitchell in 1911. (The case $q$ even was done by R.W. Hartley in 1925/6.)

Table 8.3 of the book "The Maximal Subgroups of Low-Dimensional Finite Classical Groups" by Bray, Holt and Roney-Dougal provides a convenient list. Using that it is not hard to answer your question.

For $q$ odd, the maximal subgroups of ${\rm SL}_3(q)$ of odd index are as follows:

(i) Two classes of maximal parabolic subgroups with structure ${\rm E}_q^3:{\rm GL}_2(q)$ and index $q^2+q+1$, where ${\rm E}_q$ denotes an elementary abelian group of order $q$ (the additive group of the field). These two classes are interchanged by the graph (inverse-transpose) automorphism of ${\rm SL}_3(q)$.

(ii) When $q \equiv 1 \bmod 4$, we have one class of imprimitive subgroups with structure $(q-1)^2:S_3$.

(iii) When $q = q_0^r$ for some odd prime $r$, we have $\gcd(\frac{q-1}{q_0-1},3)$ classes of subgroups with the structure ${\rm SL}_3(q_0).(\gcd(\frac{q-1}{q_0-1},3))$. When there are three such classes, they are all conjugate under a diagonal outer automorphism of ${\rm SL}_3(q)$.

The subgroups of ${\rm PSL}_3(q)$ for odd $q$ were first enumerated by H.H. Mitchell in 1911. (The case $q$ even was done by R.W. Hartley in 1925/6.)

Table 8.3 of the book "The Maximal Subgroups of Low-Dimensional Finite Classical Groups" by Bray, Holt and Roney-Dougal provides a convenient list. Using that it is not hard to answer your question.

For $q$ odd, the maximal subgroups of ${\rm SL}_3(q)$ of odd index are as follows:

(i) Two classes of maximal parabolic subgroups with structure ${\rm E}_q^2:{\rm GL}_2(q)$ and index $q^2+q+1$, where ${\rm E}_q$ denotes an elementary abelian group of order $q$ (the additive group of the field). These two classes are interchanged by the graph (inverse-transpose) automorphism of ${\rm SL}_3(q)$.

(ii) When $q \equiv 1 \bmod 4$, we have one class of imprimitive subgroups with structure $(q-1)^2:S_3$.

(iii) When $q = q_0^r$ for some odd prime $r$, we have $\gcd(\frac{q-1}{q_0-1},3)$ classes of subgroups with the structure ${\rm SL}_3(q_0).\gcd(\frac{q-1}{q_0-1},3)$. When there are three such classes, they are all conjugate under a diagonal outer automorphism of ${\rm SL}_3(q)$.

The subgroups of ${\rm PSL}_3(q)$ for odd $q$ were first enumerated by H.H. Mitchell in 1911. (The case $q$ even was done by R.W. Hartley in 1925/6.)

Table 8.3 of the book "The Maximal Subgroups of Low-Dimensional Finite Classical Groups" by Bray, Holt and Roney-Dougal provides a convenient list. Using that it is not hard to answer your question.

For $q$ odd, the maximal subgroups of ${\rm SL}_3(q)$ of odd index are as follows:

(i) Two classes of maximal parabolic subgroups with structure ${\rm E}_q^3:{\rm GL}_2(q)$ and index $q^2+q+1$, where $E_q$ denotes an elementary abelian group of order $q$ (the additive group of the field). These two classs are interchanged by the graph (inverse-transpose) automorphism of ${\rm SL}_3(q)$.

(ii) When $q \equiv 1 \bmod 4$, we have one class of imprimitive subgroups with structure $(q-1)^2:S_3$.

(iii) When $q = q_0^r$ for some odd prime $r$, we have $\gcd(\frac{q-1}{q_0-1},3)$ classes of subgroups with the structure ${\rm SL}_3(q_0).(\gcd(\frac{q-1}{q_0-1},3))$. When there are three such classes, they are all conjugate under a diagonal outer automorphism of ${\rm SL}_3(q)$.

The subgroups of ${\rm PSL}_3(q)$ for odd $q$ were first enumerated by H.H. Mitchell in 1911. (The case $q$ even was done by R.W. Hartley in 1925/6.)

Table 8.3 of the book "The Maximal Subgroups of Low-Dimensional Finite Classical Groups" by Bray, Holt and Roney-Dougal provides a convenient list. Using that it is not hard to answer your question.

For $q$ odd, the maximal subgroups of ${\rm SL}_3(q)$ of odd index are as follows:

(i) Two classes of maximal parabolic subgroups with structure ${\rm E}_q^3:{\rm GL}_2(q)$ and index $q^2+q+1$, where ${\rm E}_q$ denotes an elementary abelian group of order $q$ (the additive group of the field). These two classes are interchanged by the graph (inverse-transpose) automorphism of ${\rm SL}_3(q)$.

(ii) When $q \equiv 1 \bmod 4$, we have one class of imprimitive subgroups with structure $(q-1)^2:S_3$.

(iii) When $q = q_0^r$ for some odd prime $r$, we have $\gcd(\frac{q-1}{q_0-1},3)$ classes of subgroups with the structure ${\rm SL}_3(q_0).(\gcd(\frac{q-1}{q_0-1},3))$. When there are three such classes, they are all conjugate under a diagonal outer automorphism of ${\rm SL}_3(q)$.

The subgroups of ${\rm PSL}_3(q)$ for odd $q$ were first enumerated by H.H. Mitchell in 1911. (The case $q$ even was done by R.W. Hartley in 1925/6).

Table 8.3 of the book "The Maximal Subgroups of Low-Dimensional Finite Classical Groups" by Bray, Holt and Roney-Dougal provides a convenient list. Using that it is not hard to answer your question.

For $q$ odd, the maximal subgroups of ${\rm SL}_3(q)$ of odd index are as follows:

(i) Two classes of maximal parabolic subgroups with structure ${\rm E}_q^3:{\rm GL}_2(q)$ and index $q^2+q+1$, where $E_q$ denotes an elementary abelian group of order $q$ (the additive group of the field). These two classs are interchanged by the graph (inverse-transpose) automorphism of ${\rm SL}_3(q)$.

(ii) When $q \equiv 1 \bmod 4$, we have one class of imprimitive subgroups with structure $(q-1)^2:S_3$.

(iii) When $q = q_0^r$ for some odd prime $r$, we have $\gcd(\frac{q-1}{q_0-1},3)$ classes of subgroups with the structure ${\rm SL}_3(q_0).(\gcd(\frac{q-1}{q_0-1},3))$. When there are three such classes, they are all conjugate under a diagonal outer automorphism of ${\rm SL}_3(q)$.

The subgroups of ${\rm PSL}_3(q)$ for odd $q$ were first enumerated by H.H. Mitchell in 1911. (The case $q$ even was done by R.W. Hartley in 1925/6.)

Table 8.3 of the book "The Maximal Subgroups of Low-Dimensional Finite Classical Groups" by Bray, Holt and Roney-Dougal provides a convenient list. Using that it is not hard to answer your question.

For $q$ odd, the maximal subgroups of ${\rm SL}_3(q)$ of odd index are as follows:

(i) Two classes of maximal parabolic subgroups with structure ${\rm E}_q^3:{\rm GL}_2(q)$ and index $q^2+q+1$, where $E_q$ denotes an elementary abelian group of order $q$ (the additive group of the field). These two classs are interchanged by the graph (inverse-transpose) automorphism of ${\rm SL}_3(q)$.

(ii) When $q \equiv 1 \bmod 4$, we have one class of imprimitive subgroups with structure $(q-1)^2:S_3$.

(iii) When $q = q_0^r$ for some odd prime $r$, we have $\gcd(\frac{q-1}{q_0-1},3)$ classes of subgroups with the structure ${\rm SL}_3(q_0).(\gcd(\frac{q-1}{q_0-1},3))$. When there are three such classes, they are all conjugate under a diagonal outer automorphism of ${\rm SL}_3(q)$.