# Structure of symplectic group over finite fields

We are working over the finite field $\mathbb{F}_{q}$ of odd prime characteristic $p$ and of cardinality $q$ some power of $p$. We recall the symplectic group $Sp(4,\mathbb{F}_{q})$ as the group of transformations over $\mathbb{F}_{q}^{4}$ preserving a non degenerate alternate bilinear form, and we denote by $PSp(4,\mathbb{F}_{q})$ the quotient by its center. I would like to know all the possible maximal subgroups of this projective group or the initial one. A good reference may be of precious help.

• The following paper by Aschbacher is probably the canonical reference: Aschbacher, Michael. "On the maximal subgroups of the finite classical groups." Inventiones mathematicae 76.3 (1984): 469-514. There's also a book by Kleidman and Liebeck which gives more details. In fact, §3.5 of that book gives a list of all the subgroups as far as I can tell. – Jay Taylor Feb 14 '15 at 12:26

You can find tables of the maximal subgroups of all (almost) simple classical groups in dimensions up to $12$ in the book:

The Maximal Subgroups of the Low-Dimensional Finite Classical Groups, John N. Bray,Derek F. Holt, Colva M. Roney-Dougal, Cambridge University Press, 2013.

The maximal subgroups of ${\rm Sp}_4(q)$ for odd $q$ were first classified in:

H. H. Mitchell. The subgroups of the quaternary abelian linear group. Trans. Amer. Math. Soc. 15 (1914), 379–396.

Here is a complete list. The notation is similar to that used in the ATLAS. There is one conjugacy class of each type except where otherwise stated.

$q^{1+2}\!:\!((q-1) \times {\rm Sp}_2(q))$ (reducible)

$q^3\!:\!{\rm GL}_2(q)$ (reducible)

${\rm Sp}_2(q)^2\!:\!2$ (imprimitive)

${\rm GL}_2(q).2$ (imprimitive)

${\rm Sp}_2(q^2)\!:\!2$ (semilinear)

${\rm GU}_2(q).2$ (semilinear)

${\rm Sp}_4(q_0).(2,r)$, $(2,r)$ classes, $q=q_0^r$, $r$ prime (subfield)

$2^{1+4}.S_5$, $2$ classes, $q$ prime, $q \equiv \pm 1 \bmod 8$

$2^{1+4}.A_5$ $q$ prime, $q \equiv \pm 3 \bmod 8$

$2.A_6$, $q$ prime, $q \equiv \pm 5 \bmod 12$, $q \ne 7$

$2.S_6$, $2$ classes, $q$ prime, $q \equiv \pm 1 \bmod 12$

$2.A_7$, $q=7$

${\rm SL}_2(q)$, $p \ge 5$, $q>7$.

• Thanks for your answer. Mitchell gave another classification which is more geometrical and I would like to know the correspendance between the one you gave, especially the case where the maximal subgroup stabilizes a quadric, does he stabilze a subspace for example or does a subgroup of index $2$? – Silam Apr 1 '15 at 10:33