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. 



Structure of symplectic group over finite fields
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, $q \equiv \pm 1 \bmod 12$


$2.A\_7$, $q=7$


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


