The Maximal Subgroups of The Symplectic Group Psp(8, 2)

The purpose of this paper is to study maximal subgroups of the symplectic group PSp(8, 2). The main result is a list of maximal subgroups called "the main theorem " which has been proved by using Aschbacher’s Theorem ([1]). Thus, this work is divided into two main parts: Part (1): In this part, we will find the maximal subgroups in the classes C1 – C8 of Aschbacher’s Theorem ([1]). Part (2): In this part, we will find the maximal subgroups in the class C9 of Aschbacher’s Theorem ([1]), which are the maximal primitive subgroups H of G that have the property that the minimal normal subgroup M of H is not abelian group and simple, thus, we divided this part into two cases: Case (1): M is generated by transvections: In this case, we will use result of Kantor ([2]). Case (2): M is a finite primitive subgroup of rank three: In this case, we will use the classification of Kantor and Liebler ([8])