In this paper, we will give a proof of the complete submodule structure of Specht modules corresponding to 2-part partitions for the general linear group GL( n, q) in characteristic p coprime to q (in non-defining characteristic). The multiplicities in the Specht module S ( n− l, l) being at most 1, we introduce a partial order on the set of composition factors. Let e be the lowest integer such that p∣1+ q+⋯+ q e−1 . We explicitly construct the j, such that D ( n− j, j) is a composition factor of S ( n− l, l) , by looking at well-defined sets I of exponents of p whose coefficients in the p-adic expansion of ⌊( l− j)/ e⌋ are relevant. We thus parametrise μ I ( l)=( n− j, j). The family of such sets I then forms a partially ordered set under inclusion. We show that this is isomorphic to the poset of composition factors in S ( n− l, l) . The results in this paper are not to be confused with the results for 2-column partition Specht modules in the defining characteristic of the general linear group, obtained in [A.M. Adamovich, PhD thesis, Moscow State University, 1992] and discussed in [A. Kleshchev, J. Sheth, J. Algebra 221 (1999) 705–722]. If we were to take q≡1mod p, we would find the same submodule structure as in the corresponding Specht modules of the symmetric group (cf. [A. Kleshchev, J. Sheth, J. Algebra 221 (1999) 705–722]).