The inclusions induce an ordering on the subtoposes of $\mathcal{E}$ that makes them a lattice with a co-Heyting algebra structure i.e. the join operator has a left adjoint subtraction operator. This corresponds to a dual Heyting algebra structure on the Lawvere-Tierney topologies.

The atoms in the lattice of subtoposes of $\mathcal{E}$ are precisely the Boolean two-valued subtoposes of $\mathcal{E}$ (cf. Caramello (2009) prop.10.1 p.58).

$\mathcal{E}$ (as every Grothendieck topos over $Set$) is the classifying topos of some geometric theory$T$ and it can be shown that subtoposes of $\mathcal{E}$ correspond precisely to deductively closed quotient theories of $T$ (cf. Caramello (2009), thm.3.6 p.15) i.e. passage to a subtopos corresponds to adding further geometric axioms to $T$ - localizing geometrically amounts to theory refinement logically.

References

F. Borceux, M. Korostenski, Open Localizations , JPAA 74 (1991) pp.229-238.

O. Caramello, Lattices of theories , arXiv:0905.0299v1 (2009). (pdf)

H. Forssell, Subgroupoids and quotient theories , TAC 28 no.18 (2013) pp.541-551. (pdf)