I've been searching around for a while now and can't seem to find a clear explanation of what a subgroup lattice of a group actually is. I see the vertex set is given by the subgroups of the group, but how is the edge set related? My guess is that vertices are connected if one is a subset of the other - for example a subgroup containing just the identity would be connected to everything.
A lattice is a partially ordered set in which every two element has a least upper and a greatest lower bound (these are the lattice operations).
The elements of the subgroup lattice of a group are indeed the subgroups, and the partial order is defined by containment, so if you want to depict it you can draw an arrow from any subgroup $H_1$ to any other subgroup $H_2$ that contains it.. (If $G$ is finite, it is enough to draw down the coverings as @JackSchmidt commented, as, by transitivity it will generate the whole lattice.)
Anyway, the greatest lower bound of subgroups $H_1,\,H_2$ is $\ H_1\cap H_2$, and their least upper bound is the subgroup $\langle H_1\cup H_2\rangle$ generated by their union.