So I'm aware that $O_K$ is a lattice in Minkowski space, say in $n$ dimensions when $[K: \mathbb{Q}]=n$, and that ideals and orders are sublattices of $O_K$ of full rank (or as Neukirch calls it, complete sublattices). However, imagining $O_K$ as these discrete lattice points, it's clear that there could be a wide number of sublattices appearing in $O_K$.
My suspicion for this problem is that there are sublattices that do not correspond to any ideal since ideals are not just sublattices, but must also have the condition of multiplicatively absorbing elements of $O_K$. However, while an order also satisfies the ring properties, these are proven from their lattice properties, so I'm doubtful that a sublattice containing $1$ could not be an order.
This is primarily just a curiousity, as I've only just started learning algebraic number theory.