The lattice of primary ideals of orders in quadratic number fields

Let $O$ be an order in a quadratic number field $K$ with ring of integers $D$, such that the conductor $\f = f D$ is a prime ideal of $O$, where $f\in\Z$ is a prime. We give a complete description of the $\f$-primary ideals of $O$. They form a lattice with a particular structure by layers; the first layer, which is the core of the lattice, consists of those $\f$-primary ideals not contained in $\f^2$. We get three different cases, according to whether the prime number $f$ is split, inert or ramified in $D$.