Let $L$ be a $S_4$ or $A_4$ quartic field, and $\mathcal{O}_L$ its ring of integers. Let $K$ be its cubic resolvent field, which necessarily has the same discriminant as $L$. If $\mathcal{O}_L$ has a primitive suborder $Q$ of index $m$ then the correspondence between quartic rings and their cubic resolvent rings (necessarily unique for primitive quartic rings) implies that the cubic resolvent ring $R$ of $Q$ is a suborder of $\mathcal{O}_K$ of index $m$. In particular, $\mathcal{O}_K$ must have a suborder of index $m$ whenever $\mathcal{O}_L$ does. I do not believe the converse is true.
My question is this: for a given quartic field $L$ with Galois group isomorphic to $S_4$ or $A_4$ (in particular, it has a cubic resolvent field), does there necessarily exist a prime $p$ such that $\gcd(p, \Delta_L) = 1$ and that $\mathcal{O}_L$ has a suborder $Q$ of index $p$, and the cubic resolvent ring $R$ of $Q$ in $\mathcal{O}_K$ is monogenic?
By the work of Heath-Brown and Heath-Brown/Moroz we know that $\mathcal{O}_K$ necessarily has infinitely many monogenic suborders of prime index, but it is not clear to me whether any of them would correspond to a suborder in a given quartic field associated to it.
