Let $F$ be a finite field and $F(t)$ the rational function field. Note that all subrings $R$ of $F(t)$ that contain $F[t]$ can be described as follows. Let $P(R)$ be the set of irreducible polynomials $p(t)$ such that $1/p(t)\in R$. Then the correspondence $R\rightarrow P(R)$ is a bijection between the set of subrings of $F(t)$ that contain $F[t]$ and the collection of all subsets of irreducible polynomials over $F$. However, there are many subrings of $F(t)$ that do not contain $F[t]$.
Question: Is there a description of all subrings of $F(t)$?