Let $\mathcal H_n$ be a countable random $n$-uniform hypergraph for $n>2$, and $\P(\mathcal H_n)=\{f[\mathcal H_n]:f\colon\mathcal H_n\to \mathcal H_n \text{ is an embedding}\}$. We prove that a linear order $L$ is isomorphic to the maximal chain in the partial order $\langle\P(\mathcal H_n)\cup\{\emptyset\},\subset\rangle$ if and only if $L$ is isomorphic to the order type of a compact set of reals whose minimal element is nonisolated.