  In 1884 the German mathematician Karl Rohn published a substantial paper on
\cite{ROH} on the properties of quartic surfaces with triple points, proving
(among many other things) that the maximum number of lines contained in a
quartic monoid surface is $31$. In this paper we study in details this class of
surfaces. We prove that there exists an open subset $A \subseteq
\mathbb{P}^1_K$ ($K$ is a characteristic zero field) that parametrizes (up to a
projectivity) all the quartic monoid surfaces with $31$ lines; then we study
the action of $\mathrm{PGL}(4,K)$ on these surfaces, we show that the
stabiliser of each of them is a group isomorphic to $S_3$ except for one
surface of the family, whose stabiliser is a group isomorphic to $S_3 \times
C_3$. Finally we show that the $j$-invariant allows one to decide, also in this
situation, when two elements of $A$ give the same surface up to a projectivity.
To get our results, several computational tools, available in computer algebra
systems, are used.
