Maximal Fuchsian subgroups of the $d=2$ Bianchi group

Let $\mathrm{\Gamma }$ denote the $d=2$ Bianchi group $\mathrm{PSL}(2,\mathbb{Z}[\sqrt{-2}])$. We give an explicit description of all conjugacy classes of maximal nonelementary Fuchsian subgroups of $\mathrm{\Gamma }$ as integral orders of certain indefinite quaternion algebras over $\mathbb{Q}$. Using this description, we also provide the covolumes corresponding to each conjugacy class. As an application, we compute the limit ${\lim }_{x\to \mathrm{\infty }}\frac{\mathrm{\Pi }(x)}{x}$ where $\mathrm{\Pi }(x)$ counts the number of primitive totally geodesic immersed surfaces in the manifold $\mathrm{\Gamma }\mathrm{\backslash }{\mathbb{H}}^3$ with area less than $x$.