Let $F$ be a number field, $\mathfrak{o}$ be its ring of integers and $\mathfrak{p}$ be the associated maximal ideal. Consider the subgroup $H$ of ${GL}(3, \mathfrak{o})$ given by matrices of the forms $$\left(\matrix{ \mathfrak{o} & \mathfrak{o} & \mathfrak{o} \\ \mathfrak{p} & \mathfrak{o} & \mathfrak{o} \\ \mathfrak{p} & \mathfrak{p} & \mathfrak{o} }\right)$$
What is the index of $H$ in $GL(3, \mathfrak{o})$?
I often face this kind of questions and the answers always appear to be essentially ad hoc treatment of the specific case we are interested in. The only way I see is to compute explicitly the quotient, that is to say to find the explicit decomposition of $GL(3, \mathfrak{o})$ in left (or right) $H$-classes. Is there an algorithmic or systematic way to do so?