How many conjugacy classes of subgroups does $\mathrm{GL}(2,p)$ have?

For instance the dihedral group of order $2n$ has $\tau(n)$ cyclic normal subgroups and $\sigma(n)$ "dihedral" subgroups (as in, containing a reflection), but they fall into $\gcd(2,n) \tau(n) + \tau(n/\gcd(2,n))$ conjugacy classes. Here $\tau(n)$ is the number of divisors of $n$, and $\sigma(n)$ is their sum.

The formula is relatively compact and can be explicitly evaluated for $n$ in the millions without much work. The description is nice because it even indicates the structure of the subgroups.

The subgroups of $\mathrm{GL}(2,p)$ whose order is divisible by $p$ either have a normal Sylow $p$-subgroup or contain $\mathrm{SL}(2,p)$. The former types have conjugacy classes indexed by the subgroups of $(p-1) × (p-1)$, and the latter by subgroups of $(p-1)$. The number of the first type has some reasonable formulas at OEIS: A060724 and the latter is just $\tau(p-1)$ again.

Again the description is compact and can be explicitly evaluated for numbers into the millions without any real effort: $\mathrm{GL}(2,1000003)$ has $1000008$ conjugacy classes of subgroups of order divisible by $1000003$ and $\mathrm{GL}(2,10000019)$ has $10000024$ conjugacy classes of subgroups of order divisible by $10000019$, each number computed in under 1ms. Again the description is especially nice because it even indicates the structure of the subgroups.

What about the conjugacy classes of subgroups of $\mathrm{GL}(2,p)$ whose order is coprime to $p$?

Is there a similarly compact and easily evaluated description of their number, and even more nicely, does it also indicate the structure of the subgroups?