The density of primes $p\equiv 1\pmod{8}$ (resp. $p\equiv 7\pmod{8}$) such that the class number of $\mathbb{Q}(\sqrt{-p})$ (resp. $\mathbb{Q}(\sqrt{-2p})$) is divisible by $2^{k+2}$ is conjectured to be $2^{-k}$ for all positive integers $k$. The conjecture has been resolved for $k = 1$ by the Chebotarev Density Theorem. For the family of quadratic fields $\mathbb{Q}(\sqrt{-2p})$, we use methods of Friedlander and Iwaniec to prove the conjecture for $k = 2$. Moreover, we show that there are infinitely many primes $p$ for which the class number of $\mathbb{Q}(\sqrt{-p})$ is divisible by $16$.