  For fixed $q\in\{3,7,11,19, 43,67,163\}$, we consider the density of primes
$p$ congruent to $1$ modulo $4$ such that the class group of the number field
$\mathbb{Q}(\sqrt{-qp})$ has order divisible by $16$. We show that this density
is equal to $1/8$, in line with a more general conjecture of Gerth.
Vinogradov's method is the key analytic tool for our work.
