Imaginary quadratic number fields with class groups of small exponent

Let $D<0$ be a fundamental discriminant and denote by $E(D)$ the exponent of the ideal class group $\text{Cl}(D)$ of $K={\mathbb Q}(\sqrt{D})$. Under the assumption that no Siegel zeros exist we compute all such $D$ with $E(D)$ is a divisor of $8$. We compute all $D$ with $|D|\leq 3.1\cdot 10^{20}$ such that $E(D)\leq 8$.