For a fixed abelian group H, let \(N_H(X)\) be the number of square-free positive integers \(d\le X\) such that \(H\le \textrm{CL}(\mathbb {Q}(\sqrt{-d}))\) . We obtain asymptotic lower bounds for \(N_H(X)\) as \(X\rightarrow \infty \) in two cases: \(H=\mathbb {Z}/g_1\mathbb {Z}\times (\mathbb {Z}/2\mathbb {Z})^l\) for \(l\ge 2\) and \(2\not \mid g_1\ge 3\) , \(H=(\mathbb {Z}/g\mathbb {Z})^2\) for \(2\not \mid g\ge 5\) . More precisely, for any \(\epsilon >0\) , we show \(N_H(X)\gg X^{\frac{1}{2}+\frac{3}{2g_1+2}-\epsilon }\) when \(H=\mathbb {Z}/g_1\mathbb {Z}\times (\mathbb {Z}/2\mathbb {Z})^l\) for \(l\ge 2\) and \(2\not \mid g_1\ge 3\) . For the second case, under a well known conjecture for square-free density of integral multivariate polynomials, for any \(\epsilon >0\) , we show \(N_H(X)\gg X^{\frac{1}{g-1}-\epsilon }\) when \(H=(\mathbb {Z}/g\mathbb {Z})^2\) for odd \( g\ge 5\) . The first case is an adaptation of Soundararajan’s results for \(H=\mathbb {Z}/g\mathbb {Z}\) , and the second conditionally improves the bound \(X^{\frac{1}{g}-\epsilon }\) due to Byeon and the bound \(X^{\frac{1}{g}}/(\log X)^{2}\) due to Kulkarni and Levin.
