On imaginary quadratic fields with non-cyclic class groups

For a fixed abelian group H, let NH(X) be the number of square-free positive integers d≤ X such that H is a subgroup of CL(Q(√-d)). We obtain asymptotic lower bounds for NH(X) as X→∞ in two cases: H=Z/g₁Z× (Z/2Z)ˡ for l≥ 2 and 2 g₁≥ 3, H=(Z/gZ)² for 2 g≥ 5. More precisely, for any ε >0, we showed NH(X) X1/2+3/2g₁+2-ε when H=Z/g₁Z× (Z/2Z)ˡ for l≥ 2 and 2 g₁≥ 3. For the second case, under a well known conjecture for square-free density of integral multivariate polynomials, for any ε >0, we showed NH(X) X1/g-1-ε when H=(Z/gZ)² for g≥ 5. The first case is an adaptation of Soundararajan's results for H=Z/gZ, and the second conditionally improves the bound X1/g-ε due to Byeon and the bound X1/g/(log X)² due to Kulkarni and Levin.