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 <= CL(Q(root-d)). We obtain asymptotic lower bounds for NH (X) as X -> infinity in two cases: H = Z/g(1)Z x (Z/2Z)(l) for l >= 2 and 2 inverted iota g(1) >= 3, H = (Z/gZ)(2) for 2 inverted iota g >= 5. More precisely, for any epsilon > 0, we show N-H(X) >> X1/2+ 3/ 2g1+2-epsilon when H = Z/g(1)Z x (Z/2Z)(l) for l >= 2 and 2 inverted iota g(1) >= 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) >> X1/ 8-1-epsilon. When H =(Z/gZ)(2) for odd g >= 5. The first case is an adaptation of Soundararajan's results for H = Z/gZ, 1 and the second conditionally improves the X1/g-epsilon due to Byeon and the bound X-1/g /(log X)(2) due to Kulkarni and Levin.