  We prove that for any given positive integer $\ell$ there are infinitely many
imaginary quadratic fields with 2-class group of type $(2,2^\ell)$, and provide
a lower bound for the number of such groups with bounded discriminant for
$\ell\ge2$. This work is based on a related result for cyclic 2-class groups by
Dominguez, Miller and Wong, and our proof proceeds similarly. Our proof
requires introducing congruence conditions into Perelli's result on Goldbach
numbers represented by polynomials, which we establitish in some generality.
