Some examples of real quadratic fields with finite nonsolvable maximal unramified extensions

Let K   be a number field and KurKur be the maximal extension of K that is unramified at all places. In this article, we identify real quadratic number fields K   such that Gal(Kur/K)Gal(Kur/K) is a finite nonsolvable group under the assumption of the Generalized Riemann Hypothesis. We also explicitly calculate their Galois groups.