Subgroups of General Linear Group containing Special Linear Group, from 3rd isomorphism theorem

Question

In this one proof I was reading, I don't see how the following statement follows from the third isomorphism theorem: "The third isomorphism theorem also gives a full description of the subgroups of $GL_2(\mathbb{R})$ containing $SL_2(\mathbb{R})$ ; they consist of the matrices whose determinants lie in a subgroup of $\mathbb{R}^*$." I understand why subgroups containing $SL_2(\mathbb{R})$ are normal (follows from 3rd iso thm), and that $GL_2(\mathbb{R})$/$SL_2(\mathbb{R})\cong \mathbb{R}^*$ (follows from 1st iso thm).