Possible indices for the Galois image of elliptic curves over Q

For a non-CM elliptic curve $E$ over the rationals, the Galois action on its torsion points can be expressed in terms of a Galois representation $\rho_E : G \to GL_2(\hat{$\mathbb{Z}$})$,where$G$ is the absolute Galois group of the rationals. A well-known theorem of Serre says that the image of ${\rho }_E$ is open and hence has finite index in $G{L}_2(\widehat{\mathbb{Z}})$. We will study what indices are possible assuming that we are willing to exclude a finite number of possible $j$-invariants from consideration. For example, we will show that there is a finite set $J$ of rational numbers such that if $E\mathrm{/}\mathbb{Q}$ is a non-CM elliptic curve with $j$-invariant not in $J$ and with surjective mod $\mathrm{\ell }$ representations for all $\mathrm{\ell }>37$ (which conjecturally always holds), then the index $[G{L}_2(\widehat{\mathbb{Z}}):{\rho }_E(G)]$ lies in the set $$I:=\left\{\begin{array}{c}2,4,6,8,10,12,16,20,24,30,32,36,40,48,54,60,72,84,96,108,112,120,144,\\ 192,220,240,288,336,360,384,504,576,768,864,1152,1200,1296,1536\end{array}\right\}\mathrm{.}$$ Moreover, $I$ is the minimal set with this property.