  Let $N$ be a positive integer. Let $\operatorname{H}$ be a group of level $N$
and let $E$ be an elliptic curve defined over the rationals with
$\textit{j}_{E} \neq 0, 1728$. Then the image
$\overline{\rho}_{E,N}\left(\operatorname{Gal}\left(\overline{\mathbb{Q}}/\mathbb{Q}\right)\right)$,
of the mod-$N$ Galois representation attached to $E$, is conjugate to a
subgroup of $\operatorname{H}$ if and only if $E$ corresponds to a non-cuspidal
rational point on the modular curve $\operatorname{X}_{\operatorname{H}}$
generated by $\operatorname{H}$. In this article, we are interested when
$\overline{\rho}_{E,N}(G_{\mathbb{Q}})$ is precisely $\operatorname{H}$. More
precisely, we classify all groups $\operatorname{H}$ that are direct products
of subgroups $\operatorname{H}$ for which $\operatorname{X}_{\operatorname{H}}$
contains infinitely many non-cuspidal rational points but there is no elliptic
curve $E/\mathbb{Q}$ such that
$\overline{\rho}_{E,N}\left(\operatorname{Gal}\left(\overline{\mathbb{Q}}/\mathbb{Q}\right)\right)$
is conjugate to $\operatorname{H}$.
