Some notes:

(1) Any measurable proper subgroup of $\mathbb {R}^{1}$ is of measure $0$.

(2) Any non-measurable subgroup $G$ of $\mathbb {R}^{1}$ charges fully everywhere, i.e., for any interval $I$, $m^{\ast}(G \cap I)=|I|$, where $m^{\ast}(\cdot)$ denotes the outer Lebesgue measure.

(3) Non-measurable subgroup of $\mathbb {R}^{1}$ exists.

If you would like to classify the subgroups in the sense of Lebesugue measure, you may find the following facts helpful.

(1) Any measurable proper subgroup of the real line is of measure $0$.

(2) Any non-measurable subgroup $G$ of the real line charges fully everywhere, i.e., for any interval $I$, $m^{\ast}(G \cap I)=|I|$, where $m^{\ast}(\cdot)$ denotes the outer Lebesgue measure.

(3) Non-measurable subgroup of the real line exists.