For any topological space $\displaystyle (X,\tau ) $ and for any group $\displaystyle A\subseteq X$
$\displaystyle int(cl(A))=cl(int(A))$
Thanks
Alternatively:
Think about it like this. If $\displaystyle K=\left(\overline{A}\right)^{\circ}=\overline{A^{\ circ}}$ then we have that $\displaystyle \overline{K}=\overline{\overline{A^{\circ}}}=\over line{A^{\circ}}=K$ and so $\displaystyle K$ is closed. But, $\displaystyle K^{\circ}=\left(\left(\overline{A}\right)^{\circ}\ right)^{\circ}=\left(\overline{A}\right)^{\circ}=K$ and so $\displaystyle K$ is open. So, in a connected space $\displaystyle X$, $\displaystyle K=X$. But, $\displaystyle X=\left(\overline{A}\right)^{\circ}\subseteq\overl ine{A}\subseteq X$ and so $\displaystyle A$ must be dense in $\displaystyle X$. So more examples would be be $\displaystyle \mathbb{S}^1\subseteq\mathbb{R}^2$, $\displaystyle \text{SO}(n)\subseteq\text{GL}^+(n,\mathbb{R})$, etc.
The above also shows (like Focus) pointed out that your subgroup cannot have empty interior.
Also, note the above applies equally well to non-subgroup subsets.