Is it?

$\displaystyle \mathbb{Z} / 2 \mathbb{Z} = \{ \bar{0}, \bar{1} \} \subset \mathbb{Z} / 3 \mathbb{Z} = \{ \bar{0}, \bar{1}, \bar{2} \}$

It seems to be closed and there does seem to be inverses in it...?

Like $\displaystyle \bar{1} + \bar{1} = \bar{2} = \bar{0} \in \mathbb{Z} / 2 \mathbb{Z}$