= Solution
Each reflection has order two, giving exactly the three order-two <subgroups>
$$
\boxed{\langle s\rangle=\{e,s\},\quad\langle rs\rangle=\{e,rs\},\quad\langle r^2s\rangle=\{e,r^2s\}.}
$$
<Conjugation> by $r$ sends $r^js$ to $r^{j+2}s$, cycling the three reflections. None of these <subgroups> is therefore normal.
The two nonidentity rotations generate the same order-three <subgroup>:
$$
\boxed{R=\langle r\rangle=\{e,r,r^2\}\mathrel{\trianglelefteq}D_3.}
$$
It is invariant under <conjugation> by $r$, and $srs=r^{-1}$ shows invariance under $s$ as well; since $r,s$ generate the <group>, this proves normality. The complete partition into <conjugacy classes> is
$$
\boxed{\{e\},\qquad\{r,r^2\},\qquad\{s,rs,r^2s\}.}
$$
The rotations are mutually conjugate by $s$, and the reflections by $r$; no rotation is conjugate to a reflection because <conjugation> preserves the <order of a group element>.
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