  We study anomalies of non-Abelian discrete symmetries; which part of
non-Abelian group is anomaly free and which part can be anomalous. It is found
that the anomaly-free elements of the group $G$ generate a normal subgroup
$G_0$ of $G$ and the residue class group $G/G_0$, which becomes the anomalous
part of $G$, is isomorphic to a single cyclic group. The derived subgroup
$D(G)$ of $G$ is useful to study the anomaly structure. This structure also
constrains the structure of the anomaly-free subgroup; the derived subgroup
$D(G)$ should be included in the anomaly-free subgroup. We study the detail
structure of the anomaly-free subgroup from the structure of the derived
subgroup in various discrete groups. For example, when $G=S_n \simeq A_n
\rtimes Z_2$ and $G=\Delta(6n^2) \simeq \Delta(3n^2) \rtimes Z_2$, in
particular, $A_n$ and $\Delta(3n^2)$ are at least included in the anomaly-free
subgroup, respectively. This result holds in any arbitrary representations.
