It's probably a well known question, so it is just a reference question. 
Let $G$ be a finite group and let $C[G]$ be a group algebra. Then we can define a bracket on $C[G]$ by $[f,h]=f\*h-h\*f$. What does $C[G]$ look like as a Lie algebra? When is it solvable?



How does the group algebra look as a Lie algebra
Assuming that your ground field $K$ has characteristic prime to the order of $G$. Then the group ring is a seminsimple algebra. Therefore, $C[G]=\bigoplus\_{i=1}^{r} Mat\_{n\_i}(R\_i)$ is a direct sum of matrix algebras, where $R\_i$ is a finite-dimensional division ring over $K$. All this is very classical and nicely explained in Procesis book on Lie groups.


Thus the Lie algebra is a sum of general linear Lie algebras over division rings. 
If $K$ is $\mathbb{C}$, then $R\_i=\mathbb{C}$. So the Lie algebra is solvable iff all $n\_i$ are $1$ (this happens iff $G$ is abelian).


For $\mathbb{R}$, you also get the quaternions. For fields like $\mathbb{Q}$ or finite characteristic dividning the order of $G$, the story is going to be way more complicated and interesting.


