Let $ Q $ be an irreducible polynomial over $\mathbb{F}_{2}$, can we find a decomposition of the group $ (\mathbb{F}_{2}[x]/(Q ^ { e }))^{*}$ into a direct product of cyclic groups ?.

We know (from $P\in \mathbb{F}_{2}[x]$ for which $(\mathbb{F}_{2}[x]/(P))^{*}$ is a cyclic group) that $ (\mathbb{F}_{2}[x]/(Q ^{e}))^{*}$ is not cyclic unless $e=1$ or $ \deg(Q) = 1$ and $e \leq3$.

If no analytical decomposition can be found is there any algorithm to find such one ? In that case do we have a bound on the number of cyclic groups in the decomposition ?