Is classification by a semidirect product complete?

Question

For example, let's say we have to classify groups of a certain order. Can all the groups of that order be expressed as some semi-direct products?

Answer 1

No, not necessarily. Semi-direct products have a normal subgroup as a factor, but finite simple groups cannot (non-trivially). The first such group you typically encounter (besides $\mathbb{Z}_p$, $p$ a prime) is the alternating group $A_n$. For $n \geq 5$ the group $A_n$ is simple and so cannot admit an interesting semi-direct decomposition into smaller factors.

Answer 2

$\mathbb{Z}_4$ is not a semidirect product of $\mathbb{Z}_2$ and $\mathbb{Z}_2$ (there is no section, there's only one subgroup of order $2$, ...), but $\mathbb{Z}_2^2$ is a semidirect product (it's a direct product).