How to prove that some specific group is not isomorphic to any member of any of the 5 families of groups?

Question

For example, let's say we have a group Q8 (Quaternion group).

How to prove that this group is not isomorphic to any member of any of the families of groups (Cyclic, Abelian, Dihedral, Symmetric, Alternating)?

What is the general approach for solving this problem for any given group?

Answer 1

It is clear that $Q_8$ is not abelian, hence not cyclic, because for example $ij=-ji$. So we can use, if we want, that we only have two nonabelian groups of order $8$, namely $Q_8$ and $D_4$. So we only need to distinguish these two groups. But this is easy. Every subgroup in $Q_8$ is normal, which is not true for $D_4$.

References:

How can you show there are only 2 nonabelian groups of order 8?

Show that every subgroup of $Q_8$ is normal.

Answer 2

It's not cyclic, since there are no elements of order $8$. Neither is it abelian, since $ij \neq ji$. It can't be Dihedral, as $D_8$has four elements of order $2$ whereas the quarternions only have one. It is neither symmetric nor alternating, since these groups have sizes of $n!$ and $\frac{n!}{2}$ respectively, and $8$ is not of these forms.