Does this mean that there are only 2 possible groups structures for groups of order 6?

Question

Does saying that "up to isomorphism, the groups of order 6 are the cyclic group $C_{6}$ and the dihedral group $D_{3}$" mean that there are only 2 possible groups structures for groups of order 6?

Answer 1

Essentially yes. Precisely, it means that there are exactly $2$ distinct group structures such that every group of order $6$ is isomorphic to one of them.

Answer 2

Yes. Let $|G|=6$. By Cauchy, there is subgroup $\langle x\rangle$ of order $3$ and a subgroup $\langle y\rangle$ of order $2$. $[G\colon \langle x\rangle]=2$ hence $\langle x\rangle\trianglelefteq G$. So $yxy^{-1}\in\langle x\rangle=\{1,x,x^2\}$. Certainly $yxy^{-1}\neq 1$ (why?)

If $yxy^{-1}=x$ then $G=\langle x\rangle \times \langle y\rangle=\mathbb{Z}_3\times \mathbb{Z}_2$.

If $yxy^{-1}=x^2$ then $G=\langle x,y\colon x^3=1, y^2=1, yxy^{-1}=x^2\rangle=D_3$.