$G$ is the dihedral group $D_6$ of order 6. I need to find all subgroups of order 2 and order 3. I have found the cyclic group $\left<b\right> = \{e, b\}$, but that's as far as I have come.
How do you list all subgroups of eg. $D_6$? and how would you present all subgroups of eg. order 2?
Note that $D_6$ is not a huge group. You can pretty much see everything in $D_6$. Observe that $$D_6=\left< a,b \mid a^3=b^2=e,\, ba=a^{-1}b\right>.$$ Now, note that both $ a$, and $a^2$ generate the same cyclic group $C_3.$
There are three reflections(order 2 elements) in $D_6$. Note that $b, ab$, and $a^2b$ are the complete list of order two elements in $D_6$. Now write the three order two subgroups generated by each of them. Can you see that these three order 2 cyclic subgroups are different? If yes, then you're done. Is there any other subgroup you can Imagine in $D_6$?
Note that by the Lagrange theorem, the only possibility for the order of non-trivial subgroups is 2 and 3. Moreover, the Cauchy theorem guarantees that there must some subgroups of order 2 and order 3. Honestly, we don't need any such theorem to solve this problem. Straightforward computation is the right thing for such a small group.