example of a subgroup of index $3$ which is not normal

Question

Could you please give an example of a subgroup of index $3$ which is not normal ?

I know every subgroup of index $2$ is normal but if index is $3$ , I have no idea whether all of the subgroups are normal or not.

Answer 1

Hint. Consider $S_3$ and the subgroup $\langle(1\ 2)\rangle$.

Answer 2

As Ian Coley point out $S_3$ is smallest counter example of it.

But it is true if $G$ has odd order.

Fact: Let $p$ be smallest prime dividing $|G|$ then any group of index $p$ is normal.

Notice that when $|G|$ is odd and $3$ divides $|G|$, then $3$ is the smallest prime dividing $G$.

For fact you can check this.

Normal subgroup of prime index