I am fully aware that this question has already been addressed here and here. The question, however, derives from a Dummit and Foote exercise (section 4.2 exercise 10 page 122) and the answers provided make use of material that appears later in the book, or, at least, I do not clearly understand them in terms of the material that I have studied already.
So, I would like to submit the following tentative proof
'from first principles' (i.e. Dummit and Foote before page 122). As I feel insecure about it, I would be grateful if you could check it.
Consider a nonabelian group G of order 6. By Cauchy's theorem, the group contains at least one element of order 2, and therefore at least one subgroup of order 2. Suppose this/these subgroups are all normal. This would imply that all elements of order 2 commute with all elements of G.
The remaining elements of group G are of order 1 (which trivially is in the center of G), or order 3 (order 6 would imply the group is cyclic and therefore abelian). Since all elements of order 2 are in the center, the order 3 elements will commute with them. The following reasoning shows that they also commute with each other.
Let $x≠y, |x|=|y|=3$. $\langle x\rangle$ and $\langle y\rangle$ are normal in $G$ because their index is $2$. Now $x\{1, y, y^2\}=\{1, y, y^2\}x$ implies, in the nonabelian case, that $x.y=y^2.x$ and $x.y^2=y.x$.
Likewise, $y\langle x\rangle=\langle x\rangle y$ implies $y.x=x^2.y$ and $y.x^2=x.y$. Manipulating these equations shows $y.x = x.y$, so the nonabelian case is impossible. Hence, order 3 elements are in the center of G as well.
Therefore, G would be abelian, contrary to the assumption. So, at least one of the subgroups of order 2 should be non-normal.
Best Answer
Basically your reasoning is correct (well done!): to write it with somewhat more "sophistication" - if $N \lhd G$, with $|N|=2$, then $N \subseteq Z(G)$, that is what you are using. Now if $x \in G$ with $ord(x)=3$, then, since $N$ is central, $N \subsetneq C_G(x)$, where the inclusion is strict, because of $|N|=2$ not divisible by $3$ $\left(x \in C_G(x) \right)$. But $|G:N|=3$, so $G=C_G(x)$, meaning $x \in Z(G)$ and hence $G$ is abelian.