Let h be a subgroup of a group G of order 6. Show that if |H|=3 then every left coset of H is a right coset. Is this necessarily true if |H|=2?

I think this is a lagrane theorem question, I understand that they are asking me if H is a normal subgroup, but I am unsure how to prove this.

I have come across a proof in my notes saying that any subgroup of order 2 is normal, will this suffice for the second part of the question?

Thank you