What is an example of a finite group G that has a subgroup H which is not normal but for which the index [G] is prime? I was thinking about the alternating group A4 which has order 12, I know that the subgroups of A4 must have order 12, 6, 4, 3 , 2 or 1. But we need a prime index so I can eliminate subgroups of order 1,2 and 3 and 12. Also A4 does not have a subgroup of index 2, so I can also eliminate subgroups of order 6. So here I am looking for a subgroup of A4 of order 4. What would that be? Or any other example that satisfies the requirements.