Correct Answer - Option 2 : G may not be cyclic, but H is always cyclic.
According to the Lagrange theorem order of subgroups must divide the order of the group.
Property of group says if a group has prime order then it is cyclic.
Since the order of G is 6. Therefore its subgroup may have order 1,2,3,6
H is one of its subgroups with condition 1< |H| <6 so H may be of order 2 or 3 which is prime
Hence H must be cyclic
The order of G is 6 which is not prime and hence it may or may not be cyclic
Therefore option 2 is correct