Give an example of a group that does have subgroups of order 1,2,3,4,5,6, but does not have subgroups of order 7 or 8

Question

I was thinking Z6 under addition mod 6 and each element is a subgroup. But then I can't find subgroups with order 4 and 5.

Answer 1

Hint What are the orders of the subgroups of the symmetric group $S_n$ on $n$ objects for small $n$? Note that any group with the given property must contain an element of order $5$ and so has order divisible by $5$; what is the smallest $n$ for which this is true for $S_n$?

Alternatively, what are the orders of the subgroups of the cyclic group $C_n$?

Answer 2

What about Z/5ZxZ/12Z. Perhaps the simplest (?) example, as your group must have an order divisible by 5, 4, 3