Group Theory Sylow Subgroup

Question

What's an example of a group $G$ and an integer $n$ dividing $|G|$ with $0 < n < |G|$ such that $G$ has no subgroup of order $n$.

Answer 1

In $A_4$,with $|A_4|=12$ are there subgroup of order $6$?

Answer 2

Suppose $|G|$ is simple non-cyclic, then $G$ is non-abelian (Use Cauchy's Theorem), then $G$ has even order (Apply Feit–Thompson theorem), hence $G$ has no subgroup of order $|G|/2$. So if $G$ is simple non-cyclic, then $|G|=2n$ and $G$ has no subgroup of order $n.$ In particular the monster group has no subgroup of order $$404008712397256437943229952480855378502877184000000000.$$