Proper condition on the dihedral group

Question

Is there a theream which is a condition on $n\in\mathbb N$ that says when the dihedral group, $D_{n}$, has non-cyclic subgroups?

After spending some time figuring a condition I tried to find some similar thread but didn't find any.

Answer 1

Dihedral group $D_n = <r, s | r^n = s^2 = 1, rs = sr^{-1}>, \forall n \ge3$.

By your question, $D_n \le D_n$. So true for $\forall n\ge3$.

But if we want a proper non-cyclic subgroup, then we have to consider some $<r^a,s>$. Hence when $n$ is composite we get a required subgroup.

Answer 2

Any cyclic group has unique subgroup of any order which is divisior of its order

Now in $D_n$ How many subgroup of order 2?

They are at least n .

SO for n>1