TITLE: Finite groups as subgroups of dihedral groups
QUESTION [9 upvotes]: It is well-known that any finite group can be realised as a subgroup of a permutation group. But is it true that any finite group can be realised as a subgroup of a dihedral group?

REPLY [12 votes]: No. Dietrich Burde's reason is quite nice, but there are easy elementary concrete counterexamples. Here's one:
$Q_8 = \{\pm 1, \pm i, \pm j, \pm k\}$ has six elements of order $4$, but no dihedral group has this property: all reflections in a dihedral group have order 2, and the only rotations of order $4$ are rotations by $\frac{\pi}{2}$ and $\frac{3\pi}{2}$ so a dihedral group has at most two elements of order $4$.