  Figure-eight solutions are solutions to planar equal mass three-body problem
under homogeneous or inhomogeneous potentials. They are known to be invariant
under the transformation group $D_6$: the dihedral group of regular hexagons.
Numerical investigation shows that each figure-eight solution has some
bifurcation points. Six bifurcation patterns are known with respect to the
symmetry of the bifurcated solution.
  In this paper we will show the followings. The variational principle of
action and group theory show that the bifurcations of every figure-eight
solution are determined by the irreducible representations of $D_6$. Each
irreducible representation has one to one correspondence to each bifurcation.
This explains numerically observed six bifurcation patterns. In general, in
Lagrangian mechanics, bifurcations of a periodic solution is determined by
irreducible representations of the transformation group that leaves this
solution invariant.
