Bifurcation mechanism of honeycomb structures is elucidated by the study of a that is the direct product of O(2) and two reflection group. A flower pattern is theoretically assessed to branch from a triple bifurcation point and is actually found by a numerical analysis of a honeycomb cellular solid. Other bifurcating patterns of interest are found in this study through the analysis of bifurcation points with the multiplicity of six and twelve. Fundamentals of group representation theory in Chap. 7 and group-theoretic bifurcation theory in Chap. 8 are foundations of this chapter.