  Dirac cones are conical singularities that occur near the degenerate points
in band structures. Such singularities result in enormous unusual phenomena of
the corresponding physical systems. This work investigates double Dirac cones
that occur in the vicinity of a fourfold degenerate point in the band
structures of certain operators. It is known that such degeneracy originates in
the symmetries of the Hamiltonian. We use two dimensional periodic
Schr\"odinger operators with novel designed symmetries as our prototype. First,
we characterize admissible potentials, termed as super honeycomb lattice
potentials. They are honeycomb lattices potentials with a key additional
translation symmetry. It is rigorously justified that Schr\"odinger operators
with such potentials almost guarantee the existence of double Dirac cones on
the bands at the {\Gamma} point, the origin of the Brillouin zone. We further
show that the additional translation symmetry is an indispensable ingredient by
a perturbation analysis. Indeed, the double cones disappear if the additional
translation symmetry is broken. Many numerical simulations are provided, which
agree well with our analysis.
