  We perform the Hamiltonian constraint analysis for a wide class of gravity
theories that are invariant under spatial diffeomorphism. With very general
setup, we show that different from the general relativity, the primary and
secondary constraints associated with the lapse function $N$ become second
class, as long as the lapse function $N$ enters the Hamiltonian nonlinearly.
This fact implies that there are three degrees of freedom are propagating, of
which two correspond to the usual tensor type transverse and traceless
gravitons, and one is the scalar type graviton. By restoring the full spacetime
diffeomorphism using the St\"{u}ckelberg trick, this type of spatially
covariant gravity theories corresponds to a large class of single field
scalar-tensor theories that possess higher order derivatives in the equations
of motion, and thus is beyond the scope of the Horndeski theory.
