  We construct a family of finite element sub-complexes of the conformal
complex on tetrahedral meshes. This complex includes vector fields and
symmetric and traceless tensor fields, interlinked through the conformal
Killing operator, the linearized Cotton-York operator, and the divergence
operator, respectively. This leads to discrete versions of transverse traceless
(TT) tensors and York splits in general relativity. We provide bubble complexes
and investigate supersmoothness to facilitate the construction. We show the
exactness of the finite element complex on contractible domains.
