On Hopf algebras over the unique $12$-dimensional Hopf algebra without the dual Chevalley property

Let $\text{mathds}k$ be an algebraically closed field of characteristic zero. We determine all finite-dimensional Hopf algebras over $\text{mathds}k$ whose Hopf coradical is isomorphic to the unique $12$-dimensional Hopf algebra $\mathcal{C}$ without the dual Chevalley property, such that the diagrams are strictly graded and the corresponding infinitesimal braidings are indecomposable objects in ${}_{\mathcal{C}}^{\mathcal{C}}\mathcal{Y}\mathcal{D}$. In particular, we obtain new Nichols algebras of dimension $18$ and $36$ and two families of new Hopf algebras of dimension $216$.