We say (according to https://ncatlab.org/nlab/show/%28infinity%2Cn%29-category+with+duals) that a symmetric monoidal $(\infty,1)$ category $\mathcal{C}$ *has duals* if its homotopy category $h\mathcal{C}$ is rigid monoidal.

I'm interested in the $\infty$-category $Sp$ of spectra. What is the largest stable $\infty$-subcategory $\mathcal{C}$ of $Sp$ such that $\mathcal{C}$ has duals? Is such a category nontrivial? If so does it have a nice description? In this case the dual object of $X \in Sp$ is $Map(X, \mathbb{S})$, where $\mathbb{S}$ is the sphere spectrum $\Sigma^{\infty} S^0$.