It is known that every nuclear C*-algebra is exact, and that exactness passes to sub-C*-algebras. It follows that every sub-C*-algebra of a nuclear C*-algebra is exact. For separable C*-algebras, the converse holds. In fact, Kirchberg’s 
-embedding theorem shows that a separable C*-algebra 
is exact if and only if it embeds into the Cuntz algebra (which is nuclear), Theorem 6.3.11 in [1].
At the Kirchberg Memorial Conference in Münster, July 2023, Simon Wassermann asked if this result can be generalized to the nonseparable case:
Question: Does every (nonseparable) exact C*-algebra embed into a nuclear C*-algebra?
- [1]M. Rørdam, E. Størmer, Classification of Nuclear C*-Algebras. Entropy in Operator Algebras, Springer Berlin Heidelberg, 2002. https://doi.org/10.1007/978-3-662-04825-2.