What symmetry elements mustn't a molecule possess to be considered achiral?

Undergraduate textbooks say that a molecule mustn't posses a reflection plane and an inversion center to be considered achiral. As mentioned by Ron in his answer to [this question](https://chemistry.stackexchange.com/questions/36914/how-to-find-the-number-of-stereoisomers-of-a-tetra-substituted-allenes-exhibiti), molecules that don't possess a plane nor center of symmetry, but do possess an $S\_4$ improper axis are considered to be achiral. Are there any other symmetry elements that need to be taken into consideration when determining whether a molecule is chiral or not?


