The short answer is that they can, provided the conduct outside of work has a bearing on the employment relationship.
Cases involving violence, sexual conduct or dishonesty are most likely to affect the employment relationship, either because of the work the employee does (for example if they work in a role which requires a certain level of responsibility or trust) or their employer's reputational concerns.
There is, however, no hard and fast rule that an employee who is being investigated, charged or convicted of a criminal offence must be dismissed. The consideration here is whether their suitability to do their job has been called into question, and whether irreparable damage has been caused to their relationship with their employer, colleagues and customers.
Conduct outside of work is of course not limited to criminal behaviour. We have increasingly seen disciplinary action being taken for employees' inappropriate use of social media outside of work. These cases will generally turn on their own facts. However, any disciplinary action taken in response to inappropriate social media use must fall within the 'range of reasonable responses' open to a reasonable employer to be lawful.