Firstly you need to have two different subtask issue-types that are not "compatible" with each other (eg use different Field Configurations).
Now edit one of these sub-tasks.
You will not be able to select a different issue-type in the edit page and it will have text saying
"There are no issue types with compatible field configuration and/or workflow associations.
The issue type can only be changed by moving this issue."

This includes a link to a URL looking like http://localhost:8090/jira/secure/MoveIssue!default.jspa?id=10060 but this URL is only valid for moving parent issues.
Subtask move needs to be something like http://localhost:8090/jira/secure/MoveSubTaskChooseOperation!default.jspa?id=10060.

Clicking the provided link will produce an error message:
"Sub-tasks cannot be moved independently of the parent issue."