Unfortunately not, Brian.
Service Desk can only be used with Standard issue types, this issue types are the only ones that can generate valid Customer Request types.
The only way to create sub-tasks is by requesting an Agent to do so.
Hope this helps!
In additional to Andre's response, we have a Suggestion Request about this goal where I would strongly suggest that you vote on the following issue to increase its popularity and add it to your watchlist for future updates:
...be more productive while being fun to use at the same time. For some, getting started can be a bit intimidating. This is especially true if Jira Service Desk is your first exposure to Atlassian...
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