You can only create the issue types defined for the project. You need to talk to the JIRA admin and have tasks added to the project. I have seen some implementations that only use Tasks as a sub-task type, tied to an Epic, Story, or Bug.
Before You'll start using JIRA take the team to a room with a whiteboard and define Your workflow, issue types including hierarchy and dependencies between them.
Afterwards go to Your Jira admin
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