And I wonder what would be the difference between:
.....maintenance is out of control of this department.
and
...maintenance is outside the control of this department.
First of all, the first makes no grammatical sense as it stands, so you have to eliminate it as an option.
The 2nd says that the department does not look after maintenance (looking after gardens, repairing, etc. whatever 'maintenance' represents). Some other department does this job.
But if you modify the first, you can correct it and it can have 2 possible different meanings.
(1)Changing '.....maintenance is out of control of this department.' to '.....maintenance is out of THE control of this department.' and it will have the same meaning as your 2nd sentence - the dept has nothing to do with maintenance (another dept does it).
(2)Changing '.....maintenance is out of control of this department.' to .....maintenance is out of control WITHIN this department.' means the maintenance group itself within the department has lost control in the sense that the maintenance group is performing badly.