Your installation is not really a sub-panel. It is a separate structure and is covered in Article 225. Many of the requirements for services apply. The 8AWG neutral is OK if load calculations show it to be sufficient for the unbalanced load. If the main lug panel is serving as the disconnecting means (225.31) it must have no more than 6 breakers. I would suggest installing a main. The GEC should be No. 6 CU or No. 4 AL
