adb permission denied to push files to subdirectry
As it appears in this comment, the file permissions and access control to SubDir differs from what ideally should be.

The problem:

# ls -l on /data/media/0/Dir
dwrxrwxr-x root root


The user group should have been media_rw instead of root.

The solution is to change to appropriate group or make the directory writable by others.


chmod 777 /data/media/0/Dir/SubDir       # making SubDir writable by others
chgrp media_rw /data/media/0/Dir/SubDir  # Alternatively, changing group from root to media_rw


Note: you may need to install Busybox or Toybox to have and use chmod and/or chgrp.