Icarus2001 - No they are not and yes they do.
Network have been a QF salaried & owned subsidiary since 2011.
Network are a contract charter company owned by QF. The trick is that their only client is Qantas. Network do not sell tickets, QF do.
https://www.networkaviation.com.au/
Guess what it says on their website?:
we are Western Australia’s premier Aircraft Charter Company for FIFO Air Charters and Private Air Charters.