I have a facebook group with over 6000 members. It is a lot of admin-work. How can I make profit out of this group?

Depending on the level of skill of the group members, I would try some affiliate offers from Clickbank. I have done some Facebook advertising of affiliate offers and I have found that photography courses and foreign language courses convert pretty well. In your position, where you have authority already as group admin, you should be able to convert a much higher percentage than I would using adverts. Look for a good course that has "rebilling" for recurring monthly compensation

In addition to the answers you've already received, I would suggest utilizing an Amazon affiliate account and pointing your audience to "great deals" on photography related equipment and resources.

This way you are staying consistent with the theme of your group and you are simply adding value by making the group aware of things they could benefit from.

I suggest creating a paid Facebook group and charging members a monthly membership fee.

You don't even need it to be too many people- target a membership of around 300 when you get started.

Create great content and offer inside info around your core topic. Interview experts, set up AMAs on the private Facebook group, have weekly accountability threads- basically any information that will improve the members' skills. Often you won't even have to rack your brain- your members will tell you what they want the most.

Keep engagements level high and slowly open the gates as you see people getting value out of the group.

To ease your admin burdens you can appoint a few managers in the paid group, and give them a share of the revenue.