As someone who still does LAN weekends with a few friends a couple of times a year, this is a hard one especially if you have 6+ players, however here are my suggestions.

Supreme Commander using the FAF Client - huge scale RTS with up to 16 people in a single map (8v8).

Left4Dead Versus - 8 people go 4v4 so two teams in your case. Dedicated server recommend.

Unreal Tournament 2005/3 - lots of games modes and supports a lot of people but you may need a dedicated server.

Gary’s Mod PropHunt, Bomberman etc - Great sandbox game - can be a bit junky to setup up but excellent if you get it right

Blur - racing game which may be hard to find as it is not being sold anywhere but an excellent multiplayer one

Factorio - way slower pace than any games mentioned, but if you play Deathworld is an adrenaline rush and fun coop. Up to 500 people so 15-16 people is no sweat. Also good for long term goals for continuing play.

Human Fall Flat - funny physics puzzle game up to 4 people. Great for competition as you can make it 4 teams of 4 and have a leaderboard.

Planetside 2 - FPSMMO - epic battles and still very active

That’s a lot of typing already - let me know if you want more suggestions - happy gaming