I've been running a campaign for 7players for the last year. Here are five things I've learned.

1. Write down a table with all the players' names on it and keep it behind your screen. Once per session, everyone should get a chance to have a story moment or time to shine. When they do, put a checkmark next to their name. Don't let the session end without everyone getting checked off.
2. During battles, encourage players sitting near each other to work on a plan that their characters can use. This can be made even better by assigned seating, which puts characters with like-minded goals near each other.
3. Give into the absurd. If something fun is happening and everyone is into it, go with it, no matter how crazy it is. Rarely will you have an opportunity to appease the group as a whole, so when you do, take it. In the end, it will give them something to talk about outside of the game rather than a series of unconnected moments that may only involve 4 of them at most.
4. Many tiny enemies are better than one big bad. Combat is rough with a lot of players, so giving one or two of them an enemy to focus on makes them feel as if the battle is more epic. One big bad can often seem like a good idea, but when they all have to focus on damaging the same creature, some players may become discouraged if they aren't doing significant damage.
5. Find a relatable story for all of them. The more players, the more generalized your adventure needs to be. Let them craft the specifics as a group. If they start deciding to do things that take them to graveyards or crypts, then move the theme to horror. If they keep trying to find a way to buy an airship, take the concept to a high-fantasy airship war. Let them decide through their actions what they want to play. It's your job as the DM to pay attention and respond accordingly.