There are any number of options, and I'll recommend the following based on my experiences as a player and DM:
1. Be a leader of a team rather than just one of many individuals
This is probably the broadest solution you can employ as a player. I've used this to draw out less assertive or less interested players. Your contribution doesn't need to be an announcement of what your character alone will do-- it can also be something more along the lines of organizing other characters into a plan that the group carries out.
Whatever the challenge is, you can let every individual do their own thing and hope that at least one succeeds. Or you can work out a plan in advance, specifically suggesting actions or prompting players for input:
Okay, we need to convince Lady Obstacle to let us use her invitations to Lord Villain's Grand Plot Event Ball so that we can complete the quest. Alice, you have high stats in Persuasion and History, maybe you could talk to her and remind her about how deep the feud between House Obstacle and House Villain are, suggesting that she might not want to go to his event? Or Bob, you're a bard that she's liked before, is there something you can think of that might sway her?
Very contrived, in this example, but this allows you to fight for space in the conversation, and then offer it to the less assertive players to give them a clear chance to participate.
If you like planning things out, you can also try to persuade some of the more assertive players to take some course of action which leaves other things for the other players to do, giving them more space to participate by removing the more overbearing players from the scene.
At a minimum, you can make a point of asking other players what they think or would like to do. That's very easy to pass off in and out of character.
2. Come up with plans that require specific characters, which happen to be played by less assertive players
As more of a subset of (1), if you come up with a plan that your own PC can undertake but would be helped by participation of one of the less assertive players you can give them moments to be involved that they might otherwise not find. Being Helped by another character to gain Advantage can have a strong effect on outcomes, and the contribution can be pretty obvious (if your first roll is a 3, the Advantage clearly matters). A risk of this is that some players can be made to feel like sidekicks, so this tactic works best if it's infrequent.
3. Split the party
This is a hassle in a lot of ways, so it's also best used sparingly, but getting the overbearing players out of the picture from time to time gives the others a chance to play "normally" without being talked over and ignored.
4. Talk to the DM
Making sure that everyone is having fun is a core responsibility of the DM, and if some players are running over others to the extent that they're not playing large sections of the game that's a problem. The DM has lots of tools to address that kind of situation, perhaps most relevantly asking players what their characters are doing. They may not have anything in mind, but they'll get a chance to speak.
I've also read advice on this stack suggesting that, outside of combat, all players announce their intended actions before the DM resolves any of them, but I have no personal experience with using that technique on this sort of problem. A surefire approach is designing plot for specific characters, as those sorts of things can't just be transferred to a louder player, but that can be expensive for the DM plan and manage with such a large group.
5. Talk to the group
Some of the more assertive players may not realize the effect that their approach is having on some of the other players. They may or may not care about the issue, but bringing it up gives everyone a chance to be aware of it and talk about responses they think might be appropriate.
One particular dimension that I think is relevant is that players who are more assertive or overbearing may not be as interested in sitting on their hands, and so if too much time passes without something for their character to do they can become even more assertive and overbearing. Pointing out that more participation for them might mean far less for others gives some context to why it might be worthwhile for them to be sidelined for a few minutes.
It also happens that some players don't really want to be in the driver's seat for these sorts of things. They may prefer to be pointed at a monster and then set loose, and don't care about coming up with a freeform plan to deal with yet another NPC. Talking to the group can reveal whether or not this situation is actually a problem for the quieter players. It may be, but it could also be how they want to play.
6. This is a large group, and there will likely be some give-and-take no matter what
Seven to eight people is a lot of people to be gathered around a table for an RPG. It takes a lot of real time to go through so many characters' actions and thoughts, often with the effect that gameplay slows to a crawl. It's often just not feasible to have 7-8 plans running at once, or have people pulling in 7-8 different directions.
Sometimes it may be desirable to let some characters do little for a scene or two just to streamline things and move the game forward. That becomes a problem if it happens all the time and to the same characters, but just for perspective totally equal contributions in all cases may not be what you want.
I once tried to run a D&D 3.5e game for ten players at once and this was a big issue. More players means more competition for scarce on-camera time, because while number of players can increase freely the amount of time people have to dedicate to gaming generally does not.