Ask HN: Suggestions for managing incoming customer/vendor email?

We've got several departments that could benefit, but in particular two of our departments get hundreds of emails per day CC'd to a group of 5-8 people. There is wasted time and effort in everyone in the group reading and triaging every email.

What tools could we consider to help with this? Ideally all emails would go into a queue, with possibility to pre-filter emails to specific users based on some criteria. All members could access all threads, but more importantly, there would be a single overview where it's easy to identify whether a customer/vendor has been helped or responded to. If not, flag it yourself and take care of it.

I tried searching around, but was not very successful if there's another similar question I'll gladly check it out. Thanks!