Group Messages (for iMessage and SMS) was introduced in iOS 7. With it, you could chat with your friends as a group (instead of individual one-on-one messages). It was a new and interesting feature but the implementation sucked: there was no way to turn off or mute a group conversation.
Apple fixed this in iOS 8. You can now do a lot of things in group messages. Most importantly, you can leave a group message or if that’s too extreme a measure, you can simply mute the group message/conversation. (As a commenter points out, you can only leave a group message when all participants of the group are running iOS 8 and using iMessage).
How to Mute Group Message Chat in iOS 8
Step #1. Open the group message thread that you want to mute.
Step #2. Tap on Details from the top-right.
Step #3. Scroll down to the part where you see these.
Step #4. Switch ON Do Not Disturb to mute conversations from this group message.
Step #5. You will stop receiving any notification from the group message. No banner notifications, no lock screen notifications and no other alerts.
That Do Not Disturb switch more or fewer works like the mute switch you find on other messaging apps like Whatsapp. One major difference, however, is that there are no time limits. Once you switch it on, it stays on for as long as you want. You can turn it off to start receiving group message notifications.
What happens when you switch it on? Other than the fact that you stop receiving notifications of any type from the group message, everything else remains same. You can still take part in the conversation because you will receive the messages.
How to Leave Group Message Conversation in iOS 8
You can also quit a group message when Do Not Disturb won’t fly. To do this
Step #1. Open the Messages app.
Step #2. Tap on the group conversation/thread.
Step #3. Tap on Details from top-right.
Step #4. Scroll all the way down.
Step #5. Tap on Leave this Conversation. Confirm it and you’re out of the group conversation.