It's been my experience that, in Germany, LH does not / will not directly handle group bookings. When I needed to book a group of 12, Lufthansa directed me to contact a travel agent to arrange discount group pricing. The travel agent then works with Lufthansa to make the group booking happen.
In the U.S., LH does handle group bookings directly. We planned a group as small as seven. Handled by LH Group Sales USA. But, if it's a European departure, it's VERY doubtful that Group Sales USA will offer a competitive fare. They were consistently 1.5x to 3x more than what I could get through a German TA, even without any group discount.
In Germany, the crucial number is 10. LH does not care whether you consider your party a group. 9 or less passengers never constitute a group, 10 or more always do.
Beware that there are both advantages and disadvantages to group bookings:
Con - A group does not necessarily get a better fare than an individual. The deeply discounted fares e.g. in W class usually are not made available to groups, at least they don't get discounted even more. So, to make up an example, if you have 12 people flying from FRA to LHR, LH might offer you all seats at the Q-class fare level although W-class is even cheaper. What to do? Split your group into two or more parties smaller than 10 and book them with different travel agents or online with different user names.
Pro - When booking a group you can do so without providing names for a long time. And even after sending in a name list, before ticketing (which can happen pretty close before departure) name changes are usually free of charge and changes in the size of your group do not incur any cancellation fees. This gives you a much easier time when people change their minds or become ill and that's why a slightly higher group fare can be rectified. With individual bookings in the cheapest inventory, the tickets have to be issued immediately and after that it's "no change, no nothing".