Is there a classification of finite nonabelian 2-groups of exponent 4?


What about, finite nonabelian 3-groups of exponent 3?



Groups of exponent 4
There is no classification of finite groups of exponent 4. You might find [this paper](https://www.dropbox.com/s/jsallb6xi0rru8n/Mann%20On%20the%20orders%20of%20groups%20of%20exponent%20four.pdf?dl=0) interesting - it contains lots of information about how the group Burnside group $B(m,4)$ grows (all $m$-generator exponent-4 groups are quotients of this group).


There is also no classification of finite groups of exponent 3. However it is known that these groups must be $2$-Engel and class three. Furthermore in this case the precise size of the corresponding Burnside group is known: $B(m,3)$ is a finite group of size $3^{m + \binom{m}{2} + \binom{m}{3}}$.


