As @Benjamin said in comments above, this is a very difficult question. However if one is willing to add conditions, then there are some relevant results. Firstly:

> Chein, Orin, * Moufang Loops of Small Order*,  Trans. Amer. Math. Soc. **188** (1974), 31-51.

Chein completely classifies all **non-associative Moufang** loops of order $\leq 31$.
There are 13 such loops - one of order 12, five of order 16, one of order 20, five of order 24, and one of order 28.

In fact Chein extends this classification to give a full classification up to order $\leq 63$ in:

> Chein, Orin, *Moufang loops of small order*,
Mem. Amer. Math. Soc. **13** (1978), no. 197, iv+131 pp.

You may also be interested in [this webpage][1] which gives a full list of all **Bol** loops of order $\leq 31$.


  [1]: http://www.uwyo.edu/moorhouse/pub/bol/