What is the order of $2$ in $(\mathbb{Z}/n\mathbb{Z})^\times$? Is it there some theorem that makes a statement about the order of $2$ in the multiplicative group of integers modulo $n$ for general $n>2$? 

Let me quote from this presentation of Carl Pomerance: 
[...] the multiplicative order of $2 \pmod n$
appears to be very erratic and difficult to get hold of.
The presentation describes, however, some properties of this order. The basic facts have already been elucidated by @HagenvonEitzen in his comment.
