Every subfactor $(N \subset M)$ in this post are supposed to be finite index inclusion of ${\rm II}_1$ factors.   

*Definition* ([here][1] p64):  Such a subfactor is called extremal if $tr_{N'} = tr_M$ on $N' \cap M$ (where $N′$ is calculated on any Hilbert space on which M acts with finite $M$-dimension).  

In [this paper][2] p25, remark 6.6. states that the  $A_{\infty}^{(1)}$ subfactors, realizing every index $\ge 4$, are non-extremal.   

**Question 1**: Why the $A_{\infty}^{(1)}$ subfactors are non-extremal?    

Moreover, the $A_{\infty}^{(1)}$ subfactors are maximal, i.e. admits no non-trivial intermediate *subfactor* (see [here][3]).  
**Question 2**: What are the (smallest in index) known examples of non-$A_{\infty}^{(1)}$ non-extremal *maximal* subfactor?     

*Remark*: An irreducible subfactor (i.e. $N' \cap M = \mathbb{C}$) is *a fortiori* extremal. The finite depth subfactors are also known to be extremal. So an example should be non-irreducible, infinite depth and maximal (as $A_{\infty}^{(1)}$).   
I've added the assumption *maximal* for avoiding answers like $A_{\infty}^{(1)} \otimes X$.  


But any $A_{\infty}^{(1)}$ subfactor admits an intermediate von Neumann algebra (see also [here][3]).     
**Question 3**: Is there a non-extremal subfactor without non-trivial intermediate von Neumann algebra?

  [1]: http://arxiv.org/abs/1111.1362
  [2]: http://arxiv.org/pdf/1009.0186.pdf
  [3]: https://mathoverflow.net/q/205944/34538