Let $(N \subset M)$ be an irreducible finite index unital inclusion of hyperfinite ${\rm II}\_1$ factors.  

Let $K\_1$ and $K\_2$ be two *distinct* intermediate subfactors $N \subset K\_i \subset M$, such that $|M:K\_i| = 2$.


*Question*: Is there (at least) a third intermediate subfactor strictly between $K\_1 \cap K\_2$ and $M$?


*Remark*: It is true for the finite group-subgroup subfactors $(R \rtimes H \subset R \rtimes G)$ and $(R^G \subset R^H)$.  

*Proof using Galois correspondence*: Let $H\_1$, $H\_2$ be two intermediate subgroups $H \subset H\_i \subset G$.


* First, if $|G:H\_i| = 2$ then $H\_i$ is a normal subgroup of $G$. It follows that $L=H\_1 \cap H\_2$ is also normal and $[L,G] \simeq [1,G/L]$. If there is not a third intermediate, then $[1,G/L]$ is boolean, so by [Ore's theorem](https://mathoverflow.net/q/179555/34538) $G/L$ is cyclic, but with two subgroups of index $2$, contradiction.
* Next if $|H\_i:H| = 2$ then $H$ is a normal subgroup of $H\_i$. It follows that $H$ is also a normal subgroup of $T=\langle H\_1 , H\_2 \rangle $. If there is no third, then by the same argument $T/H$ is cyclic with two subgroups of order $2$, contradiction. $\square$



Existence of a third intermediate if there are two intermediate subfactors of index 2
The answer is yes. Moreover, $K\_1\cap K\_2 \subset M$ is a dihedral group subfactor, so the lattice of intermediate subfactors between $K\_1 \cap K\_2$ and $M$ is clear. 


Proof: Let us look at the dual lattice. Suppose $\hat{K\_1},\hat{K\_2}$ are index two intermediate subfactors of $M\subset \hat{N}$. Let the $e+p\_1, e+p\_2$ be the corresponding biprojections. The biprojection $q$ generated by the two projections is determined by the coproduct of $p\_1$ and $p\_2$, see Theorem 4.8 in [this paper](http://arxiv.org/pdf/1308.5656v2.pdf). Note that $p\_1, p\_2$ have trace 1, so their coproduct is the same as their fusion rule. We know that the quotient of $Z\_2\*Z\_2$ is a dihedral group. That means $(p\_1\otimes p\_2)^{\otimes n} \sim 1$ for some smallest $n$. So the biprojection $q$ gives a dihedral group subfactor with index $2n$. In particular, $p\_1\*p\_2$ generates a third biprojection with index $n$. 


In general, we have the following result: 


Suppose $P\_1,P\_2$ are intermediate subfactors of a finite index irreducible subfactor $N \subset M$, and $P$ is the intermediate subfactor generated by $P\_1$ and $P\_2$. If $N\subset P\_1, N\subset P\_2$ are group crossed product, then $N\subset P$ is also a group crossed product.
If $N\subset P\_1, N\subset P\_2$ are depth 2, i. e. Kac algebra crossed product, then $N\subset P$ is also depth 2, see Theorem 4.9 in [this paper](http://arxiv.org/pdf/1308.5656v2.pdf).


