If
N
⊆
P
,
Q
⊆
M
N\subseteq P,Q\subseteq M
are type II
1
_1
factors with
N
′
∩
M
=
C
i
d
N’\cap M =\mathbb C id
and
[
M
:
N
]
>
∞
[M:N]>\infty
we show that restrictions on the standard invariants of the elementary inclusions
N
⊆
P
N\subseteq P
,
N
⊆
Q
N\subseteq Q
,
P
⊆
M
P\subseteq M
and
Q
⊆
M
Q\subseteq M
imply drastic restrictions on the indices and angles between the subfactors. In particular we show that if these standard invariants are trivial and the conditional expectations onto
P
P
and
Q
Q
do not commute, then
[
M
:
N
]
[M:N]
is
6
6
or
6
+
4
2
6+4\sqrt 2
. In the former case
N
N
is the fixed point algebra for an outer action of
S
3
S_3
on
M
M
and the angle is
π
/
3
\pi /3
, and in the latter case the angle is
cos
−
1
(
2
−
1
)
\cos ^{-1}(\sqrt 2-1)
and an example may be found in the GHJ subfactor family. The techniques of proof rely heavily on planar algebras.