Let $(J,\pi)$ be a cuspidal type in $SL(2,F)$, $F$ is a non-Arch. local field and let $I$ be the Iwahori subgroup of $SL(2,F)$.


Is there any possibility that $J\subset I$ or even a subgroup? 



cuspidal types and Iwahori subgroup for $SL(2,F)$
Here is a novel, explicit answer to your question : 


Benedict Gross and Mark Reeder have recently discovered a family of supercuspidal representations, called "simple supercuspidal representations", of simply connected, split, almost simple, reductive groups. These representations are examples of what you are looking for.


Here's an example of a simple supercuspidal representation, and the general case is similar. Consider $G(\mathbb{Q}\_2) = SL(2,\mathbb{Q}\_2)$. Let $I$ be an Iwahori subgroup of $SL(2,\mathbb{Q}\_2)$, and let $I^+$ be the pro-unipotent radical of $I$. For example, we can take 


$$I = \begin{bmatrix} \mathfrak{o} & \mathfrak{o}\\\ \mathfrak{p} & \mathfrak{o} \end{bmatrix}$$


$$I^+ = \begin{bmatrix} 1 + \mathfrak{p} & \mathfrak{o}\\\ \mathfrak{p} & 1 + \mathfrak{p} \end{bmatrix}$$


Let $\chi$ be an "affine generic character" of $I^+$. Since we are dealing with $SL(2,\mathbb{Q}\_2)$, this just means the following : Let $\eta$ be the character of $\mathbb{Z}\_2$ given by $$\eta : \mathbb{Z}\_2 \rightarrow \mathbb{C}^\*$$ $$2 \mathbb{Z}\_2 \mapsto 1$$ $$1 + 2 \mathbb{Z}\_2 \mapsto -1$$ 
Then define the character


$$\chi : I^+ \rightarrow \mathbb{C}^\*$$
$$\begin{bmatrix} a & b\\\ 2c & d \end{bmatrix} \mapsto \eta(b) \eta(c)$$


Then $Ind\_{I^+}^{SL(2,\mathbb{Q}\_2)} \chi$ is a supercuspidal representation (which Gross and Reeder call a "simple supercuspidal representation", since it is so "simple" to define). The general situation is similar : If $G$ is simply connected, split, almost simple, $F$ is a $p$-adic field, $I \subset G(F)$ is an Iwahori subgroup, $I^+ \subset I$ is the pro-unipotent radical, and $\chi : Z I^+ \rightarrow \mathbb{C}^\*$ is an "affine generic character" (where $Z$ is the center of $G(F)$), then $Ind\_{Z I^+}^{G(F)} \chi$ is a supercuspidal representation (called a "simple supercuspidal representation"). For more details, check out Section 9 of <https://www2.bc.edu/~reederma/AdjointGamma.pdf>


