  We consider the invariant subspace of composition operators on Hardy space
$H^p$ where the composition operators corresponding to a function $\varphi$
that is a holomorphic self-map of $\mathbb D$. Firstly, we discuss composition
operators $C_\varphi$ on subspace $H_{\alpha,\beta}^p$ of Hardy space $H^p$. We
will explore the invariant subspaces for $C_\varphi$ in various special cases.
Secondly, we consider Beurling type invariant subspace for $C_\varphi$. When
$\theta$ is a inner function, we prove that $\theta H^p$ is invariant for
$C_\varphi$ if and only if $\displaystyle{\frac{\theta\circ\varphi}{\theta}}$
belongs to $\mathcal S(\mathbb D)$. Thirdly, we obtain that $z^nH^p$ is
nontrivial invariant subspace for Deddends algebras $\mathcal D_{C_\varphi}$
when $C_\varphi$ is a compact composition operator and $\varphi$ satisfies that
$\varphi(0)=0$ and $\parallel\varphi\parallel_\infty<1$.
