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The Morava $E$-theories, $E_{n}$, are complex-oriented $2$-periodic ringspectra, with homotopy groups $\mathbb{W}_{\mathbb{F}_{p^{n}}}[[u_{1}, u_{2},... , u_{n-1}]][u,u^{-1}]$. Here $\mathbb{W}$ denotes the Witt vector ring.$E_{n}$ is a Landweber exact spectrum and hence uniquely determined by thisring as $BP_{\ast}$-algebra. Algebraic $K$-theory of $E_{n}$ is a keyingredient towards analyzing the layers in the $p$-complete Waldhausen$K$-theory chromatic tower. One hopes to use the machinery of trace methods toget results towards $K$-theory once the computation for $THH(E_{n})$ is known.  In this paper we describe $THH(E_{2})$ as part of consecutive chain ofcofiber sequences where each cofiber sits in the next cofiber sequence and thefirst term of each cofiber sequence is describable completely in terms ofsuspensions and localizations of $E_{2}$. For these results, we first calculate$K(i)$-homology of $THH(E_{2})$ using a B\"okstedt spectral sequence and thenlift the generating classes of $K(1)$-homology to fundamental classes inhomotopy group of $THH(E_{2})$. These lifts allow us to construct terms of thecofiber sequence and explicitly understand how they map to $THH(E_{2})$.

$THH$ of the Morava $E$-theory Spectrum $E_{2}$