  We present measurements of the first six hadronic mass moments in
semileptonic $B \rightarrow X_c \ell \nu $ decays. The hadronic mass moments,
together with other observables of inclusive $B$ decays, can be used to
determine the CKM matrix element $|{V_{cb}}|$ and mass of the $b$-quark $m_b$
in the context of Heavy Quark Expansions of QCD. The Belle~II data recorded at
the $\Upsilon (4S)$ resonance in 2019 and 2020 (March-July), corresponding to
an integrated luminosity of $34.6\;\mathrm{fb}^{-1}$, is used for this
measurement. The decay $\Upsilon (4S) \rightarrow B \overline{B}$ is
reconstructed by applying the hadronic tagging algorithm provided by the Full
Event Interpretation to fully reconstruct one $B$ meson. The second $B$ meson
is reconstructed inclusively by selecting a high-momentum lepton. The $X_c$
system is identified by the remaining reconstructed tracks and clusters in the
electromagnetic calorimeter. We report preliminary results for the hadronic
mass moments $\langle M_X^n \rangle $ with $n=1,\dots,6$, measured as a
function of a lower cut on the lepton momentum in the signal $B$ rest frame.
