  In this paper, we carry out the complete $\mathcal
O(\alpha\alpha_s^{2})$-order study on the inclusive productions of
$\Upsilon(nS)$ and $\chi_b(nP)$ ($n=1,2,3$) via the Standard Model Higgs boson
decay, within the framework of nonrelativistic QCD. The feeddown effects via
the higher excited states are found to be substantial. The color-octet
$^3S_1^{[8]}$ state related processes consisting of $H^0 \to
b\bar{b}[^3S_1^{[8]}]+g$ and $H^0 \to b\bar{b}[^3S_1^{[8]}]+Q+\bar{Q}$
($Q=c,b$) play a vital role in the predictions on the decay widths. Moreover,
our newly calculated next-to-leading order QCD corrections to $H^0 \to
b\bar{b}[^3S_1^{[8]}]+g$ can enhance its leading-order result by 3-4 times,
subsequently magnifying the total $^3S_1^{[8]}$ contributions by about $40\%$.
Such a remarkable enhancement will to a large extent influence the
phenomenological conclusions. For the color-singlet $^3P_J^{[1]}$ state, in
addition to $H^{0} \to b\bar{b}[^3P_J^{[1]}]+b+\bar{b}$, the newly introduced
light hadrons associated process, $H^{0} \to b\bar{b}[^3P_J^{[1]}]+g+g$, can
also provide non-negligible contributions, especially for $^3P_2^{[1]}$.
Summing up all the contributions, we have $\mathcal B_{H^0 \to \chi_b(nP)+X}
\sim 10^{-6}-10^{-5}$ and $\mathcal B_{H^0 \to \Upsilon(nS)+X} \sim
10^{-5}-10^{-4}$, which meets marginally nowadays LHC experimental data and can
help in understanding the heavy quarkonium production mechanism as well as the
Yukawa couplings.
