An $E1+E3$ ($\ensuremath{\le}1.9% M2$) multipole mixture has been observed for the 1242-keV (${2}^{\ensuremath{-}}$ \ensuremath{\rightarrow} ${2}^{+}$) transition depopulating the ${J}^{\ensuremath{\pi}}$, $K={2}^{\ensuremath{-}},2$ octupole vibrational level at 1318 keV in $^{174}\mathrm{Yb}$ following the decay of $^{174}\mathrm{Lu}$ and $^{174}\mathrm{Lu}^{m}$. The internal-conversion c\oe{}fficient for this transition and the $\ensuremath{\gamma}\ensuremath{-}\ensuremath{\gamma}$ directional correlation for the 1242-76.5-keV cascade have been remeasured, and the results yield the unique set of multipole mixing ratios ${\ensuremath{\delta}}_{1}(\frac{M2}{E1})=0.05\ifmmode\pm\else\textpm\fi{}0.09$ and ${\ensuremath{\delta}}_{2}(\frac{E3}{E1})=0.19\ifmmode\pm\else\textpm\fi{}0.08$ which correspond to a multipole admixture of $E1({96.3}_{\ensuremath{-}2.6}^{+1.7}%)+M2({0.2}_{\ensuremath{-}0.2}^{+1.7}%)+E3({3.5}_{\ensuremath{-}2.1}^{+3.8}%)$. The $M2$ multipolarity of the 1318-keV (${2}^{\ensuremath{-}}$ \ensuremath{\rightarrow} ${0}^{+}$) transition was also verified from the measured internal-conversion coefficient. A multipole admixture of $E3$ ($<27% M2$) for the 1065-keV transition was deduced from available data, and the ratios of reduced transition probabilities for the transitions depopulating the 1318-keV level were calculated and compared with rotational-model predictions. Using the previously measured half-life of the ${2}^{\ensuremath{-}}$ level, absolute transition probabilities and hindrance factors were obtained for these transitions, and $B(E3)$ values were compared with theoretical microscopic model calculations. Partial correlation coefficients for the 992-keV transition depopulating the 1518-keV isomeric state were determined using the $\ensuremath{\gamma}\ensuremath{-}\ensuremath{\gamma}$ directional correlation data. Multipole admixtures consistent with the correlation data (and internal conversion data for the ${7}^{\ensuremath{-}}$ case) are $E2+(27\ifmmode\pm\else\textpm\fi{}8%)M1$ for the choice of ${6}^{+}$ and $E1(68\ifmmode\pm\else\textpm\fi{}7%)+M2(5\ifmmode\pm\else\textpm\fi{}2%)+E3(27\ifmmode\pm\else\textpm\fi{}7%)$ for the choice of ${7}^{\ensuremath{-}}$ for the 1518-keV level. Using the reported half-life for that level, the hindrance factors were obtained for the $K$-forbidden transitions depopulating the level for each choice of spin assignment and these are compared with hindrance factors observed for known $K$-forbidden transitions.[RADIOACTIVITY $^{174}\mathrm{Lu}$ [from $^{174}\mathrm{Yb}(d,2n)$] measured ${I}_{\ensuremath{\gamma}}$, $\ensuremath{\gamma}\ensuremath{\gamma} (\ensuremath{\theta})$, $I (\mathrm{ce})$; deduced $J$, $\ensuremath{\pi}$, ICC, $\ensuremath{\gamma}$ mixing, $B(E3)$.]