We searched for the 6α-condensed state in $^{24}$Mg by measuring the ${}^{12}C{+}^{12}C$ scattering with the SAKRA Si detector array at $E_{\mathrm{cm}}=17.5$–25.0 MeV. By using the invariant-mass method for the detected 3α particles, the inclusive cross sections for the ${}^{12}C{+}^{12}C{\to }^{12}C\left(0_2^{+}\right)+X$ and ${}^{12}C\left(3_1^{-}\right)+X$ reactions were determined. In addition, the missing-mass spectroscopy was successfully utilized to determine the excitation energy of the residual ${}^{12}C$ nucleus and the exclusive cross sections for the ${}^{12}C{+}^{12}C{\to }^{12}C\left(0_2^{+}\right){+}^{12}C\left(0_1^{+}\right)$, ${}^{12}C\left(0_2^{+}\right){+}^{12}C\left(2_1^{+}\right)$, and ${}^{12}C\left(0_2^{+}\right){+}^{12}C\left(0_2^{+}\right)$ reactions. In both the inclusive ${}^{12}C\left(0_2^{+}\right)+X$ channel and the exclusive ${}^{12}C\left(0_2^{+}\right){+}^{12}C\left(0_1^{+}\right)$ channel, the cross section peaked at $E_{\mathrm{cm}}=19.4$ MeV, which correspond to the excitation energy of $E_x=33.3$ MeV in $^{24}$Mg. This 19.4-MeV state is a candidate for the 6α-condensed state because of the agreement of the excitation energy with the theoretical value and its decay property. In the exclusive ${}^{12}C\left(0_2^{+}\right){+}^{12}C\left(0_2^{+}\right)$ channel, a broad state was observed at $E_{\mathrm{cm}}=22.5$ MeV, which correspond to the excitation energy of $E_x=36.4$ MeV in $^{24}$Mg. From the angular distribution of the differential cross section, the spin and parity of this 22.5-MeV state was assigned to be $4^{+}$. In addition, a $2^{+}$ state was suggested at the low-energy side of the 22.5-MeV state. Because their excitation energies are higher than the theoretical value of the 6α-condensed state, these states might be excited states of the 6α-condensed state such as the $2_2^{+}$ and $4_1^{+}$ states in ${}^{12}C$.