We report a systematic survey of a 6.7GHz Class II methanol maser toward a sample of 448 sources selected from the Red Midcourse Space Experiment Source catalog. These sample sources are composed of high-mass star-forming region (HMSFR) candidates and have been studied as tracers of HMSFRs, such as water masers or radio continuum emission of ultracompact HII region. The survey was conducted using the Shanghai Tianma Radio Telescope. Through the observations, we simultaneously studied the 4.7 and 6.0GHz excited-state interstellar hydroxyl (OH) maser lines and 10 hydrogen radio recombination lines (RRLs) in the C band. In total, we detected 6.7GHz methanol masers and RRLs from 102 and 116 sources, respectively. In addition, 4, 3, and 10 sources exhibit OH masers at 4765.56, 6030.75, and 6035.09MHz transitions, respectively. Through the survey, we identified four new 6.7GHz methanol maser sources and three new excited-state OH maser sources (one at 4750MHz and two at 6035MHz). The statistical analysis demonstrated that there is a positive correlation of luminosity between 6.7GHz methanol masers and RRLs. A good correlation of integrated luminosity between radio continuum emission and the 6.7GHz methanol masers is presented with respect to the RRL emission sources. The average of the integrated luminosities of the RRLs in the sources with 6.7GHz masers is greater than those without the 6.7GHz masers; similarly, the average of integrated luminosities of the 6.7GHz methanol masers in the sources with RRLs is greater than those without the RRLs. Moreover, we found that the averages of the emission measure and electron temperature of HII regions associated with 6.7GHz methanol masers are larger than those without the 6.7GHz methanol masers. This suggests that the masers are most likely produced in high gas density and luminous regions with brighter RRLs and higher radio continuum emission.