Is it possible to do a contrast in a 2x2 (or more) contingency table or am I completely missing the point?

I have the following 2x2 contingency table of observed frequencies: Contingency Table ------------------------------------------ Categories -------------- Cat 1 Cat 2 Total ------------------------------------------ Group 1 135 16 151 Participants Group 2 241 40 281 Total 376 56 432 ----------------------------------------- The Chi-squared test result is the following: χ2 Tests -------------------------------- Value df p -------------------------------- χ2 1.153 1 0.283 N 432 -------------------------------- As we can see, the p-value is > .05, therefore there is no statistically significant difference between Group and Type. Therefore, if I'm not mistaken, Group and Type seem not to be related. However, there might be a statistically significant difference between participants in Group 1 and Group 2 separately in Cat 1 and in Cat 2 (e.g., 135 vs. 241 on the one hand, and between 16 and 40 on the other hand). I am also interested to know if the difference is statistically significant in Group 1 between Cat 1 and Cat 2, and in Group 2 between Cat 1 and Cat 2 (e.g., 135 vs. 16 and 241 vs. 40). My questions: How do I test the differences within each variable? Is there an easy (relatively) way to do it? Would those be similar to ANOVA's contrasts or am I completely missing the point? Technical details: R 4.2.1 GUI 1.79 High Sierra build RStudio 2022.07.1 Build 554 for Intel macOS macOS Monterey Version 12.5 R/RStudio code (inspired by an example from ETHZ in Switzerland): tab