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Fisher exact test is not restricted to 2x2 contingency tables ; calculations just become harder and harder with larger tables !

you could try the following url :

http://www.physics.csbsju.edu/stats/exact_NROW_NCOLUMN_form.html

(up to 6x6 tables, sounds good for you, but no cell count above 100 ...)

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Fisher exact test is also available with R for general tables (of reasonable size, but not restricted to 2x2)

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Fisher exact test is not restricted to 2x2 contingency tables ; calculations just become harder and harder with larger tables !

you could try the following url :

http://www.physics.csbsju.edu/stats/exact_NROW_NCOLUMN_form.html

(up to 6x6 tables, sounds good for you, but no cell count above 100 ...)

.

Fisher exact test is also available with R for general tables (of reasonable size, but not restricted to 2x2)

.

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 Fisher exact  test is convenient in the case of a 2 × 2 contingency table. However the principle of the test can be extended to the  situation of an m × n table.

check out:

http://mathworld.wolfram.com/FishersExactTest.html

try

https://sites.google.com/a/lakeheadu.ca/bweaver/Home/statistics/notes/chisqr_assumptions

The previous answers are correct. However, sometimes it is possible to reduce the table dimension grouping 2 or more meaningful cathegories or, if the proportions are the same in a column, remove the column from the table, since the proportion is constant in both groups. However, it is usually better regrouping to avoid loosing information if possible.

Quite some time ago Albert Verbeek and I wrote a program Fisher which could carry out  Fisher's exact test in contingency tables larger than 2x2. I have appended the latest manual for this program which I can supply free of charge. It has a number of nice features in terms of test statistics, residual analysis, etc.

Note that SPSS and SAS have routines to do such tests either by enumeration or by Monte Carlo procedures.

Note, however, that the program is DOS console version, but it works under Windows 7.

Thank you all so much for your answers, it is really helpful! :)

Using software like SPSS Fisher's exact test for 2*2 or more than 2*2 or m*n can be calculated. 

But in SPSS it never gives an automatically Fisher's exact test, when your data have more than 20% of expected counts are less than 5. You should provide option to calculated it. 

Analyze>Descriptive>crosstabs> Exact test> click Exact> 

Now you should select chi-square test in "statistics" and run the test. You can get Fisher exact test results also in the output table.

Please have a look at my recent paper on this issue. It provide one numerical answer to question posed here.

Article The Tale of Cochran's Rule: My Contingency Table has so Many...

The Tale of Cochran's Rule: My Contingency Table has so Many Expected Values Smaller than 5, What Am I to Do?

https://amstat.tandfonline.com/doi/abs/10.1080/00031305.2017.1286260

Abstract:

In an informal way, some dilemmas in connection with hypothesis testing in contingency tables are discussed. The body of the paper concerns the numerical evaluation of Cochran's Rule about the minimum expected value in r×c contingency tables with fixed margins when testing independence with Pearson's X2 statistic using the chi-squared distribution.