Because you (and your family) are asylees, all of you may apply for a green card/permanent resident status one (1) year after being granted asylum if you have been physically present in the U.S. for at least 1 year after being granted asylum (which you have); continue to meet the definition of a asylee (or continue to be the spouse or child of such asylee); have not abandoned your asylee status; are not firmly resettled in any foreign country; and continue to be admissible to the U.S. (Note: waiver may be available if you are now inadmissible). Note that each case is fact specific and the answer may change depending on your particular circumstances.
Answered on Nov 11th, 2011 at 12:34 PM