This is an example of why you should always Back Up Your Data Before You Lose It, and buy and keep a high-quality print of the photograph.
There are several problems with what you want. The first problem is that you have a digital image size that won't even make a 4x6 inch print. Where's the print? If you had the print, it could be scanned and enlarged.
The second problem is that the digital file that you do have appears to have been ruined by image compression, which results in severe image artifacts even at small size.
The only way to get what you want is to rewrite all the pixel data and manually rebuild the image. To do that, you would need to know more about the missing details that were lost. Even worse, the image looks like it was done with a tiny pocket camera. The blown highlights on the faces and the lack of depth-of-field markers between the people and the background will make it even harder to extract and rebuild.
There are software programs that are designed to enlarge the pixel resolution of an image, but they won't do well with your problem. Garbage in, garbage out. If you know what you are doing with the missing details, it would probably take me about four weeks to rebuild it, going through each pixel group one at a time and essentially making a new image layer over the old one. Since the image appears to have no real significance, I suggest dropping it. Surely there are professional photographs that can substitute for this low-quality snapshot.
