Yes, they do. I recently flew Qatar (Y cabin) and selected my seat online - use the "Manage A Booking" option. Interestingly, I wasn't able to add my AAdvantage number online so had to call them to do that.
I booked J on the QR website and had no problems using MMB. But I haven't done a codeshare booking yet. I've read where apparently Row 1 can be blocked out for (elite?) Qataris at checkin, who can also have priority for your seat if desired.
It's interesting that unlike someother OW carriers QR allow people without any significant ff status to select the front rows of whY which looks like a mini-cabin with a 2-4-2 config. Other OW airlines eg CX/AY/LA may show available seats however not allow people without elite status to select them.
I am OWE and it does not appear I can choose the front row in J.
Flying Boeing 777-300ER in J. Any suggestion of best seats? Front or second cabin? Vaguely remember the back half of J was quieter (but it was several years ago I flew QR in J). Thank you in advance.
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