Plot hole: Burke is murdered so that Doward will be sent out to replace him, as both hold the same rate of CPO sonar operator. But there are many CPO sonar operators in the Royal Navy. How can the GRU be so sure that Doward will be the man chosen to replace him?
Factual error: As the Royal Navy has its own police (including detectives), there would be no need for a civilian detective to be airlifted onto a submarine, especially to investigate a sudden death that was not originally believed to be a murder.
Suggested correction: The Royal Navy police are not equipped to handle murder; local police are usually used for more serious offences so no claim of a cover up. There are no MoD police attached to a submarine either, so in theory somebody would have to go to the boat. (However they still wouldn't risk surfacing).
Initially there is no suspicion of murder, only an unexplained sudden death. There would be absolutely no need to airlift a civilian detective aboard a top-secret submarine. It wouldn't be the first time a sudden death had occurred aboard a Royal Navy vessel.
Episode #1.4 - S1-E4
Continuity mistake: The captain of the submarine has the rank of Commander (3 solid stripes) and half-way through episode 4 the captain suddenly has the rank of Captain (4 solid stripes) as shown on his shoulder epaulettes.
Factual error: The ID cards for the chief petty officers all show them in junior rates' round hats and 'sailor suits'. CPOs (and petty officers) wear officer-style peaked caps and jackets and ties. It is unlikely that they've never had their photos changed since they were junior rates.
Suggested correction: So you are wrong in the sense that ID card photos aren't always updated that quickly; you would find a lot of POs and CPOs that have the same idea card from when they were junior rates, however it's a passport type photo and hats are not worn and you should only see the head not the uniform so it shouldn't matter.
All service personnel have their identity cards changed when they become senior rates (Army and RAF sergeants and naval petty officers). This verifies their right to enter senior rates' messes when on land.
Factual error: The insignia worn by the submarine crew is inaccurate, including the lack of a white patch behind the trade badges and a strange made-up version of the White Ensign. Also the Submarine service emblem of two heraldic dolphins has them facing the other way.
Factual error: The Marine Engineering Officer on a submarine would be an experienced lieutenant-commander, not a young lieutenant.
Character mistake: Robertson is usually referred to as Detective Superintendent, but sometimes as Detective Chief Superintendent.
Character mistake: The ship's doctor is sometimes addressed as "lieutenant surgeon" instead of the correct "surgeon lieutenant."
Factual error: None of the naval officers wear medal ribbons. Even if none of them have been in a war zone (unlikely) or received an honour (also unlikely, especially given one is an admiral), the more senior would still have jubilee medals and the long service medal.
Suggested correction: Not all officers wear uniform with medal ribbons, so have a best and a work uniform. In fact most of the time at sea all crew and officers would be wearing working blue uniform, not a mix like on the show.
Obviously no ribbons are worn on working dress, but none of them are wearing medal ribbons on No.1 Dress, where they most certainly would. Wearing this without appropriate medal ribbons is wearing it incorrectly, which no military officer would do.
Factual error: The coxswain, who is a warrant officer (as it says in the last episode), wears a chief petty officer's rank badge.
Factual error: Most of the crew seem to be Scottish. Even though the submarine is based in Scotland, the Royal Navy is a national service and its crew's composition would reflect that of the UK in general: the majority would therefore be English.
Continuity mistake: The captain (who holds the rank of commander) is in some scenes wearing four rank rings instead of the correct three he wears for the rest of the series.
Factual error: Some of the ranks are wrong. There are too many chief petty officers (and they're too young). The coxswain would be a warrant officer rather than a chief. The marine engineering officer would be a lieutenant-commander rather than an inexperienced lieutenant.
Continuity mistake: Kirsten is referred to as a DC (Detective Constable) in the first half of the series and then as a DS (Detective Sergeant) in the second half. As these later episodes include flashbacks to before the previous episodes, she obviously hasn't suddenly been promoted.