Flandre is Remilia's younger sister. Flandre is stated to respect her sister, and Remilia cares for Flandre as shown where she is considering getting her a good tutor in Imperishable Night, but it is unknown how often the two interact. Perfect Memento in Strict Sense states that it's rare (but not nonexistent) to see Flandre and Remilia together. The only time in an official work that they are depicted together is in Bohemian Archive in Japanese Red, which also shows that Remilia will at least allow Aya to interview Flandre in the first place. The interview shows that although Flandre might respect her older sister, it's to the extent you'd expect a snarky child to. She is willing to outright state when she thinks Remilia is just spouting nonsense, at which point Remilia comes in with a comment that she shouldn't be talking about her older sister like that.
On top of serving Remilia, she presumably serves her younger sister, Flandre Scarlet who is usually not allowed out of Scarlet Devil Mansion due to her emotional instability. However, this is not stated, implied, or depicted in any official work. Someone prepares Flandre's food, which are humans prepared in the shape of pastries and other pleasant things, but it is never stated to be Sakuya.
<Patchouli Knowledge >
Not much is known about the relationship between Patchouli and Flandre. All that is certain is that Patchouli is aware of Flandre's existence and the need to keep her confined to the mansion, as Remilia noted Patchouli was probably causing a rainstorm around the mansion to keep Flandre from getting out.
They presumably serve Flandre as well.
In fanworks, Hong Meiling seems to be some sort of big sister or playmate to Flandre (rather than Remilia Scarlet). In comedy works, it can be to throw Meiling in the way of more danger. In serious ones, her laid back approach to her job and uncontrollably friendly nature make her a natural choice for reaching out to an otherwise isolated Flandre. Many fan works depict the two going out of their way to help each other on numerous occasions.
This stems from the fact that they both possess greater powers than those of their sisters Satori Komeiji and Remilia Scarlet. Also, Flandre and Koishi are the younger sister to their sister, who both own a mansion. Some fans think Koishi's ability would make it easier for her and Flandre to communicate.
Depictions with Nue Houjuu is also common. This stems from the fact that Nue's theme ("Heian Alien") sounds similar to Flandre's theme to some.
Depictions with Fujiwara no Mokou has dropped with popularity due to the two above, but are still common. The name Mokou means "Scarlet (younger) Sister" and according to ZUN, it can also mean "Paint me red", but it has no connection to Flandre. Another reason for the pairing (platonic usually) is that Mokou is immortal and has little to fear from Flandre's destructive strength.
Depictions with Suwako Moriya are rather uncommon. The only known relation between Suwako and Flandre is that Suwako also has a spell card to multiply herself.