Subgroups — GU(3,𝔽₂) #10620014

Order 648.

41 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	SU(3,𝔽₂)	C₃	1	1	yes	maximal, characteristic, complemented
2	#4174	—	4	1	no	maximal
3	#1088	—	3	1	no	—
4	AΣL(1,𝔽₂₇)	—	4	1	no	Sylow
5	C₃×SL(2,𝔽₃)	—	9	1	no	maximal
6	C₃²×D₃	—	12	1	no	—
7	#272	A₄	1	1	yes	characteristic
8	C₃³	—	4	1	no	—
9	3₋¹⁺²	—	8	1	no	—
10	H(3,𝔽₃)	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
11	C₃×Q₈	—	9	1	no	—
12	SL(2,𝔽₃)	—	9	3	no	—
13	C₂×C₃²	—	36	1	no	—
14	C₃×D₃	—	12	1	no	—
15	C₃×D₃	—	12	3	no	—
16	C₃×C₄	—	27	1	no	—
17	C₃²	—	4	1	no	—
18	C₃²	—	12	3	no	—
19	C₃²	—	12	1	no	—
20	C₉	—	24	1	no	—
21	Q₈	—	9	1	no	Sylow
22	C₂×C₃	—	9	1	no	—
23	C₂×C₃	—	36	3	no	—
24	D₃	—	12	1	no	—
25	C₄	—	27	1	no	—
26	C₃	—	12	1	no	—
27	C₃	—	12	3	no	—
28	C₃	PSU(3,𝔽₂)⋊C₃	1	1	yes	characteristic
29	C₂	—	9	1	no	—

	subgroup	G/H
center, Frattini, socle, hypercenter	C₃	PSU(3,𝔽₂)⋊C₃
commutator, cosocle, hypocenter	SU(3,𝔽₂)	C₃
Fitting	H(3,𝔽₃)	SL(2,𝔽₃)
LCS	G ▷ SU(3,𝔽₂)
UCS	C₁ ◁ C₃

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. SU(3,𝔽₂)	C₃	semidirect
10. H(3,𝔽₃)	SL(2,𝔽₃)	semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.