Subgroups — C₂×C₃×PSU(3,𝔽₂) #89110

Order 432.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#157	C₂	1	3	yes	maximal
2	C₃×PSU(3,𝔽₂)	C₂	1	4	yes	maximal, direct product factor
3	C₂×PSU(3,𝔽₂)	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₂×C₃×C₃⋊D₃	C₂²	1	1	yes	characteristic
5	C₃×#157	C₂²	1	6	yes	—
6	C₂×#157	C₂×C₃	1	3	yes	—
7	PSU(3,𝔽₂)	C₂×C₃	1	4	yes	direct product factor
8	C₂×C₃³	Q₈	1	1	yes	characteristic, complemented
9	C₃×C₃⋊D₃	Q₈	1	1	yes	characteristic, complemented
10	C₃×C₃⋊D₃	C₂³	1	1	yes	characteristic
11	C₂×C₃×Q₈	—	9	1	no	maximal
12	C₂×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic
13	C₂×C₃×D₃	—	12	1	no	—
14	#157	C₂²×C₃	1	6	yes	—
15	C₃³	C₂×Q₈	1	1	yes	characteristic, complemented, Sylow
16	C₃×Q₈	—	9	4	no	—
17	C₂×C₃×C₄	—	9	3	no	—
18	C₂×C₃²	—	4	1	no	—
19	C₂×C₃²	—	8	1	no	—
20	C₂×C₃²	C₃×Q₈	1	1	yes	characteristic, complemented
21	C₃⋊D₃	C₃×Q₈	1	1	yes	characteristic, complemented
22	C₃⋊D₃	C₂³×C₃	1	1	yes	characteristic
23	C₃×D₃	—	12	1	no	—
24	C₃×D₃	—	12	1	no	—
25	C₂×Q₈	—	9	1	no	Sylow
26	C₂²×C₃	—	9	1	no	—
27	C₂×D₃	—	12	1	no	—
28	C₃×C₄	—	9	6	no	—
29	C₃²	—	4	1	no	—
30	C₃²	—	8	1	no	—
31	C₃²	C₂×C₃×Q₈	1	1	yes	characteristic, complemented
32	Q₈	—	9	4	no	—
33	C₂×C₄	—	9	3	no	—
34	C₂×C₃	—	4	1	no	—
35	C₂×C₃	—	8	1	no	—
36	C₂×C₃	—	9	1	no	—
37	C₂×C₃	—	9	1	no	—
38	C₂×C₃	PSU(3,𝔽₂)	1	1	yes	characteristic, direct product factor
39	D₃	—	12	1	no	—
40	D₃	—	12	1	no	—
41	C₂²	—	9	1	no	—
42	C₄	—	9	6	no	—
43	C₃	—	4	1	no	—
44	C₃	—	8	1	no	—
45	C₃	C₂×PSU(3,𝔽₂)	1	1	yes	characteristic, direct product factor
46	C₂	—	9	1	no	—
47	C₂	—	9	1	no	—
48	C₂	C₃×PSU(3,𝔽₂)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃	PSU(3,𝔽₂)
commutator, cosocle	C₃⋊D₃	C₂³×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₃³	Q₈
prime radical	C₂×C₃×C₃⋊D₃	C₂²
hypocenter	C₃²	C₂×C₃×Q₈
LCS	G ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
2. C₃×PSU(3,𝔽₂)	C₂	direct (N × H)
3. C₂×PSU(3,𝔽₂)	C₃	direct (N × H)
7. PSU(3,𝔽₂)	C₂×C₃	direct (N × H)
8. C₂×C₃³	Q₈	semidirect
9. C₃×C₃⋊D₃	Q₈	semidirect
15. C₃³	C₂×Q₈	semidirect
20. C₂×C₃²	C₃×Q₈	semidirect
21. C₃⋊D₃	C₃×Q₈	semidirect
31. C₃²	C₂×C₃×Q₈	semidirect
38. C₂×C₃	PSU(3,𝔽₂)	direct (N × H)
45. C₃	C₂×PSU(3,𝔽₂)	direct (N × H)
48. C₂	C₃×PSU(3,𝔽₂)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#157	2
C₃×PSU(3,𝔽₂)	2
C₂×PSU(3,𝔽₂)	3
C₂×C₃×Q₈	9

Subgroups — C2×C3×PSU(3,𝔽2) #89110

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×PSU(3,𝔽₂) #89110