Subgroups — C₂³×SU(3,𝔽₂) #13180077

Order 1728.

1409 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#1088	C₂	1	3	yes	maximal
2	C₂²×SU(3,𝔽₂)	C₂	1	28	yes	maximal, direct product factor
3	C₂³×#272	C₂²	1	1	yes	characteristic
4	C₂²×#1088	C₂²	1	42	yes	—
5	C₂×SU(3,𝔽₂)	C₂²	1	112	yes	direct product factor
6	C₂³×H(3,𝔽₃)	Q₈	1	1	yes	characteristic, complemented
7	C₂²×#272	Q₈	1	7	yes	complemented
8	C₂²×#272	C₂³	1	7	yes	—
9	C₂×#1088	C₂³	1	84	yes	—
10	SU(3,𝔽₂)	C₂³	1	64	yes	direct product factor
11	C₂³×C₃×Q₈	—	9	1	no	maximal
12	C₂³×C₃×D₃	—	12	1	no	—
13	C₂²×H(3,𝔽₃)	C₂×Q₈	1	7	yes	complemented
14	C₂×#272	C₂×Q₈	1	21	yes	complemented
15	C₂×#272	C₂⁴	1	7	yes	—
16	#1088	C₂⁴	1	24	yes	—
17	C₂²×C₃×Q₈	—	9	28	no	—
18	C₂³×C₃×C₄	—	9	3	no	—
19	C₂³×C₃²	—	4	1	no	—
20	C₂²×C₃×D₃	—	12	7	no	—
21	C₂²×C₃×D₃	—	12	7	no	—
22	C₂³×Q₈	—	9	1	no	Sylow
23	C₂×H(3,𝔽₃)	C₂²×Q₈	1	7	yes	complemented
24	#272	C₂²×Q₈	1	7	yes	complemented
25	#272	C₂⁵	1	1	yes	characteristic
26	C₂⁴×C₃	—	9	1	no	—
27	C₂³×D₃	—	12	1	no	—
28	C₂×C₃×Q₈	—	9	112	no	—
29	C₂²×C₃×C₄	—	9	42	no	—
30	C₂²×C₃²	—	4	7	no	—
31	C₂×C₃×D₃	—	12	21	no	—
32	C₂×C₃×D₃	—	12	7	no	—
33	C₂²×Q₈	—	9	28	no	—
34	C₂³×C₄	—	9	3	no	—
35	H(3,𝔽₃)	C₂³×Q₈	1	1	yes	characteristic, complemented, Sylow
36	C₂³×C₃	—	9	7	no	—
37	C₂³×C₃	—	9	7	no	—
38	C₂³×C₃	—	12	1	no	—
39	C₂³×C₃	PSU(3,𝔽₂)	1	1	yes	characteristic
40	C₂²×D₃	—	12	7	no	—
41	C₂²×D₃	—	12	7	no	—
42	C₃×Q₈	—	9	64	no	—
43	C₂×C₃×C₄	—	9	84	no	—
44	C₂×C₃²	—	4	7	no	—
45	C₃×D₃	—	12	7	no	—
46	C₃×D₃	—	12	1	no	—
47	C₂⁴	—	9	1	no	—
48	C₂×Q₈	—	9	112	no	—
49	C₂²×C₄	—	9	42	no	—
50	C₂²×C₃	—	9	7	no	—
51	C₂²×C₃	—	9	21	no	—
52	C₂²×C₃	—	12	7	no	—
53	C₂²×C₃	C₂×PSU(3,𝔽₂)	1	7	yes	—
54	C₂×D₃	—	12	7	no	—
55	C₂×D₃	—	12	21	no	—
56	C₃×C₄	—	9	24	no	—
57	C₃²	—	4	1	no	—
58	C₂³	—	9	7	no	—
59	C₂³	—	9	7	no	—
60	C₂³	SU(3,𝔽₂)	1	1	yes	characteristic, direct product factor
61	Q₈	—	9	64	no	—
62	C₂×C₄	—	9	84	no	—
63	C₂×C₃	—	9	1	no	—
64	C₂×C₃	—	9	7	no	—
65	C₂×C₃	—	12	7	no	—
66	C₂×C₃	C₂²×PSU(3,𝔽₂)	1	7	yes	—
67	D₃	—	12	1	no	—
68	D₃	—	12	7	no	—
69	C₂²	—	9	7	no	—
70	C₂²	—	9	21	no	—
71	C₂²	C₂×SU(3,𝔽₂)	1	7	yes	direct product factor
72	C₄	—	9	24	no	—
73	C₃	—	12	1	no	—
74	C₃	C₂³×PSU(3,𝔽₂)	1	1	yes	characteristic
75	C₂	—	9	1	no	—
76	C₂	—	9	7	no	—
77	C₂	C₂²×SU(3,𝔽₂)	1	7	yes	direct product factor

	subgroup	G/H
center, socle, hypercenter	C₂³×C₃	PSU(3,𝔽₂)
commutator, cosocle	#272	C₂⁵
Frattini	C₃	C₂³×PSU(3,𝔽₂)
Fitting	C₂³×H(3,𝔽₃)	Q₈
prime radical	C₂³×#272	C₂²
hypocenter	H(3,𝔽₃)	C₂³×Q₈
LCS	G ▷ #272 ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂³×C₃

Complements

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

N	H ≅ G/N	type
2. C₂²×SU(3,𝔽₂)	C₂	direct (N × H)
5. C₂×SU(3,𝔽₂)	C₂²	direct (N × H)
6. C₂³×H(3,𝔽₃)	Q₈	semidirect
7. C₂²×#272	Q₈	semidirect
10. SU(3,𝔽₂)	C₂³	direct (N × H)
13. C₂²×H(3,𝔽₃)	C₂×Q₈	semidirect
14. C₂×#272	C₂×Q₈	semidirect
23. C₂×H(3,𝔽₃)	C₂²×Q₈	semidirect
24. #272	C₂²×Q₈	semidirect
35. H(3,𝔽₃)	C₂³×Q₈	semidirect
60. C₂³	SU(3,𝔽₂)	direct (N × H)
71. C₂²	C₂×SU(3,𝔽₂)	direct (N × H)
77. C₂	C₂²×SU(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
C₂³×#1088	2
C₂²×SU(3,𝔽₂)	2
C₂³×C₃×Q₈	9

Subgroups — C23×SU(3,𝔽2) #13180077

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×SU(3,𝔽₂) #13180077