Group #11721792 — subgroups

Subgroups — D₈×#272 #11721792

Order 864.

418 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×#272	C₂	1	2	yes	maximal, complemented
2	D₈×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	#88545	C₂	1	2	yes	maximal, complemented
4	#88529	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×#272	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×D₃×D₈	—	3	4	no	maximal
7	C₂²×#272	—	2	2	no	—
8	D₄×H(3,𝔽₃)	C₂²	1	2	yes	complemented
9	#6235	—	2	2	no	—
10	C₄⋊#272	C₂²	1	2	yes	complemented
11	C₄×#272	C₂²	1	1	yes	characteristic
12	C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
13	#6180	C₂²	1	1	yes	characteristic, complemented
14	C₃×D₃×D₄	—	3	8	no	—
15	C₃²×D₈	D₃	1	4	yes	complemented
16	C₃×#213	—	3	8	no	—
17	C₃×D₂₄	—	3	4	no	—
18	C₃×D₃×C₈	—	3	4	no	—
19	C₂²×H(3,𝔽₃)	—	2	2	no	—
20	C₂×#272	—	2	2	no	—
21	C₂×#272	—	4	4	no	—
22	C₂×#272	D₄	1	1	yes	characteristic
23	C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
24	#1084	D₄	1	1	yes	characteristic
25	C₂×C₃×D₈	—	9	1	no	—
26	D₃×D₈	—	3	4	no	—
27	C₂²×C₃×D₃	—	6	8	no	—
28	C₃²×D₄	C₂×D₃	1	8	yes	complemented
29	C₃×#67	—	6	8	no	—
30	C₃×D₁₂	—	3	8	no	—
31	C₃×C₄×D₃	—	3	4	no	—
32	C₃²×C₈	C₂×D₃	1	4	yes	complemented
33	C₃×C₃⋊C₈	—	3	4	no	—
34	C₂×H(3,𝔽₃)	—	4	2	no	—
35	C₂×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
36	#272	—	4	2	no	—
37	#272	D₈	1	2	yes	direct product factor
38	C₂×C₃×D₄	—	9	2	no	—
39	D₃×D₄	—	3	8	no	—
40	C₃×D₈	—	3	4	no	—
41	C₃×D₈	—	9	2	no	—
42	C₃×D₈	—	9	1	no	—
43	C₃×D₈	C₃⋊D₃	1	1	yes	characteristic
44	C₂×C₃×C₈	—	9	1	no	—
45	#213	—	3	8	no	—
46	D₂₄	—	3	4	no	—
47	D₃×C₈	—	3	4	no	—
48	C₂²×C₃²	—	2	8	no	—
49	C₂×C₃×D₃	—	3	4	no	—
50	C₂×C₃×D₃	—	6	8	no	—
51	C₂×C₃×D₃	—	12	16	no	—
52	C₃²×C₄	C₂²×D₃	1	4	yes	—
53	C₃×Dic₃	—	3	4	no	—
54	C₂×D₈	—	9	1	no	Sylow
55	H(3,𝔽₃)	C₂×D₈	1	1	yes	characteristic, complemented, Sylow
56	C₂³×C₃	—	18	2	no	—
57	C₂²×D₃	—	6	8	no	—
58	C₃×D₄	—	3	8	no	—
59	C₃×D₄	—	9	2	no	—
60	C₃×D₄	—	18	2	no	—
61	C₃×D₄	C₂×C₃⋊D₃	1	2	yes	—
62	C₂×C₃×C₄	—	9	1	no	—
63	#67	—	6	8	no	—
64	D₁₂	—	3	8	no	—
65	C₄×D₃	—	3	4	no	—
66	C₃×C₈	—	3	4	no	—
67	C₃×C₈	—	9	1	no	—
68	C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
69	C₃⋊C₈	—	3	4	no	—
70	C₂×C₃²	—	4	8	no	—
71	C₂×C₃²	D₃×D₄	1	4	yes	—
72	C₃×D₃	—	3	8	no	—
73	C₃×D₃	—	12	8	no	—
74	C₂×D₄	—	9	2	no	—
75	D₈	—	9	2	no	—
76	D₈	—	9	1	no	—
77	D₈	#272	1	1	yes	characteristic, direct product factor
78	C₂×C₈	—	9	1	no	—
79	C₂²×C₃	—	2	2	no	—
80	C₂²×C₃	—	6	8	no	—
81	C₂²×C₃	—	9	1	no	—
82	C₂²×C₃	—	18	2	no	—
83	C₂²×C₃	—	36	4	no	—
84	C₂×D₃	—	3	4	no	—
85	C₂×D₃	—	6	8	no	—
86	C₂×D₃	—	12	16	no	—
87	C₃×C₄	—	3	4	no	—
88	C₃×C₄	—	9	1	no	—
89	C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
90	Dic₃	—	3	4	no	—
91	C₃²	D₃×D₈	1	4	yes	complemented
92	C₂³	—	18	2	no	—
93	D₄	—	9	2	no	—
94	D₄	—	18	2	no	—
95	D₄	C₂×#272	1	2	yes	complemented
96	C₂×C₄	—	9	1	no	—
97	C₈	—	9	1	no	—
98	C₈	C₂×#272	1	1	yes	characteristic, complemented
99	C₂×C₃	—	3	4	no	—
100	C₂×C₃	—	4	2	no	—
101	C₂×C₃	—	9	2	no	—
102	C₂×C₃	—	12	8	no	—
103	C₂×C₃	—	36	2	no	—
104	C₂×C₃	D₄×C₃⋊D₃	1	1	yes	characteristic
105	D₃	—	3	8	no	—
106	D₃	—	12	8	no	—
107	C₂²	—	2	2	no	—
108	C₂²	—	9	1	no	—
109	C₂²	—	18	2	no	—
110	C₂²	—	36	4	no	—
111	C₄	—	9	1	no	—
112	C₄	C₂²×#272	1	1	yes	characteristic
113	C₃	—	3	4	no	—
114	C₃	D₈×C₃⋊D₃	1	1	yes	characteristic
115	C₂	—	4	2	no	—
116	C₂	—	9	2	no	—
117	C₂	—	36	2	no	—
118	C₂	D₄×#272	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₄×#272

C₂

yes

maximal, complemented

D₈×H(3,𝔽₃)

C₂

yes

maximal, characteristic, complemented

#88545

C₂

yes

maximal, complemented

#88529

C₂

yes

maximal, characteristic, complemented

C₈×#272

C₂

yes

maximal, characteristic, complemented

C₃×D₃×D₈

—

maximal

C₂²×#272

—

D₄×H(3,𝔽₃)

C₂²

yes

complemented

#6235

—

C₄⋊#272

C₂²

yes

complemented

C₄×#272

C₂²

yes

characteristic

C₈×H(3,𝔽₃)

C₂²

yes

characteristic, complemented

#6180

C₂²

yes

characteristic, complemented

C₃×D₃×D₄

—

C₃²×D₈

D₃

yes

complemented

C₃×#213

—

C₃×D₂₄

—

C₃×D₃×C₈

—

C₂²×H(3,𝔽₃)

—

C₂×#272

—

C₂×#272

—

C₂×#272

D₄

yes

characteristic

C₄×H(3,𝔽₃)

C₂³

yes

characteristic

#1084

D₄

yes

characteristic

C₂×C₃×D₈

—

D₃×D₈

—

C₂²×C₃×D₃

—

C₃²×D₄

C₂×D₃

yes

complemented

C₃×#67

—

C₃×D₁₂

—

C₃×C₄×D₃

—

C₃²×C₈

C₂×D₃

yes

complemented

C₃×C₃⋊C₈

—

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

C₂×D₄

yes

characteristic

#272

—

#272

D₈

yes

direct product factor

C₂×C₃×D₄

—

D₃×D₄

—

C₃×D₈

—

C₃×D₈

—

C₃×D₈

—

C₃×D₈

C₃⋊D₃

yes

characteristic

C₂×C₃×C₈

—

#213

—

D₂₄

—

D₃×C₈

—

C₂²×C₃²

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₃²×C₄

C₂²×D₃

yes

—

C₃×Dic₃

—

C₂×D₈

—

Sylow

H(3,𝔽₃)

C₂×D₈

yes

characteristic, complemented, Sylow

C₂³×C₃

—

C₂²×D₃

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

C₂×C₃⋊D₃

yes

—

C₂×C₃×C₄

—

#67

—

D₁₂

—

C₄×D₃

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

C₂×C₃⋊D₃

yes

characteristic

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

D₃×D₄

yes

—

C₃×D₃

—

C₃×D₃

—

C₂×D₄

—

D₈

—

D₈

—

D₈

#272

yes

characteristic, direct product factor

C₂×C₈

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂²×C₃⋊D₃

yes

characteristic

Dic₃

—

C₃²

D₃×D₈

yes

complemented

C₂³

—

D₄

—

D₄

—

D₄

C₂×#272

yes

complemented

C₂×C₄

—

C₈

—

C₈

C₂×#272

yes

characteristic, complemented

C₂×C₃

—

100

C₂×C₃

—

101

C₂×C₃

—

102

C₂×C₃

—

103

C₂×C₃

—

104

C₂×C₃

D₄×C₃⋊D₃

yes

characteristic

105

D₃

—

106

D₃

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₄

—

112

C₄

C₂²×#272

yes

characteristic

113

C₃

—

114

C₃

D₈×C₃⋊D₃

yes

characteristic

115

C₂

—

116

C₂

—

117

C₂

—

118

C₂

D₄×#272

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₃	D₄×C₃⋊D₃
commutator, cosocle	C₄×H(3,𝔽₃)	C₂³
Frattini	C₃×C₄	C₂²×C₃⋊D₃
Fitting	D₈×H(3,𝔽₃)	C₂
hypercenter	C₃×D₈	C₃⋊D₃
hypocenter	H(3,𝔽₃)	C₂×D₈
LCS	G ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×D₈

subgroup

G/H

center, socle

C₂×C₃

D₄×C₃⋊D₃

commutator, cosocle

C₄×H(3,𝔽₃)

C₂³

Frattini

C₃×C₄

C₂²×C₃⋊D₃

Fitting

D₈×H(3,𝔽₃)

C₂

hypercenter

C₃×D₈

C₃⋊D₃

hypocenter

H(3,𝔽₃)

C₂×D₈

LCS

G ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×D₈

N	H ≅ G/N	type
1. D₄×#272	C₂	semidirect
2. D₈×H(3,𝔽₃)	C₂	semidirect
3. #88545	C₂	semidirect
4. #88529	C₂	semidirect
5. C₈×#272	C₂	semidirect
8. D₄×H(3,𝔽₃)	C₂²	semidirect
10. C₄⋊#272	C₂²	semidirect
12. C₈×H(3,𝔽₃)	C₂²	semidirect
13. #6180	C₂²	semidirect
15. C₃²×D₈	D₃	semidirect
28. C₃²×D₄	C₂×D₃	semidirect
32. C₃²×C₈	C₂×D₃	semidirect
37. #272	D₈	direct (N × H)
55. H(3,𝔽₃)	C₂×D₈	semidirect
77. D₈	#272	direct (N × H)
91. C₃²	D₃×D₈	semidirect
95. D₄	C₂×#272	semidirect
98. C₈	C₂×#272	semidirect

H ≅ G/N

type

1. D₄×#272

C₂

semidirect

2. D₈×H(3,𝔽₃)

C₂

semidirect

3. #88545

C₂

semidirect

4. #88529

C₂

semidirect

5. C₈×#272

C₂

semidirect

8. D₄×H(3,𝔽₃)

C₂²

semidirect

10. C₄⋊#272

C₂²

semidirect

12. C₈×H(3,𝔽₃)

C₂²

semidirect

13. #6180

C₂²

semidirect

15. C₃²×D₈

D₃

semidirect

28. C₃²×D₄

C₂×D₃

semidirect

32. C₃²×C₈

C₂×D₃

semidirect

37. #272

D₈

direct (N × H)

55. H(3,𝔽₃)

C₂×D₈

semidirect

77. D₈

#272

direct (N × H)

91. C₃²

D₃×D₈

semidirect

95. D₄

C₂×#272

semidirect

98. C₈

C₂×#272

semidirect

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

subgroup	index
D₄×#272	2
D₈×H(3,𝔽₃)	2
#88545	2
#88529	2
C₈×#272	2
C₃×D₃×D₈	3

subgroup

index

D₈×H(3,𝔽₃)

C₃×D₃×D₈

Subgroups — D₈×#272 #11721792

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D8×#272 #11721792

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₈×#272 #11721792