Group #3273 — subgroups

Subgroups — #3273

Order 128.

260 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 94 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	C₂×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
2	#546	C₂	1	2	yes	maximal, complemented
3	#495	C₂	1	1	yes	maximal, characteristic, complemented
4	#468	C₂	1	2	yes	maximal, complemented
5	#463	C₂	1	2	yes	maximal, complemented
6	#452	C₂	1	2	yes	maximal, complemented
7	#447	C₂	1	2	yes	maximal, complemented
8	C₄×D₈	C₂	1	1	yes	maximal, characteristic, complemented
9	#436	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41	C₂²	1	2	yes	—
11	C₂²×D₄	C₂²	1	2	yes	complemented
12	AGL(1,ℤ/8)	—	2	8	no	—
13	C₂×Q₁₆	C₂²	1	1	yes	characteristic, complemented
14	C₂×SD₁₆	C₂²	1	2	yes	complemented
15	C₂×SD₁₆	C₂²	1	2	yes	complemented
16	C₂×D₈	C₂²	1	2	yes	—
17	C₂×D₈	C₂²	1	1	yes	characteristic
18	C₂×D₈	C₂²	1	1	yes	characteristic
19	C₂×M₁₆	C₂²	1	2	yes	complemented
20	#124	C₂²	1	2	yes	complemented
21	#123	—	2	2	no	—
22	C₂²⋊Q₈	C₂²	1	2	yes	complemented
23	#121	—	2	2	no	—
24	#121	C₂²	1	2	yes	—
25	#120	—	2	2	no	—
26	C₄×D₄	C₂²	1	1	yes	characteristic
27	C₄×D₄	C₂²	1	2	yes	complemented
28	C₂×C₂²⋊C₄	—	2	2	no	—
29	#107	C₂²	1	1	yes	characteristic, complemented
30	C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
31	#103	C₂²	1	2	yes	complemented
32	#102	C₂²	1	2	yes	complemented
33	#102	C₂²	1	2	yes	complemented
34	C₂²⋊C₈	C₂²	1	2	yes	complemented
35	C₄×C₈	C₂²	1	1	yes	characteristic, complemented
36	C₂⁴	—	2	2	no	—
37	#41	—	2	4	no	—
38	#41	—	4	2	no	—
39	C₂×Q₈	C₂³	1	2	yes	—
40	C₂×D₄	—	2	4	no	—
41	C₂×D₄	—	2	2	no	—
42	C₂×D₄	—	2	2	no	—
43	C₂×D₄	—	2	2	no	—
44	C₂×D₄	—	4	2	no	—
45	C₂×D₄	D₄	1	1	yes	characteristic
46	C₂×D₄	C₂³	1	2	yes	—
47	C₂×D₄	C₂³	1	2	yes	—
48	C₂²×C₄	—	2	2	no	—
49	C₂²×C₄	—	2	2	no	—
50	C₂²×C₄	C₂³	1	2	yes	—
51	Q₁₆	—	4	1	no	—
52	SD₁₆	—	2	4	no	—
53	SD₁₆	—	2	4	no	—
54	D₈	—	2	4	no	—
55	D₈	—	4	1	no	—
56	D₈	D₄	1	4	yes	complemented
57	M₁₆	—	2	4	no	—
58	C₂×C₈	C₂³	1	1	yes	characteristic
59	C₂×C₈	C₂³	1	2	yes	—
60	C₂×C₈	C₂³	1	1	yes	characteristic
61	C₄⋊C₄	—	2	2	no	—
62	C₄⋊C₄	D₄	1	1	yes	characteristic
63	C₄⋊C₄	C₂³	1	2	yes	—
64	C₂²⋊C₄	—	2	2	no	—
65	C₂²⋊C₄	—	2	2	no	—
66	C₂²⋊C₄	—	2	2	no	—
67	C₂²⋊C₄	—	2	2	no	—
68	C₂²⋊C₄	—	2	2	no	—
69	C₂²⋊C₄	D₄	1	2	yes	—
70	C₄²	C₂³	1	1	yes	characteristic
71	C₂³	—	2	2	no	—
72	C₂³	—	2	2	no	—
73	C₂³	—	4	4	no	—
74	C₂³	—	4	4	no	—
75	C₂³	—	8	2	no	—
76	C₂³	C₂×D₄	1	2	yes	—
77	Q₈	—	2	2	no	—
78	Q₈	—	2	2	no	—
79	D₄	—	2	2	no	—
80	D₄	—	2	2	no	—
81	D₄	—	2	2	no	—
82	D₄	—	4	1	no	—
83	D₄	—	4	4	no	—
84	D₄	—	4	4	no	—
85	D₄	—	8	2	no	—
86	D₄	C₂×D₄	1	4	yes	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	4	4	no	—
92	C₂×C₄	—	8	2	no	—
93	C₂×C₄	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄	C₂×D₄	1	2	yes	—
96	C₂×C₄	C₂⁴	1	1	yes	characteristic
97	C₈	—	2	2	no	—
98	C₈	—	2	1	no	—
99	C₈	C₂×D₄	1	2	yes	—
100	C₂²	—	2	2	no	—
101	C₂²	—	2	4	no	—
102	C₂²	—	2	2	no	—
103	C₂²	—	2	2	no	—
104	C₂²	—	4	2	no	—
105	C₂²	—	4	2	no	—
106	C₂²	—	4	2	no	—
107	C₂²	—	4	2	no	—
108	C₂²	—	4	2	no	—
109	C₂²	—	8	8	no	—
110	C₂²	C₂²×D₄	1	1	yes	characteristic
111	C₄	—	2	2	no	—
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	4	2	no	—
115	C₄	—	4	2	no	—
116	C₄	C₂²×D₄	1	1	yes	characteristic
117	C₄	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂	—	4	2	no	—
119	C₂	—	4	4	no	—
120	C₂	—	8	2	no	—
121	C₂	D₄²	1	1	yes	characteristic
122	C₂	#575	1	1	yes	characteristic
123	C₂	#576	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×AGL(1,ℤ/8)

C₂

yes

maximal, complemented

#546

C₂

yes

maximal, complemented

#495

C₂

yes

maximal, characteristic, complemented

#468

C₂

yes

maximal, complemented

#463

C₂

yes

maximal, complemented

#452

C₂

yes

maximal, complemented

#447

C₂

yes

maximal, complemented

C₄×D₈

C₂

yes

maximal, characteristic, complemented

#436

C₂

yes

maximal, characteristic, complemented

C₂×#41

C₂²

yes

—

C₂²×D₄

C₂²

yes

complemented

AGL(1,ℤ/8)

—

C₂×Q₁₆

C₂²

yes

characteristic, complemented

C₂×SD₁₆

C₂²

yes

complemented

C₂×SD₁₆

C₂²

yes

complemented

C₂×D₈

C₂²

yes

—

C₂×D₈

C₂²

yes

characteristic

C₂×D₈

C₂²

yes

characteristic

C₂×M₁₆

C₂²

yes

complemented

#124

C₂²

yes

complemented

#123

—

C₂²⋊Q₈

C₂²

yes

complemented

#121

—

#121

C₂²

yes

—

#120

—

C₄×D₄

C₂²

yes

characteristic

C₄×D₄

C₂²

yes

complemented

C₂×C₂²⋊C₄

—

#107

C₂²

yes

characteristic, complemented

C₄⋊C₈

C₂²

yes

characteristic, complemented

#103

C₂²

yes

complemented

#102

C₂²

yes

complemented

#102

C₂²

yes

complemented

C₂²⋊C₈

C₂²

yes

complemented

C₄×C₈

C₂²

yes

characteristic, complemented

C₂⁴

—

#41

—

#41

—

C₂×Q₈

C₂³

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₄

yes

characteristic

C₂×D₄

C₂³

yes

—

C₂×D₄

C₂³

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂³

yes

—

Q₁₆

—

SD₁₆

—

SD₁₆

—

D₈

—

D₈

—

D₈

D₄

yes

complemented

M₁₆

—

C₂×C₈

C₂³

yes

characteristic

C₂×C₈

C₂³

yes

—

C₂×C₈

C₂³

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

D₄

yes

characteristic

C₄⋊C₄

C₂³

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

D₄

yes

—

C₄²

C₂³

yes

characteristic

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂×D₄

yes

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

C₂×D₄

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄

yes

characteristic

C₂×C₄

C₂×D₄

yes

characteristic

C₂×C₄

C₂×D₄

yes

—

C₂×C₄

C₂⁴

yes

characteristic

C₈

—

C₈

—

C₈

C₂×D₄

yes

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

C₂²×D₄

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

C₂²×D₄

yes

characteristic

117

C₄

H(5,𝔽₂)

yes

characteristic

118

C₂

—

119

C₂

—

120

C₂

—

121

C₂

D₄²

yes

characteristic

122

C₂

#575

yes

characteristic

123

C₂

#576

yes

characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄×D₄ ◁ G

LECS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. C₂×AGL(1,ℤ/8)	C₂	semidirect
2. #546	C₂	semidirect
3. #495	C₂	semidirect
4. #468	C₂	semidirect
5. #463	C₂	semidirect
6. #452	C₂	semidirect
7. #447	C₂	semidirect
8. C₄×D₈	C₂	semidirect
9. #436	C₂	semidirect
11. C₂²×D₄	C₂²	semidirect
13. C₂×Q₁₆	C₂²	semidirect
14. C₂×SD₁₆	C₂²	semidirect
15. C₂×SD₁₆	C₂²	semidirect
19. C₂×M₁₆	C₂²	semidirect
20. #124	C₂²	semidirect
22. C₂²⋊Q₈	C₂²	semidirect
27. C₄×D₄	C₂²	semidirect
29. #107	C₂²	semidirect
30. C₄⋊C₈	C₂²	semidirect
31. #103	C₂²	semidirect
32. #102	C₂²	semidirect
33. #102	C₂²	semidirect
34. C₂²⋊C₈	C₂²	semidirect
35. C₄×C₈	C₂²	semidirect
56. D₈	D₄	semidirect

H ≅ G/N

type

1. C₂×AGL(1,ℤ/8)

C₂

semidirect

2. #546

C₂

semidirect

3. #495

C₂

semidirect

4. #468

C₂

semidirect

5. #463

C₂

semidirect

6. #452

C₂

semidirect

7. #447

C₂

semidirect

8. C₄×D₈

C₂

semidirect

9. #436

C₂

semidirect

11. C₂²×D₄

C₂²

semidirect

13. C₂×Q₁₆

C₂²

semidirect

14. C₂×SD₁₆

C₂²

semidirect

15. C₂×SD₁₆

C₂²

semidirect

19. C₂×M₁₆

C₂²

semidirect

20. #124

C₂²

semidirect

22. C₂²⋊Q₈

C₂²

semidirect

27. C₄×D₄

C₂²

semidirect

29. #107

C₂²

semidirect

30. C₄⋊C₈

C₂²

semidirect

31. #103

C₂²

semidirect

32. #102

C₂²

semidirect

33. #102

C₂²

semidirect

34. C₂²⋊C₈

C₂²

semidirect

35. C₄×C₈

C₂²

semidirect

56. D₈

D₄

semidirect

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

subgroup	index
C₂×AGL(1,ℤ/8)	2
#546	2
#495	2
#468	2
#463	2
#452	2
#447	2
C₄×D₈	2
#436	2

subgroup

index

C₄×D₈

Subgroups — #3273

Conjugacy classes of subgroups

Complements

Maximal subgroups