Group #3288 — subgroups

Subgroups — #3288

Order 128.

232 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 92 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	#577	C₂	1	2	yes	maximal, complemented
2	#576	C₂	1	2	yes	maximal, complemented
3	#496	C₂	1	1	yes	maximal, characteristic, complemented
4	#472	C₂	1	2	yes	maximal, complemented
5	#471	C₂	1	2	yes	maximal, complemented
6	#455	C₂	1	1	yes	maximal, characteristic, complemented
7	#454	C₂	1	1	yes	maximal, characteristic, complemented
8	#453	C₂	1	2	yes	maximal, complemented
9	#444	C₂	1	2	yes	maximal, complemented
10	2₋¹⁺⁴	C₂²	1	1	yes	characteristic, complemented
11	H(5,𝔽₂)	C₂²	1	1	yes	characteristic
12	H(5,𝔽₂)	C₂²	1	2	yes	complemented
13	#137	—	2	2	no	—
14	#137	C₂²	1	2	yes	complemented
15	AGL(1,ℤ/8)	—	2	2	no	—
16	AGL(1,ℤ/8)	—	2	2	no	—
17	AGL(1,ℤ/8)	—	2	2	no	—
18	AGL(1,ℤ/8)	C₂²	1	2	yes	—
19	AGL(1,ℤ/8)	C₂²	1	2	yes	complemented
20	AGL(1,ℤ/8)	C₂²	1	2	yes	complemented
21	#135	—	2	2	no	—
22	#135	—	2	2	no	—
23	#135	C₂²	1	2	yes	—
24	C₂×SD₁₆	—	2	2	no	—
25	C₂×SD₁₆	C₂²	1	2	yes	complemented
26	C₂×D₈	—	2	2	no	—
27	C₂×D₈	C₂²	1	2	yes	complemented
28	#131	C₂²	1	2	yes	complemented
29	#131	C₂²	1	2	yes	complemented
30	#127	C₂²	1	1	yes	characteristic, complemented
31	#124	C₂²	1	1	yes	characteristic, complemented
32	#108	C₂²	1	2	yes	complemented
33	#104	C₂²	1	2	yes	complemented
34	#104	C₂²	1	2	yes	complemented
35	#101	C₂²	1	1	yes	characteristic, complemented
36	#100	C₂²	1	2	yes	complemented
37	#100	C₂²	1	1	yes	characteristic, complemented
38	#97	C₂²	1	1	yes	characteristic, complemented
39	#41	—	2	2	no	—
40	#41	—	2	2	no	—
41	#41	—	2	2	no	—
42	#41	—	2	2	no	—
43	#41	—	4	1	no	—
44	#41	C₂³	1	2	yes	—
45	#41	C₂³	1	2	yes	—
46	C₂×Q₈	—	2	2	no	—
47	C₂×Q₈	C₂³	1	1	yes	characteristic
48	C₂×D₄	—	2	2	no	—
49	C₂×D₄	—	2	2	no	—
50	C₂×D₄	—	2	2	no	—
51	C₂×D₄	—	4	1	no	—
52	C₂×D₄	—	4	1	no	—
53	C₂×D₄	—	4	2	no	—
54	C₂×D₄	C₂³	1	1	yes	characteristic
55	C₂×D₄	C₂³	1	2	yes	—
56	Q₁₆	—	2	2	no	—
57	Q₁₆	D₄	1	2	yes	complemented
58	SD₁₆	—	2	2	no	—
59	SD₁₆	—	2	2	no	—
60	SD₁₆	—	2	2	no	—
61	SD₁₆	—	2	2	no	—
62	SD₁₆	—	2	2	no	—
63	SD₁₆	—	2	2	no	—
64	SD₁₆	D₄	1	2	yes	complemented
65	SD₁₆	D₄	1	2	yes	complemented
66	D₈	—	2	2	no	—
67	D₈	—	2	2	no	—
68	D₈	—	2	2	no	—
69	D₈	—	2	2	no	—
70	D₈	—	2	2	no	—
71	D₈	D₄	1	2	yes	complemented
72	M₁₆	—	2	2	no	—
73	M₁₆	—	2	2	no	—
74	M₁₆	C₂³	1	2	yes	—
75	M₁₆	C₂³	1	2	yes	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	C₂³	1	2	yes	—
79	C₂²⋊C₄	—	4	1	no	—
80	C₄²	C₂³	1	1	yes	characteristic
81	C₂³	—	2	1	no	—
82	C₂³	—	2	2	no	—
83	C₂³	—	4	2	no	—
84	C₂³	—	4	1	no	—
85	Q₈	—	2	2	no	—
86	Q₈	—	4	1	no	—
87	Q₈	C₂×D₄	1	2	yes	—
88	Q₈	C₂×D₄	1	2	yes	—
89	D₄	—	2	2	no	—
90	D₄	—	2	2	no	—
91	D₄	—	2	2	no	—
92	D₄	—	2	2	no	—
93	D₄	—	2	2	no	—
94	D₄	—	4	2	no	—
95	D₄	—	4	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	1	no	—
98	D₄	—	8	2	no	—
99	D₄	C₂×D₄	1	2	yes	—
100	D₄	C₂×D₄	1	2	yes	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	C₂⁴	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	2	2	no	—
111	C₈	C₂×D₄	1	2	yes	—
112	C₈	C₂×D₄	1	2	yes	—
113	C₂²	—	2	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	—	4	1	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	8	2	no	—
118	C₂²	—	8	1	no	—
119	C₂²	—	16	2	no	—
120	C₂²	—	16	1	no	—
121	C₂²	H(5,𝔽₂)	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	4	1	no	—
125	C₄	—	4	1	no	—
126	C₄	C₂²×D₄	1	2	yes	—
127	C₂	—	2	1	no	—
128	C₂	—	4	2	no	—
129	C₂	—	4	2	no	—
130	C₂	—	8	2	no	—
131	C₂	—	8	1	no	—
132	C₂	D₄²	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#577

C₂

yes

maximal, complemented

#576

C₂

yes

maximal, complemented

#496

C₂

yes

maximal, characteristic, complemented

#472

C₂

yes

maximal, complemented

#471

C₂

yes

maximal, complemented

#455

C₂

yes

maximal, characteristic, complemented

#454

C₂

yes

maximal, characteristic, complemented

#453

C₂

yes

maximal, complemented

#444

C₂

yes

maximal, complemented

2₋¹⁺⁴

C₂²

yes

characteristic, complemented

H(5,𝔽₂)

C₂²

yes

characteristic

H(5,𝔽₂)

C₂²

yes

complemented

#137

—

#137

C₂²

yes

—

—

—

C₂²

yes

—

AGL(1,ℤ/8)

C₂²

yes

complemented

AGL(1,ℤ/8)

C₂²

yes

—

—

C₂²

yes

—

C₂×SD₁₆

—

C₂×SD₁₆

C₂²

yes

complemented

C₂×D₈

—

C₂×D₈

C₂²

yes

complemented

#131

C₂²

yes

complemented

#131

C₂²

yes

complemented

#127

C₂²

yes

characteristic, complemented

#124

C₂²

yes

characteristic, complemented

#108

C₂²

yes

complemented

#104

C₂²

yes

complemented

#104

C₂²

yes

complemented

#101

C₂²

yes

characteristic, complemented

#100

C₂²

yes

complemented

#100

C₂²

yes

characteristic, complemented

#97

C₂²

yes

characteristic, complemented

—

—

—

—

—

C₂³

yes

—

#41

C₂³

yes

—

C₂×Q₈

—

C₂×Q₈

C₂³

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂³

yes

characteristic

C₂×D₄

C₂³

yes

—

Q₁₆

—

Q₁₆

D₄

yes

complemented

SD₁₆

—

SD₁₆

—

SD₁₆

—

SD₁₆

—

SD₁₆

—

SD₁₆

—

SD₁₆

D₄

yes

complemented

SD₁₆

D₄

yes

complemented

D₈

—

D₈

—

D₈

—

D₈

—

D₈

—

D₈

D₄

yes

complemented

M₁₆

—

M₁₆

—

M₁₆

C₂³

yes

—

M₁₆

C₂³

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂³

yes

—

C₂²⋊C₄

—

C₄²

C₂³

yes

characteristic

C₂³

—

C₂³

—

C₂³

—

C₂³

—

Q₈

—

Q₈

—

Q₈

C₂×D₄

yes

—

Q₈

C₂×D₄

yes

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

C₂×D₄

yes

—

100

D₄

C₂×D₄

yes

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂⁴

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

C₂×D₄

yes

—

112

C₈

C₂×D₄

yes

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

H(5,𝔽₂)

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

C₂²×D₄

yes

—

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

—

131

C₂

—

132

C₂

D₄²

yes

characteristic

	subgroup	G/H
center, socle	C₂	D₄²
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂

D₄²

commutator, Frattini, cosocle

C₂×C₄

C₂⁴

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ G

LECS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. #577	C₂	semidirect
2. #576	C₂	semidirect
3. #496	C₂	semidirect
4. #472	C₂	semidirect
5. #471	C₂	semidirect
6. #455	C₂	semidirect
7. #454	C₂	semidirect
8. #453	C₂	semidirect
9. #444	C₂	semidirect
10. 2₋¹⁺⁴	C₂²	semidirect
12. H(5,𝔽₂)	C₂²	semidirect
14. #137	C₂²	semidirect
19. AGL(1,ℤ/8)	C₂²	semidirect
20. AGL(1,ℤ/8)	C₂²	semidirect
25. C₂×SD₁₆	C₂²	semidirect
27. C₂×D₈	C₂²	semidirect
28. #131	C₂²	semidirect
29. #131	C₂²	semidirect
30. #127	C₂²	semidirect
31. #124	C₂²	semidirect
32. #108	C₂²	semidirect
33. #104	C₂²	semidirect
34. #104	C₂²	semidirect
35. #101	C₂²	semidirect
36. #100	C₂²	semidirect
37. #100	C₂²	semidirect
38. #97	C₂²	semidirect
57. Q₁₆	D₄	semidirect
64. SD₁₆	D₄	semidirect
65. SD₁₆	D₄	semidirect
71. D₈	D₄	semidirect

H ≅ G/N

type

1. #577

C₂

semidirect

2. #576

C₂

semidirect

3. #496

C₂

semidirect

4. #472

C₂

semidirect

5. #471

C₂

semidirect

6. #455

C₂

semidirect

7. #454

C₂

semidirect

8. #453

C₂

semidirect

9. #444

C₂

semidirect

10. 2₋¹⁺⁴

C₂²

semidirect

12. H(5,𝔽₂)

C₂²

semidirect

14. #137

C₂²

semidirect

19. AGL(1,ℤ/8)

C₂²

semidirect

20. AGL(1,ℤ/8)

C₂²

semidirect

25. C₂×SD₁₆

C₂²

semidirect

27. C₂×D₈

C₂²

semidirect

28. #131

C₂²

semidirect

29. #131

C₂²

semidirect

30. #127

C₂²

semidirect

31. #124

C₂²

semidirect

32. #108

C₂²

semidirect

33. #104

C₂²

semidirect

34. #104

C₂²

semidirect

35. #101

C₂²

semidirect

36. #100

C₂²

semidirect

37. #100

C₂²

semidirect

38. #97

C₂²

semidirect

57. Q₁₆

D₄

semidirect

64. SD₁₆

D₄

semidirect

65. SD₁₆

D₄

semidirect

71. D₈

D₄

semidirect

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

subgroup	index
#577	2
#576	2
#496	2
#472	2
#471	2
#455	2
#454	2
#453	2
#444	2

subgroup

index

Subgroups — #3288

Conjugacy classes of subgroups

Complements

Maximal subgroups