Group #3018 — subgroups

Subgroups — #3018

Order 128.

362 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 106 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	#585	C₂	1	1	yes	maximal, characteristic, complemented
2	#575	C₂	1	2	yes	maximal, complemented
3	#532	C₂	1	1	yes	maximal, characteristic, complemented
4	#456	C₂	1	2	yes	maximal, complemented
5	#455	C₂	1	2	yes	maximal, complemented
6	#454	C₂	1	2	yes	maximal, complemented
7	#453	C₂	1	2	yes	maximal, complemented
8	C₂×#104	C₂	1	2	yes	maximal, complemented
9	#413	C₂	1	1	yes	maximal, characteristic, complemented
10	2₋¹⁺⁴	—	2	2	no	—
11	2₋¹⁺⁴	C₂²	1	2	yes	—
12	H(5,𝔽₂)	—	2	2	no	—
13	H(5,𝔽₂)	—	2	2	no	—
14	H(5,𝔽₂)	C₂²	1	2	yes	—
15	C₂×#41	—	2	4	no	—
16	C₂×#41	—	2	2	no	—
17	C₂×#41	C₂²	1	2	yes	—
18	C₂×#41	C₂²	1	1	yes	characteristic
19	C₂×#41	C₂²	1	1	yes	characteristic, complemented
20	#137	—	2	2	no	—
21	#137	C₂²	1	4	yes	complemented
22	AGL(1,ℤ/8)	—	2	2	no	—
23	AGL(1,ℤ/8)	C₂²	1	4	yes	complemented
24	#135	—	2	4	no	—
25	C₂×M₁₆	C₂²	1	2	yes	complemented
26	C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
27	#127	C₂²	1	1	yes	characteristic, complemented
28	#124	C₂²	1	1	yes	characteristic, complemented
29	#124	C₂²	1	1	yes	characteristic, complemented
30	#121	—	2	2	no	—
31	C₄×D₄	—	2	2	no	—
32	C₂×C₄²	C₂²	1	1	yes	characteristic, complemented
33	#104	C₂²	1	4	yes	complemented
34	#104	C₂²	1	4	yes	complemented
35	#101	C₂²	1	2	yes	complemented
36	#100	C₂²	1	2	yes	complemented
37	#41	—	2	4	no	—
38	#41	—	2	8	no	—
39	#41	—	2	4	no	—
40	#41	—	2	4	no	—
41	#41	—	2	4	no	—
42	#41	—	2	2	no	—
43	#41	—	2	2	no	—
44	#41	—	2	4	no	—
45	#41	—	2	2	no	—
46	#41	—	2	2	no	—
47	#41	—	2	2	no	—
48	#41	—	2	2	no	—
49	#41	D₄	1	4	yes	complemented
50	#41	C₂³	1	4	yes	—
51	C₂×Q₈	—	2	2	no	—
52	C₂×Q₈	—	2	4	no	—
53	C₂×Q₈	D₄	1	2	yes	complemented
54	C₂×Q₈	C₂³	1	1	yes	characteristic
55	C₂×Q₈	C₂³	1	1	yes	characteristic
56	C₂×D₄	—	2	8	no	—
57	C₂×D₄	—	2	2	no	—
58	C₂×D₄	—	2	4	no	—
59	C₂×D₄	—	2	2	no	—
60	C₂×D₄	—	2	4	no	—
61	C₂×D₄	—	2	2	no	—
62	C₂×D₄	—	2	1	no	—
63	C₂×D₄	D₄	1	2	yes	—
64	C₂×D₄	D₄	1	2	yes	complemented
65	C₂×D₄	C₂³	1	1	yes	characteristic
66	C₂×D₄	C₂³	1	1	yes	characteristic
67	C₂²×C₄	—	2	4	no	—
68	C₂²×C₄	—	2	2	no	—
69	C₂²×C₄	—	2	1	no	—
70	C₂²×C₄	—	2	1	no	—
71	C₂²×C₄	D₄	1	2	yes	—
72	C₂²×C₄	C₂³	1	1	yes	characteristic
73	Q₁₆	—	2	4	no	—
74	SD₁₆	—	2	4	no	—
75	SD₁₆	—	2	4	no	—
76	D₈	—	2	4	no	—
77	M₁₆	—	2	2	no	—
78	M₁₆	C₂³	1	4	yes	—
79	C₂×C₈	—	2	2	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₂²⋊C₄	—	2	2	no	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₄²	—	2	1	no	—
84	C₄²	C₂³	1	1	yes	characteristic
85	C₄²	C₂³	1	1	yes	characteristic
86	C₂³	—	2	4	no	—
87	C₂³	—	2	2	no	—
88	C₂³	—	2	1	no	—
89	C₂³	C₂×D₄	1	2	yes	—
90	C₂³	C₂×D₄	1	1	yes	characteristic
91	Q₈	—	2	4	no	—
92	Q₈	—	2	2	no	—
93	Q₈	—	2	2	no	—
94	Q₈	—	2	2	no	—
95	Q₈	C₂×D₄	1	4	yes	—
96	D₄	—	2	8	no	—
97	D₄	—	2	4	no	—
98	D₄	—	2	4	no	—
99	D₄	—	2	4	no	—
100	D₄	—	2	4	no	—
101	D₄	—	2	2	no	—
102	D₄	—	2	2	no	—
103	D₄	—	2	2	no	—
104	D₄	—	4	4	no	—
105	D₄	—	4	2	no	—
106	D₄	C₂×D₄	1	4	yes	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	8	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	C₂×D₄	1	2	yes	—
121	C₂×C₄	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄	C₂×D₄	1	2	yes	—
123	C₂×C₄	C₂×D₄	1	2	yes	—
124	C₂×C₄	C₂⁴	1	1	yes	characteristic
125	C₈	—	2	4	no	—
126	C₂²	—	2	2	no	—
127	C₂²	—	2	4	no	—
128	C₂²	—	4	1	no	—
129	C₂²	—	4	2	no	—
130	C₂²	—	4	2	no	—
131	C₂²	—	4	1	no	—
132	C₂²	—	8	2	no	—
133	C₂²	—	8	4	no	—
134	C₂²	#120	1	2	yes	—
135	C₂²	C₂²×D₄	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	2	4	no	—
138	C₄	—	2	2	no	—
139	C₄	—	4	1	no	—
140	C₄	—	4	1	no	—
141	C₄	#120	1	1	yes	characteristic
142	C₄	#120	1	1	yes	characteristic
143	C₄	C₂²×D₄	1	2	yes	—
144	C₂	—	2	1	no	—
145	C₂	—	2	2	no	—
146	C₂	—	4	2	no	—
147	C₂	—	4	4	no	—
148	C₂	—	8	1	no	—
149	C₂	C₂×#120	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#585

C₂

yes

maximal, characteristic, complemented

#575

C₂

yes

maximal, complemented

#532

C₂

yes

maximal, characteristic, complemented

#456

C₂

yes

maximal, complemented

#455

C₂

yes

maximal, complemented

#454

C₂

yes

maximal, complemented

#453

C₂

yes

maximal, complemented

C₂×#104

C₂

yes

maximal, complemented

#413

C₂

yes

maximal, characteristic, complemented

2₋¹⁺⁴

—

2₋¹⁺⁴

C₂²

yes

—

H(5,𝔽₂)

—

H(5,𝔽₂)

—

H(5,𝔽₂)

C₂²

yes

—

C₂×#41

—

C₂×#41

—

C₂×#41

C₂²

yes

—

C₂×#41

C₂²

yes

characteristic

C₂×#41

C₂²

yes

characteristic, complemented

#137

—

#137

C₂²

yes

complemented

AGL(1,ℤ/8)

—

AGL(1,ℤ/8)

C₂²

yes

complemented

#135

—

C₂×M₁₆

C₂²

yes

complemented

C₄⋊Q₈

C₂²

yes

characteristic, complemented

#127

C₂²

yes

characteristic, complemented

#124

C₂²

yes

characteristic, complemented

#124

C₂²

yes

characteristic, complemented

#121

—

C₄×D₄

—

C₂×C₄²

C₂²

yes

characteristic, complemented

#104

C₂²

yes

complemented

#104

C₂²

yes

complemented

#101

C₂²

yes

complemented

#100

C₂²

yes

—

—

—

—

—

—

—

—

—

—

—

—

D₄

yes

complemented

#41

C₂³

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₄

yes

complemented

C₂×Q₈

C₂³

yes

characteristic

C₂×Q₈

C₂³

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₄

yes

—

C₂×D₄

D₄

yes

complemented

C₂×D₄

C₂³

yes

characteristic

C₂×D₄

C₂³

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄

yes

—

C₂²×C₄

C₂³

yes

characteristic

Q₁₆

—

SD₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

M₁₆

C₂³

yes

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂³

yes

characteristic

C₄²

C₂³

yes

characteristic

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂×D₄

yes

—

C₂³

C₂×D₄

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

C₂×D₄

yes

—

D₄

—

D₄

—

D₄

—

D₄

—

100

D₄

—

101

D₄

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

106

D₄

C₂×D₄

yes

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₂×D₄

yes

—

121

C₂×C₄

C₂×D₄

yes

characteristic

122

C₂×C₄

C₂×D₄

yes

—

123

C₂×C₄

C₂×D₄

yes

—

124

C₂×C₄

C₂⁴

yes

characteristic

125

C₈

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

#120

yes

—

135

C₂²

C₂²×D₄

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#120

yes

characteristic

142

C₄

#120

yes

characteristic

143

C₄

C₂²×D₄

yes

—

144

C₂

—

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

—

149

C₂

C₂×#120

yes

characteristic

	subgroup	G/H
center	C₄	#120
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
socle	C₂	C₂×#120
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄

#120

commutator, Frattini, cosocle

C₂×C₄

C₂⁴

C₂

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ G

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. #585	C₂	semidirect
2. #575	C₂	semidirect
3. #532	C₂	semidirect
4. #456	C₂	semidirect
5. #455	C₂	semidirect
6. #454	C₂	semidirect
7. #453	C₂	semidirect
8. C₂×#104	C₂	semidirect
9. #413	C₂	semidirect
19. C₂×#41	C₂²	semidirect
21. #137	C₂²	semidirect
23. AGL(1,ℤ/8)	C₂²	semidirect
25. C₂×M₁₆	C₂²	semidirect
26. C₄⋊Q₈	C₂²	semidirect
27. #127	C₂²	semidirect
28. #124	C₂²	semidirect
29. #124	C₂²	semidirect
32. C₂×C₄²	C₂²	semidirect
33. #104	C₂²	semidirect
34. #104	C₂²	semidirect
35. #101	C₂²	semidirect
36. #100	C₂²	semidirect
49. #41	D₄	semidirect
53. C₂×Q₈	D₄	semidirect
64. C₂×D₄	D₄	semidirect

H ≅ G/N

type

1. #585

C₂

semidirect

2. #575

C₂

semidirect

3. #532

C₂

semidirect

4. #456

C₂

semidirect

5. #455

C₂

semidirect

6. #454

C₂

semidirect

7. #453

C₂

semidirect

8. C₂×#104

C₂

semidirect

9. #413

C₂

semidirect

19. C₂×#41

C₂²

semidirect

21. #137

C₂²

semidirect

23. AGL(1,ℤ/8)

C₂²

semidirect

25. C₂×M₁₆

C₂²

semidirect

26. C₄⋊Q₈

C₂²

semidirect

27. #127

C₂²

semidirect

28. #124

C₂²

semidirect

29. #124

C₂²

semidirect

32. C₂×C₄²

C₂²

semidirect

33. #104

C₂²

semidirect

34. #104

C₂²

semidirect

35. #101

C₂²

semidirect

36. #100

C₂²

semidirect

49. #41

D₄

semidirect

53. C₂×Q₈

D₄

semidirect

64. C₂×D₄

D₄

semidirect

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

subgroup	index
#585	2
#575	2
#532	2
#456	2
#455	2
#454	2
#453	2
C₂×#104	2
#413	2

subgroup

index

Subgroups — #3018

Conjugacy classes of subgroups

Complements

Maximal subgroups