Group #9734 — subgroups

Subgroups — #9734

Order 256.

327 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 327 conjugacy classes of subgroups — too many to draw. Only its 95 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×#493	C₂	1	1	yes	maximal, characteristic
2	#1982	C₂	1	2	yes	maximal, complemented
3	C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#335	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#331	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈	—	2	1	no	—
7	C₂²×D₈	—	2	1	no	—
8	C₂×#127	C₂²	1	2	yes	—
9	#493	—	2	2	no	—
10	#493	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×#102	—	2	4	no	—
14	C₂×C₄×C₈	—	2	1	no	—
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	#336	—	2	1	no	—
18	#335	—	2	2	no	—
19	#331	—	2	4	no	—
20	C₂²×D₄	—	2	2	no	—
21	C₂²×D₄	—	2	2	no	—
22	C₂²×D₄	—	2	2	no	—
23	C₂×D₈	—	2	4	no	—
24	C₂×D₈	—	2	4	no	—
25	C₂×D₈	—	4	2	no	—
26	C₂×D₈	—	4	2	no	—
27	C₂×D₈	—	4	2	no	—
28	C₂×D₈	—	4	2	no	—
29	C₂²×C₈	—	2	1	no	—
30	C₂²×C₈	—	2	2	no	—
31	C₂²×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₈	D₄	1	1	yes	characteristic
33	#127	—	2	2	no	—
34	#127	—	2	2	no	—
35	#127	—	2	2	no	—
36	#127	C₂×C₄	1	4	yes	—
37	C₂×C₄⋊C₄	D₄	1	2	yes	—
38	C₂×C₄²	—	2	1	no	—
39	C₂×C₄²	C₂³	1	1	yes	characteristic
40	C₄⋊C₈	—	2	2	no	—
41	C₄⋊C₈	—	2	2	no	—
42	C₄⋊C₈	—	2	1	no	—
43	C₄⋊C₈	D₄	1	2	yes	—
44	#102	—	4	8	no	—
45	C₄×C₈	—	4	2	no	—
46	C₄×C₈	C₂×C₄	1	2	yes	—
47	C₄×C₈	D₄	1	2	yes	—
48	#95	—	2	1	no	—
49	C₂⁴	—	4	2	no	—
50	C₂×D₄	—	2	4	no	—
51	C₂×D₄	—	2	4	no	—
52	C₂×D₄	—	2	4	no	—
53	C₂×D₄	—	4	2	no	—
54	C₂×D₄	—	4	2	no	—
55	C₂×D₄	—	4	2	no	—
56	C₂×D₄	—	4	4	no	—
57	C₂×D₄	—	4	2	no	—
58	C₂×D₄	—	4	2	no	—
59	C₂×D₄	—	4	4	no	—
60	C₂×D₄	—	4	2	no	—
61	C₂×D₄	—	4	2	no	—
62	C₂×D₄	—	8	4	no	—
63	C₂²×C₄	—	2	2	no	—
64	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄	#41	1	1	yes	characteristic
67	D₈	—	4	8	no	—
68	D₈	—	4	8	no	—
69	C₂×C₈	—	2	1	no	—
70	C₂×C₈	—	2	1	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	4	1	no	—
73	C₂×C₈	—	4	1	no	—
74	C₂×C₈	—	4	2	no	—
75	C₂×C₈	—	4	4	no	—
76	C₂×C₈	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈	D₈	1	2	yes	—
79	C₂×C₈	SD₁₆	1	2	yes	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	D₈	1	4	yes	complemented
82	C₄²	—	4	1	no	—
83	C₄²	—	4	1	no	—
84	C₄²	C₂²×C₄	1	1	yes	characteristic
85	C₄²	C₂×D₄	1	1	yes	characteristic
86	C₄²	C₂×D₄	1	1	yes	characteristic
87	C₄²	#41	1	1	yes	characteristic
88	C₂³	—	4	4	no	—
89	C₂³	—	8	2	no	—
90	C₂³	—	8	2	no	—
91	C₂³	—	8	4	no	—
92	C₂³	—	8	4	no	—
93	C₂³	#121	1	1	yes	characteristic
94	D₄	—	4	4	no	—
95	D₄	—	4	4	no	—
96	D₄	—	4	8	no	—
97	D₄	—	8	4	no	—
98	D₄	—	8	4	no	—
99	D₄	—	8	2	no	—
100	D₄	—	8	2	no	—
101	D₄	—	8	2	no	—
102	D₄	—	8	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄	#121	1	1	yes	characteristic
111	C₂×C₄	#124	1	1	yes	characteristic
112	C₂×C₄	#127	1	1	yes	characteristic
113	C₂×C₄	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄	C₂×D₈	1	2	yes	—
115	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
116	C₂×C₄	#135	1	2	yes	—
117	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₈	—	2	2	no	—
120	C₈	—	4	2	no	—
121	C₈	—	8	2	no	—
122	C₈	#102	1	4	yes	complemented
123	C₂²	—	8	4	no	—
124	C₂²	—	8	4	no	—
125	C₂²	—	8	4	no	—
126	C₂²	—	16	4	no	—
127	C₂²	—	16	4	no	—
128	C₂²	#390	1	1	yes	characteristic
129	C₂²	#459	1	1	yes	characteristic
130	C₂²	#460	1	1	yes	characteristic
131	C₂²	#466	1	2	yes	—
132	C₂²	#472	1	2	yes	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	4	4	no	—
136	C₄	C₂×#102	1	1	yes	characteristic
137	C₄	#418	1	1	yes	characteristic
138	C₄	C₄×D₈	1	2	yes	—
139	C₄	#459	1	1	yes	characteristic
140	C₄	#460	1	1	yes	characteristic
141	C₄	#493	1	1	yes	characteristic
142	C₄	#495	1	1	yes	characteristic
143	C₂	—	16	4	no	—
144	C₂	#1667	1	2	yes	—
145	C₂	#1668	1	2	yes	—
146	C₂	#1970	1	1	yes	characteristic
147	C₂	#1978	1	1	yes	characteristic
148	C₂	#1982	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#493

C₂

yes

maximal, characteristic

#1982

C₂

yes

maximal, complemented

C₈×C₄⋊C₄

C₂

yes

maximal, characteristic, complemented

C₂×#335

C₂

yes

maximal, characteristic, complemented

C₂×#331

C₂

yes

maximal, complemented

C₂²×D₈

—

C₂²×D₈

—

C₂×#127

C₂²

yes

—

#493

—

#493

C₄

yes

complemented

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×#102

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₄×C₄⋊C₄

C₂²

yes

—

—

—

C₂²×D₄

—

C₂²×D₄

—

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊C₄

D₄

yes

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₄

yes

—

#102

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

#95

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

D₈

yes

—

C₂×C₈

SD₁₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

D₈

yes

complemented

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

#41

yes

characteristic

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

#121

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

100

D₄

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

108

C₂×C₄

C₄×D₄

yes

characteristic

109

C₂×C₄

C₄×D₄

yes

characteristic

110

C₂×C₄

#121

yes

characteristic

111

C₂×C₄

#124

yes

characteristic

112

C₂×C₄

#127

yes

characteristic

113

C₂×C₄

C₂×D₈

yes

characteristic

114

C₂×C₄

C₂×D₈

yes

—

115

C₂×C₄

C₂×SD₁₆

yes

characteristic

116

C₂×C₄

#135

yes

—

117

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

118

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

#102

yes

complemented

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

#390

yes

characteristic

129

C₂²

#459

yes

characteristic

130

C₂²

#460

yes

characteristic

131

C₂²

#466

yes

—

132

C₂²

#472

yes

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

C₂×#102

yes

characteristic

137

C₄

#418

yes

characteristic

138

C₄

C₄×D₈

yes

—

139

C₄

#459

yes

characteristic

140

C₄

#460

yes

characteristic

141

C₄

#493

yes

characteristic

142

C₄

#495

yes

characteristic

143

C₂

—

144

C₂

#1667

yes

—

145

C₂

#1668

yes

—

146

C₂

#1970

yes

characteristic

147

C₂

#1978

yes

characteristic

148

C₂

#1982

yes

characteristic

	subgroup	G/H
center, socle	C₂³	#121
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂³

#121

commutator

C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

C₂

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
2. #1982	C₂	semidirect
3. C₈×C₄⋊C₄	C₂	semidirect
4. C₂×#335	C₂	semidirect
5. C₂×#331	C₂	semidirect
10. #493	C₄	semidirect
81. C₄⋊C₄	D₈	semidirect
122. C₈	#102	semidirect

H ≅ G/N

type

2. #1982

C₂

semidirect

3. C₈×C₄⋊C₄

C₂

semidirect

4. C₂×#335

C₂

semidirect

5. C₂×#331

C₂

semidirect

10. #493

C₄

semidirect

81. C₄⋊C₄

D₈

semidirect

122. C₈

#102

semidirect

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

subgroup	index
C₂×#493	2
#1982	2
C₈×C₄⋊C₄	2
C₂×#335	2
C₂×#331	2

subgroup

index

C₈×C₄⋊C₄

Subgroups — #9734

Conjugacy classes of subgroups

Complements

Maximal subgroups