Group #9657 — subgroups

Subgroups — #9657

Order 256.

303 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 303 conjugacy classes of subgroups — too many to draw. Only its 103 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×C₄×D₈	C₂	1	1	yes	maximal, characteristic
2	#1954	C₂	1	2	yes	maximal, complemented
3	C₂×#335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄	C₂²	1	2	yes	—
8	C₄×D₈	—	2	2	no	—
9	C₄×D₈	—	2	2	no	—
10	C₂×#107	C₄	1	1	yes	characteristic
11	C₂×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×#102	C₄	1	2	yes	—
13	C₂×C₂²⋊C₈	—	2	4	no	—
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
17	#335	—	2	2	no	—
18	D₄⋊C₈	—	2	4	no	—
19	C₈²	—	2	2	no	—
20	C₂²×D₄	C₂×C₄	1	2	yes	—
21	C₂³×C₄	—	2	2	no	—
22	C₂×D₈	—	2	2	no	—
23	C₂×D₈	—	2	2	no	—
24	C₂×D₈	C₈	1	4	yes	complemented
25	C₂²×C₈	—	2	1	no	—
26	C₂²×C₈	—	2	2	no	—
27	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈	D₄	1	2	yes	—
30	C₄×D₄	—	2	2	no	—
31	C₄×D₄	—	2	2	no	—
32	C₄×D₄	—	2	2	no	—
33	C₄×D₄	—	2	2	no	—
34	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
35	C₂×C₂²⋊C₄	—	2	2	no	—
36	C₂×C₄²	C₂³	1	1	yes	characteristic
37	#107	—	2	2	no	—
38	C₄⋊C₈	—	2	2	no	—
39	C₄⋊C₈	—	2	2	no	—
40	#102	—	2	4	no	—
41	C₂²⋊C₈	—	4	8	no	—
42	C₄×C₈	—	2	1	no	—
43	C₄×C₈	—	2	1	no	—
44	C₄×C₈	—	2	1	no	—
45	C₄×C₈	D₄	1	2	yes	—
46	C₄×C₈	D₄	1	2	yes	—
47	C₄×C₈	D₄	1	2	yes	—
48	C₂⁴	—	2	2	no	—
49	C₂×D₄	—	2	2	no	—
50	C₂×D₄	—	2	2	no	—
51	C₂×D₄	—	2	2	no	—
52	C₂×D₄	—	2	2	no	—
53	C₂×D₄	—	2	2	no	—
54	C₂×D₄	C₂×C₈	1	4	yes	—
55	C₂²×C₄	—	2	2	no	—
56	C₂²×C₄	—	4	2	no	—
57	C₂²×C₄	—	4	2	no	—
58	C₂²×C₄	—	4	4	no	—
59	C₂²×C₄	—	4	4	no	—
60	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄	#41	1	1	yes	characteristic
63	D₈	—	2	8	no	—
64	C₂×C₈	—	2	4	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	—	2	2	no	—
67	C₂×C₈	—	2	1	no	—
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	4	2	no	—
71	C₂×C₈	—	4	4	no	—
72	C₂×C₈	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈	C₂×C₈	1	2	yes	—
75	C₂×C₈	M₁₆	1	2	yes	—
76	C₂×C₈	D₈	1	4	yes	complemented
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	—	2	2	no	—
79	C₂²⋊C₄	—	4	2	no	—
80	C₂²⋊C₄	—	4	2	no	—
81	C₄²	C₂×D₄	1	1	yes	characteristic
82	C₄²	C₂×D₄	1	1	yes	characteristic
83	C₄²	C₂×D₄	1	1	yes	characteristic
84	C₄²	#41	1	1	yes	characteristic
85	C₂³	—	2	4	no	—
86	C₂³	—	4	2	no	—
87	C₂³	—	4	2	no	—
88	C₂³	—	4	4	no	—
89	C₂³	—	4	4	no	—
90	C₂³	C₄×D₄	1	1	yes	characteristic
91	D₄	—	2	4	no	—
92	D₄	—	2	4	no	—
93	D₄	—	4	2	no	—
94	D₄	—	4	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	4	4	no	—
98	C₂×C₄	—	8	4	no	—
99	C₂×C₄	—	8	2	no	—
100	C₂×C₄	—	8	2	no	—
101	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄	#121	1	1	yes	characteristic
104	C₂×C₄	#121	1	1	yes	characteristic
105	C₂×C₄	#124	1	1	yes	characteristic
106	C₂×C₄	#127	1	1	yes	characteristic
107	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄	#131	1	1	yes	characteristic
110	C₂×C₄	#131	1	1	yes	characteristic
111	C₂×C₄	C₂×D₈	1	2	yes	—
112	C₂×C₄	#135	1	2	yes	—
113	C₈	—	2	4	no	—
114	C₈	—	2	2	no	—
115	C₈	—	2	4	no	—
116	C₈	—	8	2	no	—
117	C₈	C₂²⋊C₈	1	4	yes	complemented
118	C₂²	—	4	8	no	—
119	C₂²	—	4	4	no	—
120	C₂²	—	8	4	no	—
121	C₂²	—	8	2	no	—
122	C₂²	—	8	2	no	—
123	C₂²	#390	1	1	yes	characteristic
124	C₂²	C₈×D₄	1	1	yes	characteristic
125	C₂²	#436	1	1	yes	characteristic
126	C₂²	C₄×D₈	1	2	yes	—
127	C₂²	#443	1	2	yes	—
128	C₄	—	2	1	no	—
129	C₄	—	2	1	no	—
130	C₄	—	8	2	no	—
131	C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
132	C₄	#408	1	1	yes	characteristic
133	C₄	C₈×D₄	1	1	yes	characteristic
134	C₄	#436	1	1	yes	characteristic
135	C₄	#466	1	2	yes	—
136	C₄	#493	1	1	yes	characteristic
137	C₄	#495	1	1	yes	characteristic
138	C₂	—	8	4	no	—
139	C₂	C₈×D₈	1	2	yes	—
140	C₂	#1589	1	2	yes	—
141	C₂	#1954	1	1	yes	characteristic
142	C₂	#1970	1	1	yes	characteristic
143	C₂	#1974	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×D₈

C₂

yes

maximal, characteristic

#1954

C₂

yes

maximal, complemented

C₂×#335

C₂

yes

maximal, characteristic, complemented

C₂×D₄⋊C₈

C₂

yes

maximal, complemented

C₂×C₈²

C₂

yes

maximal, characteristic, complemented

C₂²×D₈

C₄

yes

characteristic

C₂×C₄×D₄

C₂²

yes

—

C₄×D₈

—

C₄×D₈

—

C₂×#107

C₄

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

—

C₂×#102

C₄

yes

—

C₂×C₂²⋊C₈

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#335

—

D₄⋊C₈

—

C₈²

—

C₂²×D₄

C₂×C₄

yes

—

C₂³×C₄

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

C₈

yes

complemented

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

D₄

yes

—

C₄×D₄

—

C₄×D₄

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂³

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×C₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

D₈

—

C₂×C₈

—

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

yes

—

C₂×C₈

M₁₆

yes

—

C₂×C₈

D₈

yes

complemented

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

#41

yes

characteristic

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₄×D₄

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

102

C₂×C₄

C₄×D₄

yes

characteristic

103

C₂×C₄

#121

yes

characteristic

104

C₂×C₄

#121

yes

characteristic

105

C₂×C₄

#124

yes

characteristic

106

C₂×C₄

#127

yes

characteristic

107

C₂×C₄

C₂²×C₈

yes

characteristic

108

C₂×C₄

C₂×M₁₆

yes

characteristic

109

C₂×C₄

#131

yes

characteristic

110

C₂×C₄

#131

yes

characteristic

111

C₂×C₄

C₂×D₈

yes

—

112

C₂×C₄

#135

yes

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

C₂²⋊C₈

yes

complemented

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

#390

yes

characteristic

124

C₂²

C₈×D₄

yes

characteristic

125

C₂²

#436

yes

characteristic

126

C₂²

C₄×D₈

yes

—

127

C₂²

#443

yes

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

C₂×C₂²⋊C₈

yes

characteristic

132

C₄

#408

yes

characteristic

133

C₄

C₈×D₄

yes

characteristic

134

C₄

#436

yes

characteristic

135

C₄

#466

yes

—

136

C₄

#493

yes

characteristic

137

C₄

#495

yes

characteristic

138

C₂

—

139

C₂

C₈×D₈

yes

—

140

C₂

#1589

yes

—

141

C₂

#1954

yes

characteristic

142

C₂

#1970

yes

characteristic

143

C₂

#1974

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂³	C₄×D₄
prime radical	C₂²×D₈	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂²×C₄

C₂×D₄

commutator

C₂×C₄

C₂²×C₈

Frattini, cosocle

C₂×C₄²

C₂³

socle

C₂³

C₄×D₄

prime radical

C₂²×D₈

C₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
2. #1954	C₂	semidirect
3. C₂×#335	C₂	semidirect
4. C₂×D₄⋊C₈	C₂	semidirect
5. C₂×C₈²	C₂	semidirect
24. C₂×D₈	C₈	semidirect
76. C₂×C₈	D₈	semidirect
117. C₈	C₂²⋊C₈	semidirect

H ≅ G/N

type

2. #1954

C₂

semidirect

3. C₂×#335

C₂

semidirect

4. C₂×D₄⋊C₈

C₂

semidirect

5. C₂×C₈²

C₂

semidirect

24. C₂×D₈

C₈

semidirect

76. C₂×C₈

D₈

semidirect

117. C₈

C₂²⋊C₈

semidirect

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

subgroup	index
C₂×C₄×D₈	2
#1954	2
C₂×#335	2
C₂×D₄⋊C₈	2
C₂×C₈²	2

subgroup

index

C₂×C₄×D₈

#1954

C₂×#335

C₂×D₄⋊C₈

C₂×C₈²

Subgroups — #9657

Conjugacy classes of subgroups

Complements

Maximal subgroups