Group #9741 — subgroups

Subgroups — #9741

Order 256.

229 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 229 conjugacy classes of subgroups — too many to draw. Only its 91 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×#498	C₂	1	1	yes	maximal, characteristic
2	#1987	C₂	1	1	yes	maximal, characteristic
3	#1981	C₂	1	1	yes	maximal, characteristic
4	#1770	C₂	1	1	yes	maximal, characteristic
5	C₂×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×#333	C₂	1	1	yes	maximal, characteristic
7	C₂×#332	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	#498	—	2	2	no	—
11	#498	C₄	1	4	yes	complemented
12	C₂×#106	—	2	1	no	—
13	C₂×#106	—	2	1	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	C₂×#103	—	2	2	no	—
18	C₂×#97	—	2	1	no	—
19	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
20	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	#340	—	2	2	no	—
22	#336	—	2	1	no	—
23	#335	—	2	2	no	—
24	#333	—	2	2	no	—
25	#332	—	2	2	no	—
26	C₂²×Q₈	—	2	1	no	—
27	C₂²×Q₈	—	2	1	no	—
28	C₂²×C₈	—	2	1	no	—
29	C₂²×C₈	—	2	1	no	—
30	C₂²×C₈	—	2	1	no	—
31	C₂²×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₈	D₄	1	1	yes	characteristic
33	C₄⋊Q₈	—	2	1	no	—
34	C₄⋊Q₈	—	2	1	no	—
35	C₄⋊Q₈	—	2	1	no	—
36	C₄⋊Q₈	—	2	1	no	—
37	C₄⋊Q₈	—	2	1	no	—
38	C₄⋊Q₈	—	2	1	no	—
39	C₄⋊Q₈	C₂×C₄	1	2	yes	—
40	C₄⋊Q₈	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊C₄	—	2	1	no	—
42	C₂×C₄⋊C₄	—	2	1	no	—
43	C₂×C₄⋊C₄	—	2	1	no	—
44	C₂×C₄⋊C₄	—	2	1	no	—
45	C₂×C₄⋊C₄	D₄	1	2	yes	—
46	C₂×C₄²	—	2	1	no	—
47	C₂×C₄²	C₂³	1	1	yes	characteristic
48	#106	—	2	4	no	—
49	#106	—	2	4	no	—
50	C₄⋊C₈	—	2	1	no	—
51	C₄⋊C₈	—	2	1	no	—
52	C₄⋊C₈	—	2	1	no	—
53	C₄⋊C₈	—	2	1	no	—
54	C₄⋊C₈	—	2	1	no	—
55	C₄⋊C₈	D₄	1	2	yes	—
56	#103	—	4	4	no	—
57	#97	—	4	2	no	—
58	C₄×C₈	C₂×C₄	1	2	yes	—
59	C₄×C₈	D₄	1	2	yes	—
60	#95	—	2	1	no	—
61	C₂×Q₈	—	2	2	no	—
62	C₂×Q₈	—	2	2	no	—
63	C₂×Q₈	—	4	1	no	—
64	C₂×Q₈	—	4	1	no	—
65	C₂×Q₈	—	4	2	no	—
66	C₂×Q₈	—	4	2	no	—
67	C₂×Q₈	—	8	2	no	—
68	C₂²×C₄	—	2	2	no	—
69	C₂²×C₄	—	4	1	no	—
70	C₂²×C₄	—	4	1	no	—
71	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄	#41	1	1	yes	characteristic
74	C₂×C₈	—	2	1	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	4	1	no	—
78	C₂×C₈	—	4	1	no	—
79	C₂×C₈	—	4	1	no	—
80	C₂×C₈	—	4	1	no	—
81	C₂×C₈	—	4	2	no	—
82	C₂×C₈	—	4	2	no	—
83	C₂×C₈	C₂²⋊C₄	1	2	yes	—
84	C₂×C₈	C₂²⋊C₄	1	2	yes	—
85	C₂×C₈	SD₁₆	1	2	yes	—
86	C₂×C₈	Q₁₆	1	2	yes	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	4	1	no	—
94	C₄⋊C₄	—	4	1	no	—
95	C₄⋊C₄	—	4	1	no	—
96	C₄⋊C₄	—	4	1	no	—
97	C₄²	—	4	1	no	—
98	C₄²	—	4	1	no	—
99	C₄²	C₂²×C₄	1	1	yes	characteristic
100	C₄²	C₂×D₄	1	1	yes	characteristic
101	C₄²	C₂×D₄	1	1	yes	characteristic
102	C₄²	#41	1	1	yes	characteristic
103	C₂³	#121	1	1	yes	characteristic
104	Q₈	—	4	4	no	—
105	Q₈	—	8	2	no	—
106	Q₈	—	8	2	no	—
107	Q₈	—	8	1	no	—
108	Q₈	—	8	1	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	8	2	no	—
116	C₂×C₄	—	8	2	no	—
117	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄	#121	1	1	yes	characteristic
121	C₂×C₄	#124	1	1	yes	characteristic
122	C₂×C₄	#127	1	1	yes	characteristic
123	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
125	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
126	C₂×C₄	#137	1	1	yes	characteristic
127	C₂×C₄	#137	1	1	yes	characteristic
128	C₂×C₄	#137	1	2	yes	—
129	C₈	—	2	2	no	—
130	C₈	—	4	2	no	—
131	C₈	—	8	1	no	—
132	C₈	—	8	1	no	—
133	C₈	#103	1	4	yes	complemented
134	C₂²	#390	1	1	yes	characteristic
135	C₂²	#461	1	1	yes	characteristic
136	C₂²	#462	1	1	yes	characteristic
137	C₂²	#468	1	2	yes	—
138	C₂²	#471	1	2	yes	—
139	C₄	—	2	1	no	—
140	C₄	—	2	1	no	—
141	C₄	—	8	2	no	—
142	C₄	—	8	2	no	—
143	C₄	—	8	2	no	—
144	C₄	C₂×#103	1	1	yes	characteristic
145	C₄	#419	1	1	yes	characteristic
146	C₄	#440	1	2	yes	—
147	C₄	#461	1	1	yes	characteristic
148	C₄	#462	1	1	yes	characteristic
149	C₄	#496	1	1	yes	characteristic
150	C₄	#497	1	1	yes	characteristic
151	C₂	#1691	1	2	yes	—
152	C₂	#1692	1	2	yes	—
153	C₂	#1973	1	1	yes	characteristic
154	C₂	#1977	1	1	yes	characteristic
155	C₂	#1981	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#498

C₂

yes

maximal, characteristic

#1987

C₂

yes

maximal, characteristic

#1981

C₂

yes

maximal, characteristic

#1770

C₂

yes

maximal, characteristic

C₂×#335

C₂

yes

maximal, characteristic

C₂×#333

C₂

yes

maximal, characteristic

C₂×#332

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈

C₂²

yes

characteristic

C₂×C₄⋊Q₈

C₂²

yes

characteristic

#498

—

#498

C₄

yes

complemented

C₂×#106

—

C₂×#106

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×#103

—

C₂×#97

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₄×C₄⋊C₄

C₂²

yes

—

—

—

—

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×C₄

yes

—

C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄

yes

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

#106

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₄

yes

—

#103

—

#97

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

C₂×C₈

—

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

SD₁₆

yes

—

C₂×C₈

Q₁₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

100

C₄²

C₂×D₄

yes

characteristic

101

C₄²

C₂×D₄

yes

characteristic

102

C₄²

#41

yes

characteristic

103

C₂³

#121

yes

characteristic

104

Q₈

—

105

Q₈

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

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

C₂×C₂²⋊C₄

yes

characteristic

118

C₂×C₄

C₄×D₄

yes

characteristic

119

C₂×C₄

C₄×D₄

yes

characteristic

120

C₂×C₄

#121

yes

characteristic

121

C₂×C₄

#124

yes

characteristic

122

C₂×C₄

#127

yes

characteristic

123

C₂×C₄

C₂×SD₁₆

yes

characteristic

124

C₂×C₄

C₂×Q₁₆

yes

characteristic

125

C₂×C₄

AGL(1,ℤ/8)

yes

—

126

C₂×C₄

#137

yes

characteristic

127

C₂×C₄

#137

yes

characteristic

128

C₂×C₄

#137

yes

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

#103

yes

complemented

134

C₂²

#390

yes

characteristic

135

C₂²

#461

yes

characteristic

136

C₂²

#462

yes

characteristic

137

C₂²

#468

yes

—

138

C₂²

#471

yes

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

C₂×#103

yes

characteristic

145

C₄

#419

yes

characteristic

146

C₄

#440

yes

—

147

C₄

#461

yes

characteristic

148

C₄

#462

yes

characteristic

149

C₄

#496

yes

characteristic

150

C₄

#497

yes

characteristic

151

C₂

#1691

yes

—

152

C₂

#1692

yes

—

153

C₂

#1973

yes

characteristic

154

C₂

#1977

yes

characteristic

155

C₂

#1981

yes

characteristic

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

subgroup

G/H

center, socle, prime radical

C₂³

#121

commutator

C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
11. #498	C₄	semidirect
133. C₈	#103	semidirect

H ≅ G/N

type

11. #498

C₄

semidirect

133. C₈

#103

semidirect

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

subgroup	index
C₂×#498	2
#1987	2
#1981	2
#1770	2
C₂×#335	2
C₂×#333	2
C₂×#332	2

subgroup

index

Subgroups — #9741

Conjugacy classes of subgroups

Complements

Maximal subgroups