Group #11780971 — subgroups

Subgroups — #11780971

Order 1152.

468 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 141 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	C₃²×#494	C₂	1	1	yes	maximal, characteristic
2	#10592793	C₂	1	2	yes	maximal
3	C₈×#634	C₂	1	1	yes	maximal, characteristic
4	#10589413	C₂	1	2	yes	maximal
5	#10589324	C₂	1	1	yes	maximal, characteristic
6	#71308	—	3	4	no	maximal
7	C₂×C₃²×Q₁₆	—	2	1	no	—
8	C₂×C₃²×Q₁₆	—	2	1	no	—
9	C₃²×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₄×#634	C₂²	1	1	yes	characteristic
11	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
12	#63641	—	2	1	no	—
13	C₈×C₃⋊Dic₃	—	2	1	no	—
14	#63637	—	2	2	no	—
15	#63635	—	2	2	no	—
16	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	2	yes	—
17	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
18	C₃×#494	D₃	1	4	yes	—
19	#5094	—	3	8	no	—
20	C₈×Dic₆	—	3	4	no	—
21	#4547	—	3	8	no	—
22	#4458	—	3	4	no	—
23	C₂×C₃²×Q₈	—	2	2	no	—
24	C₂×#634	D₄	1	1	yes	characteristic
25	C₃²×Q₁₆	—	4	2	no	—
26	C₃²×Q₁₆	—	4	2	no	—
27	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
28	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
29	C₃²×C₄⋊C₄	—	2	2	no	—
30	C₃²×C₄⋊C₄	—	2	2	no	—
31	C₃²×C₄²	C₂³	1	1	yes	characteristic
32	C₄⋊(C₃⋊Dic₃)	D₄	1	1	yes	characteristic
33	#3750	—	2	1	no	—
34	C₄×C₃⋊Dic₃	—	2	1	no	—
35	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
36	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
37	#1670	—	9	1	no	Sylow
38	C₂×C₃×Q₁₆	—	2	4	no	—
39	C₂×C₃×Q₁₆	—	2	4	no	—
40	C₃×C₄⋊Q₈	C₂×D₃	1	8	yes	—
41	C₄×Dic₆	—	3	4	no	—
42	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
43	Dic₃⋊C₈	—	6	4	no	—
44	C₈×Dic₃	—	6	4	no	—
45	#810	—	6	8	no	—
46	#808	—	6	8	no	—
47	C₄⋊(C₃⋊C₈)	—	3	4	no	—
48	C₄⋊(C₃⋊C₈)	—	3	8	no	—
49	C₃²×Q₈	—	4	2	no	—
50	C₃²×Q₈	—	4	2	no	—
51	C₂×C₃²×C₄	—	4	2	no	—
52	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
53	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
54	C₂×C₃²×C₄	#41	1	1	yes	characteristic
55	C₂×C₃⋊Dic₃	—	2	1	no	—
56	C₂×C₃⋊Dic₃	—	2	1	no	—
57	#634	—	2	1	no	—
58	#634	Q₁₆	1	2	yes	complemented
59	C₃²×C₈	—	2	1	no	—
60	C₃²×C₈	SD₁₆	1	2	yes	—
61	C₃⋊(C₃⋊C₈)	—	4	2	no	—
62	C₃⋊(C₃⋊C₈)	—	4	1	no	—
63	#494	C₃⋊D₃	1	1	yes	characteristic
64	#475	—	9	2	no	—
65	C₈×Q₈	—	9	1	no	—
66	#335	—	9	1	no	—
67	#332	—	9	2	no	—
68	C₂×C₃×Q₈	—	2	8	no	—
69	C₂×Dic₆	—	3	4	no	—
70	C₃×Q₁₆	—	4	8	no	—
71	C₃×Q₁₆	—	4	8	no	—
72	C₂×C₃×C₈	#67	1	4	yes	—
73	C₂×C₃×C₈	#67	1	4	yes	—
74	C₃×C₄⋊C₄	—	2	8	no	—
75	C₃×C₄⋊C₄	—	2	8	no	—
76	C₃×C₄²	C₂²×D₃	1	4	yes	—
77	C₄⋊Dic₃	—	3	4	no	—
78	Dic₃⋊C₄	—	6	4	no	—
79	C₄×Dic₃	—	6	4	no	—
80	C₂×C₃⋊C₈	—	6	8	no	—
81	C₂×C₃⋊C₈	—	6	4	no	—
82	C₂²×C₃²	#121	1	1	yes	characteristic
83	C₃²×C₄	—	2	1	no	—
84	C₃²×C₄	—	8	2	no	—
85	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
86	C₃²×C₄	C₂×Q₁₆	1	1	yes	characteristic
87	C₃²×C₄	#135	1	1	yes	characteristic
88	C₃²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₃⋊Dic₃	—	2	2	no	—
90	C₃⋊Dic₃	—	4	1	no	—
91	C₂×Q₁₆	—	2	1	no	—
92	C₂×Q₁₆	—	2	1	no	—
93	C₄⋊Q₈	C₂×C₃⋊D₃	1	2	yes	—
94	C₄×Q₈	—	9	1	no	—
95	#106	—	18	2	no	—
96	C₄⋊C₈	—	9	1	no	—
97	C₄⋊C₈	—	9	2	no	—
98	C₄⋊C₈	—	18	1	no	—
99	#103	—	18	2	no	—
100	C₄×C₈	—	18	1	no	—
101	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
102	C₃×Q₈	—	4	8	no	—
103	C₃×Q₈	—	4	8	no	—
104	C₂×C₃×C₄	—	4	8	no	—
105	C₂×C₃×C₄	D₃×D₄	1	4	yes	—
106	C₂×C₃×C₄	#237	1	4	yes	—
107	C₂×C₃×C₄	C₂×#67	1	4	yes	—
108	C₂×Dic₃	—	6	4	no	—
109	C₂×Dic₃	—	6	4	no	—
110	Dic₆	—	3	8	no	—
111	Dic₆	—	6	4	no	—
112	C₃×C₈	—	2	4	no	—
113	C₃×C₈	#214	1	8	yes	—
114	C₃⋊C₈	—	12	8	no	—
115	C₃⋊C₈	—	12	4	no	—
116	C₂×C₃²	#460	1	1	yes	characteristic
117	C₂×C₃²	#467	1	1	yes	characteristic
118	C₂×C₃²	#473	1	1	yes	characteristic
119	C₂×Q₈	—	2	2	no	—
120	C₂×Q₈	—	9	1	no	—
121	Q₁₆	—	4	2	no	—
122	Q₁₆	—	4	2	no	—
123	C₂×C₈	—	18	2	no	—
124	C₂×C₈	—	18	1	no	—
125	C₂×C₈	#638	1	1	yes	characteristic
126	C₂×C₈	#638	1	1	yes	characteristic
127	C₄⋊C₄	—	2	2	no	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	9	1	no	—
130	C₄⋊C₄	—	18	1	no	—
131	C₄²	—	18	1	no	—
132	C₄²	C₂²×C₃⋊D₃	1	1	yes	characteristic
133	C₂²×C₃	#938	1	4	yes	—
134	C₃×C₄	—	2	4	no	—
135	C₃×C₄	—	8	8	no	—
136	C₃×C₄	D₃×Q₁₆	1	4	yes	—
137	C₃×C₄	#919	1	4	yes	—
138	C₃×C₄	#932	1	4	yes	—
139	C₃×C₄	C₂×#214	1	4	yes	—
140	Dic₃	—	6	8	no	—
141	Dic₃	—	12	4	no	—
142	C₃²	#1670	1	1	yes	characteristic, complemented, Sylow
143	Q₈	—	4	2	no	—
144	Q₈	—	4	2	no	—
145	Q₈	—	9	2	no	—
146	Q₈	—	18	1	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	18	1	no	—
149	C₂×C₄	—	18	1	no	—
150	C₂×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
151	C₂×C₄	#3830	1	1	yes	characteristic
152	C₂×C₄	C₂×#638	1	1	yes	characteristic
153	C₈	—	2	1	no	—
154	C₈	—	36	2	no	—
155	C₈	—	36	1	no	—
156	C₈	#3754	1	2	yes	—
157	C₂×C₃	#5075	1	4	yes	—
158	C₂×C₃	#5188	1	4	yes	—
159	C₂×C₃	#5192	1	4	yes	—
160	C₂²	#64147	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	8	2	no	—
163	C₄	—	18	2	no	—
164	C₄	—	36	1	no	—
165	C₄	Q₁₆×C₃⋊D₃	1	1	yes	characteristic
166	C₄	#64128	1	1	yes	characteristic
167	C₄	#64141	1	1	yes	characteristic
168	C₄	C₂×#3754	1	1	yes	characteristic
169	C₃	#71308	1	4	yes	complemented
170	C₂	#10592774	1	1	yes	characteristic
171	C₂	#10592887	1	1	yes	characteristic
172	C₂	#10592891	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#494

C₂

yes

maximal, characteristic

#10592793

C₂

yes

maximal

C₈×#634

C₂

yes

maximal, characteristic

#10589413

C₂

yes

maximal

#10589324

C₂

yes

maximal, characteristic

#71308

—

maximal

C₂×C₃²×Q₁₆

—

C₂×C₃²×Q₁₆

—

C₃²×C₄⋊Q₈

C₂²

yes

—

C₄×#634

C₂²

yes

characteristic

C₃²×C₄×C₈

C₂²

yes

characteristic

#63641

—

C₈×C₃⋊Dic₃

—

#63637

—

#63635

—

C₄⋊(C₃⋊(C₃⋊C₈))

C₂²

yes

—

C₄⋊(C₃⋊(C₃⋊C₈))

C₂²

yes

characteristic

C₃×#494

D₃

yes

—

#5094

—

C₈×Dic₆

—

#4547

—

#4458

—

C₂×C₃²×Q₈

—

C₂×#634

D₄

yes

characteristic

C₃²×Q₁₆

—

C₃²×Q₁₆

—

C₂×C₃²×C₈

D₄

yes

characteristic

C₂×C₃²×C₈

D₄

yes

characteristic

C₃²×C₄⋊C₄

—

C₃²×C₄⋊C₄

—

C₃²×C₄²

C₂³

yes

characteristic

C₄⋊(C₃⋊Dic₃)

D₄

yes

characteristic

#3750

—

C₄×C₃⋊Dic₃

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂×C₃⋊(C₃⋊C₈)

—

#1670

—

Sylow

C₂×C₃×Q₁₆

—

C₂×C₃×Q₁₆

—

C₃×C₄⋊Q₈

C₂×D₃

yes

—

C₄×Dic₆

—

C₃×C₄×C₈

C₂×D₃

yes

—

Dic₃⋊C₈

—

C₈×Dic₃

—

#810

—

#808

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

—

C₃²×Q₈

—

C₃²×Q₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃²×C₄

#41

yes

characteristic

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

#634

—

#634

Q₁₆

yes

complemented

C₃²×C₈

—

C₃²×C₈

SD₁₆

yes

—

C₃⋊(C₃⋊C₈)

—

C₃⋊(C₃⋊C₈)

—

#494

C₃⋊D₃

yes

characteristic

#475

—

C₈×Q₈

—

#335

—

#332

—

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₃×Q₁₆

—

C₃×Q₁₆

—

C₂×C₃×C₈

#67

yes

—

C₂×C₃×C₈

#67

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂²×D₃

yes

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₃²

#121

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₂×SD₁₆

yes

characteristic

C₃²×C₄

C₂×Q₁₆

yes

C₃²×C₄

yes

C₃²×C₄

yes

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

C₂×C₃⋊D₃

yes

—

C₄×Q₈

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

100

C₄×C₈

—

101

C₄×C₈

C₂×C₃⋊D₃

yes

characteristic

102

C₃×Q₈

—

103

C₃×Q₈

—

104

C₂×C₃×C₄

—

105

C₂×C₃×C₄

D₃×D₄

yes

—

106

C₂×C₃×C₄

#237

yes

—

107

C₂×C₃×C₄

C₂×#67

yes

—

108

C₂×Dic₃

—

109

C₂×Dic₃

—

110

Dic₆

—

111

Dic₆

—

112

C₃×C₈

—

113

C₃×C₈

#214

yes

—

114

C₃⋊C₈

—

115

C₃⋊C₈

—

116

C₂×C₃²

#460

yes

characteristic

117

C₂×C₃²

#467

yes

characteristic

118

C₂×C₃²

#473

yes

characteristic

119

C₂×Q₈

—

120

C₂×Q₈

—

121

Q₁₆

—

122

Q₁₆

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

#638

yes

characteristic

126

C₂×C₈

#638

yes

characteristic

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄²

—

132

C₄²

C₂²×C₃⋊D₃

yes

characteristic

133

C₂²×C₃

#938

yes

—

134

C₃×C₄

—

135

C₃×C₄

—

136

C₃×C₄

D₃×Q₁₆

yes

—

137

C₃×C₄

#919

yes

—

138

C₃×C₄

#932

yes

—

139

C₃×C₄

C₂×#214

yes

—

140

Dic₃

—

141

Dic₃

—

142

C₃²

#1670

yes

characteristic, complemented, Sylow

143

Q₈

—

144

Q₈

—

145

Q₈

—

146

Q₈

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

D₄×C₃⋊D₃

yes

characteristic

151

C₂×C₄

#3830

yes

characteristic

152

C₂×C₄

C₂×#638

yes

characteristic

153

C₈

—

154

C₈

—

155

C₈

—

156

C₈

#3754

yes

—

157

C₂×C₃

#5075

yes

—

158

C₂×C₃

#5188

yes

—

159

C₂×C₃

#5192

yes

—

160

C₂²

#64147

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

Q₁₆×C₃⋊D₃

yes

characteristic

166

C₄

#64128

yes

characteristic

167

C₄

#64141

yes

characteristic

168

C₄

C₂×#3754

yes

characteristic

169

C₃

#71308

yes

complemented

170

C₂

#10592774

yes

characteristic

171

C₂

#10592887

yes

characteristic

172

C₂

#10592891

yes

characteristic

	subgroup	G/H
center	C₂²	#64147
commutator, cosocle	C₃²×C₄²	C₂³
Frattini	C₄²	C₂²×C₃⋊D₃
Fitting	C₃²×#494	C₂
socle, prime radical	C₂²×C₃²	#121
hypercenter	#494	C₃⋊D₃
hypocenter	C₃²	#1670
LCS	G ▷ C₃²×C₄² ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#64147

commutator, cosocle

C₃²×C₄²

C₂³

Frattini

C₄²

C₂²×C₃⋊D₃

Fitting

C₃²×#494

C₂

C₂²×C₃²

C₃⋊D₃

C₃²

G ▷ C₃²×C₄² ▷ C₂²×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
58. #634	Q₁₆	semidirect
142. C₃²	#1670	semidirect
169. C₃	#71308	semidirect

H ≅ G/N

type

58. #634

Q₁₆

semidirect

142. C₃²

#1670

semidirect

169. C₃

#71308

semidirect

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

subgroup	index
C₃²×#494	2
#10592793	2
C₈×#634	2
#10589413	2
#10589324	2
#71308	3

subgroup

index

C₃²×#494

Subgroups — #11780971

Conjugacy classes of subgroups

Complements

Maximal subgroups