Group #11782906 — subgroups

Subgroups — #11782906

Order 1152.

726 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 185 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	#10592394	C₂	1	2	yes	maximal, complemented
2	#10592394	C₂	1	1	yes	maximal, characteristic, complemented
3	#10592393	C₂	1	2	yes	maximal, complemented
4	#10592392	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	#68094	—	3	4	no	maximal
7	C₂×#3745	—	2	4	no	—
8	C₂×#3745	—	2	2	no	—
9	C₂×#3744	—	2	2	no	—
10	#64083	C₂²	1	1	yes	characteristic
11	#64083	C₂²	1	2	yes	—
12	#64078	C₂²	1	1	yes	characteristic
13	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₃²×C₄×C₈	C₂²	1	2	yes	—
15	#63648	—	2	4	no	—
16	#4695	—	3	4	no	—
17	#4695	—	3	8	no	—
18	#4694	—	3	8	no	—
19	#4693	—	3	4	no	—
20	C₃×#322	D₃	1	4	yes	complemented
21	C₂×C₄⋊(C₃⋊D₃)	—	2	1	no	—
22	C₂×C₄⋊(C₃⋊D₃)	—	2	2	no	—
23	C₂×C₄⋊(C₃⋊D₃)	—	2	2	no	—
24	C₂×C₄⋊(C₃⋊D₃)	—	2	2	no	—
25	C₂×C₄⋊(C₃⋊D₃)	—	2	2	no	—
26	C₂×#634	—	2	1	no	—
27	C₂×C₃²×C₈	D₄	1	4	yes	—
28	C₂×C₃²×C₈	D₄	1	2	yes	—
29	C₃²×C₄²	C₂³	1	1	yes	characteristic
30	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
31	#3745	—	4	8	no	—
32	#3745	—	4	4	no	—
33	#3744	—	4	4	no	—
34	#1721	—	9	1	no	Sylow
35	C₂×D₂₄	—	6	16	no	—
36	C₂×D₂₄	—	6	8	no	—
37	C₂×#204	—	6	8	no	—
38	#874	—	3	4	no	—
39	#874	—	3	8	no	—
40	#869	—	3	4	no	—
41	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
42	C₃×C₄×C₈	C₂×D₃	1	8	yes	—
43	#821	—	6	16	no	—
44	C₂²×C₃⋊D₃	—	4	1	no	—
45	C₂²×C₃⋊D₃	—	4	2	no	—
46	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
47	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
48	C₂×C₃⋊Dic₃	—	4	1	no	—
49	C₄⋊(C₃⋊D₃)	—	4	2	no	—
50	C₄⋊(C₃⋊D₃)	—	4	4	no	—
51	C₄⋊(C₃⋊D₃)	—	4	4	no	—
52	C₄⋊(C₃⋊D₃)	—	4	4	no	—
53	C₄⋊(C₃⋊D₃)	—	4	4	no	—
54	#634	—	4	2	no	—
55	C₃²×C₈	—	2	4	no	—
56	C₃²×C₈	D₈	1	4	yes	complemented
57	#493	—	9	2	no	—
58	#493	—	9	1	no	—
59	#492	—	9	1	no	—
60	#486	—	9	2	no	—
61	#322	C₃⋊D₃	1	1	yes	characteristic, complemented
62	C₂×D₁₂	—	6	8	no	—
63	C₂×D₁₂	—	6	4	no	—
64	C₂×D₁₂	—	6	8	no	—
65	C₂×D₁₂	—	6	8	no	—
66	C₂×D₁₂	—	6	8	no	—
67	C₂×Dic₆	—	6	4	no	—
68	C₂×C₃×C₈	D₁₂	1	16	yes	—
69	C₂×C₃×C₈	D₁₂	1	8	yes	—
70	C₃×C₄²	C₂²×D₃	1	4	yes	—
71	C₄⋊Dic₃	—	6	8	no	—
72	D₂₄	—	12	32	no	—
73	D₂₄	—	12	16	no	—
74	#204	—	12	16	no	—
75	C₂²×C₃²	#127	1	1	yes	characteristic
76	C₂×C₃⋊D₃	—	8	1	no	—
77	C₂×C₃⋊D₃	—	8	2	no	—
78	C₂×C₃⋊D₃	—	8	2	no	—
79	C₂×C₃⋊D₃	—	8	2	no	—
80	C₂×C₃⋊D₃	—	8	2	no	—
81	C₃²×C₄	C₂×D₈	1	2	yes	—
82	C₃²×C₄	AGL(1,ℤ/8)	1	4	yes	—
83	C₃⋊Dic₃	—	8	1	no	—
84	C₂×SD₁₆	—	18	2	no	—
85	C₂×D₈	—	18	4	no	—
86	C₂×D₈	—	18	2	no	—
87	C₄⋊Q₈	—	9	1	no	—
88	#127	—	9	2	no	—
89	#127	—	9	1	no	—
90	#102	—	18	4	no	—
91	C₄×C₈	C₂×C₃⋊D₃	1	2	yes	—
92	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
93	C₂²×D₃	—	12	8	no	—
94	C₂²×D₃	—	12	4	no	—
95	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
96	C₂×C₃×C₄	C₂×D₁₂	1	8	yes	—
97	C₂×Dic₃	—	12	4	no	—
98	D₁₂	—	12	8	no	—
99	D₁₂	—	12	16	no	—
100	D₁₂	—	12	16	no	—
101	D₁₂	—	12	16	no	—
102	D₁₂	—	12	16	no	—
103	Dic₆	—	12	8	no	—
104	C₃×C₈	—	2	16	no	—
105	C₃×C₈	D₂₄	1	16	yes	complemented
106	C₂×C₃²	#493	1	1	yes	characteristic
107	C₂×C₃²	#496	1	2	yes	—
108	C₃⋊D₃	—	16	1	no	—
109	C₃⋊D₃	—	16	2	no	—
110	C₂×Q₈	—	18	1	no	—
111	C₂×D₄	—	18	2	no	—
112	C₂×D₄	—	18	2	no	—
113	C₂×D₄	—	18	2	no	—
114	C₂×D₄	—	18	2	no	—
115	C₂×D₄	—	18	1	no	—
116	SD₁₆	—	36	4	no	—
117	D₈	—	36	8	no	—
118	D₈	—	36	4	no	—
119	C₂×C₈	C₄⋊(C₃⋊D₃)	1	4	yes	—
120	C₂×C₈	C₄⋊(C₃⋊D₃)	1	2	yes	—
121	C₄⋊C₄	—	18	2	no	—
122	C₄²	C₂²×C₃⋊D₃	1	1	yes	characteristic
123	C₂²×C₃	#874	1	4	yes	—
124	C₂×D₃	—	24	4	no	—
125	C₂×D₃	—	24	8	no	—
126	C₂×D₃	—	24	8	no	—
127	C₂×D₃	—	24	8	no	—
128	C₂×D₃	—	24	8	no	—
129	C₃×C₄	C₂×D₂₄	1	8	yes	—
130	C₃×C₄	#908	1	16	yes	—
131	Dic₃	—	24	4	no	—
132	C₃²	#1721	1	1	yes	characteristic, complemented, Sylow
133	C₂³	—	36	2	no	—
134	C₂³	—	36	1	no	—
135	Q₈	—	36	2	no	—
136	D₄	—	36	4	no	—
137	D₄	—	36	2	no	—
138	D₄	—	36	4	no	—
139	D₄	—	36	4	no	—
140	D₄	—	36	4	no	—
141	C₂×C₄	—	36	1	no	—
142	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
143	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	2	yes	—
144	C₈	—	2	4	no	—
145	C₈	#3745	1	4	yes	complemented
146	C₂×C₃	#4695	1	4	yes	—
147	C₂×C₃	#4712	1	8	yes	—
148	D₃	—	48	4	no	—
149	D₃	—	48	8	no	—
150	C₂²	—	72	2	no	—
151	C₂²	—	72	1	no	—
152	C₂²	—	72	2	no	—
153	C₂²	—	72	2	no	—
154	C₂²	—	72	2	no	—
155	C₂²	#64083	1	1	yes	characteristic
156	C₄	—	72	1	no	—
157	C₄	C₂×#3745	1	2	yes	—
158	C₄	#64117	1	4	yes	—
159	C₃	#68094	1	4	yes	complemented
160	C₂	—	144	2	no	—
161	C₂	—	144	1	no	—
162	C₂	#10592394	1	1	yes	characteristic
163	C₂	#10592411	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10592394

C₂

yes

maximal, complemented

#10592394

C₂

yes

maximal, characteristic, complemented

#10592393

C₂

yes

maximal, complemented

#10592392

C₂

yes

maximal, characteristic, complemented

C₃²×#322

C₂

yes

maximal, characteristic, complemented

—

—

—

—

C₂²

yes

characteristic

#64083

C₂²

yes

—

#64078

C₂²

yes

characteristic

C₃²×C₄×C₈

C₂²

yes

characteristic

C₃²×C₄×C₈

C₂²

yes

—

—

—

—

—

—

D₃

yes

complemented

C₂×C₄⋊(C₃⋊D₃)

—

C₂×C₄⋊(C₃⋊D₃)

—

C₂×C₄⋊(C₃⋊D₃)

—

C₂×C₄⋊(C₃⋊D₃)

—

C₂×C₄⋊(C₃⋊D₃)

—

C₂×#634

—

C₂×C₃²×C₈

D₄

yes

—

C₂×C₃²×C₈

D₄

yes

—

C₃²×C₄²

C₂³

yes

characteristic

C₄⋊(C₃⋊Dic₃)

—

—

—

—

—

C₂×D₂₄

—

C₂×D₂₄

—

—

—

—

—

C₃×C₄×C₈

C₂×D₃

yes

—

C₃×C₄×C₈

C₂×D₃

yes

—

#821

—

C₂²×C₃⋊D₃

—

C₂²×C₃⋊D₃

—

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃²×C₄

C₂×D₄

yes

—

C₂×C₃⋊Dic₃

—

C₄⋊(C₃⋊D₃)

—

C₄⋊(C₃⋊D₃)

—

C₄⋊(C₃⋊D₃)

—

C₄⋊(C₃⋊D₃)

—

C₄⋊(C₃⋊D₃)

—

#634

—

C₃²×C₈

—

C₃²×C₈

D₈

yes

—

—

—

—

C₃⋊D₃

yes

characteristic, complemented

C₂×D₁₂

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×Dic₆

—

C₂×C₃×C₈

D₁₂

yes

—

C₂×C₃×C₈

D₁₂

yes

—

C₃×C₄²

C₂²×D₃

yes

—

C₄⋊Dic₃

—

D₂₄

—

D₂₄

—

#204

—

C₂²×C₃²

#127

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₃²×C₄

C₂×D₈

yes

—

C₃²×C₄

AGL(1,ℤ/8)

yes

—

C₃⋊Dic₃

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

#127

—

#102

—

C₄×C₈

C₂×C₃⋊D₃

yes

—

C₄×C₈

C₂×C₃⋊D₃

yes

characteristic

C₂²×D₃

—

C₂²×D₃

—

C₂×C₃×C₄

C₂×D₁₂

yes

—

C₂×C₃×C₄

C₂×D₁₂

yes

—

C₂×Dic₃

—

D₁₂

—

D₁₂

—

100

D₁₂

—

101

D₁₂

—

102

D₁₂

—

103

Dic₆

—

104

C₃×C₈

—

105

C₃×C₈

D₂₄

yes

complemented

106

C₂×C₃²

#493

yes

characteristic

107

C₂×C₃²

#496

yes

—

108

C₃⋊D₃

—

109

C₃⋊D₃

—

110

C₂×Q₈

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

—

116

SD₁₆

—

117

D₈

—

118

D₈

—

119

C₂×C₈

C₄⋊(C₃⋊D₃)

yes

—

120

C₂×C₈

C₄⋊(C₃⋊D₃)

yes

—

121

C₄⋊C₄

—

122

C₄²

C₂²×C₃⋊D₃

yes

characteristic

123

C₂²×C₃

#874

yes

—

124

C₂×D₃

—

125

C₂×D₃

—

126

C₂×D₃

—

127

C₂×D₃

—

128

C₂×D₃

—

129

C₃×C₄

C₂×D₂₄

yes

—

130

C₃×C₄

#908

yes

—

131

Dic₃

—

132

C₃²

#1721

yes

characteristic, complemented, Sylow

133

C₂³

—

134

C₂³

—

135

Q₈

—

136

D₄

—

137

D₄

—

138

D₄

—

139

D₄

—

140

D₄

—

141

C₂×C₄

—

142

C₂×C₄

C₂×C₄⋊(C₃⋊D₃)

yes

characteristic

143

C₂×C₄

C₂×C₄⋊(C₃⋊D₃)

yes

—

144

C₈

—

145

C₈

#3745

yes

complemented

146

C₂×C₃

#4695

yes

—

147

C₂×C₃

#4712

yes

—

148

D₃

—

149

D₃

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

#64083

yes

characteristic

156

C₄

—

157

C₄

C₂×#3745

yes

—

158

C₄

#64117

yes

—

159

C₃

#68094

yes

complemented

160

C₂

—

144

—

161

C₂

—

144

—

162

C₂

#10592394

yes

characteristic

163

C₂

#10592411

yes

—

	subgroup	G/H
center	C₂²	#64083
commutator, cosocle	C₃²×C₄²	C₂³
Frattini	C₄²	C₂²×C₃⋊D₃
Fitting	C₃²×#322	C₂
socle	C₂²×C₃²	#127
hypercenter	#322	C₃⋊D₃
hypocenter	C₃²	#1721
LCS	G ▷ C₃²×C₄² ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#64083

commutator, cosocle

C₃²×C₄²

C₂³

Frattini

C₄²

C₂²×C₃⋊D₃

Fitting

C₃²×#322

C₂

C₂²×C₃²

C₃⋊D₃

C₃²

G ▷ C₃²×C₄² ▷ C₂²×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
1. #10592394	C₂	semidirect
2. #10592394	C₂	semidirect
3. #10592393	C₂	semidirect
4. #10592392	C₂	semidirect
5. C₃²×#322	C₂	semidirect
20. C₃×#322	D₃	semidirect
56. C₃²×C₈	D₈	semidirect
61. #322	C₃⋊D₃	semidirect
105. C₃×C₈	D₂₄	semidirect
132. C₃²	#1721	semidirect
145. C₈	#3745	semidirect
159. C₃	#68094	semidirect

H ≅ G/N

type

1. #10592394

C₂

semidirect

2. #10592394

C₂

semidirect

3. #10592393

C₂

semidirect

4. #10592392

C₂

semidirect

5. C₃²×#322

C₂

semidirect

20. C₃×#322

D₃

semidirect

56. C₃²×C₈

D₈

semidirect

61. #322

C₃⋊D₃

semidirect

105. C₃×C₈

D₂₄

semidirect

132. C₃²

#1721

semidirect

145. C₈

#3745

semidirect

159. C₃

#68094

semidirect

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

subgroup	index
#10592394	2
#10592393	2
#10592392	2
C₃²×#322	2
#68094	3

subgroup

index

C₃²×#322

Subgroups — #11782906

Conjugacy classes of subgroups

Complements

Maximal subgroups