Group #11857076 — subgroups

Subgroups — #11857076

Order 1152.

972 conjugacy classes of subgroups; only the 310 normal subgroups are shown.

normal subgroups only

This group has 972 conjugacy classes of subgroups — too many to draw. Only its 310 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×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#63506	C₂	1	4	yes	maximal
3	C₂×#63431	C₂	1	1	yes	maximal, characteristic
4	#10588655	C₂	1	8	yes	maximal
5	C₄²×Dic₉	C₂	1	1	yes	maximal, characteristic
6	#83253	—	3	1	no	maximal
7	C₂²×Q₈×C₉	C₂²	1	2	yes	—
8	C₂²×Dic₁₈	C₂²	1	2	yes	—
9	C₉×C₄⋊Q₈	—	2	4	no	—
10	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	—
11	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
12	#63506	—	2	16	no	—
13	C₂×C₄⋊Dic₉	C₂²	1	4	yes	—
14	C₂×Dic₉⋊C₄	C₂²	1	16	yes	—
15	C₂×C₄×Dic₉	C₂²	1	2	yes	—
16	C₂×C₄×Dic₉	C₂²	1	4	yes	—
17	#63431	—	2	4	no	—
18	C₂×C₃×C₄⋊Q₈	D₃	1	1	yes	characteristic
19	C₂×#944	—	3	4	no	—
20	C₂×#869	—	3	1	no	—
21	#4975	—	3	8	no	—
22	C₄²×Dic₃	—	3	1	no	—
23	C₂×Q₈×C₉	—	2	8	no	—
24	C₂×Q₈×C₉	—	2	4	no	—
25	C₂²×C₄×C₉	C₂³	1	4	yes	—
26	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₉	C₂³	1	2	yes	—
28	C₂²×Dic₉	C₂³	1	4	yes	—
29	C₂²×Dic₉	C₂³	1	4	yes	—
30	C₂×Dic₁₈	—	2	8	no	—
31	C₂×Dic₁₈	—	2	4	no	—
32	C₉×C₄⋊C₄	—	2	8	no	—
33	C₄²×C₉	D₄	1	4	yes	—
34	C₄⋊Dic₉	—	2	8	no	—
35	Dic₉⋊C₄	—	2	32	no	—
36	C₄×Dic₉	D₄	1	8	yes	—
37	C₄×Dic₉	Q₈	1	16	yes	complemented
38	#2868	—	9	1	no	Sylow
39	C₂²×C₃×Q₈	C₂×D₃	1	2	yes	—
40	C₂²×Dic₆	—	3	2	no	—
41	C₃×C₄⋊Q₈	—	2	4	no	—
42	C₂×C₃×C₄⋊C₄	C₂×D₃	1	4	yes	—
43	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
44	#944	—	6	16	no	—
45	C₂×C₄⋊Dic₃	—	3	4	no	—
46	C₂×Dic₃⋊C₄	—	3	16	no	—
47	C₂×C₄×Dic₃	—	3	4	no	—
48	C₂×C₄×Dic₃	—	3	2	no	—
49	#869	—	6	4	no	—
50	C₂³×C₉	C₂⁴	1	1	yes	characteristic
51	Q₈×C₉	—	4	8	no	—
52	C₂×C₄×C₉	—	2	8	no	—
53	C₂×C₄×C₉	—	2	4	no	—
54	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
55	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
57	C₂×C₄×C₉	C₂×Q₈	1	8	yes	—
58	C₂×Dic₉	—	2	8	no	—
59	C₂×Dic₉	—	2	4	no	—
60	C₂×Dic₉	C₂×D₄	1	8	yes	—
61	C₂×Dic₉	C₂×Q₈	1	16	yes	—
62	Dic₁₈	—	4	8	no	—
63	C₂×C₄⋊Q₈	—	9	4	no	—
64	C₂×C₄⋊Q₈	—	9	1	no	—
65	C₂×C₄⋊Q₈	D₉	1	1	yes	characteristic
66	#384	—	9	8	no	—
67	C₄³	—	9	1	no	—
68	C₂×C₃×Q₈	—	2	8	no	—
69	C₂×C₃×Q₈	—	2	4	no	—
70	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
71	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
72	C₂²×C₃×C₄	C₂²×D₃	1	2	yes	—
73	C₂²×Dic₃	—	3	4	no	—
74	C₂²×Dic₃	—	3	4	no	—
75	C₂×Dic₆	—	6	8	no	—
76	C₂×Dic₆	—	6	4	no	—
77	C₃×C₄⋊C₄	—	2	8	no	—
78	C₃×C₄²	#67	1	4	yes	—
79	C₄⋊Dic₃	—	6	8	no	—
80	Dic₃⋊C₄	—	6	32	no	—
81	C₄×Dic₃	—	3	8	no	—
82	C₄×Dic₃	—	3	16	no	—
83	C₂²×C₉	C₂²×D₄	1	2	yes	—
84	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
85	C₂²×C₉	C₂²×Q₈	1	4	yes	—
86	C₄×C₉	—	4	4	no	—
87	C₄×C₉	#127	1	4	yes	—
88	C₄×C₉	C₄⋊Q₈	1	8	yes	complemented
89	Dic₉	—	4	4	no	—
90	Dic₉	C₄⋊Q₈	1	16	yes	complemented
91	C₂²×Q₈	—	9	2	no	—
92	C₂²×Q₈	C₂×D₉	1	2	yes	—
93	C₄⋊Q₈	—	2	4	no	—
94	C₄⋊Q₈	—	18	16	no	—
95	C₄⋊Q₈	—	18	4	no	—
96	C₂×C₄⋊C₄	—	9	16	no	—
97	C₂×C₄⋊C₄	—	9	4	no	—
98	C₂×C₄⋊C₄	C₂×D₉	1	4	yes	—
99	C₂×C₄²	—	9	2	no	—
100	C₂×C₄²	—	9	4	no	—
101	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
102	C₂³×C₃	C₂³×D₃	1	1	yes	characteristic
103	C₃×Q₈	—	4	8	no	—
104	C₂×C₃×C₄	—	2	8	no	—
105	C₂×C₃×C₄	—	2	4	no	—
106	C₂×C₃×C₄	D₃×D₄	1	4	yes	—
107	C₂×C₃×C₄	D₃×Q₈	1	8	yes	—
108	C₂×C₃×C₄	C₂×#67	1	2	yes	—
109	C₂×C₃×C₄	C₂×#67	1	4	yes	—
110	C₂×Dic₃	—	3	8	no	—
111	C₂×Dic₃	—	3	16	no	—
112	C₂×Dic₃	—	6	8	no	—
113	C₂×Dic₃	—	6	4	no	—
114	Dic₆	—	12	8	no	—
115	C₂×C₉	C₂×#127	1	1	yes	characteristic
116	C₂×C₉	C₂×C₄⋊Q₈	1	4	yes	—
117	C₂×C₉	C₂×C₄⋊Q₈	1	2	yes	—
118	C₂×Q₈	—	2	8	no	—
119	C₂×Q₈	—	2	4	no	—
120	C₂×Q₈	—	18	8	no	—
121	C₂×Q₈	—	18	4	no	—
122	C₂²×C₄	—	9	4	no	—
123	C₂²×C₄	—	9	4	no	—
124	C₂²×C₄	C₂²×D₉	1	4	yes	—
125	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₉	1	2	yes	—
127	C₄⋊C₄	—	2	8	no	—
128	C₄⋊C₄	—	18	32	no	—
129	C₄⋊C₄	—	18	8	no	—
130	C₄²	—	9	8	no	—
131	C₄²	—	9	16	no	—
132	C₄²	#611	1	4	yes	—
133	C₂²×C₃	C₂×D₃×D₄	1	2	yes	—
134	C₂²×C₃	C₂×D₃×Q₈	1	4	yes	—
135	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
136	C₃×C₄	—	4	4	no	—
137	C₃×C₄	#940	1	4	yes	—
138	C₃×C₄	#944	1	8	yes	—
139	Dic₃	—	3	16	no	—
140	Dic₃	—	12	4	no	—
141	C₉	#2868	1	1	yes	characteristic, complemented, Sylow
142	C₂³	C₂³×D₉	1	1	yes	characteristic
143	Q₈	—	4	8	no	—
144	Q₈	—	36	8	no	—
145	C₂×C₄	—	2	8	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	9	8	no	—
148	C₂×C₄	—	9	16	no	—
149	C₂×C₄	—	18	8	no	—
150	C₂×C₄	—	18	4	no	—
151	C₂×C₄	D₄×D₉	1	4	yes	—
152	C₂×C₄	Q₈×D₉	1	8	yes	—
153	C₂×C₄	C₂×#611	1	2	yes	—
154	C₂×C₄	C₂×#611	1	4	yes	—
155	C₂×C₃	D₃×C₄⋊Q₈	1	4	yes	—
156	C₂×C₃	C₂×#940	1	1	yes	characteristic
157	C₂×C₃	C₂×#944	1	2	yes	—
158	C₂²	C₂×D₄×D₉	1	2	yes	—
159	C₂²	C₂×Q₈×D₉	1	4	yes	—
160	C₂²	C₂²×#611	1	1	yes	characteristic
161	C₄	—	4	4	no	—
162	C₄	—	9	16	no	—
163	C₄	—	36	4	no	—
164	C₄	#63502	1	4	yes	—
165	C₄	#63506	1	8	yes	complemented
166	C₃	#83253	1	1	yes	characteristic
167	C₂	D₉×C₄⋊Q₈	1	4	yes	—
168	C₂	C₂×#63502	1	1	yes	characteristic
169	C₂	C₂×#63506	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₉×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×#63506

C₂

yes

maximal

C₂×#63431

C₂

yes

maximal, characteristic

#10588655

C₂

yes

maximal

C₄²×Dic₉

C₂

yes

maximal, characteristic

#83253

—

maximal

C₂²×Q₈×C₉

C₂²

yes

—

C₂²×Dic₁₈

C₂²

yes

—

C₉×C₄⋊Q₈

—

C₂×C₉×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₉

C₂²

yes

characteristic

#63506

—

C₂×C₄⋊Dic₉

C₂²

yes

—

C₂×Dic₉⋊C₄

C₂²

yes

—

C₂×C₄×Dic₉

C₂²

yes

—

C₂×C₄×Dic₉

C₂²

yes

—

#63431

—

C₂×C₃×C₄⋊Q₈

D₃

yes

—

—

—

C₄²×Dic₃

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

—

C₂²×C₄×C₉

C₂³

yes

—

C₂²×C₄×C₉

C₂³

yes

characteristic

C₂²×C₄×C₉

C₂³

yes

—

C₂²×Dic₉

C₂³

yes

—

C₂²×Dic₉

C₂³

yes

—

C₂×Dic₁₈

—

C₂×Dic₁₈

—

C₉×C₄⋊C₄

—

C₄²×C₉

D₄

yes

—

C₄⋊Dic₉

—

Dic₉⋊C₄

—

C₄×Dic₉

D₄

yes

—

C₄×Dic₉

Q₈

yes

complemented

#2868

—

Sylow

C₂²×C₃×Q₈

C₂×D₃

yes

—

C₂²×Dic₆

—

C₃×C₄⋊Q₈

—

C₂×C₃×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₃×C₄²

C₂×D₃

yes

characteristic

#944

—

C₂×C₄⋊Dic₃

—

C₂×Dic₃⋊C₄

—

C₂×C₄×Dic₃

—

C₂×C₄×Dic₃

—

#869

—

C₂³×C₉

C₂⁴

yes

characteristic

Q₈×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×D₄

yes

—

C₂×C₄×C₉

C₂×D₄

yes

—

C₂×C₄×C₉

C₂×D₄

yes

—

C₂×C₄×C₉

C₂×Q₈

yes

—

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

C₂×D₄

yes

—

C₂×Dic₉

C₂×Q₈

yes

—

Dic₁₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₉

yes

characteristic

#384

—

C₄³

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

C₂²×D₃

yes

—

C₂²×C₃×C₄

C₂²×D₃

yes

characteristic

C₂²×C₃×C₄

C₂²×D₃

yes

—

C₂²×Dic₃

—

C₂²×Dic₃

—

C₂×Dic₆

—

C₂×Dic₆

—

C₃×C₄⋊C₄

—

C₃×C₄²

#67

yes

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂²×C₉

C₂²×D₄

yes

—

C₂²×C₉

C₂²×D₄

yes

characteristic

C₂²×C₉

C₂²×Q₈

yes

—

C₄×C₉

—

C₄×C₉

#127

yes

—

C₄×C₉

C₄⋊Q₈

yes

complemented

Dic₉

—

Dic₉

C₄⋊Q₈

yes

complemented

C₂²×Q₈

—

C₂²×Q₈

C₂×D₉

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₉

yes

—

C₂×C₄²

—

100

C₂×C₄²

—

101

C₂×C₄²

C₂×D₉

yes

characteristic

102

C₂³×C₃

C₂³×D₃

yes

characteristic

103

C₃×Q₈

—

104

C₂×C₃×C₄

—

105

C₂×C₃×C₄

—

106

C₂×C₃×C₄

D₃×D₄

yes

—

107

C₂×C₃×C₄

D₃×Q₈

yes

—

108

C₂×C₃×C₄

C₂×#67

yes

—

109

C₂×C₃×C₄

C₂×#67

yes

—

110

C₂×Dic₃

—

111

C₂×Dic₃

—

112

C₂×Dic₃

—

113

C₂×Dic₃

—

114

Dic₆

—

115

C₂×C₉

C₂×#127

yes

characteristic

116

C₂×C₉

C₂×C₄⋊Q₈

yes

—

117

C₂×C₉

C₂×C₄⋊Q₈

yes

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

C₂²×D₉

yes

—

125

C₂²×C₄

C₂²×D₉

yes

characteristic

126

C₂²×C₄

C₂²×D₉

yes

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄²

—

131

C₄²

—

132

C₄²

#611

yes

—

133

C₂²×C₃

C₂×D₃×D₄

yes

—

134

C₂²×C₃

C₂×D₃×Q₈

yes

—

135

C₂²×C₃

C₂²×#67

yes

characteristic

136

C₃×C₄

—

137

C₃×C₄

#940

yes

—

138

C₃×C₄

#944

yes

—

139

Dic₃

—

140

Dic₃

—

141

C₉

#2868

yes

characteristic, complemented, Sylow

142

C₂³

C₂³×D₉

yes

characteristic

143

Q₈

—

144

Q₈

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

D₄×D₉

yes

—

152

C₂×C₄

Q₈×D₉

yes

—

153

C₂×C₄

C₂×#611

yes

—

154

C₂×C₄

C₂×#611

yes

—

155

C₂×C₃

D₃×C₄⋊Q₈

yes

—

156

C₂×C₃

C₂×#940

yes

characteristic

157

C₂×C₃

C₂×#944

yes

—

158

C₂²

C₂×D₄×D₉

yes

—

159

C₂²

C₂×Q₈×D₉

yes

—

160

C₂²

C₂²×#611

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

#63502

yes

—

165

C₄

#63506

yes

complemented

166

C₃

#83253

yes

characteristic

167

C₂

D₉×C₄⋊Q₈

yes

—

168

C₂

C₂×#63502

yes

characteristic

169

C₂

C₂×#63506

yes

—

	subgroup	G/H
center	C₂³	C₂³×D₉
commutator, cosocle	C₂³×C₉	C₂⁴
Frattini, socle, prime radical	C₂³×C₃	C₂³×D₃
Fitting	C₂×C₉×C₄⋊Q₈	C₂
hypercenter	C₂×C₄⋊Q₈	D₉
hypocenter	C₉	#2868
LCS	G ▷ C₂³×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂³×D₉

commutator, cosocle

C₂³×C₉

C₂⁴

Frattini, socle, prime radical

C₂³×C₃

C₂³×D₃

Fitting

C₂×C₉×C₄⋊Q₈

C₂

hypercenter

C₂×C₄⋊Q₈

D₉

C₉

G ▷ C₂³×C₉ ▷ C₉

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
37. C₄×Dic₉	Q₈	semidirect
88. C₄×C₉	C₄⋊Q₈	semidirect
90. Dic₉	C₄⋊Q₈	semidirect
141. C₉	#2868	semidirect
165. C₄	#63506	semidirect

H ≅ G/N

type

37. C₄×Dic₉

Q₈

semidirect

88. C₄×C₉

C₄⋊Q₈

semidirect

90. Dic₉

C₄⋊Q₈

semidirect

141. C₉

#2868

semidirect

165. C₄

#63506

semidirect

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

subgroup	index
C₂×C₉×C₄⋊Q₈	2
C₂×#63506	2
C₂×#63431	2
#10588655	2
C₄²×Dic₉	2
#83253	3

subgroup

index

C₂×C₉×C₄⋊Q₈

C₄²×Dic₉

Subgroups — #11857076

Conjugacy classes of subgroups

Complements

Maximal subgroups