Group #11798928 — subgroups

Subgroups — #11798928

Order 1152.

462 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 139 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	C₂×#63656	C₂	1	2	yes	maximal
3	C₄×#3746	C₂	1	1	yes	maximal, characteristic
4	#10589430	C₂	1	2	yes	maximal
5	C₄⋊(C₃⋊(C₃⋊C₁₆))	C₂	1	1	yes	maximal, characteristic
6	#71316	—	3	4	no	maximal
7	C₂×C₃²×Q₁₆	—	2	1	no	—
8	C₂×C₃²×Q₁₆	C₂²	1	2	yes	—
9	C₃²×C₄⋊Q₈	—	2	1	no	—
10	C₂×#3746	C₂²	1	1	yes	characteristic
11	C₄×#634	—	2	1	no	—
12	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
13	#63656	—	2	4	no	—
14	#63645	C₂²	1	1	yes	characteristic
15	#63643	—	2	1	no	—
16	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
17	C₃×#494	D₃	1	4	yes	—
18	C₂×#829	—	3	8	no	—
19	C₄×Dic₁₂	—	3	4	no	—
20	#4564	—	3	8	no	—
21	C₄⋊(C₃⋊C₁₆)	—	3	4	no	—
22	C₂×C₃²×Q₈	—	2	2	no	—
23	C₂×#634	—	2	1	no	—
24	C₃²×Q₁₆	—	2	2	no	—
25	C₃²×Q₁₆	—	2	2	no	—
26	C₃²×Q₁₆	—	4	2	no	—
27	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
28	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
29	C₃²×C₄⋊C₄	—	4	1	no	—
30	C₃²×C₄⋊C₄	—	4	1	no	—
31	C₃²×C₄²	D₄	1	1	yes	characteristic
32	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
33	#3750	—	4	1	no	—
34	C₄×C₃⋊Dic₃	—	4	1	no	—
35	#3746	—	2	1	no	—
36	#3746	D₄	1	2	yes	—
37	C₃⋊(C₃⋊C₁₆)	—	2	2	no	—
38	#2209	—	9	1	no	Sylow
39	C₂×C₃×Q₁₆	—	2	4	no	—
40	C₂×C₃×Q₁₆	C₂×D₃	1	8	yes	—
41	C₃×C₄⋊Q₈	—	2	4	no	—
42	C₂×Dic₁₂	—	3	4	no	—
43	C₄×Dic₆	—	6	4	no	—
44	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
45	#829	—	6	16	no	—
46	#818	—	3	4	no	—
47	#816	—	6	4	no	—
48	C₂×C₃⋊C₁₆	—	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	D₈	1	1	yes	characteristic
53	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
54	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
55	C₂×C₃⋊Dic₃	—	4	1	no	—
56	C₂×C₃⋊Dic₃	—	4	1	no	—
57	#634	—	2	2	no	—
58	#634	—	4	1	no	—
59	C₃²×C₈	—	2	1	no	—
60	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
61	C₃²×C₈	#41	1	1	yes	characteristic
62	C₂×Q₃₂	—	9	2	no	—
63	#494	C₃⋊D₃	1	1	yes	characteristic
64	C₄×Q₁₆	—	9	1	no	—
65	C₄⋊C₁₆	—	9	1	no	—
66	#358	—	9	2	no	—
67	C₂×C₃×Q₈	—	2	8	no	—
68	C₂×Dic₆	—	6	4	no	—
69	C₃×Q₁₆	—	2	8	no	—
70	C₃×Q₁₆	—	2	8	no	—
71	C₃×Q₁₆	—	4	8	no	—
72	C₂×C₃×C₈	#67	1	4	yes	—
73	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
74	C₃×C₄⋊C₄	—	4	4	no	—
75	C₃×C₄⋊C₄	—	4	4	no	—
76	C₃×C₄²	#67	1	4	yes	—
77	C₄⋊Dic₃	—	6	4	no	—
78	Dic₃⋊C₄	—	12	4	no	—
79	C₄×Dic₃	—	12	4	no	—
80	Dic₁₂	—	3	8	no	—
81	Dic₁₂	—	6	4	no	—
82	C₃⋊C₁₆	—	6	8	no	—
83	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
84	C₃²×C₄	—	2	1	no	—
85	C₃²×C₄	—	8	2	no	—
86	C₃²×C₄	Q₃₂	1	2	yes	complemented
87	C₃²×C₄	#121	1	1	yes	characteristic
88	C₃²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₃⋊Dic₃	—	4	2	no	—
90	C₃⋊Dic₃	—	8	1	no	—
91	C₂×Q₁₆	—	2	1	no	—
92	C₂×Q₁₆	—	9	1	no	—
93	C₂×Q₁₆	C₂×C₃⋊D₃	1	2	yes	—
94	C₄⋊Q₈	—	2	1	no	—
95	C₄×Q₈	—	18	1	no	—
96	Q₃₂	—	18	4	no	—
97	C₂×C₁₆	—	9	2	no	—
98	#107	—	9	1	no	—
99	#103	—	18	1	no	—
100	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
101	C₃×Q₈	—	4	8	no	—
102	C₃×Q₈	—	4	8	no	—
103	C₂×C₃×C₄	—	4	8	no	—
104	C₂×C₃×C₄	#213	1	4	yes	—
105	C₂×C₃×C₄	#213	1	4	yes	—
106	C₂×C₃×C₄	C₂×#67	1	4	yes	—
107	C₂×Dic₃	—	12	4	no	—
108	C₂×Dic₃	—	12	4	no	—
109	Dic₆	—	6	8	no	—
110	Dic₆	—	12	4	no	—
111	C₃×C₈	—	2	4	no	—
112	C₃×C₈	D₃×D₄	1	4	yes	—
113	C₃×C₈	#237	1	4	yes	—
114	C₂×C₃²	#459	1	1	yes	characteristic
115	C₂×C₃²	C₂×Q₃₂	1	1	yes	characteristic
116	C₂×C₃²	#510	1	1	yes	characteristic
117	C₂×Q₈	—	2	2	no	—
118	C₂×Q₈	—	18	1	no	—
119	Q₁₆	—	2	2	no	—
120	Q₁₆	—	2	2	no	—
121	Q₁₆	—	4	2	no	—
122	Q₁₆	—	9	2	no	—
123	Q₁₆	—	18	1	no	—
124	C₂×C₈	#638	1	1	yes	characteristic
125	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
126	C₄⋊C₄	—	4	1	no	—
127	C₄⋊C₄	—	4	1	no	—
128	C₄⋊C₄	—	18	1	no	—
129	C₄⋊C₄	—	36	1	no	—
130	C₄²	—	36	1	no	—
131	C₄²	#638	1	1	yes	characteristic
132	C₁₆	—	18	2	no	—
133	C₂²×C₃	C₂×#213	1	4	yes	—
134	C₃×C₄	—	2	4	no	—
135	C₃×C₄	—	8	8	no	—
136	C₃×C₄	#829	1	8	yes	complemented
137	C₃×C₄	#932	1	4	yes	—
138	C₃×C₄	#938	1	4	yes	—
139	Dic₃	—	12	8	no	—
140	Dic₃	—	24	4	no	—
141	C₃²	#2209	1	1	yes	characteristic, complemented, Sylow
142	Q₈	—	4	2	no	—
143	Q₈	—	4	2	no	—
144	Q₈	—	18	2	no	—
145	Q₈	—	36	1	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	36	1	no	—
148	C₂×C₄	—	36	1	no	—
149	C₂×C₄	#3753	1	1	yes	characteristic
150	C₂×C₄	#3753	1	1	yes	characteristic
151	C₂×C₄	C₂×#638	1	1	yes	characteristic
152	C₈	—	2	1	no	—
153	C₈	D₄×C₃⋊D₃	1	1	yes	characteristic
154	C₈	#3830	1	1	yes	characteristic
155	C₂×C₃	#5072	1	4	yes	—
156	C₂×C₃	#5179	1	4	yes	—
157	C₂×C₃	C₂×#829	1	4	yes	—
158	C₂²	C₂×#3753	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	8	2	no	—
161	C₄	—	36	2	no	—
162	C₄	—	72	1	no	—
163	C₄	#63656	1	2	yes	complemented
164	C₄	#64141	1	1	yes	characteristic
165	C₄	#64147	1	1	yes	characteristic
166	C₃	#71316	1	4	yes	complemented
167	C₂	#10592771	1	1	yes	characteristic
168	C₂	#10592878	1	1	yes	characteristic
169	C₂	C₂×#63656	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#494

C₂

yes

maximal, characteristic

C₂×#63656

C₂

yes

maximal

C₄×#3746

C₂

yes

maximal, characteristic

#10589430

C₂

yes

maximal

C₄⋊(C₃⋊(C₃⋊C₁₆))

C₂

yes

maximal, characteristic

#71316

—

maximal

C₂×C₃²×Q₁₆

—

C₂×C₃²×Q₁₆

C₂²

yes

—

C₃²×C₄⋊Q₈

—

C₂×#3746

C₂²

yes

characteristic

C₄×#634

—

C₃²×C₄×C₈

C₂²

yes

characteristic

#63656

—

#63645

C₂²

yes

characteristic

#63643

—

C₂×C₃⋊(C₃⋊C₁₆)

C₂²

yes

—

C₃×#494

D₃

yes

—

C₂×#829

—

C₄×Dic₁₂

—

#4564

—

C₄⋊(C₃⋊C₁₆)

—

C₂×C₃²×Q₈

—

C₂×#634

—

C₃²×Q₁₆

—

C₃²×Q₁₆

—

C₃²×Q₁₆

—

C₂×C₃²×C₈

D₄

yes

characteristic

C₂×C₃²×C₈

C₂³

yes

characteristic

C₃²×C₄⋊C₄

—

C₃²×C₄⋊C₄

—

C₃²×C₄²

D₄

yes

characteristic

C₄⋊(C₃⋊Dic₃)

—

#3750

—

C₄×C₃⋊Dic₃

—

#3746

—

#3746

D₄

yes

—

C₃⋊(C₃⋊C₁₆)

—

#2209

—

Sylow

C₂×C₃×Q₁₆

—

C₂×C₃×Q₁₆

C₂×D₃

yes

—

C₃×C₄⋊Q₈

—

C₂×Dic₁₂

—

C₄×Dic₆

—

C₃×C₄×C₈

C₂×D₃

yes

—

#829

—

#818

—

#816

—

C₂×C₃⋊C₁₆

—

C₃²×Q₈

—

C₃²×Q₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

D₈

yes

characteristic

C₂×C₃²×C₄

D₈

yes

characteristic

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

#634

—

#634

—

C₃²×C₈

—

C₃²×C₈

C₂×D₄

yes

characteristic

C₃²×C₈

#41

yes

characteristic

C₂×Q₃₂

—

#494

C₃⋊D₃

yes

characteristic

C₄×Q₁₆

—

C₄⋊C₁₆

—

#358

—

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₃×Q₁₆

—

C₃×Q₁₆

—

C₃×Q₁₆

—

C₂×C₃×C₈

#67

yes

—

C₂×C₃×C₈

C₂²×D₃

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

#67

yes

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

Dic₁₂

—

Dic₁₂

—

C₃⋊C₁₆

—

C₂²×C₃²

C₂×D₈

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

Q₃₂

yes

C₃²×C₄

yes

C₃²×C₄

yes

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×C₃⋊D₃

yes

—

C₄⋊Q₈

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

#107

—

#103

—

100

C₄×C₈

C₂×C₃⋊D₃

yes

characteristic

101

C₃×Q₈

—

102

C₃×Q₈

—

103

C₂×C₃×C₄

—

104

C₂×C₃×C₄

#213

yes

—

105

C₂×C₃×C₄

#213

yes

—

106

C₂×C₃×C₄

C₂×#67

yes

—

107

C₂×Dic₃

—

108

C₂×Dic₃

—

109

Dic₆

—

110

Dic₆

—

111

C₃×C₈

—

112

C₃×C₈

D₃×D₄

yes

—

113

C₃×C₈

#237

yes

—

114

C₂×C₃²

#459

yes

characteristic

115

C₂×C₃²

C₂×Q₃₂

yes

characteristic

116

C₂×C₃²

#510

yes

characteristic

117

C₂×Q₈

—

118

C₂×Q₈

—

119

Q₁₆

—

120

Q₁₆

—

121

Q₁₆

—

122

Q₁₆

—

123

Q₁₆

—

124

C₂×C₈

#638

yes

characteristic

125

C₂×C₈

C₂²×C₃⋊D₃

yes

characteristic

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄²

—

131

C₄²

#638

yes

characteristic

132

C₁₆

—

133

C₂²×C₃

C₂×#213

yes

—

134

C₃×C₄

—

135

C₃×C₄

—

136

C₃×C₄

#829

yes

complemented

137

C₃×C₄

#932

yes

—

138

C₃×C₄

#938

yes

—

139

Dic₃

—

140

Dic₃

—

141

C₃²

#2209

yes

characteristic, complemented, Sylow

142

Q₈

—

143

Q₈

—

144

Q₈

—

145

Q₈

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

#3753

yes

characteristic

150

C₂×C₄

#3753

yes

characteristic

151

C₂×C₄

C₂×#638

yes

characteristic

152

C₈

—

153

C₈

D₄×C₃⋊D₃

yes

characteristic

154

C₈

#3830

yes

characteristic

155

C₂×C₃

#5072

yes

—

156

C₂×C₃

#5179

yes

—

157

C₂×C₃

C₂×#829

yes

—

158

C₂²

C₂×#3753

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#63656

yes

complemented

164

C₄

#64141

yes

characteristic

165

C₄

#64147

yes

characteristic

166

C₃

#71316

yes

complemented

167

C₂

#10592771

yes

characteristic

168

C₂

#10592878

yes

characteristic

169

C₂

C₂×#63656

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#3753
commutator, cosocle	C₂×C₃²×C₈	C₂³
Frattini	C₂×C₈	C₂²×C₃⋊D₃
Fitting	C₃²×#494	C₂
socle, prime radical	C₂²×C₃²	C₂×D₈
hypercenter	#494	C₃⋊D₃
hypocenter	C₃²	#2209
LCS	G ▷ C₂×C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #494

subgroup

G/H

center

C₂²

C₂×#3753

commutator, cosocle

C₂×C₃²×C₈

C₂³

Frattini

C₂×C₈

C₂²×C₃⋊D₃

Fitting

C₃²×#494

C₂

socle, prime radical

C₂²×C₃²

C₂×D₈

C₃⋊D₃

C₃²

G ▷ C₂×C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #494

N	H ≅ G/N	type
86. C₃²×C₄	Q₃₂	semidirect
136. C₃×C₄	#829	semidirect
141. C₃²	#2209	semidirect
163. C₄	#63656	semidirect
166. C₃	#71316	semidirect

H ≅ G/N

type

86. C₃²×C₄

Q₃₂

semidirect

136. C₃×C₄

#829

semidirect

141. C₃²

#2209

semidirect

163. C₄

#63656

semidirect

166. C₃

#71316

semidirect

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

subgroup	index
C₃²×#494	2
C₂×#63656	2
C₄×#3746	2
#10589430	2
C₄⋊(C₃⋊(C₃⋊C₁₆))	2
#71316	3

subgroup

index

C₃²×#494

C₄⋊(C₃⋊(C₃⋊C₁₆))

Subgroups — #11798928

Conjugacy classes of subgroups

Complements

Maximal subgroups