Group #11781800 — subgroups

Subgroups — #11781800

Order 1152.

456 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 149 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×#446	C₂	1	1	yes	maximal, characteristic
2	#10592719	C₂	1	2	yes	maximal
3	#10592398	C₂	1	1	yes	maximal, characteristic
4	#10589348	C₂	1	2	yes	maximal
5	#10589323	C₂	1	1	yes	maximal, characteristic
6	#70962	—	3	4	no	maximal
7	C₃²×C₄×Q₈	C₂²	1	1	yes	characteristic
8	C₂×#3746	—	2	1	no	—
9	C₂×#3746	—	2	1	no	—
10	#64078	C₂²	1	2	yes	—
11	C₃²×C₄⋊C₈	—	2	1	no	—
12	C₃²×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₃²×#97	—	2	1	no	—
14	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
15	#63662	—	2	2	no	—
16	#63635	—	2	2	no	—
17	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	2	yes	—
18	C₃×#446	D₃	1	4	yes	—
19	#5020	—	3	8	no	—
20	#4699	—	3	4	no	—
21	#4482	—	3	8	no	—
22	#4457	—	3	4	no	—
23	C₂×C₃²×Q₈	D₄	1	1	yes	characteristic
24	C₂×#634	—	2	2	no	—
25	C₂×C₃²×C₈	—	2	1	no	—
26	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
27	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
28	C₃²×C₄⋊C₄	—	2	1	no	—
29	C₃²×C₄⋊C₄	D₄	1	1	yes	characteristic
30	C₃²×C₄²	—	2	1	no	—
31	C₃²×C₄²	C₂³	1	1	yes	characteristic
32	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
33	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
34	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
35	#3746	—	4	2	no	—
36	#3746	—	4	2	no	—
37	#1690	—	9	1	no	Sylow
38	C₃×C₄×Q₈	C₂×D₃	1	4	yes	—
39	C₂×Dic₁₂	—	6	4	no	—
40	C₂×Dic₁₂	—	6	4	no	—
41	#869	—	3	8	no	—
42	C₃×C₄⋊C₈	—	2	4	no	—
43	C₃×C₄⋊C₈	C₂×D₃	1	4	yes	—
44	C₃×#97	—	2	4	no	—
45	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
46	#835	—	6	8	no	—
47	#808	—	6	8	no	—
48	C₄⋊(C₃⋊C₈)	—	3	8	no	—
49	C₃²×Q₈	—	2	2	no	—
50	C₂×C₃²×C₄	—	2	1	no	—
51	C₂×C₃²×C₄	—	2	1	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	—	4	2	no	—
56	#634	—	4	2	no	—
57	#634	—	4	2	no	—
58	C₃²×C₈	—	2	1	no	—
59	C₃²×C₈	—	4	1	no	—
60	C₃²×C₈	SD₁₆	1	2	yes	—
61	C₃⋊(C₃⋊C₈)	—	4	2	no	—
62	#494	—	9	1	no	—
63	#475	—	9	2	no	—
64	#446	C₃⋊D₃	1	1	yes	characteristic
65	#334	—	9	1	no	—
66	#332	—	9	2	no	—
67	C₂×C₃×Q₈	D₁₂	1	4	yes	—
68	C₂×Dic₆	—	6	8	no	—
69	C₂×C₃×C₈	—	2	4	no	—
70	C₂×C₃×C₈	#67	1	4	yes	—
71	C₂×C₃×C₈	#67	1	4	yes	—
72	C₃×C₄⋊C₄	—	2	4	no	—
73	C₃×C₄⋊C₄	D₁₂	1	4	yes	—
74	C₃×C₄²	—	2	4	no	—
75	C₃×C₄²	C₂²×D₃	1	4	yes	—
76	C₄⋊Dic₃	—	6	8	no	—
77	C₄⋊Dic₃	—	6	8	no	—
78	C₂×C₃⋊C₈	—	6	8	no	—
79	Dic₁₂	—	12	8	no	—
80	Dic₁₂	—	12	8	no	—
81	C₂²×C₃²	#121	1	1	yes	characteristic
82	C₃²×C₄	—	2	1	no	—
83	C₃²×C₄	—	4	1	no	—
84	C₃²×C₄	—	4	1	no	—
85	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
86	C₃²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
87	C₃²×C₄	#137	1	2	yes	—
88	C₃⋊Dic₃	—	8	2	no	—
89	C₂×Q₁₆	—	18	1	no	—
90	C₂×Q₁₆	—	18	1	no	—
91	C₄⋊Q₈	—	9	2	no	—
92	C₄×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
93	#106	—	18	2	no	—
94	C₄⋊C₈	—	2	1	no	—
95	C₄⋊C₈	—	9	2	no	—
96	C₄⋊C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
97	#103	—	18	2	no	—
98	#97	—	2	1	no	—
99	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
100	C₃×Q₈	—	2	8	no	—
101	C₂×C₃×C₄	—	2	4	no	—
102	C₂×C₃×C₄	—	2	4	no	—
103	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
104	C₂×C₃×C₄	#235	1	4	yes	—
105	C₂×C₃×C₄	C₂×#67	1	4	yes	—
106	C₂×Dic₃	—	12	8	no	—
107	Dic₆	—	12	8	no	—
108	Dic₆	—	12	8	no	—
109	C₃×C₈	—	2	4	no	—
110	C₃×C₈	—	4	4	no	—
111	C₃×C₈	#215	1	8	yes	—
112	C₃⋊C₈	—	12	8	no	—
113	C₂×C₃²	#460	1	1	yes	characteristic
114	C₂×C₃²	#470	1	1	yes	characteristic
115	C₂×C₃²	#473	1	1	yes	characteristic
116	C₂×Q₈	—	18	2	no	—
117	C₂×Q₈	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
118	Q₁₆	—	36	2	no	—
119	Q₁₆	—	36	2	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	18	2	no	—
122	C₂×C₈	#638	1	1	yes	characteristic
123	C₂×C₈	#638	1	1	yes	characteristic
124	C₄⋊C₄	—	2	1	no	—
125	C₄⋊C₄	—	18	2	no	—
126	C₄⋊C₄	—	18	2	no	—
127	C₄⋊C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
128	C₄²	—	2	1	no	—
129	C₄²	C₂²×C₃⋊D₃	1	1	yes	characteristic
130	C₂²×C₃	#930	1	4	yes	—
131	C₃×C₄	—	2	4	no	—
132	C₃×C₄	—	4	4	no	—
133	C₃×C₄	—	4	4	no	—
134	C₃×C₄	#909	1	8	yes	—
135	C₃×C₄	C₂×#215	1	4	yes	—
136	C₃×C₄	#949	1	4	yes	—
137	Dic₃	—	24	8	no	—
138	C₃²	#1690	1	1	yes	characteristic, complemented, Sylow
139	Q₈	—	2	2	no	—
140	Q₈	—	36	2	no	—
141	Q₈	—	36	2	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	36	2	no	—
145	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
146	C₂×C₄	#3828	1	1	yes	characteristic
147	C₂×C₄	C₂×#638	1	1	yes	characteristic
148	C₈	—	2	1	no	—
149	C₈	—	4	1	no	—
150	C₈	—	36	2	no	—
151	C₈	#3755	1	2	yes	—
152	C₂×C₃	#5027	1	4	yes	—
153	C₂×C₃	#5137	1	4	yes	—
154	C₂×C₃	#5143	1	4	yes	—
155	C₂²	#64139	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	4	1	no	—
158	C₄	—	4	1	no	—
159	C₄	—	72	2	no	—
160	C₄	#64118	1	2	yes	—
161	C₄	C₂×#3755	1	1	yes	characteristic
162	C₄	#64158	1	1	yes	characteristic
163	C₃	#70962	1	4	yes	complemented
164	C₂	#10592726	1	1	yes	characteristic
165	C₂	#10592836	1	1	yes	characteristic
166	C₂	#10592842	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#446

C₂

yes

maximal, characteristic

#10592719

C₂

yes

maximal

#10592398

C₂

yes

maximal, characteristic

#10589348

C₂

yes

maximal

#10589323

C₂

yes

maximal, characteristic

#70962

—

maximal

C₃²×C₄×Q₈

C₂²

yes

—

—

C₂²

yes

—

C₃²×C₄⋊C₈

—

C₃²×C₄⋊C₈

C₂²

yes

characteristic

C₃²×#97

—

C₃²×C₄×C₈

C₂²

yes

characteristic

#63662

—

#63635

—

C₄⋊(C₃⋊(C₃⋊C₈))

C₂²

yes

—

C₃×#446

D₃

yes

—

—

—

—

—

C₂×C₃²×Q₈

D₄

yes

characteristic

C₂×#634

—

C₂×C₃²×C₈

—

C₂×C₃²×C₈

D₄

yes

characteristic

C₂×C₃²×C₈

D₄

yes

characteristic

C₃²×C₄⋊C₄

—

C₃²×C₄⋊C₄

D₄

yes

characteristic

C₃²×C₄²

—

C₃²×C₄²

C₂³

yes

characteristic

C₄⋊(C₃⋊Dic₃)

—

C₄⋊(C₃⋊Dic₃)

—

C₂×C₃⋊(C₃⋊C₈)

—

—

—

—

C₃×C₄×Q₈

C₂×D₃

yes

—

C₂×Dic₁₂

—

C₂×Dic₁₂

—

#869

—

C₃×C₄⋊C₈

—

C₃×C₄⋊C₈

C₂×D₃

yes

—

C₃×#97

—

C₃×C₄×C₈

C₂×D₃

yes

—

#835

—

#808

—

C₄⋊(C₃⋊C₈)

—

C₃²×Q₈

—

C₂×C₃²×C₄

—

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

—

#634

—

#634

—

C₃²×C₈

—

C₃²×C₈

—

C₃²×C₈

SD₁₆

yes

—

C₃⋊(C₃⋊C₈)

—

#494

—

#475

—

#446

C₃⋊D₃

yes

characteristic

#334

—

#332

—

C₂×C₃×Q₈

D₁₂

yes

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

#67

yes

—

C₂×C₃×C₈

#67

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

D₁₂

yes

—

C₃×C₄²

—

C₃×C₄²

C₂²×D₃

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

Dic₁₂

—

Dic₁₂

—

C₂²×C₃²

#121

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₂×SD₁₆

yes

C₃²×C₄

yes

C₃²×C₄

yes

—

C₃⋊Dic₃

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

C₂×C₃⋊D₃

yes

characteristic

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₃⋊D₃

yes

characteristic

#103

—

#97

—

C₄×C₈

C₂×C₃⋊D₃

yes

characteristic

100

C₃×Q₈

—

101

C₂×C₃×C₄

—

102

C₂×C₃×C₄

—

103

C₂×C₃×C₄

C₂×D₁₂

yes

—

104

C₂×C₃×C₄

#235

yes

—

105

C₂×C₃×C₄

C₂×#67

yes

—

106

C₂×Dic₃

—

107

Dic₆

—

108

Dic₆

—

109

C₃×C₈

—

110

C₃×C₈

—

111

C₃×C₈

#215

yes

—

112

C₃⋊C₈

—

113

C₂×C₃²

#460

yes

characteristic

114

C₂×C₃²

#470

yes

characteristic

115

C₂×C₃²

#473

yes

characteristic

116

C₂×Q₈

—

117

C₂×Q₈

C₄⋊(C₃⋊D₃)

yes

characteristic

118

Q₁₆

—

119

Q₁₆

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

#638

yes

characteristic

123

C₂×C₈

#638

yes

characteristic

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

C₄⋊(C₃⋊D₃)

yes

characteristic

128

C₄²

—

129

C₄²

C₂²×C₃⋊D₃

yes

characteristic

130

C₂²×C₃

#930

yes

—

131

C₃×C₄

—

132

C₃×C₄

—

133

C₃×C₄

—

134

C₃×C₄

#909

yes

—

135

C₃×C₄

C₂×#215

yes

—

136

C₃×C₄

#949

yes

—

137

Dic₃

—

138

C₃²

#1690

yes

characteristic, complemented, Sylow

139

Q₈

—

140

Q₈

—

141

Q₈

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

C₂×C₄⋊(C₃⋊D₃)

yes

characteristic

146

C₂×C₄

#3828

yes

characteristic

147

C₂×C₄

C₂×#638

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

#3755

yes

—

152

C₂×C₃

#5027

yes

—

153

C₂×C₃

#5137

yes

—

154

C₂×C₃

#5143

yes

—

155

C₂²

#64139

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

#64118

yes

—

161

C₄

C₂×#3755

yes

characteristic

162

C₄

#64158

yes

characteristic

163

C₃

#70962

yes

complemented

164

C₂

#10592726

yes

characteristic

165

C₂

#10592836

yes

characteristic

166

C₂

#10592842

yes

characteristic

	subgroup	G/H
center	C₂²	#64139
commutator, cosocle	C₃²×C₄²	C₂³
Frattini	C₄²	C₂²×C₃⋊D₃
Fitting	C₃²×#446	C₂
socle, prime radical	C₂²×C₃²	#121
hypercenter	#446	C₃⋊D₃
hypocenter	C₃²	#1690
LCS	G ▷ C₃²×C₄² ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #446

subgroup

G/H

center

C₂²

#64139

commutator, cosocle

C₃²×C₄²

C₂³

Frattini

C₄²

C₂²×C₃⋊D₃

Fitting

C₃²×#446

C₂

C₂²×C₃²

C₃⋊D₃

C₃²

G ▷ C₃²×C₄² ▷ C₂²×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #446

N	H ≅ G/N	type
138. C₃²	#1690	semidirect
163. C₃	#70962	semidirect

H ≅ G/N

type

138. C₃²

#1690

semidirect

163. C₃

#70962

semidirect

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

subgroup	index
C₃²×#446	2
#10592719	2
#10592398	2
#10589348	2
#10589323	2
#70962	3

subgroup

index

C₃²×#446

Subgroups — #11781800

Conjugacy classes of subgroups

Complements

Maximal subgroups