Group #11782667 — subgroups

Subgroups — #11782667

Order 1152.

660 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 165 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×#498	C₂	1	1	yes	maximal, characteristic, complemented
2	#10592787	C₂	1	2	yes	maximal, complemented
3	#10592784	C₂	1	2	yes	maximal, complemented
4	#10592394	C₂	1	1	yes	maximal, characteristic
5	#10589321	C₂	1	1	yes	maximal, characteristic, complemented
6	#71336	—	3	4	no	maximal
7	C₃²×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×#3755	—	2	4	no	—
9	C₂×#3745	—	2	2	no	—
10	#64083	C₂²	1	1	yes	characteristic
11	#64083	C₂²	1	1	yes	characteristic
12	C₃²×#106	—	2	2	no	—
13	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
14	#63636	—	2	4	no	—
15	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
16	C₃×#498	D₃	1	4	yes	complemented
17	#5088	—	3	8	no	—
18	#5085	—	3	8	no	—
19	#4695	—	3	4	no	—
20	#4455	—	3	4	no	—
21	C₂×C₃²×Q₈	—	2	2	no	—
22	C₂×C₄⋊(C₃⋊D₃)	—	2	1	no	—
23	C₂×C₄⋊(C₃⋊D₃)	—	2	1	no	—
24	C₂×C₄⋊(C₃⋊D₃)	—	2	2	no	—
25	C₂×C₄⋊(C₃⋊D₃)	—	2	2	no	—
26	C₂×C₃²×C₈	D₄	1	2	yes	—
27	C₃²×C₄⋊C₄	—	2	2	no	—
28	C₃²×C₄⋊C₄	—	2	2	no	—
29	C₃²×C₄²	C₂³	1	1	yes	characteristic
30	#3755	—	4	8	no	—
31	C₂×C₃⋊(C₃⋊C₈)	D₄	1	4	yes	—
32	#3745	—	4	4	no	—
33	#1715	—	9	1	no	Sylow
34	C₃×C₄⋊Q₈	C₂×D₃	1	8	yes	—
35	C₂×#215	—	6	16	no	—
36	C₂×D₂₄	—	6	8	no	—
37	#874	—	3	4	no	—
38	#874	—	3	4	no	—
39	C₃×#106	—	2	8	no	—
40	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
41	#809	—	6	16	no	—
42	C₄×C₃⋊C₈	—	3	8	no	—
43	C₂²×C₃⋊D₃	—	4	1	no	—
44	C₂²×C₃⋊D₃	—	4	1	no	—
45	C₃²×Q₈	—	4	4	no	—
46	C₂×C₃²×C₄	—	4	2	no	—
47	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
48	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
49	C₄⋊(C₃⋊D₃)	—	4	2	no	—
50	C₄⋊(C₃⋊D₃)	—	4	2	no	—
51	C₄⋊(C₃⋊D₃)	—	4	4	no	—
52	C₄⋊(C₃⋊D₃)	—	4	4	no	—
53	C₃²×C₈	SD₁₆	1	4	yes	complemented
54	C₃⋊(C₃⋊C₈)	—	2	4	no	—
55	#498	C₃⋊D₃	1	1	yes	characteristic, complemented
56	#493	—	9	1	no	—
57	#492	—	9	2	no	—
58	#486	—	9	2	no	—
59	#322	—	9	1	no	—
60	C₂×C₃×Q₈	—	2	8	no	—
61	C₂×D₁₂	—	6	8	no	—
62	C₂×D₁₂	—	6	8	no	—
63	C₂×D₁₂	—	6	4	no	—
64	C₂×D₁₂	—	6	4	no	—
65	C₂×C₃×C₈	#67	1	8	yes	—
66	C₃×C₄⋊C₄	—	2	8	no	—
67	C₃×C₄⋊C₄	—	2	8	no	—
68	C₃×C₄²	C₂²×D₃	1	4	yes	—
69	#215	—	12	32	no	—
70	C₂×C₃⋊C₈	—	3	16	no	—
71	D₂₄	—	12	16	no	—
72	C₂²×C₃²	#127	1	1	yes	characteristic
73	C₂×C₃⋊D₃	—	8	1	no	—
74	C₂×C₃⋊D₃	—	8	1	no	—
75	C₂×C₃⋊D₃	—	8	2	no	—
76	C₂×C₃⋊D₃	—	8	2	no	—
77	C₃²×C₄	—	8	2	no	—
78	C₃²×C₄	C₂×SD₁₆	1	2	yes	—
79	C₃²×C₄	AGL(1,ℤ/8)	1	4	yes	—
80	C₂×SD₁₆	—	18	4	no	—
81	C₂×D₈	—	18	2	no	—
82	C₄⋊Q₈	C₂×C₃⋊D₃	1	2	yes	—
83	#127	—	9	1	no	—
84	#127	—	9	1	no	—
85	#106	—	2	2	no	—
86	#102	—	18	4	no	—
87	C₄×C₈	—	9	2	no	—
88	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
89	C₂²×D₃	—	12	4	no	—
90	C₂²×D₃	—	12	4	no	—
91	C₃×Q₈	—	4	16	no	—
92	C₂×C₃×C₄	—	4	8	no	—
93	C₂×C₃×C₄	D₃×D₄	1	8	yes	—
94	C₂×C₃×C₄	C₂×#67	1	4	yes	—
95	D₁₂	—	12	16	no	—
96	D₁₂	—	12	16	no	—
97	D₁₂	—	12	8	no	—
98	D₁₂	—	12	8	no	—
99	C₃×C₈	#215	1	16	yes	complemented
100	C₃⋊C₈	—	6	16	no	—
101	C₂×C₃²	#492	1	1	yes	characteristic
102	C₂×C₃²	#496	1	2	yes	—
103	C₃⋊D₃	—	16	1	no	—
104	C₃⋊D₃	—	16	1	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×D₄	—	18	2	no	—
107	C₂×D₄	—	18	2	no	—
108	C₂×D₄	—	18	1	no	—
109	C₂×D₄	—	18	1	no	—
110	SD₁₆	—	36	8	no	—
111	D₈	—	36	4	no	—
112	C₂×C₈	—	9	4	no	—
113	C₂×C₈	#638	1	2	yes	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄²	C₂²×C₃⋊D₃	1	1	yes	characteristic
117	C₂²×C₃	#940	1	4	yes	—
118	C₂×D₃	—	24	8	no	—
119	C₂×D₃	—	24	8	no	—
120	C₂×D₃	—	24	4	no	—
121	C₂×D₃	—	24	4	no	—
122	C₃×C₄	—	8	8	no	—
123	C₃×C₄	#914	1	16	yes	—
124	C₃×C₄	C₂×#215	1	8	yes	—
125	C₃²	#1715	1	1	yes	characteristic, complemented, Sylow
126	C₂³	—	36	1	no	—
127	C₂³	—	36	1	no	—
128	Q₈	—	4	4	no	—
129	D₄	—	36	4	no	—
130	D₄	—	36	4	no	—
131	D₄	—	36	2	no	—
132	D₄	—	36	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	D₄×C₃⋊D₃	1	2	yes	—
135	C₂×C₄	C₂×#638	1	1	yes	characteristic
136	C₈	—	18	4	no	—
137	C₈	#3755	1	4	yes	complemented
138	C₂×C₃	#5085	1	4	yes	—
139	C₂×C₃	#5176	1	8	yes	—
140	D₃	—	48	4	no	—
141	D₃	—	48	4	no	—
142	C₂²	—	72	1	no	—
143	C₂²	—	72	1	no	—
144	C₂²	—	72	2	no	—
145	C₂²	—	72	2	no	—
146	C₂²	#64149	1	1	yes	characteristic
147	C₄	—	8	2	no	—
148	C₄	#64123	1	4	yes	—
149	C₄	C₂×#3755	1	2	yes	—
150	C₃	#71336	1	4	yes	complemented
151	C₂	—	144	1	no	—
152	C₂	—	144	1	no	—
153	C₂	#10592784	1	1	yes	characteristic
154	C₂	#10592875	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#498

C₂

yes

maximal, characteristic, complemented

#10592787

C₂

yes

maximal, complemented

#10592784

C₂

yes

maximal, complemented

#10592394

C₂

yes

maximal, characteristic

#10589321

C₂

yes

maximal, characteristic, complemented

#71336

—

maximal

C₃²×C₄⋊Q₈

C₂²

yes

—

C₂×#3755

—

C₂×#3745

—

#64083

C₂²

yes

characteristic

#64083

C₂²

yes

characteristic

C₃²×#106

—

C₃²×C₄×C₈

C₂²

yes

characteristic

#63636

—

C₄×C₃⋊(C₃⋊C₈)

C₂²

yes

—

C₃×#498

D₃

yes

—

—

—

—

C₂×C₃²×Q₈

—

C₂×C₄⋊(C₃⋊D₃)

—

C₂×C₄⋊(C₃⋊D₃)

—

C₂×C₄⋊(C₃⋊D₃)

—

C₂×C₄⋊(C₃⋊D₃)

—

C₂×C₃²×C₈

D₄

yes

—

C₃²×C₄⋊C₄

—

C₃²×C₄⋊C₄

—

C₃²×C₄²

C₂³

yes

characteristic

#3755

—

C₂×C₃⋊(C₃⋊C₈)

D₄

yes

—

#3745

—

#1715

—

Sylow

C₃×C₄⋊Q₈

C₂×D₃

yes

—

C₂×#215

—

C₂×D₂₄

—

#874

—

#874

—

C₃×#106

—

C₃×C₄×C₈

C₂×D₃

yes

—

#809

—

C₄×C₃⋊C₈

—

C₂²×C₃⋊D₃

—

C₂²×C₃⋊D₃

—

C₃²×Q₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂×D₄

yes

—

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₄⋊(C₃⋊D₃)

—

C₄⋊(C₃⋊D₃)

—

C₄⋊(C₃⋊D₃)

—

C₄⋊(C₃⋊D₃)

—

C₃²×C₈

SD₁₆

yes

complemented

C₃⋊(C₃⋊C₈)

—

#498

C₃⋊D₃

yes

characteristic, complemented

—

—

—

—

C₂×C₃×Q₈

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×C₃×C₈

#67

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂²×D₃

yes

—

#215

—

C₂×C₃⋊C₈

—

D₂₄

—

C₂²×C₃²

#127

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₃²×C₄

—

C₃²×C₄

C₂×SD₁₆

yes

—

C₃²×C₄

AGL(1,ℤ/8)

yes

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

C₂×C₃⋊D₃

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

C₂×C₃⋊D₃

yes

characteristic

C₂²×D₃

—

C₂²×D₃

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

D₃×D₄

yes

—

C₂×C₃×C₄

C₂×#67

yes

—

D₁₂

—

D₁₂

—

D₁₂

—

D₁₂

—

C₃×C₈

#215

yes

complemented

100

C₃⋊C₈

—

101

C₂×C₃²

#492

yes

characteristic

102

C₂×C₃²

#496

yes

—

103

C₃⋊D₃

—

104

C₃⋊D₃

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

SD₁₆

—

111

D₈

—

112

C₂×C₈

—

113

C₂×C₈

#638

yes

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄²

C₂²×C₃⋊D₃

yes

characteristic

117

C₂²×C₃

#940

yes

—

118

C₂×D₃

—

119

C₂×D₃

—

120

C₂×D₃

—

121

C₂×D₃

—

122

C₃×C₄

—

123

C₃×C₄

#914

yes

—

124

C₃×C₄

C₂×#215

yes

—

125

C₃²

#1715

yes

characteristic, complemented, Sylow

126

C₂³

—

127

C₂³

—

128

Q₈

—

129

D₄

—

130

D₄

—

131

D₄

—

132

D₄

—

133

C₂×C₄

—

134

C₂×C₄

D₄×C₃⋊D₃

yes

—

135

C₂×C₄

C₂×#638

yes

characteristic

136

C₈

—

137

C₈

#3755

yes

complemented

138

C₂×C₃

#5085

yes

—

139

C₂×C₃

#5176

yes

—

140

D₃

—

141

D₃

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

#64149

yes

characteristic

147

C₄

—

148

C₄

#64123

yes

—

149

C₄

C₂×#3755

yes

—

150

C₃

#71336

yes

complemented

151

C₂

—

144

—

152

C₂

—

144

—

153

C₂

#10592784

yes

characteristic

154

C₂

#10592875

yes

—

	subgroup	G/H
center	C₂²	#64149
commutator, cosocle	C₃²×C₄²	C₂³
Frattini	C₄²	C₂²×C₃⋊D₃
Fitting	C₃²×#498	C₂
socle	C₂²×C₃²	#127
prime radical	#10592394	C₂
hypercenter	#498	C₃⋊D₃
hypocenter	C₃²	#1715
LCS	G ▷ C₃²×C₄² ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#64149

commutator, cosocle

C₃²×C₄²

C₂³

Frattini

C₄²

C₂²×C₃⋊D₃

Fitting

C₃²×#498

C₂

C₂²×C₃²

C₂

C₃⋊D₃

C₃²

G ▷ C₃²×C₄² ▷ C₂²×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
1. C₃²×#498	C₂	semidirect
2. #10592787	C₂	semidirect
3. #10592784	C₂	semidirect
5. #10589321	C₂	semidirect
16. C₃×#498	D₃	semidirect
53. C₃²×C₈	SD₁₆	semidirect
55. #498	C₃⋊D₃	semidirect
99. C₃×C₈	#215	semidirect
125. C₃²	#1715	semidirect
137. C₈	#3755	semidirect
150. C₃	#71336	semidirect

H ≅ G/N

type

1. C₃²×#498

C₂

semidirect

2. #10592787

C₂

semidirect

3. #10592784

C₂

semidirect

5. #10589321

C₂

semidirect

16. C₃×#498

D₃

semidirect

53. C₃²×C₈

SD₁₆

semidirect

55. #498

C₃⋊D₃

semidirect

99. C₃×C₈

#215

semidirect

125. C₃²

#1715

semidirect

137. C₈

#3755

semidirect

150. C₃

#71336

semidirect

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

subgroup	index
C₃²×#498	2
#10592787	2
#10592784	2
#10592394	2
#10589321	2
#71336	3

subgroup

index

C₃²×#498

Subgroups — #11782667

Conjugacy classes of subgroups

Complements

Maximal subgroups