Group #11782496 — subgroups

Subgroups — #11782496

Order 1152.

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

normal subgroups only

This group has 570 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×#492	C₂	1	1	yes	maximal, characteristic
2	#10592792	C₂	1	2	yes	maximal, complemented
3	#10592769	C₂	1	2	yes	maximal
4	#10592379	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#71256	—	3	4	no	maximal
7	C₂×C₃²×SD₁₆	—	2	2	no	—
8	C₃²×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₃²×#127	C₂²	1	1	yes	characteristic
10	#64078	C₂²	1	2	yes	—
11	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
12	#63659	—	2	4	no	—
13	#63644	—	2	2	no	—
14	#63637	—	2	4	no	—
15	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
16	C₃×#492	D₃	1	4	yes	—
17	#5093	—	3	8	no	—
18	#5070	—	3	8	no	—
19	#4680	—	3	4	no	—
20	C₈×C₃⋊C₈	—	3	4	no	—
21	C₂×C₃²×Q₈	—	2	1	no	—
22	C₂×C₃²×D₄	—	2	2	no	—
23	C₂×C₃²×D₄	—	2	1	no	—
24	C₂×#634	—	2	2	no	—
25	C₃²×SD₁₆	—	4	4	no	—
26	C₂×C₃²×C₈	D₄	1	2	yes	—
27	C₃²×C₄⋊C₄	—	2	2	no	—
28	C₃²×C₄²	C₂³	1	1	yes	characteristic
29	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
30	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
31	C₂×C₃⋊(C₃⋊C₈)	D₄	1	4	yes	—
32	#1708	—	9	1	no	Sylow
33	C₂×C₃×SD₁₆	—	2	8	no	—
34	C₃×C₄⋊Q₈	C₂×D₃	1	4	yes	—
35	C₃×#127	C₂×D₃	1	4	yes	—
36	#869	—	3	8	no	—
37	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
38	#832	—	6	16	no	—
39	#817	—	6	8	no	—
40	#810	—	6	16	no	—
41	C₄×C₃⋊C₈	—	3	8	no	—
42	C₂³×C₃²	—	4	1	no	—
43	C₃²×Q₈	—	4	2	no	—
44	C₃²×D₄	—	4	2	no	—
45	C₃²×D₄	—	4	4	no	—
46	C₂×C₃²×C₄	—	4	1	no	—
47	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
48	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
49	C₂×C₃⋊Dic₃	—	4	2	no	—
50	#634	—	4	4	no	—
51	C₃²×C₈	SD₁₆	1	4	yes	—
52	C₃⋊(C₃⋊C₈)	—	2	4	no	—
53	#498	—	9	1	no	—
54	#492	C₃⋊D₃	1	1	yes	characteristic
55	#487	—	9	2	no	—
56	#486	—	9	2	no	—
57	C₈²	—	9	1	no	—
58	C₂×C₃×Q₈	—	2	4	no	—
59	C₂×C₃×D₄	—	2	8	no	—
60	C₂×C₃×D₄	—	2	4	no	—
61	C₂×Dic₆	—	6	8	no	—
62	C₃×SD₁₆	—	4	16	no	—
63	C₂×C₃×C₈	#67	1	8	yes	—
64	C₃×C₄⋊C₄	—	2	8	no	—
65	C₃×C₄²	C₂²×D₃	1	4	yes	—
66	C₄⋊Dic₃	—	6	8	no	—
67	C₄⋊Dic₃	—	6	8	no	—
68	C₂×C₃⋊C₈	—	3	16	no	—
69	C₂²×C₃²	—	8	1	no	—
70	C₂²×C₃²	—	8	2	no	—
71	C₂²×C₃²	#127	1	1	yes	characteristic
72	C₃²×C₄	—	8	1	no	—
73	C₃²×C₄	C₂×SD₁₆	1	2	yes	—
74	C₃²×C₄	#135	1	4	yes	—
75	C₃⋊Dic₃	—	8	2	no	—
76	C₂×SD₁₆	—	2	2	no	—
77	C₄⋊Q₈	—	9	2	no	—
78	C₄⋊Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
79	#127	C₂×C₃⋊D₃	1	1	yes	characteristic
80	#106	—	18	2	no	—
81	#103	—	18	4	no	—
82	#102	—	18	4	no	—
83	C₄×C₈	—	9	2	no	—
84	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
85	C₂³×C₃	—	4	4	no	—
86	C₃×Q₈	—	4	8	no	—
87	C₃×D₄	—	4	16	no	—
88	C₃×D₄	—	4	8	no	—
89	C₂×C₃×C₄	—	4	4	no	—
90	C₂×C₃×C₄	D₃×D₄	1	8	yes	—
91	C₂×C₃×C₄	C₂×#67	1	4	yes	—
92	C₂×Dic₃	—	12	8	no	—
93	Dic₆	—	12	16	no	—
94	C₃×C₈	#214	1	16	yes	—
95	C₃⋊C₈	—	6	16	no	—
96	C₂×C₃²	—	16	1	no	—
97	C₂×C₃²	#492	1	1	yes	characteristic
98	C₂×C₃²	#495	1	2	yes	—
99	C₂×Q₈	—	2	1	no	—
100	C₂×Q₈	—	18	2	no	—
101	C₂×D₄	—	2	2	no	—
102	C₂×D₄	—	2	1	no	—
103	SD₁₆	—	4	4	no	—
104	C₂×C₈	—	9	4	no	—
105	C₂×C₈	#638	1	2	yes	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	18	2	no	—
108	C₄⋊C₄	—	18	2	no	—
109	C₄²	C₂²×C₃⋊D₃	1	1	yes	characteristic
110	C₂²×C₃	—	8	8	no	—
111	C₂²×C₃	—	8	4	no	—
112	C₂²×C₃	#940	1	4	yes	—
113	C₃×C₄	—	8	4	no	—
114	C₃×C₄	#916	1	16	yes	—
115	C₃×C₄	C₂×#214	1	8	yes	—
116	Dic₃	—	24	8	no	—
117	C₃²	#1708	1	1	yes	characteristic, complemented, Sylow
118	C₂³	—	4	1	no	—
119	Q₈	—	4	2	no	—
120	Q₈	—	36	4	no	—
121	D₄	—	4	4	no	—
122	D₄	—	4	2	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	36	2	no	—
125	C₂×C₄	D₄×C₃⋊D₃	1	2	yes	—
126	C₂×C₄	C₂×#638	1	1	yes	characteristic
127	C₈	—	18	4	no	—
128	C₈	#3754	1	4	yes	—
129	C₂×C₃	—	16	4	no	—
130	C₂×C₃	#5076	1	4	yes	—
131	C₂×C₃	#5168	1	8	yes	—
132	C₂²	—	8	1	no	—
133	C₂²	—	8	2	no	—
134	C₂²	#64149	1	1	yes	characteristic
135	C₄	—	8	1	no	—
136	C₄	—	72	2	no	—
137	C₄	#64125	1	4	yes	—
138	C₄	C₂×#3754	1	2	yes	—
139	C₃	#71256	1	4	yes	complemented
140	C₂	—	16	1	no	—
141	C₂	#10592775	1	1	yes	characteristic
142	C₂	#10592867	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#492

C₂

yes

maximal, characteristic

#10592792

C₂

yes

maximal, complemented

#10592769

C₂

yes

maximal

#10592379

C₂

yes

maximal, characteristic, complemented

C₈×C₃⋊(C₃⋊C₈)

C₂

yes

maximal, characteristic, complemented

#71256

—

maximal

C₂×C₃²×SD₁₆

—

C₃²×C₄⋊Q₈

C₂²

yes

characteristic

C₃²×#127

C₂²

yes

characteristic

#64078

C₂²

yes

—

C₃²×C₄×C₈

C₂²

yes

—

—

—

C₄×C₃⋊(C₃⋊C₈)

C₂²

yes

—

C₃×#492

D₃

yes

—

#5093

—

#5070

—

#4680

—

C₈×C₃⋊C₈

—

C₂×C₃²×Q₈

—

C₂×C₃²×D₄

—

C₂×C₃²×D₄

—

C₂×#634

—

C₃²×SD₁₆

—

C₂×C₃²×C₈

D₄

yes

—

C₃²×C₄⋊C₄

—

C₃²×C₄²

C₂³

yes

characteristic

C₄⋊(C₃⋊Dic₃)

—

C₄⋊(C₃⋊Dic₃)

—

C₂×C₃⋊(C₃⋊C₈)

D₄

yes

—

#1708

—

Sylow

C₂×C₃×SD₁₆

—

C₃×C₄⋊Q₈

C₂×D₃

yes

—

C₃×#127

C₂×D₃

yes

—

#869

—

C₃×C₄×C₈

C₂×D₃

yes

—

#832

—

#817

—

#810

—

C₄×C₃⋊C₈

—

C₂³×C₃²

—

C₃²×Q₈

—

C₃²×D₄

—

C₃²×D₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂×D₄

yes

—

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃⋊Dic₃

—

#634

—

C₃²×C₈

SD₁₆

yes

—

C₃⋊(C₃⋊C₈)

—

#498

—

#492

C₃⋊D₃

yes

characteristic

#487

—

#486

—

C₈²

—

C₂×C₃×Q₈

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×Dic₆

—

C₃×SD₁₆

—

C₂×C₃×C₈

#67

yes

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂²×D₃

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

#127

yes

characteristic

C₃²×C₄

—

C₃²×C₄

C₂×SD₁₆

yes

—

C₃²×C₄

#135

yes

—

C₃⋊Dic₃

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×C₃⋊D₃

yes

characteristic

#127

C₂×C₃⋊D₃

yes

—

—

—

C₄×C₈

—

C₄×C₈

C₂×C₃⋊D₃

yes

characteristic

C₂³×C₃

—

C₃×Q₈

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

D₃×D₄

yes

—

C₂×C₃×C₄

C₂×#67

yes

—

C₂×Dic₃

—

Dic₆

—

C₃×C₈

#214

yes

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

#492

yes

characteristic

C₂×C₃²

#495

yes

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×D₄

—

102

C₂×D₄

—

103

SD₁₆

—

104

C₂×C₈

—

105

C₂×C₈

#638

yes

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

C₂²×C₃⋊D₃

yes

characteristic

110

C₂²×C₃

—

111

C₂²×C₃

—

112

C₂²×C₃

#940

yes

—

113

C₃×C₄

—

114

C₃×C₄

#916

yes

—

115

C₃×C₄

C₂×#214

yes

—

116

Dic₃

—

117

C₃²

#1708

yes

characteristic, complemented, Sylow

118

C₂³

—

119

Q₈

—

120

Q₈

—

121

D₄

—

122

D₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

D₄×C₃⋊D₃

yes

—

126

C₂×C₄

C₂×#638

yes

characteristic

127

C₈

—

128

C₈

#3754

yes

—

129

C₂×C₃

—

130

C₂×C₃

#5076

yes

—

131

C₂×C₃

#5168

yes

—

132

C₂²

—

133

C₂²

—

134

C₂²

#64149

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

#64125

yes

—

138

C₄

C₂×#3754

yes

—

139

C₃

#71256

yes

complemented

140

C₂

—

141

C₂

#10592775

yes

characteristic

142

C₂

#10592867

yes

—

	subgroup	G/H
center	C₂²	#64149
commutator, cosocle	C₃²×C₄²	C₂³
Frattini	C₄²	C₂²×C₃⋊D₃
Fitting	C₃²×#492	C₂
socle	C₂²×C₃²	#127
prime radical	C₃²×#127	C₂²
hypercenter	#492	C₃⋊D₃
hypocenter	C₃²	#1708
LCS	G ▷ C₃²×C₄² ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#64149

commutator, cosocle

C₃²×C₄²

C₂³

Frattini

C₄²

C₂²×C₃⋊D₃

Fitting

C₃²×#492

C₂

socle

C₂²×C₃²

#127

prime radical

C₃²×#127

C₂²

C₃⋊D₃

C₃²

G ▷ C₃²×C₄² ▷ C₂²×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
2. #10592792	C₂	semidirect
4. #10592379	C₂	semidirect
5. C₈×C₃⋊(C₃⋊C₈)	C₂	semidirect
117. C₃²	#1708	semidirect
139. C₃	#71256	semidirect

H ≅ G/N

type

2. #10592792

C₂

semidirect

4. #10592379

C₂

semidirect

5. C₈×C₃⋊(C₃⋊C₈)

C₂

semidirect

117. C₃²

#1708

semidirect

139. C₃

#71256

semidirect

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

subgroup	index
C₃²×#492	2
#10592792	2
#10592769	2
#10592379	2
C₈×C₃⋊(C₃⋊C₈)	2
#71256	3

subgroup

index

C₃²×#492

C₈×C₃⋊(C₃⋊C₈)

Subgroups — #11782496

Conjugacy classes of subgroups

Complements

Maximal subgroups