Group #11776768 — subgroups

Subgroups — #11776768

Order 1152.

338 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 89 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	C₄×#3787	C₂	1	1	yes	maximal, characteristic
2	#10590331	C₂	1	2	yes	maximal
3	#10590041	C₂	1	4	yes	maximal, complemented
4	C₂×#3787	C₄	1	1	yes	characteristic
5	C₂×#3787	C₄	1	1	yes	characteristic
6	C₄²×C₃⋊D₃	C₂²	1	1	yes	characteristic
7	C₂×#3710	—	2	2	no	—
8	C₄×#622	C₄	1	2	yes	—
9	C₄×#622	C₂²	1	2	yes	—
10	C₃×C₄⋊(C₃⋊C₈)	—	2	2	no	—
11	#63572	C₂²	1	4	yes	—
12	D₃×C₄⋊C₈	—	6	2	no	—
13	C₂×C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₃⋊D₃	D₄	1	1	yes	characteristic
15	C₂×C₄×C₃⋊D₃	Q₈	1	1	yes	characteristic
16	C₂×#622	C₂×C₄	1	2	yes	—
17	C₂×#622	C₂×C₄	1	2	yes	—
18	#3787	—	2	2	no	—
19	#3787	C₈	1	4	yes	complemented
20	C₃²×C₄²	D₄	1	1	yes	characteristic
21	C₄×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₃⋊Dic₃	Q₈	1	1	yes	characteristic
23	C₄×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
24	C₂×C₃×C₃⋊C₈	—	2	4	no	—
25	#3710	—	4	4	no	—
26	C₂×#334	—	9	1	no	maximal, Sylow
27	C₂×D₃×C₈	—	6	4	no	—
28	C₄²×D₃	—	6	2	no	—
29	C₃×C₄⋊C₈	—	6	2	no	—
30	Dic₃⋊C₈	—	6	4	no	—
31	C₄⋊(C₃⋊C₈)	—	2	2	no	—
32	C₂²×C₃⋊D₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
36	C₂×C₃⋊Dic₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₃⋊Dic₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₃⋊Dic₃	C₂×Q₈	1	1	yes	characteristic
39	C₄×C₃⋊D₃	—	2	2	no	—
40	C₄×C₃⋊D₃	C₂×C₈	1	2	yes	—
41	C₄×C₃⋊D₃	M₁₆	1	2	yes	—
42	C₄×C₃⋊D₃	SD₁₆	1	4	yes	—
43	#622	—	2	2	no	—
44	#622	C₂×C₈	1	4	yes	complemented
45	C₃×C₃⋊C₈	—	4	4	no	—
46	C₂×C₄⋊C₈	—	9	2	no	—
47	C₂×C₄×C₈	—	9	1	no	—
48	#334	—	9	4	no	—
49	C₂×C₄×D₃	—	6	2	no	—
50	C₂×C₄×D₃	—	6	2	no	—
51	C₂×C₄×D₃	—	6	2	no	—
52	C₂×C₃×C₈	—	6	4	no	—
53	C₃×C₄²	—	2	2	no	—
54	C₄×Dic₃	—	6	2	no	—
55	C₄×Dic₃	—	6	2	no	—
56	C₄×Dic₃	—	6	2	no	—
57	C₂×C₃⋊C₈	—	2	4	no	—
58	D₃×C₈	—	12	8	no	—
59	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₃⋊D₃	C₄⋊C₈	1	2	yes	—
61	C₂×C₃⋊D₃	#106	1	2	yes	—
62	C₂×C₃⋊D₃	#108	1	2	yes	—
63	C₃²×C₄	—	2	1	no	—
64	C₃²×C₄	C₄⋊C₈	1	2	yes	—
65	C₃²×C₄	C₂×SD₁₆	1	2	yes	—
66	C₃⋊Dic₃	—	2	1	no	—
67	C₃⋊Dic₃	#106	1	2	yes	—
68	C₃⋊Dic₃	C₂²×C₈	1	1	yes	characteristic
69	C₃⋊Dic₃	C₂×M₁₆	1	1	yes	characteristic
70	C₂²×C₈	—	9	1	no	—
71	C₂²×C₈	—	9	1	no	—
72	C₂²×C₈	—	18	2	no	—
73	C₂×C₄²	—	9	1	no	—
74	C₄⋊C₈	—	6	2	no	—
75	C₄⋊C₈	—	9	4	no	—
76	C₄×C₈	—	9	2	no	—
77	C₄×C₈	—	9	2	no	—
78	C₂²×D₃	—	6	2	no	—
79	C₂×C₃×C₄	—	2	2	no	—
80	C₂×C₃×C₄	—	2	2	no	—
81	C₂×C₃×C₄	—	2	2	no	—
82	C₂×Dic₃	—	6	2	no	—
83	C₂×Dic₃	—	6	2	no	—
84	C₂×Dic₃	—	6	2	no	—
85	C₄×D₃	—	6	4	no	—
86	C₄×D₃	—	6	4	no	—
87	C₄×D₃	—	6	4	no	—
88	C₄×D₃	—	6	4	no	—
89	C₄×D₃	—	12	4	no	—
90	C₃×C₈	—	12	4	no	—
91	C₃⋊C₈	—	4	4	no	—
92	C₂×C₃²	C₂×C₄⋊C₈	1	1	yes	characteristic
93	C₂×C₃²	C₂×#106	1	1	yes	characteristic
94	C₂×C₃²	C₂×#108	1	1	yes	characteristic
95	C₃⋊D₃	#334	1	4	yes	complemented
96	C₂²×C₄	—	9	1	no	—
97	C₂²×C₄	—	9	1	no	—
98	C₂²×C₄	—	9	1	no	—
99	C₂×C₈	—	6	4	no	—
100	C₂×C₈	—	9	2	no	—
101	C₂×C₈	—	9	4	no	—
102	C₂×C₈	—	9	2	no	—
103	C₂×C₈	—	18	2	no	—
104	C₂×C₈	—	36	4	no	—
105	C₄²	—	9	1	no	—
106	C₄²	—	9	1	no	—
107	C₄²	—	9	1	no	—
108	C₄²	GO⁺(4,𝔽₂)	1	1	yes	characteristic
109	C₂²×C₃	—	2	2	no	—
110	C₂×D₃	—	6	4	no	—
111	C₂×D₃	—	6	4	no	—
112	C₂×D₃	—	6	4	no	—
113	C₃×C₄	—	2	2	no	—
114	C₃×C₄	—	2	2	no	—
115	C₃×C₄	—	2	4	no	—
116	C₃×C₄	—	4	2	no	—
117	Dic₃	—	6	4	no	—
118	Dic₃	—	6	2	no	—
119	Dic₃	—	6	2	no	—
120	Dic₃	—	12	2	no	—
121	C₃²	C₂×#334	1	1	yes	characteristic, complemented, Sylow
122	C₂³	—	9	1	no	—
123	C₂×C₄	—	9	1	no	—
124	C₂×C₄	—	9	1	no	—
125	C₂×C₄	—	9	4	no	—
126	C₂×C₄	—	9	2	no	—
127	C₂×C₄	—	9	2	no	—
128	C₂×C₄	—	9	1	no	—
129	C₂×C₄	—	18	2	no	—
130	C₂×C₄	#3773	1	1	yes	characteristic
131	C₂×C₄	#3773	1	1	yes	characteristic
132	C₂×C₄	C₂×GO⁺(4,𝔽₂)	1	1	yes	characteristic
133	C₈	—	9	4	no	—
134	C₈	—	12	4	no	—
135	C₈	—	18	2	no	—
136	C₂×C₃	—	2	2	no	—
137	C₂×C₃	—	2	2	no	—
138	C₂×C₃	—	2	2	no	—
139	D₃	—	6	8	no	—
140	C₂²	—	9	2	no	—
141	C₂²	—	9	2	no	—
142	C₂²	—	9	2	no	—
143	C₂²	C₂×#3773	1	1	yes	characteristic
144	C₄	—	2	1	no	—
145	C₄	—	9	1	no	—
146	C₄	—	9	1	no	—
147	C₄	—	9	2	no	—
148	C₄	—	18	1	no	—
149	C₄	#63732	1	2	yes	—
150	C₄	#64227	1	2	yes	—
151	C₃	—	2	2	no	—
152	C₂	—	9	4	no	—
153	C₂	C₂×#63732	1	1	yes	characteristic
154	C₂	#10593331	1	1	yes	characteristic
155	C₂	#10593387	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#3787

C₂

yes

maximal, characteristic

#10590331

C₂

yes

maximal

#10590041

C₂

yes

maximal, complemented

C₂×#3787

C₄

yes

characteristic

C₂×#3787

C₄

yes

characteristic

C₄²×C₃⋊D₃

C₂²

yes

characteristic

C₂×#3710

—

C₄×#622

C₄

yes

—

C₄×#622

C₂²

yes

—

C₃×C₄⋊(C₃⋊C₈)

—

#63572

C₂²

yes

—

D₃×C₄⋊C₈

—

C₂×C₄×C₃⋊D₃

C₂×C₄

yes

characteristic

C₂×C₄×C₃⋊D₃

D₄

yes

characteristic

C₂×C₄×C₃⋊D₃

Q₈

yes

characteristic

C₂×#622

C₂×C₄

yes

—

C₂×#622

C₂×C₄

yes

—

#3787

—

#3787

C₈

yes

complemented

C₃²×C₄²

D₄

yes

characteristic

C₄×C₃⋊Dic₃

C₂×C₄

yes

characteristic

C₄×C₃⋊Dic₃

Q₈

yes

characteristic

C₄×C₃⋊Dic₃

C₂³

yes

characteristic

C₂×C₃×C₃⋊C₈

—

#3710

—

C₂×#334

—

maximal, Sylow

C₂×D₃×C₈

—

C₄²×D₃

—

C₃×C₄⋊C₈

—

Dic₃⋊C₈

—

C₄⋊(C₃⋊C₈)

—

C₂²×C₃⋊D₃

C₄⋊C₄

yes

characteristic

C₂×C₃²×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃²×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃⋊Dic₃

C₄⋊C₄

yes

characteristic

C₂×C₃⋊Dic₃

C₂²×C₄

yes

characteristic

C₂×C₃⋊Dic₃

C₂×Q₈

yes

characteristic

C₄×C₃⋊D₃

—

C₄×C₃⋊D₃

C₂×C₈

yes

—

C₄×C₃⋊D₃

M₁₆

yes

—

C₄×C₃⋊D₃

SD₁₆

yes

—

#622

—

#622

C₂×C₈

yes

complemented

C₃×C₃⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

#334

—

C₂×C₄×D₃

—

C₂×C₄×D₃

—

C₂×C₄×D₃

—

C₂×C₃×C₈

—

C₃×C₄²

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

D₃×C₈

—

C₂²×C₃²

C₂×C₄⋊C₄

yes

characteristic

C₂×C₃⋊D₃

C₄⋊C₈

yes

—

C₂×C₃⋊D₃

#106

yes

—

C₂×C₃⋊D₃

#108

yes

—

C₃²×C₄

—

C₃²×C₄

C₄⋊C₈

yes

—

C₃²×C₄

C₂×SD₁₆

yes

—

C₃⋊Dic₃

—

C₃⋊Dic₃

#106

yes

—

C₃⋊Dic₃

C₂²×C₈

yes

characteristic

C₃⋊Dic₃

C₂×M₁₆

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

C₂×C₄⋊C₈

yes

C₂×C₃²

yes

C₂×C₃²

yes

C₃⋊D₃

yes

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₄²

GO⁺(4,𝔽₂)

yes

characteristic

109

C₂²×C₃

—

110

C₂×D₃

—

111

C₂×D₃

—

112

C₂×D₃

—

113

C₃×C₄

—

114

C₃×C₄

—

115

C₃×C₄

—

116

C₃×C₄

—

117

Dic₃

—

118

Dic₃

—

119

Dic₃

—

120

Dic₃

—

121

C₃²

C₂×#334

yes

characteristic, complemented, Sylow

122

C₂³

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

#3773

yes

characteristic

131

C₂×C₄

#3773

yes

characteristic

132

C₂×C₄

C₂×GO⁺(4,𝔽₂)

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₂×C₃

—

137

C₂×C₃

—

138

C₂×C₃

—

139

D₃

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×#3773

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

#63732

yes

—

150

C₄

#64227

yes

—

151

C₃

—

152

C₂

—

153

C₂

C₂×#63732

yes

characteristic

154

C₂

#10593331

yes

characteristic

155

C₂

#10593387

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×GO⁺(4,𝔽₂)
commutator	C₃⋊Dic₃	C₂²×C₈
Fitting	C₃²×C₄²	D₄
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂²×C₃⋊D₃	C₄⋊C₄
cosocle	C₄×C₃⋊Dic₃	C₂³
hypercenter	C₄²	GO⁺(4,𝔽₂)
hypocenter	C₃²	C₂×#334
LCS	G ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

subgroup

G/H

center, Frattini

C₂×C₄

C₂×GO⁺(4,𝔽₂)

commutator

C₃⋊Dic₃

C₂²×C₈

Fitting

C₃²×C₄²

D₄

socle

C₂²×C₃²

C₂×C₄⋊C₄

prime radical

C₂²×C₃⋊D₃

C₄⋊C₄

cosocle

C₄×C₃⋊Dic₃

C₂³

hypercenter

C₄²

GO⁺(4,𝔽₂)

hypocenter

C₃²

C₂×#334

LCS

G ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂×C₄ ◁ C₄²

N	H ≅ G/N	type
3. #10590041	C₂	semidirect
19. #3787	C₈	semidirect
44. #622	C₂×C₈	semidirect
95. C₃⋊D₃	#334	semidirect
121. C₃²	C₂×#334	semidirect

H ≅ G/N

type

3. #10590041

C₂

semidirect

19. #3787

C₈

semidirect

44. #622

C₂×C₈

semidirect

95. C₃⋊D₃

#334

semidirect

121. C₃²

C₂×#334

semidirect

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

subgroup	index
C₄×#3787	2
#10590331	2
#10590041	2
C₂×#334	9

subgroup

index

Subgroups — #11776768

Conjugacy classes of subgroups

Complements

Maximal subgroups