Group #69691 — subgroups

Subgroups — #69691

Order 384.

370 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₂²×#216	C₂	1	2	yes	maximal
2	C₂×#869	C₂	1	1	yes	maximal, characteristic
3	C₃×#391	C₂	1	1	yes	maximal, characteristic
4	C₂×#810	C₂	1	2	yes	maximal
5	#4524	C₂	1	1	yes	maximal, characteristic
6	#2016	—	3	1	no	maximal, Sylow
7	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
8	C₂²×Dic₆	C₂²	1	2	yes	—
9	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
11	C₂×#216	—	2	4	no	—
12	C₂×#216	—	2	4	no	—
13	C₂×#216	—	2	4	no	—
14	C₂×C₄⋊Dic₃	—	2	2	no	—
15	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
16	#869	—	2	4	no	—
17	C₃×#95	—	2	2	no	—
18	#810	—	2	4	no	—
19	C₂²×Q₁₆	—	3	2	no	—
20	C₂×C₄⋊Q₈	—	3	1	no	—
21	C₂×#103	—	3	2	no	—
22	#391	D₃	1	1	yes	characteristic
23	#336	—	3	1	no	—
24	C₂×C₃×Q₈	—	2	1	no	—
25	C₂×C₃×Q₈	—	2	2	no	—
26	C₂×C₃×Q₈	—	2	2	no	—
27	C₂×C₃×Q₈	D₄	1	2	yes	—
28	C₂²×C₃×C₄	—	2	1	no	—
29	C₂²×C₃×C₄	—	2	1	no	—
30	C₂²×C₃×C₄	D₄	1	2	yes	—
31	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
32	C₂²×Dic₃	—	2	2	no	—
33	C₂×Dic₆	—	2	2	no	—
34	C₂×Dic₆	—	2	2	no	—
35	C₂×Dic₆	—	2	2	no	—
36	C₂×Dic₆	—	2	2	no	—
37	C₂×Dic₆	—	2	2	no	—
38	C₂×Dic₆	D₄	1	4	yes	—
39	C₃×C₄⋊C₄	—	2	2	no	—
40	C₃×C₄²	—	2	2	no	—
41	#216	—	2	16	no	—
42	C₄⋊Dic₃	—	2	8	no	—
43	C₂×C₃⋊C₈	—	2	2	no	—
44	C₂×C₃⋊C₈	—	2	2	no	—
45	C₂×C₃⋊C₈	D₄	1	4	yes	—
46	C₂²×Q₈	—	3	2	no	—
47	C₂²×Q₈	C₂×D₃	1	1	yes	characteristic
48	C₂×Q₁₆	—	6	4	no	—
49	C₂×Q₁₆	—	6	4	no	—
50	C₂×Q₁₆	—	6	4	no	—
51	C₂²×C₈	—	3	2	no	—
52	C₄⋊Q₈	—	6	4	no	—
53	C₂×C₄⋊C₄	—	6	2	no	—
54	C₂×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
55	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
56	#103	—	6	4	no	—
57	#95	—	2	2	no	—
58	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
59	C₃×Q₈	—	2	4	no	—
60	C₃×Q₈	—	4	2	no	—
61	C₂×C₃×C₄	—	2	2	no	—
62	C₂×C₃×C₄	—	2	2	no	—
63	C₂×C₃×C₄	—	2	2	no	—
64	C₂×C₃×C₄	—	4	1	no	—
65	C₂×C₃×C₄	—	4	1	no	—
66	C₂×C₃×C₄	—	4	1	no	—
67	C₂×C₃×C₄	—	4	2	no	—
68	C₂×C₃×C₄	Q₁₆	1	4	yes	complemented
69	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
71	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
72	C₂×C₃×C₄	#41	1	1	yes	characteristic
73	C₂×Dic₃	—	2	4	no	—
74	C₂×Dic₃	—	4	2	no	—
75	C₂×Dic₃	—	4	2	no	—
76	Dic₆	—	2	4	no	—
77	Dic₆	—	2	4	no	—
78	Dic₆	—	4	2	no	—
79	Dic₆	—	4	2	no	—
80	C₃⋊C₈	—	2	4	no	—
81	C₂×Q₈	—	2	1	no	—
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	—	3	4	no	—
85	C₂×Q₈	—	6	2	no	—
86	C₂×Q₈	—	6	2	no	—
87	C₂×Q₈	—	6	2	no	—
88	C₂×Q₈	—	6	2	no	—
89	C₂×Q₈	—	6	2	no	—
90	C₂×Q₈	D₁₂	1	2	yes	—
91	C₂²×C₄	—	2	1	no	—
92	C₂²×C₄	—	2	1	no	—
93	C₂²×C₄	—	6	2	no	—
94	C₂²×C₄	#67	1	2	yes	—
95	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
96	Q₁₆	—	6	16	no	—
97	C₂×C₈	—	3	4	no	—
98	C₂×C₈	—	6	2	no	—
99	C₂×C₈	—	6	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	6	8	no	—
102	C₄²	—	2	2	no	—
103	C₂²×C₃	#120	1	1	yes	characteristic
104	C₂²×C₃	#121	1	1	yes	characteristic
105	C₂²×C₃	#127	1	1	yes	characteristic
106	C₂²×C₃	C₂×Q₁₆	1	2	yes	—
107	C₂²×C₃	#137	1	2	yes	—
108	C₃×C₄	—	2	4	no	—
109	C₃×C₄	—	4	2	no	—
110	C₃×C₄	—	8	1	no	—
111	C₃×C₄	#120	1	2	yes	—
112	C₃×C₄	#121	1	2	yes	—
113	Dic₃	—	4	4	no	—
114	C₂³	C₂×#67	1	1	yes	characteristic
115	Q₈	—	2	4	no	—
116	Q₈	—	4	2	no	—
117	Q₈	—	6	4	no	—
118	Q₈	—	6	4	no	—
119	Q₈	—	12	2	no	—
120	Q₈	—	12	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	6	4	no	—
129	C₂×C₄	—	12	2	no	—
130	C₂×C₄	—	12	2	no	—
131	C₂×C₄	#216	1	4	yes	complemented
132	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
133	C₂×C₄	#235	1	1	yes	characteristic
134	C₂×C₄	D₃×D₄	1	2	yes	—
135	C₂×C₄	D₃×D₄	1	2	yes	—
136	C₈	—	6	4	no	—
137	C₂×C₃	#392	1	1	yes	characteristic
138	C₂×C₃	#451	1	2	yes	—
139	C₂×C₃	#462	1	2	yes	—
140	C₂×C₃	#494	1	1	yes	characteristic
141	C₂×C₃	#497	1	1	yes	characteristic
142	C₂²	#930	1	1	yes	characteristic
143	C₂²	#937	1	1	yes	characteristic
144	C₂²	#940	1	1	yes	characteristic
145	C₂²	C₂×#216	1	2	yes	—
146	C₂²	#951	1	2	yes	—
147	C₄	—	2	4	no	—
148	C₄	—	4	2	no	—
149	C₄	—	8	1	no	—
150	C₄	—	12	4	no	—
151	C₄	#882	1	2	yes	—
152	C₄	#884	1	2	yes	—
153	C₃	#2016	1	1	yes	characteristic, complemented, Sylow
154	C₂	#4960	1	1	yes	characteristic
155	C₂	#5031	1	2	yes	—
156	C₂	#5050	1	2	yes	—
157	C₂	#5060	1	1	yes	characteristic
158	C₂	#5092	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#216

C₂

yes

maximal

C₂×#869

C₂

yes

maximal, characteristic

C₃×#391

C₂

yes

maximal, characteristic

C₂×#810

C₂

yes

maximal

#4524

C₂

yes

maximal, characteristic

#2016

—

maximal, Sylow

C₂²×C₃×Q₈

C₂²

yes

characteristic

C₂²×Dic₆

C₂²

yes

—

C₂×C₃×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₃×C₄²

C₂²

yes

—

—

—

C₂×C₄⋊Dic₃

—

C₂²×C₃⋊C₈

C₂²

yes

—

#869

—

C₃×#95

—

#810

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×#103

—

#391

D₃

yes

characteristic

#336

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

D₄

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

D₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

D₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

#216

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

D₄

yes

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₃

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃

yes

characteristic

C₂×C₄²

C₂×D₃

yes

characteristic

#103

—

#95

—

C₂³×C₃

C₂×D₄

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

Q₁₆

yes

complemented

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

#41

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

Dic₆

—

Dic₆

—

C₃⋊C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₁₂

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#67

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄²

—

103

C₂²×C₃

#120

yes

characteristic

104

C₂²×C₃

#121

yes

characteristic

105

C₂²×C₃

#127

yes

characteristic

106

C₂²×C₃

C₂×Q₁₆

yes

—

107

C₂²×C₃

#137

yes

—

108

C₃×C₄

—

109

C₃×C₄

—

110

C₃×C₄

—

111

C₃×C₄

#120

yes

—

112

C₃×C₄

#121

yes

—

113

Dic₃

—

114

C₂³

C₂×#67

yes

characteristic

115

Q₈

—

116

Q₈

—

117

Q₈

—

118

Q₈

—

119

Q₈

—

120

Q₈

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

#216

yes

complemented

132

C₂×C₄

C₂×D₁₂

yes

characteristic

133

C₂×C₄

#235

yes

characteristic

134

C₂×C₄

D₃×D₄

yes

—

135

C₂×C₄

D₃×D₄

yes

—

136

C₈

—

137

C₂×C₃

#392

yes

characteristic

138

C₂×C₃

#451

yes

—

139

C₂×C₃

#462

yes

—

140

C₂×C₃

#494

yes

characteristic

141

C₂×C₃

#497

yes

characteristic

142

C₂²

#930

yes

characteristic

143

C₂²

#937

yes

characteristic

144

C₂²

#940

yes

characteristic

145

C₂²

C₂×#216

yes

—

146

C₂²

#951

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

#882

yes

—

152

C₄

#884

yes

—

153

C₃

#2016

yes

characteristic, complemented, Sylow

154

C₂

#4960

yes

characteristic

155

C₂

#5031

yes

—

156

C₂

#5050

yes

—

157

C₂

#5060

yes

characteristic

158

C₂

#5092

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#67
commutator, cosocle	C₂²×C₃×C₄	C₂³
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#391	C₂
socle, prime radical	C₂³×C₃	C₂×D₄
hypercenter	#391	D₃
hypocenter	C₃	#2016
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×#67

commutator, cosocle

C₂²×C₃×C₄

C₂³

Frattini

C₂²×C₄

C₂²×D₃

Fitting

C₃×#391

C₂

socle, prime radical

C₂³×C₃

C₂×D₄

D₃

C₃

G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
68. C₂×C₃×C₄	Q₁₆	semidirect
131. C₂×C₄	#216	semidirect
153. C₃	#2016	semidirect

H ≅ G/N

type

68. C₂×C₃×C₄

Q₁₆

semidirect

131. C₂×C₄

#216

semidirect

153. C₃

#2016

semidirect

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

subgroup	index
C₂²×#216	2
C₂×#869	2
C₃×#391	2
C₂×#810	2
#4524	2
#2016	3

subgroup

index

C₂²×#216

Subgroups — #69691

Conjugacy classes of subgroups

Complements

Maximal subgroups