Group #71052 — subgroups

Subgroups — #71052

Order 384.

150 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 57 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	C₃×#462	C₂	1	1	yes	maximal, characteristic
2	#5080	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#5021	C₂	1	1	yes	maximal, characteristic
5	#4547	C₂	1	1	yes	maximal, characteristic
6	#4481	C₂	1	1	yes	maximal, characteristic
7	#4480	C₂	1	1	yes	maximal, characteristic
8	#1672	—	3	1	no	maximal, Sylow
9	C₂×C₃×Q₁₆	—	2	1	no	—
10	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
12	#869	C₂²	1	1	yes	characteristic
13	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₃×#103	—	2	1	no	—
15	#835	—	2	1	no	—
16	#808	—	2	1	no	—
17	#808	—	2	1	no	—
18	#807	—	2	1	no	—
19	C₄⋊(C₃⋊C₈)	—	2	1	no	—
20	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
21	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
22	C₄×C₃⋊C₈	—	2	1	no	—
23	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	#498	—	3	1	no	—
25	#477	—	3	1	no	—
26	#462	D₃	1	1	yes	characteristic
27	C₈×Q₈	—	3	1	no	—
28	#334	—	3	1	no	—
29	#333	—	3	1	no	—
30	#332	—	3	1	no	—
31	C₂×C₃×Q₈	—	2	1	no	—
32	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
33	C₂×Dic₆	—	2	1	no	—
34	C₃×Q₁₆	—	4	2	no	—
35	C₂×C₃×C₈	—	2	1	no	—
36	C₃×C₄⋊C₄	—	2	1	no	—
37	C₃×C₄⋊C₄	—	2	1	no	—
38	C₃×C₄⋊C₄	—	2	1	no	—
39	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
40	C₃×C₄²	—	2	1	no	—
41	C₃×C₄²	C₂³	1	1	yes	characteristic
42	C₄⋊Dic₃	—	2	1	no	—
43	C₄⋊Dic₃	—	2	1	no	—
44	C₂×C₃⋊C₈	—	2	1	no	—
45	C₂×C₃⋊C₈	—	2	1	no	—
46	C₂×C₃⋊C₈	D₄	1	1	yes	characteristic
47	C₂×C₃⋊C₈	D₄	1	1	yes	characteristic
48	C₂×Q₁₆	—	2	1	no	—
49	C₄⋊Q₈	—	3	1	no	—
50	C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
51	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
52	#107	—	6	1	no	—
53	#106	—	6	1	no	—
54	#106	—	6	1	no	—
55	C₄⋊C₈	—	3	1	no	—
56	C₄⋊C₈	—	3	1	no	—
57	C₄⋊C₈	—	6	1	no	—
58	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
59	#103	—	2	1	no	—
60	#103	—	6	1	no	—
61	C₄×C₈	—	3	1	no	—
62	C₄×C₈	—	6	1	no	—
63	C₃×Q₈	—	2	1	no	—
64	C₃×Q₈	—	4	1	no	—
65	C₃×Q₈	—	4	1	no	—
66	C₃×Q₈	SD₁₆	1	2	yes	—
67	C₂×C₃×C₄	—	2	1	no	—
68	C₂×C₃×C₄	—	2	1	no	—
69	C₂×C₃×C₄	—	4	1	no	—
70	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
71	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃×C₄	#41	1	1	yes	characteristic
73	C₂×Dic₃	—	4	1	no	—
74	Dic₆	—	8	1	no	—
75	C₃×C₈	—	4	1	no	—
76	C₃⋊C₈	—	2	1	no	—
77	C₃⋊C₈	—	4	1	no	—
78	C₃⋊C₈	—	4	1	no	—
79	C₃⋊C₈	Q₁₆	1	2	yes	complemented
80	C₂×Q₈	—	2	1	no	—
81	C₂×Q₈	—	6	1	no	—
82	C₂×Q₈	#67	1	1	yes	characteristic
83	Q₁₆	—	4	2	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	3	1	no	—
86	C₂×C₈	—	3	1	no	—
87	C₂×C₈	—	6	1	no	—
88	C₂×C₈	—	6	1	no	—
89	C₄⋊C₄	—	2	1	no	—
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	2	1	no	—
92	C₄⋊C₄	—	6	1	no	—
93	C₄⋊C₄	—	6	1	no	—
94	C₄⋊C₄	#67	1	1	yes	characteristic
95	C₄²	—	2	1	no	—
96	C₄²	C₂²×D₃	1	1	yes	characteristic
97	C₂²×C₃	#121	1	1	yes	characteristic
98	C₃×C₄	—	2	2	no	—
99	C₃×C₄	—	2	1	no	—
100	C₃×C₄	—	4	1	no	—
101	C₃×C₄	—	8	1	no	—
102	C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
103	C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
104	C₃×C₄	#135	1	1	yes	characteristic
105	C₃×C₄	#137	1	1	yes	characteristic
106	Dic₃	—	8	1	no	—
107	Q₈	—	2	1	no	—
108	Q₈	—	4	1	no	—
109	Q₈	—	4	1	no	—
110	Q₈	—	24	1	no	—
111	Q₈	#214	1	2	yes	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	12	1	no	—
116	C₂×C₄	D₃×D₄	1	1	yes	characteristic
117	C₂×C₄	#237	1	1	yes	characteristic
118	C₂×C₄	C₂×#67	1	1	yes	characteristic
119	C₈	—	3	2	no	—
120	C₈	—	4	1	no	—
121	C₈	—	6	1	no	—
122	C₈	—	12	1	no	—
123	C₈	—	12	1	no	—
124	C₂×C₃	#462	1	1	yes	characteristic
125	C₂×C₃	#465	1	1	yes	characteristic
126	C₂×C₃	#471	1	1	yes	characteristic
127	C₂²	#939	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	2	2	no	—
130	C₄	—	4	1	no	—
131	C₄	—	8	1	no	—
132	C₄	—	24	1	no	—
133	C₄	D₃×Q₁₆	1	1	yes	characteristic
134	C₄	#918	1	1	yes	characteristic
135	C₄	C₂×#214	1	1	yes	characteristic
136	C₄	#950	1	1	yes	characteristic
137	C₃	#1672	1	1	yes	characteristic, complemented, Sylow
138	C₂	#5041	1	1	yes	characteristic
139	C₂	#5182	1	1	yes	characteristic
140	C₂	#5203	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#462

C₂

yes

maximal, characteristic

#5080

C₂

yes

maximal, characteristic

Q₈×C₃⋊C₈

C₂

yes

maximal, characteristic

#5021

C₂

yes

maximal, characteristic

#4547

C₂

yes

maximal, characteristic

#4481

C₂

yes

maximal, characteristic

#4480

C₂

yes

maximal, characteristic

#1672

—

maximal, Sylow

C₂×C₃×Q₁₆

—

C₃×C₄⋊Q₈

C₂²

yes

characteristic

C₃×C₄×Q₈

C₂²

yes

characteristic

#869

C₂²

yes

characteristic

C₃×C₄⋊C₈

C₂²

yes

—

—

—

—

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₃⋊C₈)

C₂²

yes

characteristic

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

C₂²

yes

—

—

D₃

yes

characteristic

C₈×Q₈

—

#334

—

#333

—

#332

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

D₄

yes

characteristic

C₂×Dic₆

—

C₃×Q₁₆

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

D₄

yes

characteristic

C₃×C₄²

—

C₃×C₄²

C₂³

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

D₄

yes

characteristic

C₂×C₃⋊C₈

D₄

yes

characteristic

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃

yes

characteristic

C₄×Q₈

C₂×D₃

yes

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₃

yes

characteristic

#103

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

SD₁₆

yes

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

#41

yes

characteristic

C₂×Dic₃

—

Dic₆

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₃⋊C₈

Q₁₆

yes

complemented

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

#67

yes

characteristic

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

#67

yes

characteristic

C₄²

—

C₄²

C₂²×D₃

yes

characteristic

C₂²×C₃

#121

yes

characteristic

C₃×C₄

—

C₃×C₄

—

100

C₃×C₄

—

101

C₃×C₄

—

102

C₃×C₄

C₂×SD₁₆

yes

characteristic

103

C₃×C₄

C₂×Q₁₆

yes

characteristic

104

C₃×C₄

#135

yes

characteristic

105

C₃×C₄

#137

yes

characteristic

106

Dic₃

—

107

Q₈

—

108

Q₈

—

109

Q₈

—

110

Q₈

—

111

Q₈

#214

yes

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

D₃×D₄

yes

characteristic

117

C₂×C₄

#237

yes

characteristic

118

C₂×C₄

C₂×#67

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₂×C₃

#462

yes

characteristic

125

C₂×C₃

#465

yes

characteristic

126

C₂×C₃

#471

yes

characteristic

127

C₂²

#939

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

D₃×Q₁₆

yes

characteristic

134

C₄

#918

yes

characteristic

135

C₄

C₂×#214

yes

characteristic

136

C₄

#950

yes

characteristic

137

C₃

#1672

yes

characteristic, complemented, Sylow

138

C₂

#5041

yes

characteristic

139

C₂

#5182

yes

characteristic

140

C₂

#5203

yes

characteristic

	subgroup	G/H
center	C₂²	#939
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#462	C₂
socle, prime radical	C₂²×C₃	#121
hypercenter	#462	D₃
hypocenter	C₃	#1672
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

subgroup

G/H

center

C₂²

#939

commutator, cosocle

C₃×C₄²

C₂³

Frattini

C₄²

C₂²×D₃

C₂

C₂²×C₃

D₃

C₃

G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #462

N	H ≅ G/N	type
79. C₃⋊C₈	Q₁₆	semidirect
137. C₃	#1672	semidirect

H ≅ G/N

type

79. C₃⋊C₈

Q₁₆

semidirect

137. C₃

#1672

semidirect

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

subgroup	index
C₃×#462	2
#5080	2
Q₈×C₃⋊C₈	2
#5021	2
#4547	2
#4481	2
#4480	2
#1672	3

subgroup

index

Q₈×C₃⋊C₈

Subgroups — #71052

Conjugacy classes of subgroups

Complements

Maximal subgroups