Group #71117 — subgroups

Subgroups — #71117

Order 384.

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

normal subgroups only

This group has 156 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×#475	C₂	1	1	yes	maximal, characteristic
2	#5092	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#5020	C₂	1	1	yes	maximal, characteristic
5	#4547	C₂	1	1	yes	maximal, characteristic
6	#4486	C₂	1	1	yes	maximal, characteristic
7	#4479	C₂	1	1	yes	maximal, characteristic
8	#1670	—	3	1	no	maximal, Sylow
9	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
11	C₂×#216	—	2	1	no	—
12	C₂×#216	—	2	1	no	—
13	#869	C₂²	1	1	yes	characteristic
14	C₃×#106	—	2	1	no	—
15	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₃×#103	—	2	1	no	—
17	#835	—	2	1	no	—
18	#808	—	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	#494	—	3	1	no	—
25	#475	—	3	1	no	—
26	#475	D₃	1	1	yes	characteristic
27	C₈×Q₈	—	3	1	no	—
28	#335	—	3	1	no	—
29	#332	—	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	C₂×C₃×C₈	—	2	1	no	—
35	C₃×C₄⋊C₄	—	2	1	no	—
36	C₃×C₄⋊C₄	—	2	1	no	—
37	C₃×C₄⋊C₄	—	2	1	no	—
38	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
39	C₃×C₄²	—	2	1	no	—
40	C₃×C₄²	C₂³	1	1	yes	characteristic
41	#216	—	4	2	no	—
42	#216	—	4	2	no	—
43	C₄⋊Dic₃	—	2	1	no	—
44	C₄⋊Dic₃	—	2	1	no	—
45	C₂×C₃⋊C₈	—	2	1	no	—
46	C₂×C₃⋊C₈	—	2	1	no	—
47	C₂×C₃⋊C₈	D₄	1	1	yes	characteristic
48	C₂×C₃⋊C₈	D₄	1	1	yes	characteristic
49	C₂×Q₁₆	—	6	1	no	—
50	C₂×Q₁₆	—	6	1	no	—
51	C₄⋊Q₈	—	3	1	no	—
52	C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
53	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
54	#106	—	2	1	no	—
55	#106	—	6	1	no	—
56	C₄⋊C₈	—	3	1	no	—
57	C₄⋊C₈	—	3	1	no	—
58	C₄⋊C₈	—	6	1	no	—
59	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
60	#103	—	2	1	no	—
61	#103	—	6	1	no	—
62	C₄×C₈	—	3	1	no	—
63	C₄×C₈	—	6	1	no	—
64	C₃×Q₈	—	2	1	no	—
65	C₃×Q₈	—	4	1	no	—
66	C₃×Q₈	—	4	1	no	—
67	C₃×Q₈	Q₁₆	1	2	yes	complemented
68	C₂×C₃×C₄	—	2	1	no	—
69	C₂×C₃×C₄	—	2	1	no	—
70	C₂×C₃×C₄	—	4	1	no	—
71	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
73	C₂×C₃×C₄	#41	1	1	yes	characteristic
74	C₂×Dic₃	—	4	1	no	—
75	Dic₆	—	4	1	no	—
76	Dic₆	—	4	1	no	—
77	C₃×C₈	—	4	1	no	—
78	C₃⋊C₈	—	2	1	no	—
79	C₃⋊C₈	—	4	1	no	—
80	C₃⋊C₈	—	4	1	no	—
81	C₃⋊C₈	SD₁₆	1	2	yes	—
82	C₂×Q₈	—	2	1	no	—
83	C₂×Q₈	—	6	1	no	—
84	C₂×Q₈	#67	1	1	yes	characteristic
85	Q₁₆	—	12	2	no	—
86	Q₁₆	—	12	2	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	3	1	no	—
89	C₂×C₈	—	3	1	no	—
90	C₂×C₈	—	6	1	no	—
91	C₂×C₈	—	6	1	no	—
92	C₄⋊C₄	—	2	1	no	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	6	1	no	—
96	C₄⋊C₄	—	6	1	no	—
97	C₄⋊C₄	#67	1	1	yes	characteristic
98	C₄²	—	2	1	no	—
99	C₄²	C₂²×D₃	1	1	yes	characteristic
100	C₂²×C₃	#121	1	1	yes	characteristic
101	C₃×C₄	—	2	2	no	—
102	C₃×C₄	—	2	1	no	—
103	C₃×C₄	—	4	1	no	—
104	C₃×C₄	—	8	1	no	—
105	C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
106	C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
107	C₃×C₄	#135	1	1	yes	characteristic
108	C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
109	Dic₃	—	8	1	no	—
110	Q₈	—	2	1	no	—
111	Q₈	—	4	1	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	12	1	no	—
114	Q₈	—	12	1	no	—
115	Q₈	#216	1	2	yes	complemented
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	12	1	no	—
120	C₂×C₄	D₃×D₄	1	1	yes	characteristic
121	C₂×C₄	#237	1	1	yes	characteristic
122	C₂×C₄	C₂×#67	1	1	yes	characteristic
123	C₈	—	3	2	no	—
124	C₈	—	4	1	no	—
125	C₈	—	6	1	no	—
126	C₈	—	12	1	no	—
127	C₈	—	12	1	no	—
128	C₂×C₃	#460	1	1	yes	characteristic
129	C₂×C₃	#467	1	1	yes	characteristic
130	C₂×C₃	#473	1	1	yes	characteristic
131	C₂²	#939	1	1	yes	characteristic
132	C₄	—	2	1	no	—
133	C₄	—	2	2	no	—
134	C₄	—	4	1	no	—
135	C₄	—	8	1	no	—
136	C₄	—	24	1	no	—
137	C₄	D₃×SD₁₆	1	1	yes	characteristic
138	C₄	#914	1	1	yes	characteristic
139	C₄	C₂×#216	1	1	yes	characteristic
140	C₄	#950	1	1	yes	characteristic
141	C₃	#1670	1	1	yes	characteristic, complemented, Sylow
142	C₂	#5051	1	1	yes	characteristic
143	C₂	#5163	1	1	yes	characteristic
144	C₂	#5204	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#475

C₂

yes

maximal, characteristic

#5092

C₂

yes

maximal, characteristic

Q₈×C₃⋊C₈

C₂

yes

maximal, characteristic

#5020

C₂

yes

maximal, characteristic

#4547

C₂

yes

maximal, characteristic

#4486

C₂

yes

maximal, characteristic

#4479

C₂

yes

maximal, characteristic

#1670

—

maximal, Sylow

C₃×C₄⋊Q₈

C₂²

yes

characteristic

C₃×C₄×Q₈

C₂²

yes

—

—

C₂²

yes

characteristic

C₃×#106

—

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

—

#335

—

#332

—

#332

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

D₄

yes

characteristic

C₂×Dic₆

—

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

#216

—

#216

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

D₄

yes

characteristic

C₂×C₃⋊C₈

D₄

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃

yes

characteristic

C₄×Q₈

C₂×D₃

yes

characteristic

#106

—

#106

—

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

Q₁₆

yes

complemented

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

—

Dic₆

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₃⋊C₈

SD₁₆

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

#67

yes

characteristic

Q₁₆

—

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

100

C₂²×C₃

#121

yes

characteristic

101

C₃×C₄

—

102

C₃×C₄

—

103

C₃×C₄

—

104

C₃×C₄

—

105

C₃×C₄

C₂×SD₁₆

yes

characteristic

106

C₃×C₄

C₂×Q₁₆

yes

characteristic

107

C₃×C₄

#135

yes

characteristic

108

C₃×C₄

AGL(1,ℤ/8)

yes

characteristic

109

Dic₃

—

110

Q₈

—

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

#216

yes

complemented

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

D₃×D₄

yes

characteristic

121

C₂×C₄

#237

yes

characteristic

122

C₂×C₄

C₂×#67

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₂×C₃

#460

yes

characteristic

129

C₂×C₃

#467

yes

characteristic

130

C₂×C₃

#473

yes

characteristic

131

C₂²

#939

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

D₃×SD₁₆

yes

characteristic

138

C₄

#914

yes

characteristic

139

C₄

C₂×#216

yes

characteristic

140

C₄

#950

yes

characteristic

141

C₃

#1670

yes

characteristic, complemented, Sylow

142

C₂

#5051

yes

characteristic

143

C₂

#5163

yes

characteristic

144

C₂

#5204

yes

characteristic

	subgroup	G/H
center	C₂²	#939
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#475	C₂
socle, prime radical	C₂²×C₃	#121
hypercenter	#475	D₃
hypocenter	C₃	#1670
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

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 ◁ #475

N	H ≅ G/N	type
67. C₃×Q₈	Q₁₆	semidirect
115. Q₈	#216	semidirect
141. C₃	#1670	semidirect

H ≅ G/N

type

67. C₃×Q₈

Q₁₆

semidirect

115. Q₈

#216

semidirect

141. C₃

#1670

semidirect

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

subgroup	index
C₃×#475	2
#5092	2
Q₈×C₃⋊C₈	2
#5020	2
#4547	2
#4486	2
#4479	2
#1670	3

subgroup

index

Q₈×C₃⋊C₈

Subgroups — #71117

Conjugacy classes of subgroups

Complements

Maximal subgroups