Group #129180 — subgroups

Subgroups — #129180

Order 512.

228 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 110 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	#12676	C₂	1	1	yes	maximal, characteristic
2	#11999	C₂	1	1	yes	maximal, characteristic
3	C₂×#1313	C₂	1	1	yes	maximal, characteristic
4	#8295	C₂	1	1	yes	maximal, characteristic
5	#8294	C₂	1	1	yes	maximal, characteristic
6	#7177	C₂	1	1	yes	maximal, characteristic
7	#7177	C₂	1	1	yes	maximal, characteristic
8	C₂×#368	C₄	1	2	yes	—
9	C₂×#346	C₂²	1	1	yes	characteristic
10	C₂×#346	C₂²	1	1	yes	characteristic
11	C₄×#108	C₂²	1	1	yes	characteristic
12	C₄×#108	C₂²	1	1	yes	characteristic
13	#1564	C₂²	1	1	yes	characteristic
14	#1564	C₂²	1	1	yes	characteristic
15	#1451	—	2	1	no	—
16	C₈×M₁₆	—	2	1	no	—
17	C₂×#322	C₂²	1	1	yes	characteristic
18	#1381	C₄	1	2	yes	—
19	#1368	C₄	1	4	yes	—
20	#1313	C₄	1	4	yes	—
21	#1278	—	2	1	no	—
22	#1278	—	2	1	no	—
23	C₂²×C₁₆	—	2	1	no	—
24	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
26	#431	—	2	1	no	—
27	#431	—	2	1	no	—
28	C₂×#108	C₂×C₄	1	2	yes	—
29	C₂×#108	C₂×C₄	1	2	yes	—
30	C₄×M₁₆	—	2	1	no	—
31	C₄×M₁₆	—	2	1	no	—
32	C₂×C₄×C₈	D₄	1	1	yes	characteristic
33	C₂×C₄×C₈	D₄	1	1	yes	characteristic
34	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
35	#368	—	2	4	no	—
36	#346	—	2	1	no	—
37	#346	C₂×C₄	1	2	yes	—
38	#346	C₂×C₄	1	2	yes	—
39	#346	C₂×C₄	1	2	yes	—
40	#341	—	2	1	no	—
41	#341	—	2	1	no	—
42	#335	—	2	1	no	—
43	#335	—	2	1	no	—
44	#334	C₂×C₄	1	2	yes	—
45	#334	C₂×C₄	1	2	yes	—
46	#322	—	4	1	no	—
47	#322	—	4	2	no	—
48	#322	C₂×C₄	1	2	yes	—
49	#322	Q₈	1	2	yes	—
50	C₈²	—	4	1	no	—
51	C₂×M₁₆	—	2	2	no	—
52	C₂×M₁₆	—	2	2	no	—
53	C₂²×C₈	—	2	1	no	—
54	C₂²×C₈	—	2	1	no	—
55	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₈	#41	1	1	yes	characteristic
57	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
58	C₂×C₁₆	—	2	4	no	—
59	C₂×C₁₆	—	2	2	no	—
60	C₂×C₁₆	—	2	2	no	—
61	C₂×C₁₆	—	2	2	no	—
62	C₂×C₁₆	C₄²	1	2	yes	—
63	C₂×C₁₆	C₄²	1	2	yes	—
64	#108	—	2	2	no	—
65	#108	—	2	2	no	—
66	#108	—	2	2	no	—
67	#108	M₁₆	1	4	yes	—
68	C₄⋊C₈	—	2	2	no	—
69	C₄⋊C₈	—	2	2	no	—
70	C₂²⋊C₈	—	4	1	no	—
71	C₂²⋊C₈	—	4	1	no	—
72	#97	—	4	1	no	—
73	#97	—	4	1	no	—
74	C₄×C₈	—	2	1	no	—
75	C₄×C₈	—	2	1	no	—
76	C₄×C₈	—	4	1	no	—
77	C₄×C₈	—	4	1	no	—
78	C₄×C₈	—	4	1	no	—
79	C₄×C₈	—	4	1	no	—
80	C₄×C₈	C₄⋊C₄	1	2	yes	—
81	C₄×C₈	C₄⋊C₄	1	2	yes	—
82	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
83	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
85	C₄×C₈	#41	1	1	yes	characteristic
86	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
87	C₂²×C₄	#137	1	1	yes	characteristic
88	C₂²×C₄	#137	1	1	yes	characteristic
89	M₁₆	—	2	4	no	—
90	M₁₆	—	4	2	no	—
91	M₁₆	—	4	2	no	—
92	M₁₆	—	4	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	4	2	no	—
97	C₂×C₈	—	4	2	no	—
98	C₂×C₈	—	4	2	no	—
99	C₂×C₈	—	4	2	no	—
100	C₂×C₈	—	8	1	no	—
101	C₂×C₈	—	8	1	no	—
102	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₈	#117	1	1	yes	characteristic
104	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
105	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
106	C₂×C₈	C₂×M₁₆	1	2	yes	—
107	C₂×C₈	#131	1	2	yes	—
108	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₄²	C₄×D₄	1	1	yes	characteristic
110	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₁₆	—	2	4	no	—
113	C₁₆	—	2	2	no	—
114	C₁₆	—	2	2	no	—
115	C₂³	#441	1	1	yes	characteristic
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄	#428	1	1	yes	characteristic
120	C₂×C₄	#428	1	1	yes	characteristic
121	C₂×C₄	#435	1	2	yes	—
122	C₂×C₄	#440	1	1	yes	characteristic
123	C₂×C₄	#440	1	1	yes	characteristic
124	C₂×C₄	#442	1	1	yes	characteristic
125	C₂×C₄	#443	1	2	yes	—
126	C₈	—	2	1	no	—
127	C₈	—	2	1	no	—
128	C₈	—	4	2	no	—
129	C₈	—	4	2	no	—
130	C₈	—	4	4	no	—
131	C₈	—	8	2	no	—
132	C₈	—	8	2	no	—
133	C₈	—	8	2	no	—
134	C₈	C₄×M₁₆	1	1	yes	characteristic
135	C₈	#405	1	1	yes	characteristic
136	C₈	#446	1	2	yes	—
137	C₂²	—	2	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	#1587	1	2	yes	—
140	C₂²	#1776	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	#1586	1	2	yes	—
146	C₄	#1771	1	1	yes	characteristic
147	C₄	#1772	1	1	yes	characteristic
148	C₂	—	2	2	no	—
149	C₂	#8288	1	1	yes	characteristic
150	C₂	#12678	1	1	yes	characteristic
151	C₂	#12678	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12676

C₂

yes

maximal, characteristic

#11999

C₂

yes

maximal, characteristic

C₂×#1313

C₂

yes

maximal, characteristic

#8295

C₂

yes

maximal, characteristic

#8294

C₂

yes

maximal, characteristic

#7177

C₂

yes

maximal, characteristic

#7177

C₂

yes

maximal, characteristic

C₂×#368

C₄

yes

—

C₂×#346

C₂²

yes

characteristic

C₂×#346

C₂²

yes

characteristic

C₄×#108

C₂²

yes

characteristic

C₄×#108

C₂²

yes

characteristic

#1564

C₂²

yes

characteristic

#1564

C₂²

yes

characteristic

#1451

—

C₈×M₁₆

—

C₂×#322

C₂²

yes

characteristic

#1381

C₄

yes

—

#1368

C₄

yes

—

#1313

C₄

yes

—

#1278

—

#1278

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

—

—

C₂×C₄

yes

—

C₂×#108

C₂×C₄

yes

—

C₄×M₁₆

—

C₄×M₁₆

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

—

C₂×C₄

yes

—

#346

C₂×C₄

yes

—

#346

C₂×C₄

yes

—

—

—

—

—

C₂×C₄

yes

—

#334

C₂×C₄

yes

—

#322

—

#322

—

#322

C₂×C₄

yes

—

#322

Q₈

yes

—

C₈²

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄²

yes

—

C₂×C₁₆

C₄²

yes

—

—

—

—

M₁₆

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂²×C₄

C₄×D₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₂×C₄²

yes

characteristic

103

C₂×C₈

#117

yes

characteristic

104

C₂×C₈

C₄×Q₈

yes

characteristic

105

C₂×C₈

C₄×Q₈

yes

characteristic

106

C₂×C₈

C₂×M₁₆

yes

—

107

C₂×C₈

#131

yes

—

108

C₄²

C₂×C₄⋊C₄

yes

characteristic

109

C₄²

C₄×D₄

yes

characteristic

110

C₄²

AGL(1,ℤ/8)

yes

characteristic

111

C₄²

AGL(1,ℤ/8)

yes

characteristic

112

C₁₆

—

113

C₁₆

—

114

C₁₆

—

115

C₂³

#441

yes

characteristic

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₄×C₄⋊C₄

yes

characteristic

119

C₂×C₄

#428

yes

characteristic

120

C₂×C₄

#428

yes

characteristic

121

C₂×C₄

#435

yes

—

122

C₂×C₄

#440

yes

characteristic

123

C₂×C₄

#440

yes

characteristic

124

C₂×C₄

#442

yes

characteristic

125

C₂×C₄

#443

yes

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

C₄×M₁₆

yes

characteristic

135

C₈

#405

yes

characteristic

136

C₈

#446

yes

—

137

C₂²

—

138

C₂²

—

139

C₂²

#1587

yes

—

140

C₂²

#1776

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

#1586

yes

—

146

C₄

#1771

yes

characteristic

147

C₄

#1772

yes

characteristic

148

C₂

—

149

C₂

#8288

yes

characteristic

150

C₂

#12678

yes

characteristic

151

C₂

#12678

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#442
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1776
prime radical	C₂³	#441
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#442

commutator

C₂×C₈

C₂×C₄²

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322 ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#12676	2
#11999	2
C₂×#1313	2
#8295	2
#8294	2
#7177	2

subgroup

index

Subgroups — #129180

Conjugacy classes of subgroups

Maximal subgroups