Group #142126 — subgroups

Subgroups — #142126

Order 512.

288 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 98 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	#33256	C₂	1	1	yes	maximal, characteristic
2	#12900	C₂	1	1	yes	maximal, characteristic
3	#8267	C₂	1	1	yes	maximal, characteristic
4	#7278	C₂	1	2	yes	maximal
5	#7278	C₂	1	2	yes	maximal
6	#2859	—	2	1	no	—
7	C₂×#364	—	2	2	no	—
8	#2150	C₂²	1	2	yes	—
9	#1941	—	2	2	no	—
10	#1941	C₄	1	4	yes	—
11	#1844	C₂²	1	2	yes	—
12	#1834	—	2	1	no	—
13	C₄²×C₈	C₂²	1	1	yes	characteristic
14	#1564	C₂²	1	1	yes	characteristic
15	#1564	C₂²	1	1	yes	characteristic
16	#1452	—	2	1	no	—
17	#1276	—	2	1	no	—
18	C₂×#125	—	4	1	no	—
19	C₂×M₃₂	—	2	2	no	—
20	#431	—	2	1	no	—
21	#431	—	2	1	no	—
22	C₂×#106	—	2	2	no	—
23	C₂×#106	C₂×C₄	1	4	yes	—
24	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₈	D₄	1	2	yes	—
26	C₂×C₄×C₈	D₄	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
28	#389	—	2	4	no	—
29	#389	—	4	2	no	—
30	#384	—	2	2	no	—
31	#382	—	4	1	no	—
32	C₄³	D₄	1	1	yes	characteristic
33	#364	—	4	4	no	—
34	C₄⋊C₁₆	—	2	2	no	—
35	C₄⋊C₁₆	—	2	2	no	—
36	#340	—	2	4	no	—
37	#335	Q₈	1	2	yes	—
38	#335	Q₈	1	2	yes	—
39	#334	—	2	1	no	—
40	#334	—	2	1	no	—
41	#322	—	4	2	no	—
42	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
45	#125	—	8	4	no	—
46	C₂×C₄⋊C₄	—	2	4	no	—
47	C₂×C₄⋊C₄	—	4	4	no	—
48	C₂×C₄⋊C₄	—	4	2	no	—
49	C₂×C₄⋊C₄	—	4	2	no	—
50	C₂×C₄²	—	2	1	no	—
51	C₂×C₄²	—	2	1	no	—
52	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
53	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
54	M₃₂	—	4	4	no	—
55	C₂×C₁₆	—	4	2	no	—
56	#106	—	4	2	no	—
57	#106	—	4	2	no	—
58	#106	—	4	4	no	—
59	C₄⋊C₈	—	2	2	no	—
60	C₄⋊C₈	—	2	2	no	—
61	C₂²⋊C₈	—	4	1	no	—
62	C₂²⋊C₈	—	4	1	no	—
63	C₄×C₈	—	2	2	no	—
64	C₄×C₈	—	2	2	no	—
65	C₄×C₈	—	4	2	no	—
66	C₄×C₈	—	4	1	no	—
67	C₄×C₈	—	4	1	no	—
68	C₄×C₈	SD₁₆	1	2	yes	—
69	C₄×C₈	Q₁₆	1	2	yes	—
70	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
71	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
72	C₄×C₈	#41	1	1	yes	characteristic
73	C₄×C₈	#41	1	1	yes	characteristic
74	#95	—	4	4	no	—
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	—	4	4	no	—
77	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	4	2	no	—
84	C₂×C₈	—	4	2	no	—
85	C₂×C₈	#103	1	4	yes	—
86	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
87	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
88	C₂×C₈	#124	1	1	yes	characteristic
89	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
90	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
91	C₂×C₈	#137	1	2	yes	—
92	C₄⋊C₄	—	4	4	no	—
93	C₄⋊C₄	—	4	4	no	—
94	C₄⋊C₄	—	8	4	no	—
95	C₄⋊C₄	—	8	4	no	—
96	C₄⋊C₄	—	8	8	no	—
97	C₄²	—	2	4	no	—
98	C₄²	—	2	2	no	—
99	C₄²	—	2	2	no	—
100	C₄²	—	4	2	no	—
101	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
102	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
103	C₄²	C₂×SD₁₆	1	1	yes	characteristic
104	C₄²	C₂×Q₁₆	1	1	yes	characteristic
105	C₁₆	—	8	2	no	—
106	C₂³	#418	1	1	yes	characteristic
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	—	8	4	no	—
113	C₂×C₄	—	8	4	no	—
114	C₂×C₄	—	8	4	no	—
115	C₂×C₄	#391	1	1	yes	characteristic
116	C₂×C₄	#413	1	2	yes	—
117	C₂×C₄	C₂×#103	1	1	yes	characteristic
118	C₂×C₄	#417	1	2	yes	—
119	C₂×C₄	#476	1	1	yes	characteristic
120	C₂×C₄	#476	1	1	yes	characteristic
121	C₂×C₄	#477	1	1	yes	characteristic
122	C₂×C₄	#477	1	1	yes	characteristic
123	C₈	—	2	4	no	—
124	C₈	—	2	1	no	—
125	C₈	—	2	1	no	—
126	C₈	—	8	2	no	—
127	C₈	—	8	2	no	—
128	C₈	#490	1	1	yes	characteristic
129	C₈	#491	1	1	yes	characteristic
130	C₈	#501	1	2	yes	—
131	C₂²	—	2	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	#1557	1	2	yes	—
134	C₂²	#1987	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	4	4	no	—
138	C₄	—	16	4	no	—
139	C₄	#1550	1	2	yes	—
140	C₄	#1992	1	1	yes	characteristic
141	C₄	#1998	1	1	yes	characteristic
142	C₄	#2230	1	2	yes	—
143	C₄	#2231	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	#9902	1	1	yes	characteristic
146	C₂	#13272	1	1	yes	characteristic
147	C₂	#13273	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33256

C₂

yes

maximal, characteristic

#12900

C₂

yes

maximal, characteristic

#8267

C₂

yes

maximal, characteristic

#7278

C₂

yes

maximal

#7278

C₂

yes

—

—

C₂²

yes

—

#1941

—

#1941

C₄

yes

—

#1844

C₂²

yes

—

#1834

—

C₄²×C₈

C₂²

yes

characteristic

#1564

C₂²

yes

characteristic

#1564

C₂²

yes

—

—

—

C₂×M₃₂

—

—

—

—

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

—

—

—

C₄³

D₄

yes

characteristic

#364

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#340

—

#335

Q₈

yes

—

#335

Q₈

yes

—

#334

—

#334

—

#322

—

C₂²×C₈

C₂²⋊C₄

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂²⋊C₄

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

#106

—

#106

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

C₄×C₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#103

yes

—

C₂×C₈

C₄×Q₈

yes

characteristic

C₂×C₈

C₄×Q₈

yes

characteristic

C₂×C₈

#124

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

AGL(1,ℤ/8)

yes

—

C₂×C₈

#137

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

100

C₄²

—

101

C₄²

C₂²⋊Q₈

yes

characteristic

102

C₄²

C₂²⋊Q₈

yes

characteristic

103

C₄²

C₂×SD₁₆

yes

characteristic

104

C₄²

C₂×Q₁₆

yes

characteristic

105

C₁₆

—

106

C₂³

#418

yes

characteristic

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

#391

yes

characteristic

116

C₂×C₄

#413

yes

—

117

C₂×C₄

C₂×#103

yes

characteristic

118

C₂×C₄

#417

yes

—

119

C₂×C₄

#476

yes

characteristic

120

C₂×C₄

#476

yes

characteristic

121

C₂×C₄

#477

yes

characteristic

122

C₂×C₄

#477

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

#490

yes

characteristic

129

C₈

#491

yes

characteristic

130

C₈

#501

yes

—

131

C₂²

—

132

C₂²

—

133

C₂²

#1557

yes

—

134

C₂²

#1987

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

#1550

yes

—

140

C₄

#1992

yes

characteristic

141

C₄

#1998

yes

characteristic

142

C₄

#2230

yes

—

143

C₄

#2231

yes

—

144

C₂

—

145

C₂

#9902

yes

characteristic

146

C₂

#13272

yes

characteristic

147

C₂

#13273

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1987
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#418
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1987

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#418

LCS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ 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
#33256	2
#12900	2
#8267	2
#7278	2

subgroup

index

Subgroups — #142126

Conjugacy classes of subgroups

Maximal subgroups