Group #146288 — subgroups

Subgroups — #146288

Order 512.

142 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 52 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	#9609	C₂	1	1	yes	maximal, characteristic
2	#8798	C₂	1	1	yes	maximal, characteristic
3	#7303	C₂	1	1	yes	maximal, characteristic
4	#7297	C₂	1	1	yes	maximal, characteristic
5	#7294	C₂	1	1	yes	maximal, characteristic
6	#7290	C₂	1	1	yes	maximal, characteristic
7	#7147	C₂	1	1	yes	maximal, characteristic
8	#2163	C₂²	1	1	yes	characteristic
9	#2162	C₂²	1	1	yes	characteristic
10	C₄×#108	C₂²	1	1	yes	characteristic
11	#1568	C₂²	1	1	yes	characteristic
12	#1568	C₂²	1	1	yes	characteristic
13	#1565	—	2	1	no	—
14	C₂×#334	C₂²	1	1	yes	characteristic
15	#1452	C₂²	1	1	yes	characteristic
16	#1287	—	2	1	no	—
17	#1287	—	2	1	no	—
18	#1283	—	2	1	no	—
19	#432	—	2	1	no	—
20	#432	—	2	1	no	—
21	#431	D₄	1	1	yes	characteristic
22	#431	D₄	1	1	yes	characteristic
23	C₂×#108	—	2	1	no	—
24	C₂×C₄⋊C₈	—	2	1	no	—
25	C₄×M₁₆	—	2	1	no	—
26	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
27	C₄⋊C₁₆	—	2	1	no	—
28	C₄⋊C₁₆	—	2	1	no	—
29	C₂²⋊C₁₆	—	2	1	no	—
30	C₂²⋊C₁₆	—	2	1	no	—
31	#346	—	2	1	no	—
32	C₄×C₁₆	—	2	1	no	—
33	#341	Q₈	1	1	yes	characteristic
34	#341	Q₈	1	1	yes	characteristic
35	#335	—	2	1	no	—
36	#335	—	2	1	no	—
37	#335	—	4	1	no	—
38	#334	—	2	1	no	—
39	#334	—	2	1	no	—
40	#334	—	2	1	no	—
41	#334	—	2	1	no	—
42	#334	—	4	1	no	—
43	#322	—	2	1	no	—
44	#322	—	2	1	no	—
45	C₂×M₁₆	—	2	1	no	—
46	C₂×M₁₆	—	2	1	no	—
47	C₂²×C₈	—	4	1	no	—
48	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₈	#41	1	1	yes	characteristic
50	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
51	C₂×C₁₆	—	4	1	no	—
52	C₂×C₁₆	—	4	1	no	—
53	#108	—	4	2	no	—
54	C₄⋊C₈	—	2	1	no	—
55	C₄⋊C₈	—	2	1	no	—
56	C₄⋊C₈	—	4	1	no	—
57	C₄⋊C₈	—	4	1	no	—
58	C₄⋊C₈	—	4	1	no	—
59	C₄⋊C₈	—	4	1	no	—
60	C₂²⋊C₈	—	2	1	no	—
61	C₂²⋊C₈	—	2	1	no	—
62	C₂²⋊C₈	—	4	1	no	—
63	C₂²⋊C₈	—	4	1	no	—
64	#97	—	4	1	no	—
65	#97	—	4	1	no	—
66	#97	—	4	1	no	—
67	C₄×C₈	—	2	1	no	—
68	C₄×C₈	—	2	1	no	—
69	C₄×C₈	—	4	1	no	—
70	C₄×C₈	#41	1	1	yes	characteristic
71	C₄×C₈	#41	1	1	yes	characteristic
72	C₄×C₈	#41	1	1	yes	characteristic
73	C₄×C₈	#41	1	1	yes	characteristic
74	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂²×C₄	#135	1	1	yes	characteristic
76	C₂²×C₄	#137	1	1	yes	characteristic
77	M₁₆	—	2	2	no	—
78	M₁₆	—	4	1	no	—
79	M₁₆	—	4	1	no	—
80	M₁₆	—	4	1	no	—
81	C₂×C₈	—	4	1	no	—
82	C₂×C₈	—	4	1	no	—
83	C₂×C₈	—	4	1	no	—
84	C₂×C₈	—	4	1	no	—
85	C₂×C₈	—	4	1	no	—
86	C₂×C₈	—	4	1	no	—
87	C₂×C₈	—	8	1	no	—
88	C₂×C₈	—	8	1	no	—
89	C₂×C₈	—	8	1	no	—
90	C₂×C₈	—	16	1	no	—
91	C₂×C₈	#125	1	1	yes	characteristic
92	C₂×C₈	#125	1	1	yes	characteristic
93	C₂×C₈	#126	1	1	yes	characteristic
94	C₂×C₈	#126	1	1	yes	characteristic
95	C₄²	#123	1	1	yes	characteristic
96	C₄²	#124	1	1	yes	characteristic
97	C₄²	#135	1	1	yes	characteristic
98	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
99	C₁₆	—	8	1	no	—
100	C₁₆	—	8	1	no	—
101	C₂³	#479	1	1	yes	characteristic
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	#400	1	1	yes	characteristic
107	C₂×C₄	#478	1	1	yes	characteristic
108	C₂×C₄	#482	1	1	yes	characteristic
109	C₂×C₄	#485	1	1	yes	characteristic
110	C₂×C₄	#488	1	1	yes	characteristic
111	C₂×C₄	#489	1	1	yes	characteristic
112	C₈	—	2	1	no	—
113	C₈	—	2	1	no	—
114	C₈	—	2	1	no	—
115	C₈	—	2	1	no	—
116	C₈	—	4	2	no	—
117	C₈	—	8	1	no	—
118	C₈	—	8	1	no	—
119	C₈	—	8	1	no	—
120	C₈	—	16	1	no	—
121	C₈	—	16	1	no	—
122	C₈	—	16	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	—	2	1	no	—
125	C₂²	—	2	1	no	—
126	C₂²	#2101	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	2	1	no	—
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	—	2	1	no	—
132	C₄	#2102	1	1	yes	characteristic
133	C₄	#2103	1	1	yes	characteristic
134	C₂	—	4	1	no	—
135	C₂	#10689	1	1	yes	characteristic
136	C₂	#13544	1	1	yes	characteristic
137	C₂	#13545	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#9609

C₂

yes

maximal, characteristic

#8798

C₂

yes

maximal, characteristic

#7303

C₂

yes

maximal, characteristic

#7297

C₂

yes

maximal, characteristic

#7294

C₂

yes

maximal, characteristic

#7290

C₂

yes

maximal, characteristic

#7147

C₂

yes

maximal, characteristic

#2163

C₂²

yes

characteristic

#2162

C₂²

yes

characteristic

C₄×#108

C₂²

yes

characteristic

#1568

C₂²

yes

characteristic

#1568

C₂²

yes

characteristic

#1565

—

C₂×#334

C₂²

yes

characteristic

#1452

C₂²

yes

—

—

—

—

—

D₄

yes

characteristic

#431

D₄

yes

characteristic

C₂×#108

—

C₂×C₄⋊C₈

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

C₄×C₁₆

—

#341

Q₈

yes

characteristic

#341

Q₈

yes

—

—

—

—

—

—

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

yes

C₄×C₈

yes

C₄×C₈

yes

C₄×C₈

yes

C₂²×C₄

C₂²⋊Q₈

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

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄²

yes

C₄²

yes

C₄²

yes

C₄²

yes

C₁₆

—

100

C₁₆

—

101

C₂³

#479

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

#400

yes

characteristic

107

C₂×C₄

#478

yes

characteristic

108

C₂×C₄

#482

yes

characteristic

109

C₂×C₄

#485

yes

characteristic

110

C₂×C₄

#488

yes

characteristic

111

C₂×C₄

#489

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

#2101

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

#2102

yes

characteristic

133

C₄

#2103

yes

characteristic

134

C₂

—

135

C₂

#10689

yes

characteristic

136

C₂

#13544

yes

characteristic

137

C₂

#13545

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#2101
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#479
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2101

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#479

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452 ◁ 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
#9609	2
#8798	2
#7303	2
#7297	2
#7294	2
#7290	2
#7147	2

subgroup

index

Subgroups — #146288

Conjugacy classes of subgroups

Maximal subgroups