Group #144030 — subgroups

Subgroups — #144030

Order 512.

334 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 86 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	C₂×#2254	C₂	1	1	yes	maximal, characteristic
2	#19209	C₂	1	2	yes	maximal
3	#13266	C₂	1	1	yes	maximal, characteristic
4	#11940	C₂	1	2	yes	maximal
5	#7155	C₂	1	1	yes	maximal, characteristic
6	#3230	—	2	2	no	—
7	C₂×#500	C₂²	1	2	yes	—
8	#2960	C₂²	1	1	yes	characteristic
9	#2254	—	2	4	no	—
10	C₂×#367	—	2	2	no	—
11	#2150	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	#1696	—	2	4	no	—
14	#1676	—	2	4	no	—
15	#1564	C₂²	1	2	yes	—
16	#1548	—	2	2	no	—
17	#1379	—	2	2	no	—
18	#1364	—	2	4	no	—
19	#533	D₄	1	1	yes	characteristic
20	C₂×C₄⋊Q₈	—	2	2	no	—
21	C₂×M₃₂	—	2	1	no	—
22	C₂²×C₁₆	D₄	1	2	yes	—
23	#500	—	2	2	no	—
24	#500	—	2	2	no	—
25	#500	—	2	2	no	—
26	#500	—	4	2	no	—
27	#500	D₄	1	4	yes	—
28	#487	—	4	2	no	—
29	#484	—	4	4	no	—
30	#477	—	2	8	no	—
31	#446	—	2	2	no	—
32	C₈×Q₈	—	2	2	no	—
33	#431	—	2	2	no	—
34	C₂×#107	—	2	2	no	—
35	C₂×#107	—	2	2	no	—
36	#424	—	2	1	no	—
37	#423	D₄	1	1	yes	characteristic
38	#405	—	2	1	no	—
39	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
40	#367	—	2	8	no	—
41	C₄⋊C₁₆	—	2	2	no	—
42	C₄×C₁₆	—	2	2	no	—
43	#335	D₄	1	4	yes	—
44	#334	—	2	2	no	—
45	#333	—	2	8	no	—
46	C₂²×Q₈	—	4	2	no	—
47	C₂×M₁₆	—	2	1	no	—
48	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₈	#41	1	1	yes	characteristic
50	C₄⋊Q₈	—	2	4	no	—
51	C₄⋊Q₈	—	4	2	no	—
52	C₄⋊Q₈	—	4	2	no	—
53	C₄⋊Q₈	—	4	2	no	—
54	C₄⋊Q₈	D₈	1	2	yes	—
55	#125	—	2	1	no	—
56	C₂²⋊Q₈	—	2	2	no	—
57	C₄×Q₈	—	2	2	no	—
58	#117	—	2	1	no	—
59	C₂×C₄⋊C₄	—	4	2	no	—
60	C₂×C₄⋊C₄	—	4	2	no	—
61	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
62	M₃₂	—	4	2	no	—
63	C₂×C₁₆	—	2	2	no	—
64	C₂×C₁₆	—	2	2	no	—
65	C₂×C₁₆	—	4	1	no	—
66	C₂×C₁₆	Q₁₆	1	4	yes	—
67	#107	—	2	4	no	—
68	#107	—	2	4	no	—
69	#107	—	4	8	no	—
70	#107	—	4	2	no	—
71	#107	—	4	2	no	—
72	C₄⋊C₈	—	2	2	no	—
73	C₄⋊C₈	—	2	4	no	—
74	C₄⋊C₈	—	4	2	no	—
75	#103	—	4	8	no	—
76	C₂²⋊C₈	—	4	2	no	—
77	#97	—	4	1	no	—
78	C₄×C₈	—	4	1	no	—
79	C₄×C₈	—	4	2	no	—
80	C₄×C₈	C₂×D₄	1	2	yes	—
81	C₄×C₈	C₂×D₄	1	2	yes	—
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	—	4	4	no	—
84	C₂×Q₈	—	8	2	no	—
85	C₂×Q₈	—	16	4	no	—
86	C₂²×C₄	—	8	2	no	—
87	C₂²×C₄	#121	1	1	yes	characteristic
88	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
89	M₁₆	—	4	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	4	1	no	—
92	C₂×C₈	—	4	4	no	—
93	C₂×C₈	#120	1	1	yes	characteristic
94	C₂×C₈	#121	1	2	yes	—
95	C₂×C₈	#127	1	1	yes	characteristic
96	C₂×C₈	C₂×Q₁₆	1	2	yes	—
97	C₂×C₈	#137	1	2	yes	—
98	C₄⋊C₄	—	2	2	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	4	4	no	—
102	C₄⋊C₄	—	4	4	no	—
103	C₄⋊C₄	—	8	2	no	—
104	C₄⋊C₄	—	8	2	no	—
105	C₂²⋊C₄	—	4	1	no	—
106	C₄²	—	4	1	no	—
107	C₄²	#120	1	2	yes	—
108	C₄²	C₂×D₈	1	1	yes	characteristic
109	C₄²	#135	1	1	yes	characteristic
110	C₁₆	—	2	4	no	—
111	C₁₆	—	8	1	no	—
112	C₂³	#469	1	1	yes	characteristic
113	Q₈	—	4	4	no	—
114	Q₈	—	16	4	no	—
115	Q₈	—	16	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	8	4	no	—
119	C₂×C₄	—	16	4	no	—
120	C₂×C₄	#392	1	1	yes	characteristic
121	C₂×C₄	#447	1	2	yes	—
122	C₂×C₄	#449	1	2	yes	—
123	C₂×C₄	#462	1	2	yes	—
124	C₂×C₄	#466	1	1	yes	characteristic
125	C₂×C₄	#473	1	2	yes	—
126	C₈	—	2	2	no	—
127	C₈	—	8	4	no	—
128	C₈	—	8	1	no	—
129	C₈	#451	1	2	yes	—
130	C₈	#494	1	1	yes	characteristic
131	C₈	#497	1	1	yes	characteristic
132	C₂²	—	2	2	no	—
133	C₂²	#1694	1	2	yes	—
134	C₂²	#1999	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	8	2	no	—
137	C₄	—	16	4	no	—
138	C₄	#1674	1	2	yes	—
139	C₄	#2016	1	1	yes	characteristic
140	C₄	#2020	1	1	yes	characteristic
141	C₄	#2193	1	4	yes	—
142	C₂	—	2	2	no	—
143	C₂	#10265	1	1	yes	characteristic
144	C₂	#13320	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2254

C₂

yes

maximal, characteristic

#19209

C₂

yes

maximal

#13266

C₂

yes

maximal, characteristic

#11940

C₂

yes

maximal

#7155

C₂

yes

maximal, characteristic

#3230

—

C₂×#500

C₂²

yes

—

#2960

C₂²

yes

—

—

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

—

—

C₂²

yes

—

—

—

—

D₄

yes

characteristic

C₂×C₄⋊Q₈

—

C₂×M₃₂

—

C₂²×C₁₆

D₄

yes

—

—

—

—

—

D₄

yes

—

—

—

—

—

C₈×Q₈

—

—

—

—

—

D₄

yes

characteristic

#405

—

C₂×C₄×C₈

C₂³

yes

characteristic

#367

—

C₄⋊C₁₆

—

C₄×C₁₆

—

#335

D₄

yes

—

#334

—

#333

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₈

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

Q₁₆

yes

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

C₂×D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#120

yes

characteristic

C₂×C₈

#121

yes

—

C₂×C₈

#127

yes

characteristic

C₂×C₈

C₂×Q₁₆

yes

—

C₂×C₈

#137

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₂²⋊C₄

—

106

C₄²

—

107

C₄²

#120

yes

—

108

C₄²

C₂×D₈

yes

characteristic

109

C₄²

#135

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₂³

#469

yes

characteristic

113

Q₈

—

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

#392

yes

characteristic

121

C₂×C₄

#447

yes

—

122

C₂×C₄

#449

yes

—

123

C₂×C₄

#462

yes

—

124

C₂×C₄

#466

yes

characteristic

125

C₂×C₄

#473

yes

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

#451

yes

—

130

C₈

#494

yes

characteristic

131

C₈

#497

yes

characteristic

132

C₂²

—

133

C₂²

#1694

yes

—

134

C₂²

#1999

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#1674

yes

—

139

C₄

#2016

yes

characteristic

140

C₄

#2020

yes

characteristic

141

C₄

#2193

yes

—

142

C₂

—

143

C₂

#10265

yes

characteristic

144

C₂

#13320

yes

—

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#469
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1999

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#469

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ 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
C₂×#2254	2
#19209	2
#13266	2
#11940	2
#7155	2

subgroup

index

Subgroups — #144030

Conjugacy classes of subgroups

Maximal subgroups