Group #140484 — subgroups

Subgroups — #140484

Order 512.

183 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 183 conjugacy classes of subgroups — too many to draw. Only its 82 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	#12023	C₂	1	1	yes	maximal, characteristic
2	#12008	C₂	1	1	yes	maximal, characteristic
3	C₁₆×M₁₆	C₂	1	1	yes	maximal, characteristic
4	#9634	C₂	1	2	yes	maximal
5	#7291	C₂	1	2	yes	maximal
6	#2162	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
8	#1574	—	2	2	no	—
9	#1567	C₂²	1	2	yes	—
10	#1564	C₂²	1	2	yes	—
11	C₈×M₁₆	C₂²	1	1	yes	characteristic
12	#1372	—	2	1	no	—
13	#1372	—	2	1	no	—
14	#1372	C₄	1	2	yes	—
15	#1371	C₄	1	2	yes	—
16	#1312	—	2	1	no	—
17	C₈×C₁₆	—	2	1	no	—
18	#1299	—	2	2	no	—
19	C₂²×C₁₆	Q₈	1	1	yes	characteristic
20	C₂²×C₁₆	Q₈	1	1	yes	characteristic
21	#431	—	2	2	no	—
22	#431	D₄	1	1	yes	characteristic
23	#423	—	2	2	no	—
24	C₄×M₁₆	D₄	1	1	yes	characteristic
25	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
26	C₄⋊C₁₆	—	2	1	no	—
27	C₂²⋊C₁₆	Q₈	1	2	yes	—
28	C₄×C₁₆	—	2	1	no	—
29	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
30	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
31	C₄×C₁₆	D₄	1	1	yes	characteristic
32	C₄×C₁₆	D₄	1	1	yes	characteristic
33	#335	—	2	2	no	—
34	#335	—	2	2	no	—
35	#335	—	4	2	no	—
36	#335	C₂×C₄	1	4	yes	—
37	#334	—	2	2	no	—
38	#334	—	4	2	no	—
39	#322	—	2	1	no	—
40	C₈²	—	2	1	no	—
41	C₂×M₁₆	—	2	1	no	—
42	C₂×M₁₆	—	4	2	no	—
43	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
46	C₂×C₁₆	—	2	2	no	—
47	C₂×C₁₆	—	2	2	no	—
48	C₂×C₁₆	—	4	1	no	—
49	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
53	C₄⋊C₈	—	2	4	no	—
54	C₄⋊C₈	—	4	2	no	—
55	C₄⋊C₈	—	4	2	no	—
56	C₄⋊C₈	D₈	1	2	yes	—
57	C₂²⋊C₈	—	2	1	no	—
58	C₂²⋊C₈	—	4	2	no	—
59	#97	Q₁₆	1	2	yes	—
60	C₄×C₈	—	2	1	no	—
61	C₄×C₈	—	2	1	no	—
62	C₄×C₈	—	4	2	no	—
63	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
64	C₄×C₈	C₂×D₄	1	1	yes	characteristic
65	C₄×C₈	#41	1	1	yes	characteristic
66	C₄×C₈	#41	1	1	yes	characteristic
67	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
68	C₂²×C₄	#135	1	1	yes	characteristic
69	C₂²×C₄	#135	1	1	yes	characteristic
70	M₁₆	—	2	4	no	—
71	M₁₆	—	16	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	2	1	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	4	1	no	—
77	C₂×C₈	—	4	4	no	—
78	C₂×C₈	—	8	2	no	—
79	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
81	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
82	C₂×C₈	#125	1	1	yes	characteristic
83	C₄²	C₄×D₄	1	1	yes	characteristic
84	C₄²	#121	1	1	yes	characteristic
85	C₄²	C₂×D₈	1	1	yes	characteristic
86	C₄²	C₂×Q₁₆	1	1	yes	characteristic
87	C₁₆	—	2	1	no	—
88	C₁₆	—	2	1	no	—
89	C₁₆	—	2	1	no	—
90	C₁₆	—	4	2	no	—
91	C₁₆	#108	1	2	yes	—
92	C₂³	#499	1	1	yes	characteristic
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	#389	1	1	yes	characteristic
98	C₂×C₄	C₄×D₈	1	1	yes	characteristic
99	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
100	C₂×C₄	#466	1	1	yes	characteristic
101	C₂×C₄	#467	1	1	yes	characteristic
102	C₂×C₄	#500	1	1	yes	characteristic
103	C₈	—	2	4	no	—
104	C₈	—	2	1	no	—
105	C₈	—	2	1	no	—
106	C₈	—	2	1	no	—
107	C₈	—	4	2	no	—
108	C₈	—	8	4	no	—
109	C₈	—	16	2	no	—
110	C₈	C₂×#108	1	1	yes	characteristic
111	C₈	#430	1	1	yes	characteristic
112	C₂²	—	2	1	no	—
113	C₂²	—	2	1	no	—
114	C₂²	—	2	1	no	—
115	C₂²	#1942	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	2	1	no	—
119	C₄	#1950	1	1	yes	characteristic
120	C₄	#1953	1	1	yes	characteristic
121	C₄	#2178	1	2	yes	—
122	C₄	#2223	1	2	yes	—
123	C₂	—	4	1	no	—
124	C₂	#9624	1	1	yes	characteristic
125	C₂	#13169	1	1	yes	characteristic
126	C₂	#13172	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12023

C₂

yes

maximal, characteristic

#12008

C₂

yes

maximal, characteristic

C₁₆×M₁₆

C₂

yes

maximal, characteristic

#9634

C₂

yes

maximal

#7291

C₂

yes

maximal

#2162

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1574

—

#1567

C₂²

yes

—

#1564

C₂²

yes

—

C₈×M₁₆

C₂²

yes

—

—

C₄

yes

—

#1371

C₄

yes

—

#1312

—

C₈×C₁₆

—

#1299

—

C₂²×C₁₆

Q₈

yes

characteristic

C₂²×C₁₆

Q₈

yes

characteristic

#431

—

#431

D₄

yes

characteristic

#423

—

C₄×M₁₆

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

Q₈

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

D₄

yes

—

—

—

C₂×C₄

yes

—

#334

—

#334

—

#322

—

C₈²

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄⋊C₄

yes

characteristic

C₂×C₁₆

C₄⋊C₄

yes

characteristic

C₂×C₁₆

C₄⋊C₄

yes

characteristic

C₂×C₁₆

C₄⋊C₄

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₈

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

Q₁₆

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

C₄×C₈

yes

C₄×C₈

yes

C₂²×C₄

C₄⋊Q₈

yes

C₂²×C₄

yes

C₂²×C₄

yes

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

C₂×C₈

C₄×Q₈

yes

characteristic

C₂×C₈

C₂²⋊Q₈

yes

characteristic

C₂×C₈

#125

yes

characteristic

C₄²

C₄×D₄

yes

characteristic

C₄²

#121

yes

characteristic

C₄²

C₂×D₈

yes

characteristic

C₄²

C₂×Q₁₆

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

#108

yes

—

C₂³

#499

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#389

yes

characteristic

C₂×C₄

C₄×D₈

yes

characteristic

C₂×C₄

C₄×Q₁₆

yes

characteristic

100

C₂×C₄

#466

yes

characteristic

101

C₂×C₄

#467

yes

characteristic

102

C₂×C₄

#500

yes

characteristic

103

C₈

—

104

C₈

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

C₂×#108

yes

characteristic

111

C₈

#430

yes

characteristic

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

#1942

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

#1950

yes

characteristic

120

C₄

#1953

yes

characteristic

121

C₄

#2178

yes

—

122

C₄

#2223

yes

—

123

C₂

—

124

C₂

#9624

yes

characteristic

125

C₂

#13169

yes

characteristic

126

C₂

#13172

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1942
prime radical	C₂³	#499
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#466

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ 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
#12023	2
#12008	2
C₁₆×M₁₆	2
#9634	2
#7291	2

subgroup

index

C₁₆×M₁₆

Subgroups — #140484

Conjugacy classes of subgroups

Maximal subgroups