Group #140480 — subgroups

Subgroups — #140480

Order 512.

198 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 92 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	C₂×#1372	C₂	1	1	yes	maximal, characteristic
3	C₁₆×M₁₆	C₂	1	1	yes	maximal, characteristic
4	#9624	C₂	1	2	yes	maximal
5	#7288	C₂	1	2	yes	maximal
6	#2162	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
8	#1950	—	2	2	no	—
9	#1567	C₂²	1	2	yes	—
10	C₂×#335	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	#1372	C₄	1	2	yes	—
16	#1312	—	2	1	no	—
17	C₈×C₁₆	—	2	1	no	—
18	#1301	—	2	2	no	—
19	C₂²×C₁₆	Q₈	1	1	yes	characteristic
20	C₂²×C₁₆	Q₈	1	1	yes	characteristic
21	#431	D₄	1	1	yes	characteristic
22	C₂×#108	—	4	2	no	—
23	#423	—	2	2	no	—
24	C₂×C₄⋊C₈	—	2	2	no	—
25	C₄×M₁₆	D₄	1	1	yes	characteristic
26	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
27	C₄⋊C₁₆	—	2	1	no	—
28	C₂²⋊C₁₆	Q₈	1	2	yes	—
29	C₄×C₁₆	—	2	1	no	—
30	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
31	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
32	C₄×C₁₆	D₄	1	1	yes	characteristic
33	C₄×C₁₆	D₄	1	1	yes	characteristic
34	#341	—	4	2	no	—
35	#335	—	2	2	no	—
36	#335	—	2	2	no	—
37	#335	—	2	2	no	—
38	#335	C₂×C₄	1	4	yes	—
39	#322	—	2	1	no	—
40	C₈²	—	2	1	no	—
41	C₂×M₁₆	—	4	2	no	—
42	C₂×M₁₆	D₈	1	2	yes	—
43	C₂²×C₈	—	4	2	no	—
44	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
47	C₂×C₁₆	—	2	2	no	—
48	C₂×C₁₆	—	2	2	no	—
49	C₂×C₁₆	—	4	1	no	—
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	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
54	#108	—	8	4	no	—
55	C₄⋊C₈	—	2	4	no	—
56	C₄⋊C₈	—	2	1	no	—
57	C₄⋊C₈	—	4	2	no	—
58	C₄⋊C₈	—	4	2	no	—
59	C₄⋊C₈	—	4	2	no	—
60	C₂²⋊C₈	Q₁₆	1	2	yes	—
61	#97	—	2	1	no	—
62	C₄×C₈	—	2	1	no	—
63	C₄×C₈	—	2	1	no	—
64	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
65	C₄×C₈	C₂×D₄	1	1	yes	characteristic
66	C₄×C₈	#41	1	1	yes	characteristic
67	C₄×C₈	#41	1	1	yes	characteristic
68	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
69	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
70	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	8	2	no	—
73	M₁₆	—	8	2	no	—
74	M₁₆	SD₃₂	1	4	yes	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	4	1	no	—
80	C₂×C₈	—	4	4	no	—
81	C₂×C₈	—	8	2	no	—
82	C₂×C₈	—	8	2	no	—
83	C₂×C₈	—	8	2	no	—
84	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
86	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×C₈	#125	1	1	yes	characteristic
88	C₄²	C₄×D₄	1	1	yes	characteristic
89	C₄²	#121	1	1	yes	characteristic
90	C₄²	#135	1	1	yes	characteristic
91	C₄²	#135	1	1	yes	characteristic
92	C₁₆	—	2	1	no	—
93	C₁₆	—	2	1	no	—
94	C₁₆	—	2	1	no	—
95	C₁₆	—	4	2	no	—
96	C₁₆	#108	1	2	yes	complemented
97	C₂³	#500	1	1	yes	characteristic
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	#389	1	1	yes	characteristic
102	C₂×C₄	C₄×D₈	1	1	yes	characteristic
103	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
104	C₂×C₄	#466	1	1	yes	characteristic
105	C₂×C₄	#467	1	1	yes	characteristic
106	C₂×C₄	#499	1	1	yes	characteristic
107	C₂×C₄	C₂×SD₃₂	1	2	yes	—
108	C₂×C₄	#508	1	2	yes	—
109	C₈	—	2	1	no	—
110	C₈	—	2	1	no	—
111	C₈	—	2	1	no	—
112	C₈	—	2	4	no	—
113	C₈	—	4	2	no	—
114	C₈	—	8	4	no	—
115	C₈	—	16	2	no	—
116	C₈	C₂×#108	1	1	yes	characteristic
117	C₈	#430	1	1	yes	characteristic
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	#1942	1	1	yes	characteristic
121	C₂²	#2253	1	1	yes	characteristic
122	C₂²	#2254	1	1	yes	characteristic
123	C₄	—	2	1	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	#1950	1	1	yes	characteristic
127	C₄	#1953	1	1	yes	characteristic
128	C₄	C₄×SD₃₂	1	2	yes	—
129	C₄	#2217	1	2	yes	—
130	C₂	—	2	1	no	—
131	C₂	—	2	1	no	—
132	C₂	#9624	1	1	yes	characteristic
133	C₂	#13166	1	1	yes	characteristic
134	C₂	#13173	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12023

C₂

yes

maximal, characteristic

C₂×#1372

C₂

yes

maximal, characteristic

C₁₆×M₁₆

C₂

yes

maximal, characteristic

#9624

C₂

yes

maximal

#7288

C₂

yes

maximal

#2162

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1950

—

#1567

C₂²

yes

—

C₂×#335

C₂²

yes

—

C₈×M₁₆

C₂²

yes

—

—

C₄

yes

—

#1372

C₄

yes

—

#1312

—

C₈×C₁₆

—

#1301

—

C₂²×C₁₆

Q₈

yes

characteristic

C₂²×C₁₆

Q₈

yes

characteristic

#431

D₄

yes

characteristic

C₂×#108

—

#423

—

C₂×C₄⋊C₈

—

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

—

#322

—

C₈²

—

C₂×M₁₆

—

C₂×M₁₆

D₈

yes

—

C₂²×C₈

—

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

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

Q₁₆

yes

—

#97

—

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

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

C₂×Q₁₆

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

SD₃₂

yes

—

C₂×C₈

—

C₂×C₈

—

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

C₄²

yes

C₄²

yes

C₄²

yes

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

yes

C₂³

yes

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

#389

yes

characteristic

102

C₂×C₄

C₄×D₈

yes

characteristic

103

C₂×C₄

C₄×Q₁₆

yes

characteristic

104

C₂×C₄

#466

yes

characteristic

105

C₂×C₄

#467

yes

characteristic

106

C₂×C₄

#499

yes

characteristic

107

C₂×C₄

C₂×SD₃₂

yes

—

108

C₂×C₄

#508

yes

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

C₂×#108

yes

characteristic

117

C₈

#430

yes

characteristic

118

C₂²

—

119

C₂²

—

120

C₂²

#1942

yes

characteristic

121

C₂²

#2253

yes

characteristic

122

C₂²

#2254

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#1950

yes

characteristic

127

C₄

#1953

yes

characteristic

128

C₄

C₄×SD₃₂

yes

—

129

C₄

#2217

yes

—

130

C₂

—

131

C₂

—

132

C₂

#9624

yes

characteristic

133

C₂

#13166

yes

characteristic

134

C₂

#13173

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	#500
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

N	H ≅ G/N	type
96. C₁₆	#108	semidirect

H ≅ G/N

type

96. C₁₆

#108

semidirect

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

subgroup	index
#12023	2
C₂×#1372	2
C₁₆×M₁₆	2
#9624	2
#7288	2

subgroup

index

C₁₆×M₁₆

Subgroups — #140480

Conjugacy classes of subgroups

Complements

Maximal subgroups