Group #152990 — subgroups

Subgroups — #152990

Order 512.

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

normal subgroups only

This group has 368 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	C₂×#2246	C₂	1	1	yes	maximal, characteristic
2	#13627	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#1419	C₂	1	4	yes	maximal, complemented
4	C₄×#368	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₁₆	—	2	1	no	—
6	C₂²×D₁₆	C₂²	1	2	yes	—
7	C₂×#493	—	2	1	no	—
8	C₂×M₆₄	C₂²	1	2	yes	—
9	#2246	—	2	2	no	—
10	#2246	C₄	1	4	yes	—
11	C₂×#368	C₂²	1	2	yes	—
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	C₄×#108	—	2	1	no	—
14	C₄⋊C₃₂	—	2	2	no	—
15	#1419	—	2	8	no	—
16	#1381	—	2	1	no	—
17	C₂²×D₈	—	2	2	no	—
18	C₂²×D₈	—	4	1	no	—
19	C₂×#127	—	4	1	no	—
20	C₂×D₁₆	—	2	4	no	—
21	C₂×D₁₆	—	2	4	no	—
22	C₂×D₁₆	—	2	2	no	—
23	C₂×D₁₆	—	2	4	no	—
24	C₂×D₁₆	—	4	4	no	—
25	C₂×D₁₆	C₂×C₄	1	4	yes	—
26	C₂²×C₁₆	D₄	1	1	yes	characteristic
27	C₂²×C₁₆	C₂³	1	1	yes	characteristic
28	#493	—	2	4	no	—
29	#493	—	4	2	no	—
30	C₂×#108	—	2	2	no	—
31	C₄×M₁₆	—	4	1	no	—
32	C₂×C₄×C₈	D₄	1	1	yes	characteristic
33	M₆₄	—	2	4	no	—
34	C₂×C₃₂	—	2	2	no	—
35	#368	—	2	2	no	—
36	#368	D₄	1	4	yes	complemented
37	C₄×C₁₆	C₂×C₄	1	2	yes	—
38	C₄×C₁₆	D₄	1	2	yes	—
39	#341	—	4	1	no	—
40	C₂²×D₄	—	4	2	no	—
41	C₂²×D₄	—	8	2	no	—
42	C₂×D₈	—	2	4	no	—
43	C₂×D₈	—	4	4	no	—
44	C₂×D₈	—	4	4	no	—
45	C₂×D₈	—	4	2	no	—
46	C₂×D₈	—	4	4	no	—
47	C₂×D₈	—	8	4	no	—
48	C₂×M₁₆	—	4	2	no	—
49	C₂²×C₈	D₈	1	1	yes	characteristic
50	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
51	#127	—	4	4	no	—
52	#127	—	8	1	no	—
53	#127	—	8	1	no	—
54	C₂×C₄²	D₈	1	1	yes	characteristic
55	D₁₆	—	2	8	no	—
56	D₁₆	—	4	8	no	—
57	D₁₆	—	4	2	no	—
58	D₁₆	—	4	2	no	—
59	C₂×C₁₆	—	2	2	no	—
60	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
61	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₁₆	C₂×D₄	1	2	yes	—
63	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
64	C₂×C₁₆	#41	1	2	yes	—
65	#108	—	2	4	no	—
66	#108	—	4	2	no	—
67	#97	—	8	1	no	—
68	C₄×C₈	—	8	1	no	—
69	C₄×C₈	C₂²⋊C₄	1	2	yes	—
70	C₄×C₈	SD₁₆	1	2	yes	—
71	C₃₂	—	4	2	no	—
72	C₂⁴	—	8	2	no	—
73	C₂×D₄	—	4	4	no	—
74	C₂×D₄	—	8	4	no	—
75	C₂×D₄	—	8	4	no	—
76	C₂×D₄	—	8	4	no	—
77	C₂×D₄	—	8	2	no	—
78	C₂×D₄	—	8	4	no	—
79	C₂×D₄	—	16	8	no	—
80	C₂²×C₄	SD₃₂	1	2	yes	—
81	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
82	D₈	—	4	4	no	—
83	D₈	—	4	4	no	—
84	D₈	—	8	8	no	—
85	D₈	—	8	4	no	—
86	M₁₆	—	4	4	no	—
87	M₁₆	—	8	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	8	2	no	—
90	C₂×C₈	#102	1	2	yes	—
91	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈	#121	1	2	yes	—
93	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
95	C₄²	#102	1	2	yes	—
96	C₄²	D₁₆	1	1	yes	characteristic
97	C₄²	D₁₆	1	1	yes	characteristic
98	C₁₆	—	2	2	no	—
99	C₁₆	C₄×D₄	1	2	yes	—
100	C₁₆	#124	1	1	yes	characteristic
101	C₁₆	#127	1	1	yes	characteristic
102	C₂³	—	8	4	no	—
103	C₂³	—	16	8	no	—
104	C₂³	—	16	4	no	—
105	C₂³	C₂×SD₃₂	1	1	yes	characteristic
106	D₄	—	8	4	no	—
107	D₄	—	8	4	no	—
108	D₄	—	16	8	no	—
109	D₄	—	16	8	no	—
110	D₄	—	16	4	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	#357	1	2	yes	—
113	C₂×C₄	#357	1	2	yes	—
114	C₂×C₄	C₂×#102	1	1	yes	characteristic
115	C₂×C₄	#459	1	2	yes	—
116	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
117	C₂×C₄	#509	1	2	yes	—
118	C₈	—	2	2	no	—
119	C₈	—	8	4	no	—
120	C₈	#390	1	1	yes	characteristic
121	C₈	#418	1	1	yes	characteristic
122	C₈	#460	1	2	yes	—
123	C₂²	—	2	2	no	—
124	C₂²	—	16	8	no	—
125	C₂²	—	16	4	no	—
126	C₂²	—	32	8	no	—
127	C₂²	C₂×#357	1	1	yes	characteristic
128	C₂²	#2207	1	2	yes	—
129	C₄	—	2	2	no	—
130	C₄	#1419	1	4	yes	complemented
131	C₄	#1982	1	1	yes	characteristic
132	C₄	#2140	1	1	yes	characteristic
133	C₄	#2206	1	2	yes	—
134	C₂	—	2	2	no	—
135	C₂	—	32	4	no	—
136	C₂	#13252	1	1	yes	characteristic
137	C₂	C₂×#1419	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2246

C₂

yes

maximal, characteristic

#13627

C₂

yes

maximal, characteristic, complemented

C₂×#1419

C₂

yes

maximal, complemented

C₄×#368

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₆

—

C₂²×D₁₆

C₂²

yes

—

C₂×#493

—

C₂×M₆₄

C₂²

yes

—

#2246

—

#2246

C₄

yes

—

C₂×#368

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₄×#108

—

C₄⋊C₃₂

—

#1419

—

#1381

—

C₂²×D₈

—

C₂²×D₈

—

C₂×#127

—

C₂×D₁₆

—

C₂×D₁₆

—

C₂×D₁₆

—

C₂×D₁₆

—

C₂×D₁₆

—

C₂×D₁₆

C₂×C₄

yes

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

C₄×M₁₆

—

C₂×C₄×C₈

D₄

yes

characteristic

M₆₄

—

C₂×C₃₂

—

#368

—

#368

D₄

yes

complemented

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

D₄

yes

—

#341

—

C₂²×D₄

—

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×M₁₆

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

—

—

—

C₂×C₄²

D₈

yes

characteristic

D₁₆

—

D₁₆

—

D₁₆

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

—

C₂×C₁₆

C₂²×C₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

#41

yes

—

#108

—

#108

—

#97

—

C₄×C₈

—

C₄×C₈

C₂²⋊C₄

yes

—

C₄×C₈

SD₁₆

yes

—

C₃₂

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

SD₃₂

yes

—

C₂²×C₄

C₂×D₈

yes

characteristic

D₈

—

D₈

—

D₈

—

D₈

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#102

yes

—

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₈

#121

yes

—

C₂×C₈

C₂×SD₁₆

yes

characteristic

C₂×C₈

AGL(1,ℤ/8)

yes

—

C₄²

#102

yes

—

C₄²

D₁₆

yes

characteristic

C₄²

D₁₆

yes

characteristic

C₁₆

—

C₁₆

C₄×D₄

yes

—

100

C₁₆

#124

yes

characteristic

101

C₁₆

#127

yes

characteristic

102

C₂³

—

103

C₂³

—

104

C₂³

—

105

C₂³

C₂×SD₃₂

yes

characteristic

106

D₄

—

107

D₄

—

108

D₄

—

109

D₄

—

110

D₄

—

111

C₂×C₄

—

112

C₂×C₄

#357

yes

—

113

C₂×C₄

#357

yes

—

114

C₂×C₄

C₂×#102

yes

characteristic

115

C₂×C₄

#459

yes

—

116

C₂×C₄

C₂×D₁₆

yes

characteristic

117

C₂×C₄

#509

yes

—

118

C₈

—

119

C₈

—

120

C₈

#390

yes

characteristic

121

C₈

#418

yes

characteristic

122

C₈

#460

yes

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

C₂×#357

yes

characteristic

128

C₂²

#2207

yes

—

129

C₄

—

130

C₄

#1419

yes

complemented

131

C₄

#1982

yes

characteristic

132

C₄

#2140

yes

characteristic

133

C₄

#2206

yes

—

134

C₂

—

135

C₂

—

136

C₂

#13252

yes

characteristic

137

C₂

C₂×#1419

yes

—

	subgroup	G/H
center, socle	C₂²	C₂×#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#2246	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂×#357

commutator

C₂×C₁₆

C₂²×C₄

Frattini, cosocle

C₂²×C₁₆

C₂³

prime radical

C₂×#2246

C₂

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #13627	C₂	semidirect
3. C₂×#1419	C₂	semidirect
4. C₄×#368	C₂	semidirect
36. #368	D₄	semidirect
130. C₄	#1419	semidirect

H ≅ G/N

type

2. #13627

C₂

semidirect

3. C₂×#1419

C₂

semidirect

4. C₄×#368

C₂

semidirect

36. #368

D₄

semidirect

130. C₄

#1419

semidirect

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

subgroup	index
C₂×#2246	2
#13627	2
C₂×#1419	2
C₄×#368	2

subgroup

index

Subgroups — #152990

Conjugacy classes of subgroups

Complements

Maximal subgroups