Group #151986 — subgroups

Subgroups — #151986

Order 512.

218 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 146 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×#1422	C₂	1	1	yes	maximal, characteristic
2	#13549	C₂	1	2	yes	maximal
3	C₂×C₄×C₃₂	C₂	1	1	yes	maximal, characteristic
4	#12664	C₂	1	1	yes	maximal, characteristic
5	#7470	C₂	1	2	yes	maximal
6	C₂×M₆₄	C₂²	1	2	yes	—
7	C₂²×C₃₂	C₂²	1	1	yes	characteristic
8	C₂²×C₃₂	C₂²	1	1	yes	characteristic
9	C₂×#364	C₄	1	1	yes	characteristic
10	#2108	C₂²	1	2	yes	—
11	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	#1772	C₄	1	1	yes	characteristic
13	#1422	C₄	1	4	yes	complemented
14	#1397	—	2	2	no	—
15	C₄×C₃₂	—	2	2	no	—
16	C₄×C₃₂	C₄	1	4	yes	complemented
17	#1379	C₄	1	2	yes	—
18	C₂×M₃₂	C₂×C₄	1	2	yes	—
19	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₁₆	C₂³	1	1	yes	characteristic
21	C₂×#108	C₈	1	1	yes	characteristic
22	#427	C₈	1	1	yes	characteristic
23	#405	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
25	M₆₄	—	2	2	no	—
26	M₆₄	C₂×C₄	1	4	yes	—
27	C₂×C₃₂	—	2	2	no	—
28	C₂×C₃₂	—	2	1	no	—
29	C₂×C₃₂	—	2	1	no	—
30	C₂×C₃₂	C₂×C₄	1	2	yes	—
31	C₂×C₃₂	C₂×C₄	1	2	yes	—
32	C₂×C₃₂	D₄	1	1	yes	characteristic
33	C₂×C₃₂	D₄	1	1	yes	characteristic
34	C₂×C₃₂	Q₈	1	2	yes	—
35	#364	C₈	1	4	yes	—
36	#346	—	2	2	no	—
37	C₄×C₁₆	—	2	2	no	—
38	C₄×C₁₆	C₂×C₄	1	2	yes	—
39	C₄×C₁₆	C₂×C₄	1	2	yes	—
40	#339	C₈	1	2	yes	—
41	C₂×M₁₆	C₂×C₈	1	2	yes	—
42	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
44	#117	C₂×C₈	1	2	yes	—
45	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
46	M₃₂	—	2	2	no	—
47	M₃₂	C₂×C₈	1	4	yes	—
48	C₂×C₁₆	—	2	2	no	—
49	C₂×C₁₆	—	2	1	no	—
50	C₂×C₁₆	—	2	1	no	—
51	C₂×C₁₆	C₄²	1	2	yes	—
52	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
55	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₁₆	#41	1	1	yes	characteristic
57	C₂×C₁₆	#41	1	1	yes	characteristic
58	#108	C₁₆	1	4	yes	—
59	#107	C₁₆	1	2	yes	—
60	#106	C₁₆	1	2	yes	—
61	#97	—	2	2	no	—
62	C₄×C₈	—	2	2	no	—
63	C₄×C₈	C₄²	1	2	yes	—
64	C₄×C₈	C₂×C₈	1	2	yes	—
65	C₃₂	—	2	2	no	—
66	C₃₂	—	2	4	no	—
67	C₃₂	C₄⋊C₄	1	2	yes	—
68	C₃₂	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄	—	2	1	no	—
70	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	C₂×C₁₆	1	4	yes	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	C₄×C₈	1	2	yes	—
75	C₂×C₈	C₂×C₁₆	1	1	yes	characteristic
76	C₂×C₈	C₂×C₁₆	1	1	yes	characteristic
77	C₂×C₈	C₂×C₁₆	1	2	yes	—
78	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₈	C₄×D₄	1	1	yes	characteristic
80	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
81	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₈	#131	1	1	yes	characteristic
83	C₂×C₈	#131	1	1	yes	characteristic
84	C₄⋊C₄	C₂×C₁₆	1	4	yes	—
85	C₂²⋊C₄	—	2	2	no	—
86	C₄²	—	2	2	no	—
87	C₄²	—	2	1	no	—
88	C₄²	C₄×C₈	1	2	yes	—
89	C₁₆	—	2	2	no	—
90	C₁₆	—	2	4	no	—
91	C₁₆	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₁₆	#117	1	1	yes	characteristic
93	C₁₆	C₄×D₄	1	1	yes	characteristic
94	C₁₆	C₄×Q₈	1	1	yes	characteristic
95	C₂³	#504	1	1	yes	characteristic
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
100	C₂×C₄	C₈×D₄	1	1	yes	characteristic
101	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
102	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
103	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
104	C₂×C₄	#504	1	1	yes	characteristic
105	C₈	—	2	4	no	—
106	C₈	C₄×C₁₆	1	2	yes	—
107	C₈	C₄×C₁₆	1	2	yes	—
108	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
109	C₈	#405	1	1	yes	characteristic
110	C₈	C₈×D₄	1	1	yes	characteristic
111	C₈	C₈×Q₈	1	1	yes	characteristic
112	C₂²	—	2	2	no	—
113	C₂²	#2108	1	1	yes	characteristic
114	C₂²	D₄×C₁₆	1	1	yes	characteristic
115	C₂²	Q₈×C₁₆	1	1	yes	characteristic
116	C₄	—	2	4	no	—
117	C₄	—	4	4	no	—
118	C₄	C₈×C₄⋊C₄	1	1	yes	characteristic
119	C₄	C₂×C₄×C₁₆	1	1	yes	characteristic
120	C₄	D₄×C₁₆	1	1	yes	characteristic
121	C₄	Q₈×C₁₆	1	1	yes	characteristic
122	C₂	—	2	1	no	—
123	C₂	—	2	1	no	—
124	C₂	C₁₆×C₄⋊C₄	1	1	yes	characteristic
125	C₂	#13647	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1422

C₂

yes

maximal, characteristic

#13549

C₂

yes

maximal

C₂×C₄×C₃₂

C₂

yes

maximal, characteristic

#12664

C₂

yes

maximal, characteristic

#7470

C₂

yes

maximal

C₂×M₆₄

C₂²

yes

—

C₂²×C₃₂

C₂²

yes

characteristic

C₂²×C₃₂

C₂²

yes

characteristic

C₂×#364

C₄

yes

characteristic

#2108

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1772

C₄

yes

characteristic

#1422

C₄

yes

complemented

#1397

—

C₄×C₃₂

—

C₄×C₃₂

C₄

yes

complemented

#1379

C₄

yes

—

C₂×M₃₂

C₂×C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

characteristic

C₂×#108

C₈

yes

characteristic

#427

C₈

yes

characteristic

#405

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

M₆₄

—

M₆₄

C₂×C₄

yes

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂×C₄

yes

—

C₂×C₃₂

C₂×C₄

yes

—

C₂×C₃₂

D₄

yes

characteristic

C₂×C₃₂

D₄

yes

characteristic

C₂×C₃₂

Q₈

yes

—

#364

C₈

yes

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

—

#339

C₈

yes

—

C₂×M₁₆

C₂×C₈

yes

—

C₂²×C₈

C₂×C₈

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

#117

C₂×C₈

yes

—

C₂×C₄²

C₂×C₈

yes

characteristic

M₃₂

—

M₃₂

C₂×C₈

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄²

yes

—

C₂×C₁₆

C₂²×C₄

yes

characteristic

C₂×C₁₆

C₂²×C₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×Q₈

yes

C₂×C₁₆

yes

C₂×C₁₆

yes

C₁₆

yes

—

#107

C₁₆

yes

—

#106

C₁₆

yes

—

#97

—

C₄×C₈

—

C₄×C₈

C₄²

yes

—

C₄×C₈

C₂×C₈

yes

—

C₃₂

—

C₃₂

—

C₃₂

C₄⋊C₄

yes

—

C₃₂

C₄⋊C₄

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²×C₈

yes

characteristic

M₁₆

—

M₁₆

C₂×C₁₆

yes

—

C₂×C₈

—

C₂×C₈

C₄×C₈

yes

—

C₂×C₈

C₂×C₁₆

yes

characteristic

C₂×C₈

C₂×C₁₆

yes

characteristic

C₂×C₈

C₂×C₁₆

yes

—

C₂×C₈

C₂×C₄²

yes

characteristic

C₂×C₈

C₄×D₄

yes

characteristic

C₂×C₈

C₄×Q₈

yes

characteristic

C₂×C₈

C₂²×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

C₂×C₁₆

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₄×C₈

yes

—

C₁₆

—

C₁₆

—

C₁₆

C₂×C₄⋊C₄

yes

characteristic

C₁₆

#117

yes

characteristic

C₁₆

C₄×D₄

yes

characteristic

C₁₆

C₄×Q₈

yes

characteristic

C₂³

#504

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄×C₈

yes

characteristic

100

C₂×C₄

C₈×D₄

yes

characteristic

101

C₂×C₄

C₈×Q₈

yes

characteristic

102

C₂×C₄

C₂²×C₁₆

yes

characteristic

103

C₂×C₄

C₂²×C₁₆

yes

characteristic

104

C₂×C₄

#504

yes

characteristic

105

C₈

—

106

C₈

C₄×C₁₆

yes

—

107

C₈

C₄×C₁₆

yes

—

108

C₈

C₄×C₄⋊C₄

yes

characteristic

109

C₈

#405

yes

characteristic

110

C₈

C₈×D₄

yes

characteristic

111

C₈

C₈×Q₈

yes

characteristic

112

C₂²

—

113

C₂²

#2108

yes

characteristic

114

C₂²

D₄×C₁₆

yes

characteristic

115

C₂²

Q₈×C₁₆

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

C₈×C₄⋊C₄

yes

characteristic

119

C₄

C₂×C₄×C₁₆

yes

characteristic

120

C₄

D₄×C₁₆

yes

characteristic

121

C₄

Q₈×C₁₆

yes

characteristic

122

C₂

—

123

C₂

—

124

C₂

C₁₆×C₄⋊C₄

yes

characteristic

125

C₂

#13647

yes

—

	subgroup	G/H
center	C₂×C₃₂	D₄
commutator	C₄	C₂×C₄×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	D₄×C₁₆
prime radical	C₂³	#504
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₂ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₃₂

D₄

commutator

C₄

C₂×C₄×C₁₆

Frattini, cosocle

C₂²×C₁₆

C₂³

socle

C₂²

D₄×C₁₆

C₂³

G ▷ C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂×C₃₂ ◁ C₂²×C₃₂ ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
13. #1422	C₄	semidirect
16. C₄×C₃₂	C₄	semidirect

H ≅ G/N

type

13. #1422

C₄

semidirect

16. C₄×C₃₂

C₄

semidirect

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

subgroup	index
C₂×#1422	2
#13549	2
C₂×C₄×C₃₂	2
#12664	2
#7470	2

subgroup

index

C₂×C₄×C₃₂

Subgroups — #151986

Conjugacy classes of subgroups

Complements

Maximal subgroups