Group #153385 — subgroups

Subgroups — #153385

Order 512.

165 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 165 conjugacy classes of subgroups — too many to draw. Only its 70 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	#13638	C₂	1	2	yes	maximal
2	#13170	C₂	1	1	yes	maximal, characteristic
3	#12910	C₂	1	1	yes	maximal, characteristic
4	#7476	C₂	1	2	yes	maximal
5	#7466	C₂	1	1	yes	maximal, characteristic
6	C₂×M₆₄	C₂²	1	2	yes	—
7	C₂×#368	—	2	1	no	—
8	C₂×#368	C₂²	1	1	yes	characteristic
9	C₂×#368	C₂²	1	2	yes	—
10	#2158	C₂²	1	1	yes	characteristic
11	C₂×#346	C₂²	1	1	yes	characteristic
12	#1945	—	2	1	no	—
13	#1848	—	2	1	no	—
14	#1426	—	2	4	no	—
15	#1382	—	2	1	no	—
16	C₂²×C₁₆	D₄	1	1	yes	characteristic
17	C₂²×C₁₆	C₂³	1	1	yes	characteristic
18	C₂×#108	—	2	2	no	—
19	C₂×#108	—	2	1	no	—
20	C₂×#108	—	4	1	no	—
21	#427	—	2	1	no	—
22	#424	—	4	1	no	—
23	#423	—	4	1	no	—
24	C₂×C₄×C₈	D₄	1	1	yes	characteristic
25	M₆₄	—	2	2	no	—
26	M₆₄	—	2	2	no	—
27	C₂×C₃₂	Q₈	1	4	yes	—
28	#368	—	2	2	no	—
29	#368	—	2	2	no	—
30	#368	—	2	2	no	—
31	#368	—	4	2	no	—
32	#367	D₄	1	2	yes	—
33	#366	—	2	1	no	—
34	#346	—	2	2	no	—
35	#339	—	4	1	no	—
36	C₂×M₁₆	—	4	1	no	—
37	C₂×M₁₆	—	4	2	no	—
38	C₂²×C₈	D₈	1	1	yes	characteristic
39	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
40	#117	—	4	1	no	—
41	C₂×C₄²	D₈	1	1	yes	characteristic
42	C₂×C₁₆	—	2	1	no	—
43	C₂×C₁₆	—	2	1	no	—
44	C₂×C₁₆	Q₁₆	1	2	yes	—
45	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
46	C₂×C₁₆	C₂×Q₈	1	2	yes	—
47	C₂×C₁₆	#41	1	2	yes	—
48	C₂×C₁₆	#41	1	1	yes	characteristic
49	#108	—	4	2	no	—
50	#108	—	4	1	no	—
51	#108	—	4	2	no	—
52	#108	—	4	1	no	—
53	#108	—	8	2	no	—
54	#107	—	2	2	no	—
55	#106	—	4	1	no	—
56	C₄⋊C₈	—	8	2	no	—
57	C₄⋊C₈	—	8	2	no	—
58	C₄×C₈	—	2	1	no	—
59	C₄×C₈	—	2	1	no	—
60	C₃₂	—	2	4	no	—
61	C₂²×C₄	—	2	1	no	—
62	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
63	M₁₆	—	8	2	no	—
64	M₁₆	—	8	1	no	—
65	M₁₆	—	8	2	no	—
66	M₁₆	—	8	1	no	—
67	C₂×C₈	—	2	1	no	—
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	8	2	no	—
70	C₂×C₈	—	8	1	no	—
71	C₂×C₈	SD₃₂	1	2	yes	—
72	C₂×C₈	#121	1	1	yes	characteristic
73	C₂×C₈	#123	1	1	yes	characteristic
74	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂×C₈	#137	1	1	yes	characteristic
78	C₄⋊C₄	—	4	2	no	—
79	C₂²⋊C₄	—	8	1	no	—
80	C₄²	—	2	1	no	—
81	C₄²	—	8	1	no	—
82	C₄²	SD₃₂	1	2	yes	—
83	C₁₆	—	2	2	no	—
84	C₁₆	C₂²⋊Q₈	1	2	yes	—
85	C₁₆	#125	1	2	yes	—
86	C₂³	#510	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	8	1	no	—
89	C₂×C₄	—	8	2	no	—
90	C₂×C₄	#459	1	1	yes	characteristic
91	C₂×C₄	#480	1	1	yes	characteristic
92	C₂×C₄	#500	1	1	yes	characteristic
93	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
94	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
95	C₂×C₄	#509	1	1	yes	characteristic
96	C₈	—	2	2	no	—
97	C₈	—	16	2	no	—
98	C₈	—	16	1	no	—
99	C₈	#398	1	1	yes	characteristic
100	C₈	#462	1	1	yes	characteristic
101	C₈	#484	1	1	yes	characteristic
102	C₈	#501	1	1	yes	characteristic
103	C₂²	—	4	1	no	—
104	C₂²	#2208	1	1	yes	characteristic
105	C₂²	#2235	1	1	yes	characteristic
106	C₂²	#2255	1	1	yes	characteristic
107	C₄	—	4	2	no	—
108	C₄	—	16	2	no	—
109	C₄	#2083	1	1	yes	characteristic
110	C₄	#2207	1	1	yes	characteristic
111	C₄	#2233	1	1	yes	characteristic
112	C₄	#2254	1	1	yes	characteristic
113	C₂	—	2	1	no	—
114	C₂	—	2	1	no	—
115	C₂	—	2	1	no	—
116	C₂	#13506	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13638

C₂

yes

maximal

#13170

C₂

yes

maximal, characteristic

#12910

C₂

yes

maximal, characteristic

#7476

C₂

yes

maximal

#7466

C₂

yes

maximal, characteristic

C₂×M₆₄

C₂²

yes

—

C₂×#368

—

C₂×#368

C₂²

yes

characteristic

C₂×#368

C₂²

yes

—

#2158

C₂²

yes

characteristic

C₂×#346

C₂²

yes

—

—

—

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

M₆₄

—

M₆₄

—

C₂×C₃₂

Q₈

yes

—

—

—

—

—

D₄

yes

—

#366

—

#346

—

#339

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#117

—

C₂×C₄²

D₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

Q₁₆

yes

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×Q₈

yes

—

C₂×C₁₆

#41

yes

—

C₂×C₁₆

yes

—

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₃₂

—

C₂²×C₄

—

C₂²×C₄

C₂×D₈

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

SD₃₂

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

C₂×Q₁₆

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

SD₃₂

yes

—

C₁₆

—

C₁₆

C₂²⋊Q₈

yes

—

C₁₆

#125

yes

—

C₂³

#510

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×SD₃₂

yes

characteristic

C₂×C₄

C₂×SD₃₂

yes

characteristic

C₂×C₄

#509

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

#398

yes

characteristic

100

C₈

#462

yes

characteristic

101

C₈

#484

yes

characteristic

102

C₈

#501

yes

characteristic

103

C₂²

—

104

C₂²

#2208

yes

characteristic

105

C₂²

#2235

yes

characteristic

106

C₂²

#2255

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

#2083

yes

characteristic

110

C₄

#2207

yes

characteristic

111

C₄

#2233

yes

characteristic

112

C₄

#2254

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

—

116

C₂

#13506

yes

characteristic

	subgroup	G/H
center	C₄	#2254
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13506
prime radical	C₂³	#510
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄

#2254

commutator, Frattini, cosocle

C₂²×C₁₆

C₂³

C₂

C₂³

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#13638	2
#13170	2
#12910	2
#7476	2
#7466	2

subgroup

index

Subgroups — #153385

Conjugacy classes of subgroups

Maximal subgroups