Group #146064 — subgroups

Subgroups — #146064

Order 512.

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

normal subgroups only

This group has 196 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	#13267	C₂	1	1	yes	maximal, characteristic
2	#12892	C₂	1	1	yes	maximal, characteristic
3	#8683	C₂	1	2	yes	maximal
4	#7280	C₂	1	2	yes	maximal
5	#7156	C₂	1	1	yes	maximal, characteristic
6	#2960	C₂²	1	1	yes	characteristic
7	#2150	C₂²	1	1	yes	characteristic
8	C₂×#346	C₂²	1	1	yes	characteristic
9	#1848	C₂²	1	2	yes	—
10	C₂×#334	C₂²	1	2	yes	—
11	#1550	—	2	2	no	—
12	#1381	—	2	2	no	—
13	#1379	—	2	2	no	—
14	#1292	—	2	2	no	—
15	#533	D₄	1	1	yes	characteristic
16	C₂×M₃₂	—	2	1	no	—
17	C₂²×C₁₆	D₄	1	2	yes	—
18	#446	—	2	2	no	—
19	C₈×Q₈	—	2	2	no	—
20	C₂×#108	—	2	2	no	—
21	C₂×#108	—	2	2	no	—
22	#424	—	2	1	no	—
23	#423	—	2	2	no	—
24	#423	D₄	1	1	yes	characteristic
25	C₂×C₄⋊C₈	—	2	2	no	—
26	#405	—	2	1	no	—
27	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
28	C₄⋊C₁₆	—	2	2	no	—
29	#346	—	2	2	no	—
30	#334	—	2	2	no	—
31	#334	—	2	2	no	—
32	#333	—	4	2	no	—
33	#332	—	4	2	no	—
34	C₂×M₁₆	—	2	1	no	—
35	C₂×M₁₆	—	4	2	no	—
36	C₂²×C₈	—	4	2	no	—
37	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₈	#41	1	1	yes	characteristic
39	C₄⋊Q₈	D₈	1	2	yes	—
40	#125	—	2	1	no	—
41	C₂²⋊Q₈	—	2	2	no	—
42	C₄×Q₈	—	2	2	no	—
43	#117	—	2	1	no	—
44	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
45	M₃₂	—	4	2	no	—
46	C₂×C₁₆	—	2	2	no	—
47	C₂×C₁₆	—	2	2	no	—
48	C₂×C₁₆	—	2	2	no	—
49	C₂×C₁₆	—	4	1	no	—
50	#108	—	4	2	no	—
51	#108	—	4	2	no	—
52	#108	—	8	2	no	—
53	C₄⋊C₈	—	2	2	no	—
54	C₄⋊C₈	—	4	2	no	—
55	C₄⋊C₈	—	4	2	no	—
56	C₄⋊C₈	—	4	2	no	—
57	C₄⋊C₈	—	4	2	no	—
58	C₄⋊C₈	—	4	2	no	—
59	#97	—	4	1	no	—
60	C₄×C₈	—	4	1	no	—
61	C₄×C₈	#41	1	2	yes	—
62	C₄×C₈	#41	1	2	yes	—
63	C₂×Q₈	—	2	2	no	—
64	C₂²×C₄	#121	1	1	yes	characteristic
65	C₂²×C₄	#137	1	2	yes	—
66	M₁₆	—	4	2	no	—
67	M₁₆	—	8	2	no	—
68	M₁₆	—	8	2	no	—
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	4	1	no	—
71	C₂×C₈	—	8	2	no	—
72	C₂×C₈	—	8	2	no	—
73	C₂×C₈	—	16	2	no	—
74	C₂×C₈	#123	1	1	yes	characteristic
75	C₂×C₈	#124	1	1	yes	characteristic
76	C₂×C₈	#126	1	2	yes	—
77	C₂×C₈	#135	1	2	yes	—
78	C₂×C₈	#137	1	2	yes	—
79	C₄⋊C₄	—	2	2	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	4	2	no	—
82	C₂²⋊C₄	—	4	1	no	—
83	C₄²	—	4	1	no	—
84	C₄²	#123	1	2	yes	—
85	C₄²	C₂×D₈	1	1	yes	characteristic
86	C₄²	#135	1	1	yes	characteristic
87	C₁₆	—	4	2	no	—
88	C₁₆	—	8	1	no	—
89	C₂³	#470	1	1	yes	characteristic
90	Q₈	—	8	2	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	#397	1	1	yes	characteristic
94	C₂×C₄	#464	1	2	yes	—
95	C₂×C₄	#466	1	1	yes	characteristic
96	C₂×C₄	#471	1	2	yes	—
97	C₂×C₄	#480	1	2	yes	—
98	C₂×C₄	#485	1	2	yes	—
99	C₈	—	2	2	no	—
100	C₈	—	8	1	no	—
101	C₈	—	16	2	no	—
102	C₈	—	16	2	no	—
103	C₈	#485	1	2	yes	—
104	C₈	#488	1	1	yes	characteristic
105	C₈	#491	1	1	yes	characteristic
106	C₂²	—	2	2	no	—
107	C₂²	#1704	1	2	yes	—
108	C₂²	#2075	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	—	2	2	no	—
111	C₄	—	8	2	no	—
112	C₄	#1682	1	2	yes	—
113	C₄	#2074	1	1	yes	characteristic
114	C₄	#2079	1	1	yes	characteristic
115	C₂	—	2	2	no	—
116	C₂	#10623	1	1	yes	characteristic
117	C₂	#13497	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13267

C₂

yes

maximal, characteristic

#12892

C₂

yes

maximal, characteristic

#8683

C₂

yes

maximal

#7280

C₂

yes

maximal

#7156

C₂

yes

maximal, characteristic

#2960

C₂²

yes

characteristic

#2150

C₂²

yes

characteristic

C₂×#346

C₂²

yes

characteristic

#1848

C₂²

yes

—

C₂×#334

C₂²

yes

—

—

—

—

—

D₄

yes

characteristic

C₂×M₃₂

—

C₂²×C₁₆

D₄

yes

—

#446

—

C₈×Q₈

—

—

—

—

—

D₄

yes

characteristic

C₂×C₄⋊C₈

—

#405

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

—

—

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₄⋊Q₈

D₈

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#108

—

#108

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

#41

yes

—

C₄×C₈

#41

yes

—

C₂×Q₈

—

C₂²×C₄

#121

yes

characteristic

C₂²×C₄

#137

yes

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

#135

yes

—

C₂×C₈

#137

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

#123

yes

—

C₄²

C₂×D₈

yes

characteristic

C₄²

#135

yes

characteristic

C₁₆

—

C₁₆

—

C₂³

#470

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#397

yes

characteristic

C₂×C₄

#464

yes

—

C₂×C₄

#466

yes

characteristic

C₂×C₄

#471

yes

—

C₂×C₄

#480

yes

—

C₂×C₄

#485

yes

—

C₈

—

100

C₈

—

101

C₈

—

102

C₈

—

103

C₈

#485

yes

—

104

C₈

#488

yes

characteristic

105

C₈

#491

yes

characteristic

106

C₂²

—

107

C₂²

#1704

yes

—

108

C₂²

#2075

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

#1682

yes

—

113

C₄

#2074

yes

characteristic

114

C₄

#2079

yes

characteristic

115

C₂

—

116

C₂

#10623

yes

characteristic

117

C₂

#13497

yes

—

	subgroup	G/H
center, socle	C₂²	#2075
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#470
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2075

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#470

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ 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
#13267	2
#12892	2
#8683	2
#7280	2
#7156	2

subgroup

index

Subgroups — #146064

Conjugacy classes of subgroups

Maximal subgroups