Group #129126 — subgroups

Subgroups — #129126

Order 512.

232 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 114 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	#12591	C₂	1	1	yes	maximal, characteristic
2	#11999	C₂	1	2	yes	maximal
3	#8267	C₂	1	1	yes	maximal, characteristic
4	#8266	C₂	1	1	yes	maximal, characteristic
5	#7157	C₂	1	2	yes	maximal
6	C₂×#346	C₂²	1	1	yes	characteristic
7	C₂×#346	C₂²	1	1	yes	characteristic
8	C₄²×C₈	C₂²	1	1	yes	characteristic
9	#1564	C₂²	1	2	yes	—
10	#1564	C₂²	1	2	yes	—
11	#1452	—	2	1	no	—
12	#1450	—	2	1	no	—
13	#1374	C₄	1	4	yes	—
14	#1368	C₄	1	8	yes	complemented
15	#1276	—	2	1	no	—
16	#1276	—	2	1	no	—
17	C₂²×C₁₆	C₂×C₄	1	2	yes	—
18	C₂²×C₁₆	C₂×C₄	1	2	yes	—
19	#431	—	2	2	no	—
20	#431	—	2	2	no	—
21	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₈	D₄	1	1	yes	characteristic
23	C₂×C₄×C₈	Q₈	1	2	yes	—
24	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
25	C₄³	D₄	1	1	yes	characteristic
26	#346	—	2	2	no	—
27	#346	C₂×C₄	1	4	yes	—
28	#335	—	2	2	no	—
29	#335	—	2	2	no	—
30	#334	C₂×C₄	1	4	yes	—
31	#334	C₂×C₄	1	4	yes	—
32	#322	—	4	1	no	—
33	#322	—	4	2	no	—
34	C₈²	—	4	1	no	—
35	C₂²×C₈	C₄⋊C₄	1	2	yes	—
36	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄²	—	2	2	no	—
39	C₂×C₄²	C₄⋊C₄	1	2	yes	—
40	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
41	C₂×C₁₆	—	2	4	no	—
42	C₂×C₁₆	—	2	2	no	—
43	C₂×C₁₆	—	2	4	no	—
44	C₂×C₁₆	C₄²	1	4	yes	—
45	C₄⋊C₈	—	2	4	no	—
46	C₄⋊C₈	—	2	4	no	—
47	C₂²⋊C₈	—	4	2	no	—
48	C₂²⋊C₈	—	4	2	no	—
49	C₄×C₈	—	2	4	no	—
50	C₄×C₈	—	4	2	no	—
51	C₄×C₈	—	4	2	no	—
52	C₄×C₈	—	4	2	no	—
53	C₄×C₈	SD₁₆	1	1	yes	characteristic
54	C₄×C₈	SD₁₆	1	1	yes	characteristic
55	C₄×C₈	SD₁₆	1	2	yes	—
56	C₄×C₈	C₂²×C₄	1	2	yes	—
57	C₄×C₈	#41	1	2	yes	—
58	C₂²×C₄	—	2	2	no	—
59	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄	#135	1	2	yes	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	4	4	no	—
65	C₂×C₈	—	4	4	no	—
66	C₂×C₈	#106	1	2	yes	—
67	C₂×C₈	#106	1	2	yes	—
68	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₈	#117	1	1	yes	characteristic
70	C₂×C₈	C₄×Q₈	1	2	yes	—
71	C₂×C₈	#131	1	4	yes	—
72	C₄²	—	2	2	no	—
73	C₄²	—	2	2	no	—
74	C₄²	—	4	4	no	—
75	C₄²	C₄×D₄	1	2	yes	—
76	C₄²	C₂×SD₁₆	1	1	yes	characteristic
77	C₄²	C₂×SD₁₆	1	1	yes	characteristic
78	C₁₆	—	2	4	no	—
79	C₁₆	—	4	2	no	—
80	C₂³	#427	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	4	2	no	—
84	C₂×C₄	—	4	2	no	—
85	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄	#424	1	2	yes	—
87	C₂×C₄	C₂×#106	1	1	yes	characteristic
88	C₂×C₄	#430	1	2	yes	—
89	C₂×C₄	C₄×SD₁₆	1	2	yes	—
90	C₂×C₄	C₄×SD₁₆	1	2	yes	—
91	C₈	—	2	2	no	—
92	C₈	—	2	2	no	—
93	C₈	—	2	2	no	—
94	C₈	—	8	4	no	—
95	C₈	—	8	4	no	—
96	C₈	#405	1	2	yes	—
97	C₈	#433	1	2	yes	—
98	C₂²	—	2	2	no	—
99	C₂²	#1572	1	2	yes	—
100	C₂²	C₄×#106	1	1	yes	characteristic
101	C₄	—	2	2	no	—
102	C₄	—	2	2	no	—
103	C₄	—	4	4	no	—
104	C₄	#1570	1	2	yes	—
105	C₄	#1768	1	1	yes	characteristic
106	C₄	#1778	1	1	yes	characteristic
107	C₂	—	2	2	no	—
108	C₂	#8277	1	1	yes	characteristic
109	C₂	#12668	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#12591

C₂

yes

maximal, characteristic

#11999

C₂

yes

maximal

#8267

C₂

yes

maximal, characteristic

#8266

C₂

yes

maximal, characteristic

#7157

C₂

yes

maximal

C₂×#346

C₂²

yes

characteristic

C₂×#346

C₂²

yes

characteristic

C₄²×C₈

C₂²

yes

characteristic

#1564

C₂²

yes

—

#1564

C₂²

yes

—

#1452

—

#1450

—

#1374

C₄

yes

—

#1368

C₄

yes

complemented

#1276

—

#1276

—

C₂²×C₁₆

C₂×C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

—

#431

—

#431

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

Q₈

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

D₄

yes

characteristic

#346

—

#346

C₂×C₄

yes

—

#335

—

#335

—

#334

C₂×C₄

yes

—

#334

C₂×C₄

yes

—

#322

—

#322

—

C₈²

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₄⋊C₄

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄²

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

—

C₄×C₈

C₂²×C₄

yes

—

C₄×C₈

#41

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

#135

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#106

yes

—

C₂×C₈

#106

yes

—

C₂×C₈

C₂×C₄²

yes

characteristic

C₂×C₈

#117

yes

characteristic

C₂×C₈

C₄×Q₈

yes

—

C₂×C₈

#131

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄×D₄

yes

—

C₄²

C₂×SD₁₆

yes

characteristic

C₄²

C₂×SD₁₆

yes

characteristic

C₁₆

—

C₁₆

—

C₂³

#427

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×C₄⋊C₄

yes

characteristic

C₂×C₄

#424

yes

—

C₂×C₄

C₂×#106

yes

characteristic

C₂×C₄

#430

yes

—

C₂×C₄

C₄×SD₁₆

yes

—

C₂×C₄

C₄×SD₁₆

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

#405

yes

—

C₈

#433

yes

—

C₂²

—

C₂²

#1572

yes

—

100

C₂²

C₄×#106

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

#1570

yes

—

105

C₄

#1768

yes

characteristic

106

C₄

#1778

yes

characteristic

107

C₂

—

108

C₂

#8277

yes

characteristic

109

C₂

#12668

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×#106
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₄×#106
prime radical	C₂³	#427
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

C₂×#106

commutator

C₂×C₈

C₂×C₄²

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
14. #1368	C₄	semidirect

H ≅ G/N

type

14. #1368

C₄

semidirect

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

subgroup	index
#12591	2
#11999	2
#8267	2
#8266	2
#7157	2

subgroup

index

Subgroups — #129126

Conjugacy classes of subgroups

Complements

Maximal subgroups