Group #129178 — subgroups

Subgroups — #129178

Order 512.

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

normal subgroups only

This group has 222 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	#12676	C₂	1	1	yes	maximal, characteristic
2	#11999	C₂	1	1	yes	maximal, characteristic
3	C₂×#1313	C₂	1	1	yes	maximal, characteristic
4	#8295	C₂	1	2	yes	maximal
5	#7177	C₂	1	2	yes	maximal
6	C₂×#368	C₄	1	2	yes	—
7	C₂×#346	C₂²	1	1	yes	characteristic
8	C₂×#346	C₂²	1	1	yes	characteristic
9	C₄×#108	C₂²	1	2	yes	—
10	#1564	C₂²	1	2	yes	—
11	#1451	—	2	2	no	—
12	C₂×#322	C₂²	1	1	yes	characteristic
13	#1381	C₄	1	2	yes	—
14	#1368	C₄	1	4	yes	—
15	#1313	C₄	1	4	yes	—
16	#1278	—	2	2	no	—
17	C₂²×C₁₆	—	2	1	no	—
18	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
20	#431	—	2	2	no	—
21	C₂×#108	C₂×C₄	1	4	yes	—
22	C₄×M₁₆	—	2	2	no	—
23	C₂×C₄×C₈	D₄	1	2	yes	—
24	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
25	#368	—	2	4	no	—
26	#346	—	2	1	no	—
27	#346	C₂×C₄	1	2	yes	—
28	#346	C₂×C₄	1	2	yes	—
29	#346	C₂×C₄	1	2	yes	—
30	#341	—	2	2	no	—
31	#335	—	2	2	no	—
32	#334	C₂×C₄	1	4	yes	—
33	#322	—	4	4	no	—
34	#322	C₂×C₄	1	2	yes	—
35	#322	Q₈	1	2	yes	—
36	C₂×M₁₆	—	2	4	no	—
37	C₂²×C₈	—	2	2	no	—
38	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₈	#41	1	1	yes	characteristic
40	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
41	C₂×C₁₆	—	2	2	no	—
42	C₂×C₁₆	—	2	2	no	—
43	C₂×C₁₆	—	4	2	no	—
44	C₂×C₁₆	C₄²	1	2	yes	—
45	C₂×C₁₆	C₄²	1	2	yes	—
46	C₂×C₁₆	M₁₆	1	4	yes	—
47	#108	—	2	4	no	—
48	#108	—	2	4	no	—
49	C₄⋊C₈	—	2	4	no	—
50	C₂²⋊C₈	—	4	2	no	—
51	#97	—	4	2	no	—
52	C₄×C₈	—	2	2	no	—
53	C₄×C₈	—	4	2	no	—
54	C₄×C₈	—	4	2	no	—
55	C₄×C₈	C₄⋊C₄	1	4	yes	—
56	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
57	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
58	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
59	C₄×C₈	#41	1	1	yes	characteristic
60	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₄	#137	1	2	yes	—
62	M₁₆	—	4	4	no	—
63	M₁₆	—	4	4	no	—
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	2	4	no	—
66	C₂×C₈	—	4	4	no	—
67	C₂×C₈	—	4	4	no	—
68	C₂×C₈	—	8	2	no	—
69	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₈	#117	1	1	yes	characteristic
71	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₈	C₂×M₁₆	1	2	yes	—
74	C₂×C₈	#131	1	2	yes	—
75	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₄²	C₄×D₄	1	1	yes	characteristic
77	C₄²	AGL(1,ℤ/8)	1	2	yes	—
78	C₁₆	—	2	2	no	—
79	C₁₆	—	4	2	no	—
80	C₁₆	#97	1	4	yes	—
81	C₂³	#441	1	1	yes	characteristic
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄	#428	1	2	yes	—
85	C₂×C₄	#436	1	2	yes	—
86	C₂×C₄	#440	1	2	yes	—
87	C₂×C₄	#442	1	1	yes	characteristic
88	C₂×C₄	#444	1	2	yes	—
89	C₈	—	2	1	no	—
90	C₈	—	2	1	no	—
91	C₈	—	4	4	no	—
92	C₈	—	8	4	no	—
93	C₈	—	8	4	no	—
94	C₈	C₂×#97	1	1	yes	characteristic
95	C₈	#405	1	1	yes	characteristic
96	C₈	#446	1	2	yes	—
97	C₂²	—	2	2	no	—
98	C₂²	#1593	1	2	yes	—
99	C₂²	#1776	1	1	yes	characteristic
100	C₄	—	2	2	no	—
101	C₄	—	2	2	no	—
102	C₄	#1594	1	2	yes	—
103	C₄	#1770	1	1	yes	characteristic
104	C₄	#1773	1	1	yes	characteristic
105	C₂	—	2	2	no	—
106	C₂	#8287	1	1	yes	characteristic
107	C₂	#12678	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#12676

C₂

yes

maximal, characteristic

#11999

C₂

yes

maximal, characteristic

C₂×#1313

C₂

yes

maximal, characteristic

#8295

C₂

yes

maximal

#7177

C₂

yes

maximal

C₂×#368

C₄

yes

—

C₂×#346

C₂²

yes

characteristic

C₂×#346

C₂²

yes

characteristic

C₄×#108

C₂²

yes

—

#1564

C₂²

yes

—

#1451

—

C₂×#322

C₂²

yes

characteristic

#1381

C₄

yes

—

#1368

C₄

yes

—

#1313

C₄

yes

—

#1278

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

#431

—

C₂×#108

C₂×C₄

yes

—

C₄×M₁₆

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

C₂³

yes

—

—

C₂×C₄

yes

—

#346

C₂×C₄

yes

—

#346

C₂×C₄

yes

—

#341

—

#335

—

#334

C₂×C₄

yes

—

#322

—

#322

C₂×C₄

yes

—

#322

Q₈

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄²

yes

—

C₂×C₁₆

C₄²

yes

—

C₂×C₁₆

M₁₆

yes

—

#108

—

#108

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂²×C₄

C₄×D₄

yes

characteristic

C₂²×C₄

#137

yes

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄²

yes

characteristic

C₂×C₈

#117

yes

characteristic

C₂×C₈

C₄×Q₈

yes

characteristic

C₂×C₈

C₄×Q₈

yes

characteristic

C₂×C₈

C₂×M₁₆

yes

—

C₂×C₈

#131

yes

—

C₄²

C₂×C₄⋊C₄

yes

characteristic

C₄²

C₄×D₄

yes

characteristic

C₄²

AGL(1,ℤ/8)

yes

—

C₁₆

—

C₁₆

—

C₁₆

#97

yes

—

C₂³

#441

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄×C₄⋊C₄

yes

characteristic

C₂×C₄

#428

yes

—

C₂×C₄

#436

yes

—

C₂×C₄

#440

yes

—

C₂×C₄

#442

yes

characteristic

C₂×C₄

#444

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

C₂×#97

yes

C₈

yes

C₈

yes

—

C₂²

—

C₂²

#1593

yes

—

C₂²

#1776

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

#1594

yes

—

103

C₄

#1770

yes

characteristic

104

C₄

#1773

yes

characteristic

105

C₂

—

106

C₂

#8287

yes

characteristic

107

C₂

#12678

yes

—

	subgroup	G/H
center	C₂×C₄	#442
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1776
prime radical	C₂³	#441
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#442

commutator

C₂×C₈

C₂×C₄²

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322 ◁ 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
#12676	2
#11999	2
C₂×#1313	2
#8295	2
#7177	2

subgroup

index

Subgroups — #129178

Conjugacy classes of subgroups

Maximal subgroups