Group #146243 — subgroups

Subgroups — #146243

Order 512.

246 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 92 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	#13286	C₂	1	2	yes	maximal
2	#12910	C₂	1	2	yes	maximal
3	#9861	C₂	1	1	yes	maximal, characteristic
4	#8826	C₂	1	1	yes	maximal, characteristic
5	#7137	C₂	1	1	yes	maximal, characteristic
6	#3160	C₂²	1	1	yes	characteristic
7	C₂×#368	—	2	4	no	—
8	C₂×#346	C₂²	1	2	yes	—
9	#1848	C₂²	1	1	yes	characteristic
10	#1848	C₂²	1	2	yes	—
11	#1567	—	2	2	no	—
12	#1493	—	2	2	no	—
13	C₂×#322	C₂²	1	1	yes	characteristic
14	#1382	—	2	4	no	—
15	#533	—	2	1	no	—
16	C₂²×C₁₆	D₄	1	4	yes	—
17	#500	Q₈	1	2	yes	—
18	#499	—	2	2	no	—
19	#498	—	2	1	no	—
20	C₂×#108	—	2	2	no	—
21	C₂×#108	—	2	2	no	—
22	C₂×#108	—	2	2	no	—
23	#427	—	2	2	no	—
24	#423	—	2	2	no	—
25	#423	—	2	1	no	—
26	C₂×C₄×C₈	D₄	1	2	yes	—
27	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
28	#368	—	4	8	no	—
29	#346	—	2	2	no	—
30	#346	—	2	2	no	—
31	#335	—	4	4	no	—
32	Q₈⋊C₈	—	4	4	no	—
33	#322	—	2	2	no	—
34	C₂×M₁₆	—	4	2	no	—
35	C₂×M₁₆	—	4	1	no	—
36	C₂²×C₈	—	2	2	no	—
37	C₂²×C₈	C₂×D₄	1	2	yes	—
38	C₄⋊Q₈	—	2	2	no	—
39	#125	—	4	1	no	—
40	C₂²⋊Q₈	—	4	2	no	—
41	C₄×Q₈	—	4	2	no	—
42	#117	—	4	1	no	—
43	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
44	C₂×C₁₆	—	2	4	no	—
45	C₂×C₁₆	—	2	4	no	—
46	C₂×C₁₆	Q₁₆	1	8	yes	—
47	#108	—	4	4	no	—
48	#108	—	4	4	no	—
49	#108	—	4	4	no	—
50	#107	—	2	4	no	—
51	#106	—	4	2	no	—
52	C₄⋊C₈	—	4	2	no	—
53	C₄⋊C₈	—	4	4	no	—
54	C₄×C₈	—	2	2	no	—
55	C₄×C₈	D₈	1	4	yes	—
56	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
57	C₄×C₈	#41	1	1	yes	characteristic
58	C₄×C₈	#41	1	2	yes	—
59	C₂×Q₈	—	4	2	no	—
60	C₂²×C₄	#127	1	1	yes	characteristic
61	C₂²×C₄	#137	1	2	yes	—
62	M₁₆	—	8	2	no	—
63	M₁₆	—	8	2	no	—
64	M₁₆	—	8	2	no	—
65	C₂×C₈	—	2	4	no	—
66	C₂×C₈	—	8	2	no	—
67	C₂×C₈	—	8	1	no	—
68	C₂×C₈	—	8	2	no	—
69	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
70	C₂×C₈	#123	1	2	yes	—
71	C₂×C₈	C₂×Q₁₆	1	4	yes	—
72	C₂×C₈	AGL(1,ℤ/8)	1	4	yes	—
73	C₄⋊C₄	—	4	2	no	—
74	C₄⋊C₄	—	4	2	no	—
75	C₄⋊C₄	—	8	2	no	—
76	C₂²⋊C₄	—	8	1	no	—
77	C₄²	—	8	1	no	—
78	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
79	C₄²	#123	1	1	yes	characteristic
80	C₄²	C₂×D₈	1	2	yes	—
81	C₁₆	—	2	8	no	—
82	C₂³	#497	1	1	yes	characteristic
83	Q₈	—	8	4	no	—
84	C₂×C₄	—	4	2	no	—
85	C₂×C₄	—	8	2	no	—
86	C₂×C₄	#399	1	1	yes	characteristic
87	C₂×C₄	#474	1	2	yes	—
88	C₂×C₄	#480	1	2	yes	—
89	C₂×C₄	#493	1	1	yes	characteristic
90	C₂×C₄	#494	1	2	yes	—
91	C₂×C₄	#496	1	2	yes	—
92	C₈	—	4	4	no	—
93	C₈	—	16	1	no	—
94	C₈	—	16	2	no	—
95	C₈	#477	1	4	yes	—
96	C₈	#484	1	4	yes	—
97	C₂²	—	2	2	no	—
98	C₂²	#1723	1	2	yes	—
99	C₂²	#2093	1	1	yes	characteristic
100	C₄	—	2	2	no	—
101	C₄	—	2	2	no	—
102	C₄	—	16	2	no	—
103	C₄	#1713	1	1	yes	characteristic
104	C₄	#1721	1	1	yes	characteristic
105	C₄	#2095	1	2	yes	—
106	C₂	—	2	2	no	—
107	C₂	#10674	1	1	yes	characteristic
108	C₂	#13532	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13286

C₂

yes

maximal

#12910

C₂

yes

maximal

#9861

C₂

yes

maximal, characteristic

#8826

C₂

yes

maximal, characteristic

#7137

C₂

yes

maximal, characteristic

#3160

C₂²

yes

characteristic

C₂×#368

—

C₂×#346

C₂²

yes

—

#1848

C₂²

yes

characteristic

#1848

C₂²

yes

—

#1567

—

#1493

—

C₂×#322

C₂²

yes

characteristic

#1382

—

#533

—

C₂²×C₁₆

D₄

yes

—

#500

Q₈

yes

—

—

—

—

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

C₂³

yes

—

—

—

—

Q₈⋊C₈

—

#322

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

Q₁₆

yes

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

D₈

yes

—

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

C₄×C₈

yes

—

C₂×Q₈

—

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

#137

yes

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊Q₈

yes

—

C₂×C₈

#123

yes

—

C₂×C₈

C₂×Q₁₆

yes

—

C₂×C₈

AGL(1,ℤ/8)

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²⋊Q₈

yes

characteristic

C₄²

#123

yes

characteristic

C₄²

C₂×D₈

yes

—

C₁₆

—

C₂³

#497

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#399

yes

characteristic

C₂×C₄

#474

yes

—

C₂×C₄

#480

yes

—

C₂×C₄

#493

yes

characteristic

C₂×C₄

#494

yes

—

C₂×C₄

#496

yes

—

C₈

—

C₈

—

C₈

—

C₈

#477

yes

—

C₈

#484

yes

—

C₂²

—

C₂²

#1723

yes

—

C₂²

#2093

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

#1713

yes

characteristic

104

C₄

#1721

yes

characteristic

105

C₄

#2095

yes

—

106

C₂

—

107

C₂

#10674

yes

characteristic

108

C₂

#13532

yes

—

	subgroup	G/H
center	C₂×C₄	#493
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#2093
prime radical	C₂³	#497
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×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

#493

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₂×C₄×C₈ ▷ C₂×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
#13286	2
#12910	2
#9861	2
#8826	2
#7137	2

subgroup

index

Subgroups — #146243

Conjugacy classes of subgroups

Maximal subgroups