Group #146279 — subgroups

Subgroups — #146279

Order 512.

266 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 96 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	#13287	C₂	1	2	yes	maximal
2	#12903	C₂	1	2	yes	maximal
3	#9855	C₂	1	1	yes	maximal, characteristic
4	#8815	C₂	1	1	yes	maximal, characteristic
5	#7138	C₂	1	1	yes	maximal, characteristic
6	#3160	C₂²	1	1	yes	characteristic
7	C₂×#365	—	2	4	no	—
8	C₄×M₃₂	C₂²	1	2	yes	—
9	#1848	C₂²	1	1	yes	characteristic
10	#1844	C₂²	1	2	yes	—
11	#1834	—	2	2	no	—
12	#1493	—	2	2	no	—
13	C₂×C₈²	C₂²	1	1	yes	characteristic
14	#1386	—	2	4	no	—
15	#533	—	2	1	no	—
16	C₂×M₃₂	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	#427	—	2	2	no	—
22	C₂×#106	—	2	2	no	—
23	C₂×#106	—	2	2	no	—
24	#423	—	2	1	no	—
25	C₂×C₄×C₈	D₄	1	2	yes	—
26	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
27	#384	—	2	2	no	—
28	#365	—	4	8	no	—
29	#346	—	2	2	no	—
30	C₄×C₁₆	—	2	2	no	—
31	#340	—	4	4	no	—
32	Q₈⋊C₈	—	4	4	no	—
33	C₈²	—	2	2	no	—
34	C₂×M₁₆	—	4	1	no	—
35	C₂²×C₈	—	2	2	no	—
36	C₂²×C₈	C₂×D₄	1	2	yes	—
37	C₄⋊Q₈	—	2	2	no	—
38	#125	—	4	1	no	—
39	C₂²⋊Q₈	—	4	2	no	—
40	C₄×Q₈	—	4	2	no	—
41	#117	—	4	1	no	—
42	C₂×C₄⋊C₄	—	4	2	no	—
43	C₂×C₄⋊C₄	—	4	4	no	—
44	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
45	M₃₂	—	2	4	no	—
46	M₃₂	SD₁₆	1	8	yes	—
47	C₂×C₁₆	—	2	4	no	—
48	#108	—	4	4	no	—
49	#107	—	2	4	no	—
50	#106	—	4	2	no	—
51	#106	—	4	4	no	—
52	#106	—	4	4	no	—
53	C₄⋊C₈	—	4	2	no	—
54	C₄×C₈	—	2	2	no	—
55	C₄×C₈	SD₁₆	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	—	8	2	no	—
61	C₂²×C₄	#127	1	1	yes	characteristic
62	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
63	M₁₆	—	8	2	no	—
64	C₂×C₈	—	2	4	no	—
65	C₂×C₈	—	4	4	no	—
66	C₂×C₈	—	8	1	no	—
67	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
68	C₂×C₈	#123	1	2	yes	—
69	C₂×C₈	C₂×SD₁₆	1	4	yes	—
70	C₂×C₈	#137	1	4	yes	—
71	C₄⋊C₄	—	4	2	no	—
72	C₄⋊C₄	—	4	2	no	—
73	C₄⋊C₄	—	8	8	no	—
74	C₄⋊C₄	—	8	2	no	—
75	C₄⋊C₄	—	8	2	no	—
76	C₄⋊C₄	—	8	2	no	—
77	C₂²⋊C₄	—	8	1	no	—
78	C₄²	—	8	1	no	—
79	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
80	C₄²	#123	1	1	yes	characteristic
81	C₄²	C₂×SD₁₆	1	2	yes	—
82	C₁₆	—	2	8	no	—
83	C₂³	#496	1	1	yes	characteristic
84	Q₈	—	8	4	no	—
85	C₂×C₄	—	2	4	no	—
86	C₂×C₄	—	8	2	no	—
87	C₂×C₄	—	16	4	no	—
88	C₂×C₄	—	16	2	no	—
89	C₂×C₄	#399	1	1	yes	characteristic
90	C₂×C₄	#476	1	2	yes	—
91	C₂×C₄	#481	1	2	yes	—
92	C₂×C₄	#492	1	2	yes	—
93	C₂×C₄	#492	1	1	yes	characteristic
94	C₂×C₄	#497	1	2	yes	—
95	C₈	—	4	4	no	—
96	C₈	—	16	1	no	—
97	C₈	#475	1	4	yes	—
98	C₈	#483	1	4	yes	—
99	C₂²	—	2	2	no	—
100	C₂²	#1714	1	2	yes	—
101	C₂²	#2094	1	1	yes	characteristic
102	C₄	—	2	2	no	—
103	C₄	—	16	2	no	—
104	C₄	—	32	2	no	—
105	C₄	#1705	1	1	yes	characteristic
106	C₄	#1716	1	1	yes	characteristic
107	C₄	#2096	1	2	yes	—
108	C₄	#2231	1	4	yes	—
109	C₂	—	2	2	no	—
110	C₂	#10685	1	1	yes	characteristic
111	C₂	#13534	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13287

C₂

yes

maximal

#12903

C₂

yes

maximal

#9855

C₂

yes

maximal, characteristic

#8815

C₂

yes

maximal, characteristic

#7138

C₂

yes

maximal, characteristic

#3160

C₂²

yes

characteristic

C₂×#365

—

C₄×M₃₂

C₂²

yes

—

#1848

C₂²

yes

characteristic

#1844

C₂²

yes

—

#1834

—

#1493

—

C₂×C₈²

C₂²

yes

characteristic

#1386

—

#533

—

C₂×M₃₂

D₄

yes

—

#500

Q₈

yes

—

—

—

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

C₂³

yes

—

—

—

C₄×C₁₆

—

#340

—

Q₈⋊C₈

—

C₈²

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

SD₁₆

yes

—

C₂×C₁₆

—

—

—

—

—

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

C₄×C₈

yes

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊Q₈

yes

—

C₂×C₈

#123

yes

—

C₂×C₈

C₂×SD₁₆

yes

—

C₂×C₈

#137

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²⋊Q₈

yes

characteristic

C₄²

#123

yes

characteristic

C₄²

C₂×SD₁₆

yes

—

C₁₆

—

C₂³

#496

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#399

yes

characteristic

C₂×C₄

#476

yes

—

C₂×C₄

#481

yes

—

C₂×C₄

#492

yes

—

C₂×C₄

#492

yes

characteristic

C₂×C₄

#497

yes

—

C₈

—

C₈

—

C₈

#475

yes

—

C₈

#483

yes

—

C₂²

—

100

C₂²

#1714

yes

—

101

C₂²

#2094

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#1705

yes

characteristic

106

C₄

#1716

yes

characteristic

107

C₄

#2096

yes

—

108

C₄

#2231

yes

—

109

C₂

—

110

C₂

#10685

yes

characteristic

111

C₂

#13534

yes

—

	subgroup	G/H
center, socle	C₂²	#2094
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#496
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2094

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#496

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ 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
#13287	2
#12903	2
#9855	2
#8815	2
#7138	2

subgroup

index

Subgroups — #146279

Conjugacy classes of subgroups

Maximal subgroups