Group #146256 — subgroups

Subgroups — #146256

Order 512.

279 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 279 conjugacy classes of subgroups — too many to draw. Only its 88 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	#13212	C₂	1	2	yes	maximal
2	#12886	C₂	1	2	yes	maximal
3	#9839	C₂	1	1	yes	maximal, characteristic
4	#8825	C₂	1	1	yes	maximal, characteristic
5	#7139	C₂	1	1	yes	maximal, characteristic
6	C₂×#494	C₂²	1	1	yes	characteristic
7	C₂×#362	—	2	4	no	—
8	C₄×M₃₂	C₂²	1	2	yes	—
9	#1990	—	2	2	no	—
10	#1848	C₂²	1	2	yes	—
11	#1844	C₂²	1	1	yes	characteristic
12	#1566	—	2	2	no	—
13	C₂×#322	C₂²	1	1	yes	characteristic
14	#1383	—	2	4	no	—
15	C₂²×Q₁₆	—	2	2	no	—
16	C₂×C₄⋊Q₈	—	2	1	no	—
17	C₂×M₃₂	D₄	1	4	yes	—
18	#494	—	2	1	no	—
19	#494	—	2	2	no	—
20	#494	Q₈	1	2	yes	—
21	C₂×#108	—	2	2	no	—
22	C₂×#108	—	2	2	no	—
23	C₂×#106	—	2	2	no	—
24	#423	—	2	2	no	—
25	C₂×C₄×C₈	D₄	1	2	yes	—
26	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
27	#384	—	2	1	no	—
28	#362	—	4	8	no	—
29	#346	—	2	2	no	—
30	C₄×C₁₆	—	2	2	no	—
31	#340	—	4	4	no	—
32	#334	—	4	4	no	—
33	#322	—	2	2	no	—
34	C₂²×Q₈	—	4	1	no	—
35	C₂×Q₁₆	—	2	4	no	—
36	C₂×Q₁₆	—	4	2	no	—
37	C₂×Q₁₆	—	4	4	no	—
38	C₂×Q₁₆	—	4	4	no	—
39	C₂×M₁₆	—	4	2	no	—
40	C₂²×C₈	SD₁₆	1	4	yes	—
41	C₂²×C₈	C₂×D₄	1	2	yes	—
42	C₄⋊Q₈	—	2	2	no	—
43	C₄⋊Q₈	—	4	1	no	—
44	C₄⋊Q₈	—	4	2	no	—
45	C₂×C₄⋊C₄	—	4	2	no	—
46	C₂×C₄⋊C₄	—	4	1	no	—
47	C₂×C₄⋊C₄	—	4	2	no	—
48	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
49	M₃₂	—	2	4	no	—
50	M₃₂	—	2	4	no	—
51	C₂×C₁₆	—	2	4	no	—
52	#108	—	4	4	no	—
53	#108	—	4	4	no	—
54	#106	—	4	4	no	—
55	C₄⋊C₈	—	4	4	no	—
56	C₄×C₈	—	2	2	no	—
57	C₄×C₈	—	2	2	no	—
58	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
59	C₄×C₈	#41	1	1	yes	characteristic
60	C₄×C₈	#41	1	2	yes	—
61	C₂×Q₈	—	4	2	no	—
62	C₂×Q₈	—	8	1	no	—
63	C₂×Q₈	—	8	2	no	—
64	C₂×Q₈	—	8	2	no	—
65	C₂²×C₄	—	8	1	no	—
66	C₂²×C₄	—	8	1	no	—
67	C₂²×C₄	#127	1	1	yes	characteristic
68	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
69	Q₁₆	—	4	8	no	—
70	Q₁₆	—	8	4	no	—
71	M₁₆	—	8	2	no	—
72	M₁₆	—	8	2	no	—
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	—	4	4	no	—
75	C₂×C₈	—	8	2	no	—
76	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
77	C₂×C₈	#123	1	2	yes	—
78	C₂×C₈	#135	1	4	yes	—
79	C₂×C₈	AGL(1,ℤ/8)	1	4	yes	—
80	C₄⋊C₄	—	4	4	no	—
81	C₄⋊C₄	—	8	4	no	—
82	C₄⋊C₄	—	8	2	no	—
83	C₄⋊C₄	—	8	2	no	—
84	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
85	C₄²	#123	1	1	yes	characteristic
86	C₄²	AGL(1,ℤ/8)	1	2	yes	—
87	C₁₆	—	4	4	no	—
88	C₂³	#492	1	1	yes	characteristic
89	Q₈	—	8	4	no	—
90	Q₈	—	16	2	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	8	2	no	—
93	C₂×C₄	—	16	1	no	—
94	C₂×C₄	—	16	1	no	—
95	C₂×C₄	—	16	2	no	—
96	C₂×C₄	—	16	1	no	—
97	C₂×C₄	#399	1	1	yes	characteristic
98	C₂×C₄	#476	1	2	yes	—
99	C₂×C₄	#481	1	2	yes	—
100	C₂×C₄	#495	1	2	yes	—
101	C₂×C₄	#496	1	2	yes	—
102	C₂×C₄	#496	1	1	yes	characteristic
103	C₈	—	4	4	no	—
104	C₈	—	16	2	no	—
105	C₈	#478	1	4	yes	—
106	C₈	#482	1	4	yes	—
107	C₂²	—	2	2	no	—
108	C₂²	#1708	1	1	yes	characteristic
109	C₂²	#1715	1	1	yes	characteristic
110	C₂²	#2094	1	1	yes	characteristic
111	C₄	—	2	2	no	—
112	C₄	—	16	2	no	—
113	C₄	—	32	1	no	—
114	C₄	#1717	1	2	yes	—
115	C₄	#2092	1	2	yes	—
116	C₄	#2239	1	4	yes	—
117	C₂	—	2	1	no	—
118	C₂	—	2	1	no	—
119	C₂	#10677	1	1	yes	characteristic
120	C₂	#13536	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13212

C₂

yes

maximal

#12886

C₂

yes

maximal

#9839

C₂

yes

maximal, characteristic

#8825

C₂

yes

maximal, characteristic

#7139

C₂

yes

maximal, characteristic

C₂×#494

C₂²

yes

characteristic

C₂×#362

—

C₄×M₃₂

C₂²

yes

—

#1990

—

#1848

C₂²

yes

—

#1844

C₂²

yes

characteristic

#1566

—

C₂×#322

C₂²

yes

characteristic

#1383

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×M₃₂

D₄

yes

—

#494

—

#494

—

#494

Q₈

yes

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

C₂³

yes

—

—

—

C₄×C₁₆

—

#340

—

#334

—

#322

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

SD₁₆

yes

—

C₂²×C₈

C₂×D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#108

—

#108

—

#106

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

C₄×C₈

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

—

Q₁₆

—

Q₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊Q₈

yes

—

C₂×C₈

#123

yes

—

C₂×C₈

#135

yes

—

C₂×C₈

AGL(1,ℤ/8)

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊Q₈

yes

C₄²

yes

C₄²

yes

—

C₁₆

—

C₂³

#492

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#399

yes

characteristic

C₂×C₄

#476

yes

—

C₂×C₄

#481

yes

—

100

C₂×C₄

#495

yes

—

101

C₂×C₄

#496

yes

—

102

C₂×C₄

#496

yes

characteristic

103

C₈

—

104

C₈

—

105

C₈

#478

yes

—

106

C₈

#482

yes

—

107

C₂²

—

108

C₂²

#1708

yes

characteristic

109

C₂²

#1715

yes

characteristic

110

C₂²

#2094

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

#1717

yes

—

115

C₄

#2092

yes

—

116

C₄

#2239

yes

—

117

C₂

—

118

C₂

—

119

C₂

#10677

yes

characteristic

120

C₂

#13536

yes

—

	subgroup	G/H
center, socle	C₂²	#2094
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#492
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ 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

#492

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ 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
#13212	2
#12886	2
#9839	2
#8825	2
#7139	2

subgroup

index

Subgroups — #146256

Conjugacy classes of subgroups

Maximal subgroups