Group #145877 — subgroups

Subgroups — #145877

Order 512.

225 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 225 conjugacy classes of subgroups — too many to draw. Only its 74 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	#29230	C₂	1	1	yes	maximal, characteristic
2	#13188	C₂	1	2	yes	maximal
3	#8823	C₂	1	1	yes	maximal, characteristic
4	#7273	C₂	1	1	yes	maximal, characteristic
5	#7179	C₂	1	2	yes	maximal
6	#2155	—	2	2	no	—
7	C₂×#364	C₂²	1	2	yes	—
8	#2102	—	2	2	no	—
9	#1953	—	2	2	no	—
10	#1848	C₂²	1	1	yes	characteristic
11	#1848	C₂²	1	1	yes	characteristic
12	#1833	—	2	2	no	—
13	#1772	C₂²	1	2	yes	—
14	#1567	—	2	2	no	—
15	C₂×C₈²	C₂²	1	1	yes	characteristic
16	#1388	—	2	2	no	—
17	C₂×M₃₂	—	2	2	no	—
18	#430	—	2	2	no	—
19	C₂×#108	—	2	2	no	—
20	C₂×#108	—	2	2	no	—
21	C₂×#108	D₄	1	2	yes	—
22	#427	D₄	1	2	yes	—
23	#423	—	2	1	no	—
24	#423	D₄	1	2	yes	—
25	#405	—	2	2	no	—
26	C₂×C₄×C₈	—	2	1	no	—
27	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
28	#365	—	4	4	no	—
29	#364	—	2	2	no	—
30	#364	Q₈	1	4	yes	—
31	#344	—	2	4	no	—
32	#339	—	2	2	no	—
33	H(3,ℤ/4)	—	2	2	no	—
34	#335	—	4	4	no	—
35	C₈²	—	2	1	no	—
36	C₈²	Q₈	1	1	yes	characteristic
37	C₈²	Q₈	1	1	yes	characteristic
38	C₂×M₁₆	—	2	2	no	—
39	C₂×M₁₆	—	2	2	no	—
40	C₂×M₁₆	—	4	1	no	—
41	C₂²×C₈	—	2	2	no	—
42	C₂²×C₈	C₂×D₄	1	2	yes	—
43	#117	—	2	2	no	—
44	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
45	M₃₂	—	2	4	no	—
46	M₃₂	—	4	2	no	—
47	C₂×C₁₆	—	4	2	no	—
48	#108	—	2	2	no	—
49	#108	—	2	2	no	—
50	#108	—	4	2	no	—
51	#108	—	4	2	no	—
52	#108	—	4	2	no	—
53	#108	—	4	2	no	—
54	#108	—	8	2	no	—
55	#107	—	2	2	no	—
56	#106	—	2	2	no	—
57	C₄⋊C₈	—	4	2	no	—
58	C₄⋊C₈	—	4	2	no	—
59	#97	—	4	2	no	—
60	C₄×C₈	—	2	2	no	—
61	C₄×C₈	—	2	1	no	—
62	C₄×C₈	—	2	1	no	—
63	C₄×C₈	—	4	2	no	—
64	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
65	C₄×C₈	C₂×Q₈	1	2	yes	—
66	C₄×C₈	#41	1	1	yes	characteristic
67	C₂²×C₄	—	2	1	no	—
68	C₂²×C₄	#120	1	1	yes	characteristic
69	M₁₆	—	4	4	no	—
70	M₁₆	—	4	2	no	—
71	M₁₆	—	4	2	no	—
72	M₁₆	—	8	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	4	4	no	—
76	C₂×C₈	—	4	2	no	—
77	C₂×C₈	—	4	2	no	—
78	C₂×C₈	—	8	1	no	—
79	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
80	C₂×C₈	C₄⋊Q₈	1	2	yes	—
81	C₂×C₈	#135	1	2	yes	—
82	C₂×C₈	#135	1	2	yes	—
83	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
84	C₂×C₈	#137	1	2	yes	—
85	C₄⋊C₄	—	4	2	no	—
86	C₂²⋊C₄	—	4	2	no	—
87	C₄²	—	2	1	no	—
88	C₄²	—	4	2	no	—
89	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
90	C₄²	C₄⋊Q₈	1	1	yes	characteristic
91	C₁₆	—	4	4	no	—
92	C₂³	#456	1	1	yes	characteristic
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	8	2	no	—
97	C₂×C₄	#395	1	1	yes	characteristic
98	C₂×C₄	#449	1	2	yes	—
99	C₂×C₄	#452	1	2	yes	—
100	C₂×C₄	#453	1	1	yes	characteristic
101	C₈	—	2	4	no	—
102	C₈	—	4	2	no	—
103	C₈	—	8	2	no	—
104	C₈	—	8	2	no	—
105	C₈	—	16	1	no	—
106	C₈	#478	1	2	yes	—
107	C₈	#479	1	2	yes	—
108	C₈	#499	1	2	yes	—
109	C₈	#501	1	2	yes	—
110	C₂²	—	4	1	no	—
111	C₂²	#1650	1	2	yes	—
112	C₂²	#2060	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	—	4	1	no	—
115	C₄	—	16	2	no	—
116	C₄	#1620	1	1	yes	characteristic
117	C₄	#1634	1	1	yes	characteristic
118	C₄	#2059	1	2	yes	—
119	C₂	—	2	1	no	—
120	C₂	—	2	2	no	—
121	C₂	#10576	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#29230

C₂

yes

maximal, characteristic

#13188

C₂

yes

maximal

#8823

C₂

yes

maximal, characteristic

#7273

C₂

yes

maximal, characteristic

#7179

C₂

yes

maximal

#2155

—

C₂×#364

C₂²

yes

—

#2102

—

#1953

—

#1848

C₂²

yes

characteristic

#1848

C₂²

yes

characteristic

#1833

—

#1772

C₂²

yes

—

#1567

—

C₂×C₈²

C₂²

yes

characteristic

#1388

—

C₂×M₃₂

—

—

—

—

D₄

yes

—

#427

D₄

yes

—

#423

—

#423

D₄

yes

—

#405

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂³

yes

—

—

Q₈

yes

—

—

—

—

—

C₈²

—

C₈²

Q₈

yes

characteristic

C₈²

Q₈

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄

yes

—

#117

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

—

—

—

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

#41

yes

characteristic

C₂²×C₄

—

C₂²×C₄

#120

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊Q₈

yes

—

C₂×C₈

C₄⋊Q₈

yes

—

C₂×C₈

#135

yes

—

C₂×C₈

#135

yes

—

C₂×C₈

AGL(1,ℤ/8)

yes

—

C₂×C₈

#137

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊Q₈

yes

characteristic

C₄²

C₄⋊Q₈

yes

characteristic

C₁₆

—

C₂³

#456

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#395

yes

characteristic

C₂×C₄

#449

yes

—

C₂×C₄

#452

yes

—

100

C₂×C₄

#453

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

—

106

C₈

#478

yes

—

107

C₈

#479

yes

—

108

C₈

#499

yes

—

109

C₈

#501

yes

—

110

C₂²

—

111

C₂²

#1650

yes

—

112

C₂²

#2060

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

#1620

yes

characteristic

117

C₄

#1634

yes

characteristic

118

C₄

#2059

yes

—

119

C₂

—

120

C₂

—

121

C₂

#10576

yes

characteristic

	subgroup	G/H
center	C₄	#1634
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#10576
prime radical	C₂³	#456
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄

#1634

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂

C₂³

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ 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
#29230	2
#13188	2
#8823	2
#7273	2
#7179	2

subgroup

index

Subgroups — #145877

Conjugacy classes of subgroups

Maximal subgroups