Group #135648 — subgroups

Subgroups — #135648

Order 512.

280 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 154 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	C₄×#346	C₂	1	1	yes	maximal, characteristic
2	#8805	C₂	1	2	yes	maximal
3	#7156	C₂	1	4	yes	maximal
4	C₂×#346	C₄	1	2	yes	—
5	C₂×#346	C₂²	1	2	yes	—
6	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
7	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
8	#1842	—	2	2	no	—
9	C₄²×C₈	C₂²	1	1	yes	characteristic
10	C₂×#334	—	2	2	no	—
11	C₂×#334	C₂²	1	4	yes	—
12	C₂²×C₁₆	C₂×C₄	1	2	yes	—
13	C₂²×C₁₆	C₂×C₄	1	2	yes	—
14	C₂×C₄⋊C₈	—	2	4	no	—
15	C₂×C₄⋊C₈	—	4	4	no	—
16	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₈	D₄	1	1	yes	characteristic
19	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
20	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
21	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
22	C₄³	D₄	1	1	yes	characteristic
23	#346	—	2	4	no	—
24	#346	—	2	4	no	—
25	C₄×C₁₆	—	2	2	no	—
26	C₄×C₁₆	C₈	1	4	yes	complemented
27	#334	—	2	8	no	—
28	#334	—	4	4	no	—
29	C₂²×C₈	—	4	4	no	—
30	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄²	SD₁₆	1	4	yes	—
37	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
38	C₂×C₁₆	—	2	2	no	—
39	C₂×C₁₆	—	2	2	no	—
40	C₂×C₁₆	—	2	2	no	—
41	C₂×C₁₆	—	2	2	no	—
42	C₂×C₁₆	C₂×C₈	1	4	yes	—
43	C₂×C₁₆	M₁₆	1	4	yes	—
44	C₄⋊C₈	—	4	4	no	—
45	C₄⋊C₈	—	4	4	no	—
46	C₄⋊C₈	—	8	8	no	—
47	C₄×C₈	—	2	2	no	—
48	C₄×C₈	—	2	2	no	—
49	C₄×C₈	—	2	2	no	—
50	C₄×C₈	C₂×C₈	1	2	yes	—
51	C₄×C₈	M₁₆	1	2	yes	—
52	C₄×C₈	D₈	1	2	yes	—
53	C₄×C₈	Q₁₆	1	2	yes	—
54	C₄×C₈	#41	1	4	yes	—
55	C₂²×C₄	#106	1	2	yes	—
56	C₂²×C₄	#106	1	2	yes	—
57	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
59	C₂×C₈	—	2	1	no	—
60	C₂×C₈	—	2	1	no	—
61	C₂×C₈	—	2	1	no	—
62	C₂×C₈	—	2	1	no	—
63	C₂×C₈	—	8	8	no	—
64	C₂×C₈	—	8	4	no	—
65	C₂×C₈	C₄⋊C₈	1	2	yes	—
66	C₂×C₈	#107	1	2	yes	—
67	C₂×C₈	#107	1	2	yes	—
68	C₂×C₈	#108	1	2	yes	—
69	C₂×C₈	#117	1	2	yes	—
70	C₂×C₈	#125	1	2	yes	—
71	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₈	C₂×M₁₆	1	2	yes	—
73	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
74	C₄²	—	2	8	no	—
75	C₄²	—	2	1	no	—
76	C₄²	—	2	1	no	—
77	C₄²	C₄⋊C₈	1	2	yes	—
78	C₄²	#108	1	2	yes	—
79	C₄²	#124	1	2	yes	—
80	C₄²	C₂×D₈	1	1	yes	characteristic
81	C₄²	C₂×Q₁₆	1	1	yes	characteristic
82	C₁₆	—	2	4	no	—
83	C₁₆	—	2	4	no	—
84	C₂³	C₂×#106	1	1	yes	characteristic
85	C₂×C₄	—	2	4	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	#334	1	4	yes	—
89	C₂×C₄	#365	1	8	yes	—
90	C₂×C₄	#382	1	1	yes	characteristic
91	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
92	C₂×C₄	#423	1	1	yes	characteristic
93	C₂×C₄	C₂×#107	1	1	yes	characteristic
94	C₂×C₄	C₂×#108	1	1	yes	characteristic
95	C₂×C₄	C₂×#108	1	1	yes	characteristic
96	C₂×C₄	#486	1	2	yes	—
97	C₂×C₄	#487	1	2	yes	—
98	C₈	—	2	2	no	—
99	C₈	—	2	2	no	—
100	C₈	—	16	4	no	—
101	C₈	#335	1	4	yes	—
102	C₈	#431	1	2	yes	—
103	C₈	#432	1	2	yes	—
104	C₂²	C₂×#334	1	1	yes	characteristic
105	C₂²	#1566	1	1	yes	characteristic
106	C₂²	#1834	1	1	yes	characteristic
107	C₂²	C₂×#365	1	4	yes	—
108	C₄	—	2	2	no	—
109	C₄	—	2	2	no	—
110	C₄	—	4	2	no	—
111	C₄	#1368	1	8	yes	—
112	C₄	C₂×#335	1	1	yes	characteristic
113	C₄	#1567	1	1	yes	characteristic
114	C₄	#1831	1	1	yes	characteristic
115	C₄	#1837	1	1	yes	characteristic
116	C₂	#8784	1	1	yes	characteristic
117	C₂	C₂×#1368	1	2	yes	—
118	C₂	#12020	1	2	yes	—
119	C₂	#12886	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#346

C₂

yes

maximal, characteristic

#8805

C₂

yes

maximal

#7156

C₂

yes

maximal

C₂×#346

C₄

yes

—

C₂×#346

C₂²

yes

—

C₂×C₄×C₁₆

C₄

yes

characteristic

C₂×C₄×C₁₆

C₄

yes

characteristic

#1842

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#334

—

C₂×#334

C₂²

yes

—

C₂²×C₁₆

C₂×C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

Q₈

yes

characteristic

C₂×C₄×C₈

Q₈

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

D₄

yes

characteristic

#346

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

C₈

yes

complemented

#334

—

#334

—

C₂²×C₈

—

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

SD₁₆

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₈

yes

—

C₂×C₁₆

M₁₆

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

D₈

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

#41

yes

—

C₂²×C₄

#106

yes

—

C₂²×C₄

#106

yes

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

#107

yes

—

C₂×C₈

#107

yes

—

C₂×C₈

#108

yes

—

C₂×C₈

#117

yes

—

C₂×C₈

#125

yes

—

C₂×C₈

C₂²×C₈

yes

characteristic

C₂×C₈

C₂×M₁₆

yes

—

C₂×C₈

C₂×M₁₆

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄⋊C₈

yes

—

C₄²

#108

yes

—

C₄²

#124

yes

—

C₄²

C₂×D₈

yes

characteristic

C₄²

C₂×Q₁₆

yes

characteristic

C₁₆

—

C₁₆

—

C₂³

C₂×#106

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#334

yes

—

C₂×C₄

#365

yes

—

C₂×C₄

#382

yes

characteristic

C₂×C₄

C₂×C₄⋊C₈

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

#487

yes

—

C₈

—

C₈

—

100

C₈

—

101

C₈

#335

yes

—

102

C₈

#431

yes

—

103

C₈

#432

yes

—

104

C₂²

C₂×#334

yes

characteristic

105

C₂²

#1566

yes

characteristic

106

C₂²

#1834

yes

characteristic

107

C₂²

C₂×#365

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#1368

yes

—

112

C₄

C₂×#335

yes

characteristic

113

C₄

#1567

yes

characteristic

114

C₄

#1831

yes

characteristic

115

C₄

#1837

yes

characteristic

116

C₂

#8784

yes

characteristic

117

C₂

C₂×#1368

yes

—

118

C₂

#12020

yes

—

119

C₂

#12886

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#106
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂×#106

commutator

C₂×C₈

C₂²×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
26. C₄×C₁₆	C₈	semidirect

H ≅ G/N

type

26. C₄×C₁₆

C₈

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×#346	2
#8805	2
#7156	2

subgroup

index

C₄×#346

#8805

#7156

Subgroups — #135648

Conjugacy classes of subgroups

Complements

Maximal subgroups