Group #135658 — subgroups

Subgroups — #135658

Order 512.

298 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 146 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	#12879	C₂	1	2	yes	maximal
2	C₄×C₈²	C₂	1	1	yes	maximal, characteristic
3	#7151	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₁₆	C₂²	1	4	yes	—
5	C₄²×C₈	C₄	1	1	yes	characteristic
6	C₄²×C₈	C₄	1	1	yes	characteristic
7	C₄²×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₈²	C₄	1	2	yes	—
9	C₂×C₈²	C₂²	1	2	yes	—
10	#1374	—	2	4	no	—
11	C₂²×C₁₆	—	2	4	no	—
12	C₂×C₄×C₈	—	2	2	no	—
13	C₂×C₄×C₈	C₈	1	2	yes	—
14	C₂×C₄×C₈	C₈	1	1	yes	characteristic
15	C₂×C₄×C₈	C₈	1	1	yes	characteristic
16	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
18	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₈	D₄	1	2	yes	—
20	C₂×C₄×C₈	Q₈	1	2	yes	—
21	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
22	C₄³	C₂×C₄	1	1	yes	characteristic
23	C₄⋊C₁₆	—	2	8	no	—
24	C₈²	—	2	4	no	—
25	C₈²	—	2	4	no	—
26	C₂²×C₈	—	2	2	no	—
27	C₂²×C₈	C₄⋊C₄	1	2	yes	—
28	C₂²×C₈	C₂×C₈	1	2	yes	—
29	C₂²×C₈	C₂×C₈	1	2	yes	—
30	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄²	C₄⋊C₄	1	2	yes	—
33	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄²	M₁₆	1	2	yes	—
36	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₁₆	—	4	4	no	—
38	C₂×C₁₆	—	4	8	no	—
39	C₄×C₈	—	2	8	no	—
40	C₄×C₈	—	2	4	no	—
41	C₄×C₈	—	2	4	no	—
42	C₄×C₈	—	2	4	no	—
43	C₄×C₈	—	2	2	no	—
44	C₄×C₈	—	2	2	no	—
45	C₄×C₈	—	2	2	no	—
46	C₄×C₈	—	2	2	no	—
47	C₄×C₈	—	2	2	no	—
48	C₄×C₈	C₁₆	1	4	yes	complemented
49	C₄×C₈	M₁₆	1	4	yes	—
50	C₄×C₈	#41	1	4	yes	—
51	C₂²×C₄	C₄⋊C₈	1	2	yes	—
52	C₂²×C₄	C₄⋊C₈	1	2	yes	—
53	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
55	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
56	C₂×C₈	—	2	8	no	—
57	C₂×C₈	—	2	2	no	—
58	C₂×C₈	—	2	2	no	—
59	C₂×C₈	—	2	2	no	—
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	2	4	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	#108	1	4	yes	—
66	C₂×C₈	C₂×C₁₆	1	4	yes	—
67	C₂×C₈	M₃₂	1	4	yes	—
68	C₂×C₈	#124	1	2	yes	—
69	C₂×C₈	#125	1	2	yes	—
70	C₂×C₈	#135	1	4	yes	—
71	C₄²	—	2	4	no	—
72	C₄²	—	2	2	no	—
73	C₄²	—	2	2	no	—
74	C₄²	—	2	2	no	—
75	C₄²	C₂×C₁₆	1	2	yes	—
76	C₄²	M₃₂	1	2	yes	—
77	C₄²	#117	1	2	yes	—
78	C₄²	C₂×M₁₆	1	2	yes	—
79	C₁₆	—	8	4	no	—
80	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	C₄⋊C₁₆	1	4	yes	—
87	C₂×C₄	#364	1	4	yes	—
88	C₂×C₄	#382	1	1	yes	characteristic
89	C₂×C₄	#423	1	1	yes	characteristic
90	C₂×C₄	#427	1	2	yes	—
91	C₂×C₄	C₂×#108	1	2	yes	—
92	C₂×C₄	#431	1	2	yes	—
93	C₂×C₄	#432	1	2	yes	—
94	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
95	C₂×C₄	C₂×M₃₂	1	2	yes	—
96	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
97	C₈	—	2	8	no	—
98	C₈	—	2	4	no	—
99	C₈	—	2	2	no	—
100	C₈	—	2	4	no	—
101	C₈	—	2	4	no	—
102	C₈	#488	1	4	yes	—
103	C₂²	#1564	1	2	yes	—
104	C₂²	#1831	1	1	yes	characteristic
105	C₂²	C₂×C₄⋊C₁₆	1	1	yes	characteristic
106	C₂²	#2150	1	1	yes	characteristic
107	C₂²	C₂×#364	1	2	yes	—
108	C₄	—	2	2	no	—
109	C₄	—	2	4	no	—
110	C₄	—	2	4	no	—
111	C₄	#1568	1	2	yes	—
112	C₄	#1835	1	1	yes	characteristic
113	C₄	#1837	1	1	yes	characteristic
114	C₄	#2162	1	2	yes	—
115	C₄	#2163	1	2	yes	—
116	C₂	#8785	1	1	yes	characteristic
117	C₂	#12012	1	2	yes	—
118	C₂	#12013	1	2	yes	—
119	C₂	#12879	1	1	yes	characteristic
120	C₂	#12883	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12879

C₂

yes

maximal

C₄×C₈²

C₂

yes

maximal, characteristic

#7151

C₂

yes

maximal

C₂×C₄⋊C₁₆

C₂²

yes

—

C₄²×C₈

C₄

yes

characteristic

C₄²×C₈

C₄

yes

characteristic

C₄²×C₈

C₂²

yes

characteristic

C₂×C₈²

C₄

yes

—

C₂×C₈²

C₂²

yes

—

#1374

—

C₂²×C₁₆

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₈

yes

—

C₂×C₄×C₈

C₈

yes

characteristic

C₂×C₄×C₈

C₈

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

Q₈

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

C₂×C₄

yes

characteristic

C₄⋊C₁₆

—

C₈²

—

C₈²

—

C₂²×C₈

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂²×C₈

C₂×C₈

yes

—

C₂²×C₈

C₂×C₈

yes

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

—

C₂×C₄²

C₂×C₈

yes

characteristic

C₂×C₄²

C₂×C₈

yes

characteristic

C₂×C₄²

M₁₆

yes

—

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₁₆

yes

complemented

C₄×C₈

M₁₆

yes

—

C₄×C₈

#41

yes

—

C₂²×C₄

C₄⋊C₈

yes

—

C₂²×C₄

C₄⋊C₈

yes

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

C₂²×C₈

yes

characteristic

C₂²×C₄

C₂×M₁₆

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#108

yes

—

C₂×C₈

C₂×C₁₆

yes

—

C₂×C₈

M₃₂

yes

—

C₂×C₈

#124

yes

—

C₂×C₈

#125

yes

—

C₂×C₈

#135

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×C₁₆

yes

—

C₄²

M₃₂

yes

—

C₄²

#117

yes

—

C₄²

C₂×M₁₆

yes

—

C₁₆

—

C₂³

C₂×C₄⋊C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊C₁₆

yes

—

C₂×C₄

#364

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

C₂×#108

yes

—

C₂×C₄

#431

yes

—

C₂×C₄

#432

yes

—

C₂×C₄

C₂²×C₁₆

yes

characteristic

C₂×C₄

C₂×M₃₂

yes

—

C₂×C₄

C₂×M₃₂

yes

characteristic

C₈

—

C₈

—

C₈

—

100

C₈

—

101

C₈

—

102

C₈

#488

yes

—

103

C₂²

#1564

yes

—

104

C₂²

#1831

yes

characteristic

105

C₂²

C₂×C₄⋊C₁₆

yes

characteristic

106

C₂²

#2150

yes

characteristic

107

C₂²

C₂×#364

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#1568

yes

—

112

C₄

#1835

yes

characteristic

113

C₄

#1837

yes

characteristic

114

C₄

#2162

yes

—

115

C₄

#2163

yes

—

116

C₂

#8785

yes

characteristic

117

C₂

#12012

yes

—

118

C₂

#12013

yes

—

119

C₂

#12879

yes

characteristic

120

C₂

#12883

yes

characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂²×C₈

C₂×D₄

commutator

C₂×C₄

C₂²×C₁₆

Frattini, cosocle

C₂×C₄×C₈

C₂³

socle, prime radical

C₂³

C₂×C₄⋊C₈

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

H ≅ G/N

type

48. C₄×C₈

C₁₆

semidirect

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

subgroup	index
#12879	2
C₄×C₈²	2
#7151	2

subgroup

index

#12879

C₄×C₈²

#7151

Subgroups — #135658

Conjugacy classes of subgroups

Complements

Maximal subgroups