Group #135669 — subgroups

Subgroups — #135669

Order 512.

206 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 98 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	#12901	C₂	1	1	yes	maximal, characteristic
2	#12899	C₂	1	1	yes	maximal, characteristic
3	#8137	C₂	1	1	yes	maximal, characteristic
4	#7153	C₂	1	4	yes	maximal
5	C₂×#364	—	2	1	no	—
6	C₂×#364	—	2	1	no	—
7	#2150	C₂²	1	2	yes	—
8	C₂×C₄⋊C₁₆	—	2	1	no	—
9	C₂×C₄⋊C₁₆	—	2	1	no	—
10	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
11	#1726	C₂²	1	1	yes	characteristic
12	#1450	C₂²	1	2	yes	—
13	#1275	C₄	1	2	yes	—
14	#1274	C₄	1	2	yes	—
15	C₂×M₃₂	—	2	2	no	—
16	C₂²×C₁₆	—	2	2	no	—
17	#431	C₂×C₄	1	4	yes	—
18	C₂×#97	C₂×C₄	1	1	yes	characteristic
19	C₂×#97	Q₈	1	1	yes	characteristic
20	C₂×#97	Q₈	1	1	yes	characteristic
21	C₂×C₄×C₈	D₄	1	1	yes	characteristic
22	C₂×C₄×C₈	D₄	1	1	yes	characteristic
23	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
24	#376	C₂×C₄	1	1	yes	characteristic
25	#364	—	4	2	no	—
26	#364	—	4	2	no	—
27	C₄⋊C₁₆	—	2	4	no	—
28	C₄⋊C₁₆	—	2	4	no	—
29	C₄⋊C₁₆	—	4	2	no	—
30	C₄⋊C₁₆	—	4	2	no	—
31	#322	—	2	2	no	—
32	C₈²	—	2	2	no	—
33	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
40	M₃₂	—	4	2	no	—
41	M₃₂	—	4	2	no	—
42	C₂×C₁₆	—	4	2	no	—
43	C₂×C₁₆	—	4	2	no	—
44	C₂×C₁₆	—	4	2	no	—
45	C₂×C₁₆	—	4	2	no	—
46	C₄⋊C₈	—	2	4	no	—
47	C₂²⋊C₈	—	2	4	no	—
48	#97	—	2	2	no	—
49	#97	—	4	1	no	—
50	#97	—	4	1	no	—
51	C₄×C₈	—	2	4	no	—
52	C₄×C₈	D₈	1	2	yes	—
53	C₄×C₈	SD₁₆	1	2	yes	—
54	C₄×C₈	SD₁₆	1	2	yes	—
55	C₄×C₈	Q₁₆	1	2	yes	—
56	C₄×C₈	#41	1	2	yes	—
57	C₄×C₈	#41	1	2	yes	—
58	#95	—	2	2	no	—
59	C₂²×C₄	—	2	1	no	—
60	C₂²×C₄	—	2	1	no	—
61	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄	#131	1	2	yes	—
63	C₂×C₈	—	2	1	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	—	2	1	no	—
66	C₂×C₈	—	2	1	no	—
67	C₂×C₈	—	2	1	no	—
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	4	4	no	—
70	C₂×C₈	#106	1	2	yes	—
71	C₂×C₈	#107	1	2	yes	—
72	C₂×C₈	#124	1	1	yes	characteristic
73	C₂×C₈	#124	1	1	yes	characteristic
74	C₂×C₈	#125	1	2	yes	—
75	C₂×C₈	C₂×D₈	1	1	yes	characteristic
76	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
79	C₄²	—	2	1	no	—
80	C₄²	—	2	1	no	—
81	C₄²	#106	1	2	yes	—
82	C₄²	#107	1	2	yes	—
83	C₄²	#117	1	2	yes	—
84	C₄²	#131	1	2	yes	—
85	C₁₆	—	8	2	no	—
86	C₁₆	—	8	2	no	—
87	C₂³	#424	1	1	yes	characteristic
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	#382	1	1	yes	characteristic
94	C₂×C₄	#424	1	1	yes	characteristic
95	C₂×C₄	C₂×#106	1	1	yes	characteristic
96	C₂×C₄	C₂×#107	1	1	yes	characteristic
97	C₂×C₄	#430	1	1	yes	characteristic
98	C₂×C₄	#430	1	1	yes	characteristic
99	C₂×C₄	#433	1	4	yes	—
100	C₈	—	2	2	no	—
101	C₈	—	2	2	no	—
102	C₈	—	2	2	no	—
103	C₈	—	8	4	no	—
104	C₈	#486	1	1	yes	characteristic
105	C₈	#486	1	1	yes	characteristic
106	C₈	#487	1	1	yes	characteristic
107	C₈	#487	1	1	yes	characteristic
108	C₂²	—	2	2	no	—
109	C₂²	#1570	1	1	yes	characteristic
110	C₂²	#1571	1	1	yes	characteristic
111	C₂²	#1832	1	1	yes	characteristic
112	C₄	—	2	4	no	—
113	C₄	—	4	2	no	—
114	C₄	—	4	2	no	—
115	C₄	#1570	1	1	yes	characteristic
116	C₄	#1571	1	1	yes	characteristic
117	C₄	#1834	1	1	yes	characteristic
118	C₄	#1838	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#8787	1	1	yes	characteristic
122	C₂	#12885	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#12901

C₂

yes

maximal, characteristic

#12899

C₂

yes

maximal, characteristic

#8137

C₂

yes

maximal, characteristic

#7153

C₂

yes

—

—

C₂²

yes

—

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

C₂²

yes

—

#1726

C₂²

yes

characteristic

#1450

C₂²

yes

—

#1275

C₄

yes

—

#1274

C₄

yes

—

C₂×M₃₂

—

C₂²×C₁₆

—

#431

C₂×C₄

yes

—

C₂×#97

C₂×C₄

yes

characteristic

C₂×#97

Q₈

yes

characteristic

C₂×#97

Q₈

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

#376

C₂×C₄

yes

characteristic

#364

—

#364

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#322

—

C₈²

—

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₂²×C₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

D₈

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

#41

yes

—

C₄×C₈

#41

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

#131

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#106

yes

—

C₂×C₈

#107

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

C₂×D₈

yes

characteristic

C₂×C₈

C₂×SD₁₆

yes

characteristic

C₂×C₈

C₂×SD₁₆

yes

characteristic

C₂×C₈

C₂×Q₁₆

yes

characteristic

C₄²

—

C₄²

—

C₄²

#106

yes

—

C₄²

#107

yes

—

C₄²

#117

yes

—

C₄²

#131

yes

—

C₁₆

—

C₁₆

—

C₂³

#424

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

100

C₈

—

101

C₈

—

102

C₈

—

103

C₈

—

104

C₈

#486

yes

characteristic

105

C₈

#486

yes

characteristic

106

C₈

#487

yes

characteristic

107

C₈

#487

yes

characteristic

108

C₂²

—

109

C₂²

#1570

yes

characteristic

110

C₂²

#1571

yes

characteristic

111

C₂²

#1832

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

#1570

yes

characteristic

116

C₄

#1571

yes

characteristic

117

C₄

#1834

yes

characteristic

118

C₄

#1838

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#8787

yes

characteristic

122

C₂

#12885

yes

—

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1832
prime radical	C₂³	#424
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₈

#124

commutator

C₂×C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₈ ◁ 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
#12901	2
#12899	2
#8137	2
#7153	2

subgroup

index

Subgroups — #135669

Conjugacy classes of subgroups

Maximal subgroups