Group #135656 — subgroups

Subgroups — #135656

Order 512.

252 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 134 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	#12900	C₂	1	2	yes	maximal
2	C₄²×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#7160	C₂	1	4	yes	maximal
4	C₂×#364	—	2	2	no	—
5	C₂×#364	C₂²	1	4	yes	—
6	#2150	—	2	2	no	—
7	C₂×C₄×C₁₆	C₄	1	2	yes	—
8	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
9	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
10	C₂×C₄×C₁₆	C₂²	1	2	yes	—
11	C₄²×C₈	C₂²	1	1	yes	characteristic
12	C₂×M₃₂	—	2	4	no	—
13	C₂²×C₁₆	C₂×C₄	1	2	yes	—
14	C₂²×C₁₆	C₂×C₄	1	2	yes	—
15	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₈	D₄	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	C₂³	1	1	yes	characteristic
21	C₄³	Q₈	1	1	yes	characteristic
22	#364	—	2	8	no	—
23	#364	—	4	4	no	—
24	C₄⋊C₁₆	—	4	4	no	—
25	C₄×C₁₆	—	2	4	no	—
26	C₄×C₁₆	—	2	1	no	—
27	C₄×C₁₆	—	2	1	no	—
28	C₄×C₁₆	—	2	4	no	—
29	C₄×C₁₆	C₈	1	2	yes	—
30	C₄×C₁₆	C₈	1	2	yes	—
31	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄²	—	2	2	no	—
36	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
39	M₃₂	—	4	4	no	—
40	M₃₂	—	4	4	no	—
41	C₂×C₁₆	—	2	2	no	—
42	C₂×C₁₆	—	2	2	no	—
43	C₂×C₁₆	—	2	2	no	—
44	C₂×C₁₆	—	2	2	no	—
45	C₂×C₁₆	—	4	4	no	—
46	C₂×C₁₆	C₂×C₈	1	4	yes	—
47	C₂×C₁₆	M₁₆	1	4	yes	—
48	C₄×C₈	—	2	2	no	—
49	C₄×C₈	—	2	2	no	—
50	C₄×C₈	C₂×C₈	1	1	yes	characteristic
51	C₄×C₈	C₂×C₈	1	1	yes	characteristic
52	C₄×C₈	M₁₆	1	1	yes	characteristic
53	C₄×C₈	M₁₆	1	1	yes	characteristic
54	C₄×C₈	D₈	1	2	yes	—
55	C₄×C₈	SD₁₆	1	4	yes	—
56	C₄×C₈	Q₁₆	1	2	yes	—
57	C₄×C₈	#41	1	4	yes	—
58	C₂²×C₄	—	2	1	no	—
59	C₂²×C₄	—	2	1	no	—
60	C₂²×C₄	#108	1	2	yes	—
61	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₈	—	2	1	no	—
63	C₂×C₈	—	2	1	no	—
64	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
65	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
66	C₂×C₈	#106	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	#124	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	C₂×C₈	C₂×D₈	1	1	yes	characteristic
75	C₂×C₈	C₂×SD₁₆	1	2	yes	—
76	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
77	C₄²	—	2	8	no	—
78	C₄²	—	2	1	no	—
79	C₄²	—	2	1	no	—
80	C₄²	C₄⋊C₈	1	1	yes	characteristic
81	C₄²	C₄⋊C₈	1	1	yes	characteristic
82	C₄²	#106	1	2	yes	—
83	C₄²	#107	1	2	yes	—
84	C₄²	#125	1	2	yes	—
85	C₁₆	—	2	4	no	—
86	C₁₆	—	2	4	no	—
87	C₁₆	—	8	4	no	—
88	C₂³	C₂×#108	1	1	yes	characteristic
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	2	4	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	#335	1	2	yes	—
95	C₂×C₄	#335	1	2	yes	—
96	C₂×C₄	#382	1	1	yes	characteristic
97	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
98	C₂×C₄	#423	1	1	yes	characteristic
99	C₂×C₄	C₂×#106	1	1	yes	characteristic
100	C₂×C₄	C₂×#107	1	1	yes	characteristic
101	C₂×C₄	C₂×#108	1	1	yes	characteristic
102	C₈	—	2	2	no	—
103	C₈	#334	1	2	yes	—
104	C₈	#334	1	2	yes	—
105	C₈	#431	1	2	yes	—
106	C₈	#432	1	2	yes	—
107	C₈	#486	1	2	yes	—
108	C₈	#487	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	C₂×#335	1	1	yes	characteristic
111	C₂²	#1566	1	1	yes	characteristic
112	C₂²	#1837	1	1	yes	characteristic
113	C₄	—	2	4	no	—
114	C₄	—	2	4	no	—
115	C₄	—	2	2	no	—
116	C₄	#1373	1	4	yes	—
117	C₄	C₂×#334	1	1	yes	characteristic
118	C₄	#1567	1	1	yes	characteristic
119	C₄	#1831	1	1	yes	characteristic
120	C₄	#1834	1	1	yes	characteristic
121	C₂	—	2	1	no	—
122	C₂	—	2	1	no	—
123	C₂	#8784	1	1	yes	characteristic
124	C₂	C₂×#1373	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#12900

C₂

yes

maximal

C₄²×C₁₆

C₂

yes

maximal, characteristic

#7160

C₂

yes

maximal

C₂×#364

—

C₂×#364

C₂²

yes

—

#2150

—

C₂×C₄×C₁₆

C₄

yes

—

C₂×C₄×C₁₆

C₄

yes

characteristic

C₂×C₄×C₁₆

C₄

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

—

C₄²×C₈

C₂²

yes

characteristic

C₂×M₃₂

—

C₂²×C₁₆

C₂×C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

Q₈

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

Q₈

yes

characteristic

#364

—

#364

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₈

yes

—

C₄×C₁₆

C₈

yes

—

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₂×Q₈

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₈

yes

—

C₂×C₁₆

M₁₆

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈

yes

characteristic

C₄×C₈

C₂×C₈

yes

characteristic

C₄×C₈

M₁₆

yes

characteristic

C₄×C₈

M₁₆

yes

characteristic

C₄×C₈

D₈

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

#41

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#108

yes

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₈

yes

characteristic

C₂×C₈

C₄⋊C₈

yes

characteristic

C₂×C₈

#106

yes

—

C₂×C₈

#107

yes

—

C₂×C₈

#108

yes

—

C₂×C₈

#117

yes

—

C₂×C₈

#124

yes

—

C₂×C₈

C₂²×C₈

yes

characteristic

C₂×C₈

C₂×M₁₆

yes

—

C₂×C₈

C₂×M₁₆

yes

characteristic

C₂×C₈

C₂×D₈

yes

characteristic

C₂×C₈

C₂×SD₁₆

yes

—

C₂×C₈

C₂×Q₁₆

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄⋊C₈

yes

characteristic

C₄²

C₄⋊C₈

yes

characteristic

C₄²

#106

yes

—

C₄²

#107

yes

—

C₄²

#125

yes

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

C₂×#108

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#335

yes

—

C₂×C₄

#335

yes

—

C₂×C₄

#382

yes

characteristic

C₂×C₄

C₂×C₄⋊C₈

yes

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

C₂×#107

yes

characteristic

101

C₂×C₄

C₂×#108

yes

characteristic

102

C₈

—

103

C₈

#334

yes

—

104

C₈

#334

yes

—

105

C₈

#431

yes

—

106

C₈

#432

yes

—

107

C₈

#486

yes

—

108

C₈

#487

yes

—

109

C₂²

—

110

C₂²

C₂×#335

yes

characteristic

111

C₂²

#1566

yes

characteristic

112

C₂²

#1837

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

#1373

yes

—

117

C₄

C₂×#334

yes

characteristic

118

C₄

#1567

yes

characteristic

119

C₄

#1831

yes

characteristic

120

C₄

#1834

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

#8784

yes

characteristic

124

C₂

C₂×#1373

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₈
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#335
prime radical	C₂³	C₂×#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₈

C₂×D₈

commutator

C₂×C₈

C₂²×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×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
#12900	2
C₄²×C₁₆	2
#7160	2

subgroup

index

#12900

C₄²×C₁₆

#7160

Subgroups — #135656

Conjugacy classes of subgroups

Maximal subgroups