Group #94015 — subgroups

Subgroups — #94015

Order 512.

172 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 106 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	C₂×C₄⋊C₃₂	C₂	1	2	yes	maximal
2	C₂×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃₂	—	2	2	no	—
4	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
5	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
6	C₂×C₈²	C₄	1	1	yes	characteristic
7	C₄⋊C₃₂	—	2	2	no	—
8	C₄⋊C₃₂	C₄	1	4	yes	—
9	C₈×C₁₆	—	2	2	no	—
10	C₂²×C₁₆	—	2	1	no	—
11	C₂²×C₁₆	D₄	1	1	yes	characteristic
12	C₂²×C₁₆	Q₈	1	1	yes	characteristic
13	C₂×C₄×C₈	C₈	1	1	yes	characteristic
14	C₂×C₄×C₈	C₈	1	1	yes	characteristic
15	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃₂	—	2	4	no	—
17	C₂×C₃₂	—	4	2	no	—
18	C₂×C₃₂	—	4	2	no	—
19	C₄×C₁₆	—	2	1	no	—
20	C₄×C₁₆	C₈	1	2	yes	—
21	C₄×C₁₆	C₂×C₄	1	2	yes	—
22	C₄×C₁₆	D₄	1	2	yes	—
23	C₈²	—	2	1	no	—
24	C₈²	—	2	1	no	—
25	C₂²×C₈	—	2	1	no	—
26	C₂²×C₈	C₁₆	1	2	yes	—
27	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
28	C₂²×C₈	M₁₆	1	1	yes	characteristic
29	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
30	C₂×C₁₆	—	2	2	no	—
31	C₂×C₁₆	—	2	2	no	—
32	C₂×C₁₆	—	2	2	no	—
33	C₂×C₁₆	—	2	1	no	—
34	C₂×C₁₆	—	2	1	no	—
35	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
36	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
37	C₂×C₁₆	D₈	1	1	yes	characteristic
38	C₂×C₁₆	SD₁₆	1	2	yes	—
39	C₂×C₁₆	Q₁₆	1	1	yes	characteristic
40	C₄×C₈	—	2	1	no	—
41	C₄×C₈	—	2	1	no	—
42	C₄×C₈	—	2	1	no	—
43	C₄×C₈	C₁₆	1	2	yes	—
44	C₄×C₈	C₄²	1	1	yes	characteristic
45	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
46	C₄×C₈	C₂×C₈	1	1	yes	characteristic
47	C₄×C₈	M₁₆	1	1	yes	characteristic
48	C₃₂	—	4	4	no	—
49	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
50	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
51	C₂²×C₄	M₃₂	1	1	yes	characteristic
52	C₂×C₈	—	2	4	no	—
53	C₂×C₈	—	2	2	no	—
54	C₂×C₈	—	2	2	no	—
55	C₂×C₈	—	2	1	no	—
56	C₂×C₈	—	2	1	no	—
57	C₂×C₈	—	2	2	no	—
58	C₂×C₈	C₃₂	1	4	yes	complemented
59	C₂×C₈	#95	1	1	yes	characteristic
60	C₂×C₈	#97	1	1	yes	characteristic
61	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
62	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₈	#106	1	1	yes	characteristic
64	C₂×C₈	#107	1	1	yes	characteristic
65	C₂×C₈	#108	1	1	yes	characteristic
66	C₂×C₈	#108	1	1	yes	characteristic
67	C₄²	C₄×C₈	1	1	yes	characteristic
68	C₄²	C₂²⋊C₈	1	1	yes	characteristic
69	C₄²	C₂×C₁₆	1	1	yes	characteristic
70	C₄²	M₃₂	1	1	yes	characteristic
71	C₁₆	—	2	4	no	—
72	C₁₆	—	2	2	no	—
73	C₁₆	#102	1	2	yes	—
74	C₁₆	#103	1	2	yes	—
75	C₂³	C₄⋊C₁₆	1	1	yes	characteristic
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	#334	1	1	yes	characteristic
78	C₂×C₄	#335	1	1	yes	characteristic
79	C₂×C₄	#336	1	1	yes	characteristic
80	C₂×C₄	C₄×C₁₆	1	1	yes	characteristic
81	C₂×C₄	#346	1	1	yes	characteristic
82	C₂×C₄	C₂²⋊C₁₆	1	1	yes	characteristic
83	C₂×C₄	C₂²⋊C₁₆	1	1	yes	characteristic
84	C₂×C₄	C₄⋊C₁₆	1	1	yes	characteristic
85	C₂×C₄	#364	1	1	yes	characteristic
86	C₂×C₄	#364	1	1	yes	characteristic
87	C₂×C₄	C₂×C₃₂	1	2	yes	—
88	C₂×C₄	M₆₄	1	2	yes	—
89	C₈	—	2	4	no	—
90	C₈	—	2	1	no	—
91	C₈	—	2	1	no	—
92	C₈	—	2	4	no	—
93	C₈	#334	1	1	yes	characteristic
94	C₈	#335	1	1	yes	characteristic
95	C₈	#340	1	1	yes	characteristic
96	C₈	#341	1	1	yes	characteristic
97	C₂²	#1367	1	1	yes	characteristic
98	C₂²	#1370	1	1	yes	characteristic
99	C₂²	#1374	1	1	yes	characteristic
100	C₂²	C₄⋊C₃₂	1	2	yes	—
101	C₂²	#1422	1	2	yes	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	#1277	1	1	yes	characteristic
105	C₄	#1367	1	1	yes	characteristic
106	C₄	#1370	1	1	yes	characteristic
107	C₄	#1376	1	1	yes	characteristic
108	C₄	C₄×C₃₂	1	1	yes	characteristic
109	C₄	#1397	1	1	yes	characteristic
110	C₄	C₂²⋊C₃₂	1	2	yes	—
111	C₂	#7151	1	1	yes	characteristic
112	C₂	#7455	1	2	yes	—
113	C₂	#7458	1	2	yes	—
114	C₂	#7465	1	1	yes	characteristic
115	C₂	#7467	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊C₃₂

C₂

yes

maximal

C₂×C₈×C₁₆

C₂

yes

maximal, characteristic

C₂²×C₃₂

—

C₂×C₄×C₁₆

C₄

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₂×C₈²

C₄

yes

characteristic

C₄⋊C₃₂

—

C₄⋊C₃₂

C₄

yes

—

C₈×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

Q₈

yes

characteristic

C₂×C₄×C₈

C₈

yes

characteristic

C₂×C₄×C₈

C₈

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₄×C₁₆

—

C₄×C₁₆

C₈

yes

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

D₄

yes

—

C₈²

—

C₈²

—

C₂²×C₈

—

C₂²×C₈

C₁₆

yes

—

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

M₁₆

yes

characteristic

C₂×C₄²

C₂×C₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₂×C₁₆

D₈

yes

characteristic

C₂×C₁₆

SD₁₆

yes

—

C₂×C₁₆

Q₁₆

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₁₆

yes

—

C₄×C₈

C₄²

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂×C₈

yes

characteristic

C₄×C₈

M₁₆

yes

characteristic

C₃₂

—

C₂²×C₄

C₄⋊C₈

yes

characteristic

C₂²×C₄

C₂×C₁₆

yes

characteristic

C₂²×C₄

M₃₂

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₃₂

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₂²⋊C₈

yes

characteristic

C₂×C₈

C₄⋊C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄²

C₄×C₈

yes

characteristic

C₄²

C₂²⋊C₈

yes

characteristic

C₄²

C₂×C₁₆

yes

characteristic

C₄²

M₃₂

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

#102

yes

—

C₁₆

#103

yes

—

C₂³

C₄⋊C₁₆

yes

characteristic

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₄×C₁₆

yes

characteristic

C₂×C₄

#346

yes

characteristic

C₂×C₄

C₂²⋊C₁₆

yes

characteristic

C₂×C₄

C₂²⋊C₁₆

yes

characteristic

C₂×C₄

C₄⋊C₁₆

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×C₃₂

yes

—

C₂×C₄

M₆₄

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

yes

C₈

yes

C₈

yes

C₈

yes

C₂²

yes

C₂²

yes

C₂²

yes

100

C₂²

C₄⋊C₃₂

yes

—

101

C₂²

#1422

yes

—

102

C₄

—

103

C₄

—

104

C₄

#1277

yes

characteristic

105

C₄

#1367

yes

characteristic

106

C₄

#1370

yes

characteristic

107

C₄

#1376

yes

characteristic

108

C₄

C₄×C₃₂

yes

characteristic

109

C₄

#1397

yes

characteristic

110

C₄

C₂²⋊C₃₂

yes

—

111

C₂

#7151

yes

characteristic

112

C₂

#7455

yes

—

113

C₂

#7458

yes

—

114

C₂

#7465

yes

characteristic

115

C₂

#7467

yes

characteristic

	subgroup	G/H
center	C₂²×C₁₆	D₄
commutator	C₄	C₄×C₃₂
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
socle, prime radical	C₂³	C₄⋊C₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂²×C₁₆

D₄

commutator

C₄

C₄×C₃₂

Frattini, cosocle

C₂×C₄×C₁₆

C₂²

socle, prime radical

C₂³

C₄⋊C₁₆

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
58. C₂×C₈	C₃₂	semidirect

H ≅ G/N

type

58. C₂×C₈

C₃₂

semidirect

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

subgroup	index
C₂×C₄⋊C₃₂	2
C₂×C₈×C₁₆	2

subgroup

index

C₂×C₄⋊C₃₂

C₂×C₈×C₁₆

Subgroups — #94015

Conjugacy classes of subgroups

Complements

Maximal subgroups