Group #93268 — subgroups

Subgroups — #93268

Order 512.

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

normal subgroups only

This group has 284 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	C₄²×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₄×#335	C₂	1	2	yes	maximal
3	C₂×C₄×C₁₆	—	2	2	no	—
4	C₂×C₄×C₁₆	C₄	1	2	yes	—
5	C₄×C₄⋊C₈	—	2	2	no	—
6	C₄²×C₈	C₂²	1	1	yes	characteristic
7	C₂×#335	C₄	1	4	yes	—
8	#1277	—	2	2	no	—
9	C₂²×C₁₆	—	2	2	no	—
10	C₂×C₄⋊C₈	—	2	4	no	—
11	C₂×C₄×C₈	—	4	2	no	—
12	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
13	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₈	D₄	1	2	yes	—
15	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
16	C₄³	D₄	1	1	yes	characteristic
17	C₄×C₁₆	—	2	8	no	—
18	C₄×C₁₆	—	2	2	no	—
19	C₄×C₁₆	C₈	1	4	yes	complemented
20	#336	—	4	2	no	—
21	#335	—	2	4	no	—
22	#335	C₈	1	8	yes	complemented
23	C₂²×C₈	—	4	4	no	—
24	C₂²×C₈	C₄²	1	1	yes	characteristic
25	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
26	C₂²×C₈	C₄⋊C₄	1	2	yes	—
27	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
28	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄²	D₈	1	1	yes	characteristic
30	C₂×C₄²	SD₁₆	1	2	yes	—
31	C₂×C₄²	Q₁₆	1	1	yes	characteristic
32	C₂×C₁₆	—	2	8	no	—
33	C₂×C₁₆	—	2	4	no	—
34	C₄⋊C₈	—	2	8	no	—
35	C₄⋊C₈	—	4	4	no	—
36	C₄×C₈	—	2	4	no	—
37	C₄×C₈	—	2	2	no	—
38	C₄×C₈	—	8	2	no	—
39	C₄×C₈	—	8	2	no	—
40	C₄×C₈	C₂×C₈	1	4	yes	—
41	C₄×C₈	C₂×C₈	1	2	yes	—
42	C₄×C₈	M₁₆	1	4	yes	—
43	C₄×C₈	M₁₆	1	2	yes	—
44	C₂²×C₄	#95	1	1	yes	characteristic
45	C₂²×C₄	#102	1	2	yes	—
46	C₂²×C₄	#103	1	2	yes	—
47	C₂²×C₄	#106	1	1	yes	characteristic
48	C₂²×C₄	#107	1	1	yes	characteristic
49	C₂×C₈	—	2	2	no	—
50	C₂×C₈	—	2	2	no	—
51	C₂×C₈	—	2	2	no	—
52	C₂×C₈	—	4	8	no	—
53	C₂×C₈	—	8	4	no	—
54	C₂×C₈	—	8	4	no	—
55	C₂×C₈	C₄×C₈	1	1	yes	characteristic
56	C₂×C₈	C₄×C₈	1	2	yes	—
57	C₂×C₈	#97	1	1	yes	characteristic
58	C₂×C₈	#97	1	2	yes	—
59	C₂×C₈	C₂²⋊C₈	1	2	yes	—
60	C₂×C₈	C₄⋊C₈	1	4	yes	—
61	C₄²	—	2	4	no	—
62	C₄²	—	2	2	no	—
63	C₄²	C₂²⋊C₈	1	4	yes	—
64	C₄²	#104	1	4	yes	—
65	C₄²	C₄⋊C₈	1	2	yes	—
66	C₄²	#108	1	2	yes	—
67	C₄²	D₁₆	1	1	yes	characteristic
68	C₄²	SD₃₂	1	2	yes	—
69	C₄²	Q₃₂	1	1	yes	characteristic
70	C₁₆	—	2	8	no	—
71	C₂³	#340	1	1	yes	characteristic
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	D₄⋊C₈	1	4	yes	—
76	C₂×C₄	Q₈⋊C₈	1	4	yes	—
77	C₂×C₄	#334	1	2	yes	—
78	C₂×C₄	#335	1	2	yes	—
79	C₂×C₄	#336	1	1	yes	characteristic
80	C₂×C₄	#336	1	2	yes	—
81	C₂×C₄	#339	1	2	yes	—
82	C₂×C₄	#341	1	1	yes	characteristic
83	C₂×C₄	#357	1	2	yes	—
84	C₂×C₄	#358	1	2	yes	—
85	C₂×C₄	#359	1	4	yes	—
86	C₂×C₄	#366	1	1	yes	characteristic
87	C₂×C₄	#367	1	1	yes	characteristic
88	C₂×C₄	#368	1	2	yes	—
89	C₈	—	2	2	no	—
90	C₈	—	2	4	no	—
91	C₈	—	8	8	no	—
92	C₈	C₈²	1	1	yes	characteristic
93	C₈	#322	1	1	yes	characteristic
94	C₈	#322	1	2	yes	—
95	C₂²	#1277	1	1	yes	characteristic
96	C₂²	#1279	1	2	yes	—
97	C₂²	#1380	1	1	yes	characteristic
98	C₂²	#1381	1	1	yes	characteristic
99	C₂²	#1382	1	2	yes	—
100	C₄	—	2	4	no	—
101	C₄	—	2	2	no	—
102	C₄	#1273	1	1	yes	characteristic
103	C₄	#1276	1	1	yes	characteristic
104	C₄	#1278	1	2	yes	—
105	C₄	#1331	1	4	yes	—
106	C₄	#1332	1	4	yes	—
107	C₄	#1371	1	2	yes	—
108	C₄	#1372	1	2	yes	—
109	C₂	#7015	1	1	yes	characteristic
110	C₂	#7154	1	1	yes	characteristic
111	C₂	#7155	1	1	yes	characteristic
112	C₂	#7171	1	2	yes	—
113	C₂	#7172	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄²×C₁₆

C₂

yes

maximal, characteristic

C₄×#335

C₂

yes

maximal

C₂×C₄×C₁₆

—

C₂×C₄×C₁₆

C₄

yes

—

C₄×C₄⋊C₈

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#335

C₄

yes

—

#1277

—

C₂²×C₁₆

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

Q₈

yes

characteristic

C₄³

D₄

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₈

yes

—

—

C₈

yes

complemented

C₂²×C₈

—

C₂²×C₈

C₄²

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

—

C₂×C₄²

C₂²⋊C₄

yes

—

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

D₈

yes

characteristic

C₂×C₄²

SD₁₆

yes

—

C₂×C₄²

Q₁₆

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

M₁₆

yes

—

C₂²×C₄

#95

yes

characteristic

C₂²×C₄

#102

yes

—

C₂²×C₄

#103

yes

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×C₈

yes

characteristic

C₂×C₈

C₄×C₈

yes

—

C₂×C₈

#97

yes

characteristic

C₂×C₈

#97

yes

—

C₂×C₈

C₂²⋊C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₄²

—

C₄²

—

C₄²

C₂²⋊C₈

yes

—

C₄²

#104

yes

—

C₄²

C₄⋊C₈

yes

—

C₄²

#108

yes

—

C₄²

D₁₆

yes

characteristic

C₄²

SD₃₂

yes

—

C₄²

Q₃₂

yes

characteristic

C₁₆

—

C₂³

#340

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₄⋊C₈

yes

—

C₂×C₄

Q₈⋊C₈

yes

—

C₂×C₄

#334

yes

—

C₂×C₄

#335

yes

—

C₂×C₄

#336

yes

characteristic

C₂×C₄

#336

yes

—

C₂×C₄

#339

yes

—

C₂×C₄

#341

yes

characteristic

C₂×C₄

#357

yes

—

C₂×C₄

#358

yes

—

C₂×C₄

#359

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₈

—

C₈

—

C₈

—

C₈

C₈²

yes

C₈

yes

C₈

yes

—

C₂²

#1277

yes

characteristic

C₂²

#1279

yes

—

C₂²

yes

C₂²

yes

C₂²

yes

—

100

C₄

—

101

C₄

—

102

C₄

#1273

yes

characteristic

103

C₄

#1276

yes

characteristic

104

C₄

#1278

yes

—

105

C₄

#1331

yes

—

106

C₄

#1332

yes

—

107

C₄

#1371

yes

—

108

C₄

#1372

yes

—

109

C₂

#7015

yes

characteristic

110

C₂

#7154

yes

characteristic

111

C₂

#7155

yes

characteristic

112

C₂

#7171

yes

—

113

C₂

#7172

yes

—

	subgroup	G/H
center	C₂×C₄²	D₈
commutator	C₈	C₈²
Frattini, cosocle	C₄²×C₈	C₂²
socle, prime radical	C₂³	#340
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄²

D₈

commutator

C₈

C₈²

Frattini, cosocle

C₄²×C₈

C₂²

socle, prime radical

C₂³

#340

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄² ◁ C₄³ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
19. C₄×C₁₆	C₈	semidirect
22. #335	C₈	semidirect

H ≅ G/N

type

19. C₄×C₁₆

C₈

semidirect

22. #335

C₈

semidirect

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

subgroup	index
C₄²×C₁₆	2
C₄×#335	2

subgroup

index

C₄²×C₁₆

C₄×#335

Subgroups — #93268

Conjugacy classes of subgroups

Complements

Maximal subgroups