Group #94049 — subgroups

Subgroups — #94049

Order 512.

160 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 84 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	C₄×M₆₄	C₂	1	1	yes	maximal, characteristic
2	C₄×#368	C₂	1	1	yes	maximal, characteristic
3	C₂×#1371	C₂	1	1	yes	maximal, characteristic
4	C₂×M₆₄	—	2	1	no	—
5	C₂×#368	C₄	1	2	yes	—
6	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
7	C₄×#108	—	2	1	no	—
8	C₂×#335	—	2	1	no	—
9	#1397	C₄	1	2	yes	—
10	C₄×C₃₂	—	2	1	no	—
11	#1381	—	2	1	no	—
12	#1371	—	2	1	no	—
13	#1371	C₄	1	2	yes	—
14	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₁₆	D₄	1	1	yes	characteristic
16	C₂×#108	—	2	2	no	—
17	C₂×C₄⋊C₈	—	4	1	no	—
18	C₄×M₁₆	—	4	1	no	—
19	C₂×C₄×C₈	D₄	1	1	yes	characteristic
20	M₆₄	—	2	4	no	—
21	C₂×C₃₂	—	2	2	no	—
22	#368	—	2	2	no	—
23	#368	C₈	1	4	yes	complemented
24	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₄×C₁₆	D₄	1	1	yes	characteristic
27	C₄×C₁₆	Q₈	1	1	yes	characteristic
28	#341	—	4	1	no	—
29	#335	—	2	2	no	—
30	#335	—	4	1	no	—
31	C₂×M₁₆	—	4	2	no	—
32	C₂²×C₈	—	8	1	no	—
33	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₈	SD₁₆	1	1	yes	characteristic
35	C₂×C₄²	D₈	1	1	yes	characteristic
36	C₂×C₁₆	—	2	2	no	—
37	C₂×C₁₆	C₄²	1	1	yes	characteristic
38	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₁₆	C₂×C₈	1	2	yes	—
42	C₂×C₁₆	M₁₆	1	2	yes	—
43	#108	—	2	4	no	—
44	#108	—	4	2	no	—
45	C₄⋊C₈	—	4	2	no	—
46	C₄⋊C₈	—	8	1	no	—
47	#97	—	8	1	no	—
48	C₄×C₈	—	8	1	no	—
49	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
50	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
51	C₄×C₈	SD₁₆	1	1	yes	characteristic
52	C₄×C₈	Q₁₆	1	1	yes	characteristic
53	C₃₂	—	2	4	no	—
54	C₂²×C₄	#102	1	1	yes	characteristic
55	C₂²×C₄	SD₃₂	1	1	yes	characteristic
56	C₂²×C₄	Q₃₂	1	1	yes	characteristic
57	M₁₆	—	4	4	no	—
58	M₁₆	—	8	2	no	—
59	C₂×C₈	—	2	2	no	—
60	C₂×C₈	—	8	2	no	—
61	C₂×C₈	—	8	2	no	—
62	C₂×C₈	—	16	2	no	—
63	C₂×C₈	#95	1	1	yes	characteristic
64	C₂×C₈	C₂²⋊C₈	1	2	yes	—
65	C₂×C₈	#103	1	1	yes	characteristic
66	C₂×C₈	#103	1	1	yes	characteristic
67	C₂×C₈	#104	1	2	yes	—
68	C₂×C₈	#106	1	1	yes	characteristic
69	C₄²	#102	1	1	yes	characteristic
70	C₄²	#107	1	1	yes	characteristic
71	C₄²	D₁₆	1	1	yes	characteristic
72	C₄²	SD₃₂	1	1	yes	characteristic
73	C₁₆	—	2	1	no	—
74	C₁₆	—	2	1	no	—
75	C₁₆	C₄×C₈	1	1	yes	characteristic
76	C₁₆	#97	1	1	yes	characteristic
77	C₁₆	C₄⋊C₈	1	2	yes	—
78	C₂³	#358	1	1	yes	characteristic
79	C₂×C₄	—	2	1	no	—
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	4	1	no	—
82	C₂×C₄	D₄⋊C₈	1	2	yes	—
83	C₂×C₄	#340	1	1	yes	characteristic
84	C₂×C₄	#357	1	1	yes	characteristic
85	C₂×C₄	#357	1	1	yes	characteristic
86	C₂×C₄	#358	1	1	yes	characteristic
87	C₂×C₄	#359	1	2	yes	—
88	C₂×C₄	#366	1	1	yes	characteristic
89	C₂×C₄	#367	1	1	yes	characteristic
90	C₈	—	2	1	no	—
91	C₈	—	2	1	no	—
92	C₈	—	8	4	no	—
93	C₈	—	16	2	no	—
94	C₈	Q₈⋊C₈	1	2	yes	—
95	C₈	#336	1	1	yes	characteristic
96	C₈	#339	1	1	yes	characteristic
97	C₂²	—	2	1	no	—
98	C₂²	—	2	1	no	—
99	C₂²	#1332	1	2	yes	—
100	C₂²	#1380	1	1	yes	characteristic
101	C₄	—	2	1	no	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	#1279	1	1	yes	characteristic
105	C₄	#1331	1	2	yes	—
106	C₄	#1382	1	1	yes	characteristic
107	C₄	#1419	1	1	yes	characteristic
108	C₄	#1420	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#7171	1	1	yes	characteristic
111	C₂	#7478	1	1	yes	characteristic
112	C₂	#7479	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×M₆₄

C₂

yes

maximal, characteristic

C₄×#368

C₂

yes

maximal, characteristic

C₂×#1371

C₂

yes

maximal, characteristic

C₂×M₆₄

—

C₂×#368

C₄

yes

—

C₂×C₄×C₁₆

C₂²

yes

—

—

C₄

yes

—

C₄×C₃₂

—

#1381

—

#1371

—

#1371

C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

D₄

yes

characteristic

C₂×#108

—

C₂×C₄⋊C₈

—

C₄×M₁₆

—

C₂×C₄×C₈

D₄

yes

characteristic

M₆₄

—

C₂×C₃₂

—

#368

—

#368

C₈

yes

complemented

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

Q₈

yes

—

—

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

SD₁₆

yes

characteristic

C₂×C₄²

D₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₄²

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₄⋊C₄

yes

characteristic

C₂×C₁₆

C₄⋊C₄

yes

characteristic

C₂×C₁₆

C₂×C₈

yes

—

C₂×C₁₆

M₁₆

yes

—

#108

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

Q₁₆

yes

characteristic

C₃₂

—

C₂²×C₄

#102

yes

characteristic

C₂²×C₄

SD₃₂

yes

characteristic

C₂²×C₄

Q₃₂

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#95

yes

characteristic

C₂×C₈

C₂²⋊C₈

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

yes

C₄²

yes

C₄²

yes

C₄²

D₁₆

yes

characteristic

C₄²

SD₃₂

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

C₄×C₈

yes

characteristic

C₁₆

#97

yes

characteristic

C₁₆

C₄⋊C₈

yes

—

C₂³

#358

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₄⋊C₈

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

C₈

—

C₈

Q₈⋊C₈

yes

—

C₈

yes

C₈

yes

C₂²

—

C₂²

—

C₂²

#1332

yes

—

100

C₂²

#1380

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

#1279

yes

characteristic

105

C₄

#1331

yes

—

106

C₄

#1382

yes

characteristic

107

C₄

#1419

yes

characteristic

108

C₄

#1420

yes

characteristic

109

C₂

—

110

C₂

#7171

yes

characteristic

111

C₂

#7478

yes

characteristic

112

C₂

#7479

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#357
commutator	C₁₆	C₄×C₈
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
socle	C₂²	#1380
prime radical	C₂³	#358
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#357

commutator

C₁₆

C₄×C₈

Frattini, cosocle

C₂×C₄×C₁₆

C₂²

C₂²

C₂³

G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
23. #368	C₈	semidirect

H ≅ G/N

type

23. #368

C₈

semidirect

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

subgroup	index
C₄×M₆₄	2
C₄×#368	2
C₂×#1371	2

subgroup

index

C₄×M₆₄

C₄×#368

C₂×#1371

Subgroups — #94049

Conjugacy classes of subgroups

Complements

Maximal subgroups