Group #94165 — subgroups

Subgroups — #94165

Order 512.

134 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 70 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	#13640	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
3	#11408	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₃₂	C₄	1	2	yes	—
5	#2162	C₄	1	1	yes	characteristic
6	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
7	C₈×M₁₆	C₄	1	1	yes	characteristic
8	C₄⋊C₃₂	—	2	1	no	—
9	C₄⋊C₃₂	—	2	1	no	—
10	C₄⋊C₃₂	—	2	1	no	—
11	C₂²⋊C₃₂	C₄	1	2	yes	—
12	#1397	—	2	1	no	—
13	#1312	—	2	1	no	—
14	#1311	—	2	1	no	—
15	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
17	#431	C₈	1	1	yes	characteristic
18	C₄×M₁₆	C₈	1	1	yes	characteristic
19	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃₂	—	2	2	no	—
21	C₂×C₃₂	—	2	2	no	—
22	C₂×C₃₂	—	4	2	no	—
23	C₄⋊C₁₆	—	2	1	no	—
24	C₂²⋊C₁₆	C₈	1	2	yes	—
25	C₄×C₁₆	—	2	1	no	—
26	C₄×C₁₆	D₄	1	1	yes	characteristic
27	C₄×C₁₆	D₄	1	1	yes	characteristic
28	C₄×C₁₆	D₄	1	1	yes	characteristic
29	C₄×C₁₆	Q₈	1	1	yes	characteristic
30	#322	—	2	1	no	—
31	C₈²	—	2	1	no	—
32	C₂×M₁₆	—	2	1	no	—
33	C₂²×C₈	C₄²	1	1	yes	characteristic
34	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
36	C₂×C₁₆	—	2	1	no	—
37	C₂×C₁₆	—	2	1	no	—
38	C₂×C₁₆	—	2	2	no	—
39	C₂×C₁₆	—	4	1	no	—
40	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
44	C₄⋊C₈	C₁₆	1	2	yes	—
45	C₂²⋊C₈	—	2	1	no	—
46	#97	C₁₆	1	2	yes	—
47	C₄×C₈	—	2	1	no	—
48	C₄×C₈	—	2	1	no	—
49	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
50	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
51	C₄×C₈	M₁₆	1	1	yes	characteristic
52	C₄×C₈	M₁₆	1	1	yes	characteristic
53	C₃₂	—	4	2	no	—
54	C₃₂	—	4	2	no	—
55	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
56	C₂²×C₄	M₃₂	1	1	yes	characteristic
57	C₂²×C₄	M₃₂	1	1	yes	characteristic
58	M₁₆	—	4	2	no	—
59	C₂×C₈	—	2	1	no	—
60	C₂×C₈	—	2	1	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	4	1	no	—
64	C₂×C₈	#95	1	1	yes	characteristic
65	C₂×C₈	#97	1	1	yes	characteristic
66	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
67	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
68	C₄²	C₂²⋊C₈	1	1	yes	characteristic
69	C₄²	C₄⋊C₈	1	1	yes	characteristic
70	C₄²	C₂×C₁₆	1	1	yes	characteristic
71	C₄²	C₂×C₁₆	1	1	yes	characteristic
72	C₁₆	—	2	1	no	—
73	C₁₆	—	2	1	no	—
74	C₁₆	—	2	1	no	—
75	C₁₆	—	4	2	no	—
76	C₁₆	#100	1	1	yes	characteristic
77	C₁₆	#101	1	1	yes	characteristic
78	C₂³	#346	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	#336	1	1	yes	characteristic
84	C₂×C₄	C₄×C₁₆	1	1	yes	characteristic
85	C₂×C₄	C₂²⋊C₁₆	1	1	yes	characteristic
86	C₂×C₄	C₂²⋊C₁₆	1	1	yes	characteristic
87	C₂×C₄	C₄⋊C₁₆	1	1	yes	characteristic
88	C₂×C₄	C₄⋊C₁₆	1	1	yes	characteristic
89	C₈	—	2	1	no	—
90	C₈	—	2	1	no	—
91	C₈	—	2	1	no	—
92	C₈	—	4	2	no	—
93	C₈	—	4	2	no	—
94	C₈	#343	1	1	yes	characteristic
95	C₈	#344	1	1	yes	characteristic
96	C₂²	—	2	1	no	—
97	C₂²	—	2	1	no	—
98	C₂²	#1374	1	1	yes	characteristic
99	C₂²	#1398	1	2	yes	—
100	C₄	—	2	1	no	—
101	C₄	—	2	1	no	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	#1283	1	1	yes	characteristic
105	C₄	#1378	1	1	yes	characteristic
106	C₄	#1401	1	2	yes	—
107	C₂	—	2	2	no	—
108	C₂	#7209	1	1	yes	characteristic
109	C₂	#7466	1	1	yes	characteristic
110	C₂	#7470	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13640

C₂

yes

maximal, characteristic

C₂×C₄⋊C₃₂

C₂

yes

maximal, characteristic

#11408

C₂

yes

maximal, characteristic

C₂²×C₃₂

C₄

yes

—

#2162

C₄

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₈×M₁₆

C₄

yes

characteristic

C₄⋊C₃₂

—

C₄⋊C₃₂

—

C₄⋊C₃₂

—

C₂²⋊C₃₂

C₄

yes

—

#1397

—

#1312

—

#1311

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

#431

C₈

yes

characteristic

C₄×M₁₆

C₈

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

C₈

yes

—

C₄×C₁₆

—

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

Q₈

yes

characteristic

#322

—

C₈²

—

C₂×M₁₆

—

C₂²×C₈

C₄²

yes

characteristic

C₂²×C₈

C₂×C₈

yes

characteristic

C₂×C₄²

C₂×C₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₄⋊C₄

yes

characteristic

C₂×C₁₆

C₄⋊C₄

yes

characteristic

C₄⋊C₈

C₁₆

yes

—

C₂²⋊C₈

—

#97

C₁₆

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

M₁₆

yes

characteristic

C₄×C₈

M₁₆

yes

characteristic

C₃₂

—

C₃₂

—

C₂²×C₄

C₄×C₈

yes

characteristic

C₂²×C₄

M₃₂

yes

characteristic

C₂²×C₄

M₃₂

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₂²⋊C₈

yes

characteristic

C₂×C₈

C₄⋊C₈

yes

characteristic

C₄²

C₂²⋊C₈

yes

characteristic

C₄²

C₄⋊C₈

yes

characteristic

C₄²

C₂×C₁₆

yes

characteristic

C₄²

C₂×C₁₆

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

yes

C₁₆

yes

C₂³

yes

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#336

yes

characteristic

C₂×C₄

C₄×C₁₆

yes

characteristic

C₂×C₄

C₂²⋊C₁₆

yes

characteristic

C₂×C₄

C₂²⋊C₁₆

yes

characteristic

C₂×C₄

C₄⋊C₁₆

yes

characteristic

C₂×C₄

C₄⋊C₁₆

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

yes

C₈

yes

C₂²

—

C₂²

—

C₂²

#1374

yes

characteristic

C₂²

#1398

yes

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

#1283

yes

characteristic

105

C₄

#1378

yes

characteristic

106

C₄

#1401

yes

—

107

C₂

—

108

C₂

#7209

yes

characteristic

109

C₂

#7466

yes

characteristic

110

C₂

#7470

yes

characteristic

	subgroup	G/H
center	C₂×C₈	#95
commutator	C₂×C₄	C₄×C₁₆
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
socle	C₂²	#1374
prime radical	C₂³	#346
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₈

#95

commutator

C₂×C₄

C₄×C₁₆

Frattini, cosocle

C₂×C₄×C₁₆

C₂²

C₂²

C₂³

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#13640	2
C₂×C₄⋊C₃₂	2
#11408	2

subgroup

index

#13640

C₂×C₄⋊C₃₂

#11408

Subgroups — #94165

Conjugacy classes of subgroups

Maximal subgroups