Group #134857 — subgroups

Subgroups — #134857

Order 512.

223 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 223 conjugacy classes of subgroups — too many to draw. Only its 86 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	#33474	C₂	1	1	yes	maximal, characteristic
2	#12024	C₂	1	2	yes	maximal
3	#8633	C₂	1	2	yes	maximal
4	#7246	C₂	1	2	yes	maximal
5	#2960	C₂²	1	1	yes	characteristic
6	#2183	—	2	2	no	—
7	Q₈×C₁₆	—	2	2	no	—
8	#2151	C₄	1	2	yes	—
9	#2150	C₂²	1	1	yes	characteristic
10	#2108	C₄	1	2	yes	—
11	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	#1568	C₂²	1	4	yes	—
13	#1537	—	2	2	no	—
14	#1372	—	2	2	no	—
15	#1371	—	2	2	no	—
16	#1307	—	2	2	no	—
17	#533	D₄	1	1	yes	characteristic
18	C₂×M₃₂	C₂×C₄	1	2	yes	—
19	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
21	#446	—	2	2	no	—
22	C₈×Q₈	—	2	2	no	—
23	#432	—	2	4	no	—
24	#424	Q₈	1	2	yes	—
25	#423	D₄	1	1	yes	characteristic
26	#405	C₂×C₄	1	2	yes	—
27	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
28	C₄⋊C₁₆	—	2	4	no	—
29	C₄⋊C₁₆	—	2	2	no	—
30	#346	—	2	2	no	—
31	C₄×C₁₆	—	2	2	no	—
32	C₄×C₁₆	D₄	1	2	yes	—
33	C₄×C₁₆	D₄	1	2	yes	—
34	#335	—	2	4	no	—
35	#334	—	2	4	no	—
36	#330	—	4	4	no	—
37	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
38	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
40	C₄⋊Q₈	Q₁₆	1	2	yes	—
41	#125	D₈	1	2	yes	—
42	C₂²⋊Q₈	—	2	2	no	—
43	C₄×Q₈	—	2	2	no	—
44	#117	C₄⋊C₄	1	2	yes	—
45	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
46	M₃₂	—	4	2	no	—
47	C₂×C₁₆	—	2	2	no	—
48	C₂×C₁₆	—	2	2	no	—
49	C₂×C₁₆	—	4	1	no	—
50	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
52	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
53	C₄⋊C₈	—	2	4	no	—
54	C₄⋊C₈	—	2	2	no	—
55	C₄⋊C₈	—	4	4	no	—
56	C₂²⋊C₈	—	4	4	no	—
57	#97	—	2	2	no	—
58	#97	—	4	4	no	—
59	C₄×C₈	—	2	2	no	—
60	C₄×C₈	C₂×D₄	1	2	yes	—
61	C₄×C₈	#41	1	2	yes	—
62	C₂×Q₈	—	2	2	no	—
63	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄	#135	1	2	yes	—
65	M₁₆	—	4	2	no	—
66	C₂×C₈	—	2	1	no	—
67	C₂×C₈	—	2	1	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	4	1	no	—
70	C₂×C₈	—	8	4	no	—
71	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈	#117	1	1	yes	characteristic
73	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
74	C₄⋊C₄	—	2	4	no	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	#107	1	4	yes	—
77	C₂²⋊C₄	—	2	2	no	—
78	C₄²	—	2	2	no	—
79	C₄²	#121	1	2	yes	—
80	C₄²	C₂×D₈	1	1	yes	characteristic
81	C₄²	C₂×Q₁₆	1	1	yes	characteristic
82	C₁₆	—	2	1	no	—
83	C₁₆	—	2	2	no	—
84	C₁₆	—	4	2	no	—
85	C₁₆	#101	1	2	yes	—
86	C₂³	#427	1	1	yes	characteristic
87	Q₈	—	4	4	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	4	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	#380	1	1	yes	characteristic
93	C₂×C₄	C₂×#107	1	1	yes	characteristic
94	C₂×C₄	#466	1	2	yes	—
95	C₂×C₄	#467	1	2	yes	—
96	C₈	—	2	1	no	—
97	C₈	—	2	2	no	—
98	C₈	—	4	2	no	—
99	C₈	—	16	4	no	—
100	C₈	C₂×#101	1	1	yes	characteristic
101	C₈	#413	1	1	yes	characteristic
102	C₂²	—	2	2	no	—
103	C₂²	—	2	1	no	—
104	C₂²	#1808	1	1	yes	characteristic
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	2	1	no	—
108	C₄	—	4	4	no	—
109	C₄	#1801	1	1	yes	characteristic
110	C₄	#1813	1	1	yes	characteristic
111	C₂	—	4	1	no	—
112	C₂	#8617	1	1	yes	characteristic
113	C₂	#12841	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33474

C₂

yes

maximal, characteristic

#12024

C₂

yes

maximal

#8633

C₂

yes

maximal

#7246

C₂

yes

maximal

#2960

C₂²

yes

characteristic

#2183

—

Q₈×C₁₆

—

#2151

C₄

yes

—

#2150

C₂²

yes

characteristic

#2108

C₄

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1568

C₂²

yes

—

—

—

—

—

D₄

yes

characteristic

C₂×M₃₂

C₂×C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

#446

—

C₈×Q₈

—

#432

—

#424

Q₈

yes

—

#423

D₄

yes

characteristic

#405

C₂×C₄

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

D₄

yes

—

C₄×C₁₆

D₄

yes

—

#335

—

#334

—

#330

—

C₂×M₁₆

C₄⋊C₄

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₄⋊Q₈

Q₁₆

yes

—

#125

D₈

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₄⋊C₄

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

—

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

#41

yes

—

C₂×Q₈

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

#135

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₈

#117

yes

characteristic

C₂×C₈

C₂²⋊Q₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

#107

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

#121

yes

—

C₄²

C₂×D₈

yes

characteristic

C₄²

C₂×Q₁₆

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

#101

yes

—

C₂³

#427

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

#467

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

100

C₈

C₂×#101

yes

characteristic

101

C₈

#413

yes

characteristic

102

C₂²

—

103

C₂²

—

104

C₂²

#1808

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

#1801

yes

characteristic

110

C₄

#1813

yes

characteristic

111

C₂

—

112

C₂

#8617

yes

characteristic

113

C₂

#12841

yes

—

	subgroup	G/H
center, socle	C₂²	#1808
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#427
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1808

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#427

LCS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ 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
#33474	2
#12024	2
#8633	2
#7246	2

subgroup

index

Subgroups — #134857

Conjugacy classes of subgroups

Maximal subgroups