Group #135702 — subgroups

Subgroups — #135702

Order 512.

189 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 189 conjugacy classes of subgroups — too many to draw. Only its 90 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	#11411	C₂	1	1	yes	maximal, characteristic
2	#8824	C₂	1	1	yes	maximal, characteristic
3	#8822	C₂	1	1	yes	maximal, characteristic
4	#7219	C₂	1	4	yes	maximal
5	#2162	C₂²	1	2	yes	—
6	#1848	—	2	1	no	—
7	#1837	C₂²	1	2	yes	—
8	#1567	—	2	1	no	—
9	#1566	—	2	1	no	—
10	#1566	—	2	1	no	—
11	#1566	C₂²	1	2	yes	—
12	C₂×C₈²	C₂²	1	1	yes	characteristic
13	#1312	C₄	1	2	yes	—
14	#1311	C₄	1	2	yes	—
15	C₂×#108	—	4	2	no	—
16	#423	—	2	2	no	—
17	#423	—	2	2	no	—
18	C₂×C₄×C₈	D₄	1	1	yes	characteristic
19	C₂×C₄×C₈	D₄	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
21	C₄⋊C₁₆	—	2	2	no	—
22	C₂²⋊C₁₆	—	2	2	no	—
23	C₄×C₁₆	C₂×C₄	1	4	yes	—
24	#341	—	2	4	no	—
25	#335	—	4	2	no	—
26	#334	—	2	4	no	—
27	#334	—	4	2	no	—
28	#334	—	4	2	no	—
29	C₈²	C₂×C₄	1	1	yes	characteristic
30	C₈²	C₂×C₄	1	1	yes	characteristic
31	C₈²	Q₈	1	1	yes	characteristic
32	C₈²	Q₈	1	1	yes	characteristic
33	C₂×M₁₆	—	4	2	no	—
34	C₂×M₁₆	—	4	2	no	—
35	C₂²×C₈	SD₁₆	1	2	yes	—
36	C₂²×C₈	Q₁₆	1	2	yes	—
37	C₂²×C₈	#41	1	2	yes	—
38	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
39	C₂×C₁₆	—	2	4	no	—
40	#108	—	8	4	no	—
41	C₄⋊C₈	—	4	2	no	—
42	C₄⋊C₈	—	4	2	no	—
43	C₄⋊C₈	—	4	2	no	—
44	C₄⋊C₈	—	4	2	no	—
45	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
46	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
47	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
48	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
49	C₄×C₈	D₈	1	2	yes	—
50	C₄×C₈	SD₁₆	1	2	yes	—
51	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
52	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
53	C₄×C₈	#41	1	2	yes	—
54	C₂²×C₄	#124	1	1	yes	characteristic
55	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
56	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
57	M₁₆	—	8	4	no	—
58	M₁₆	—	8	4	no	—
59	C₂×C₈	—	2	4	no	—
60	C₂×C₈	—	4	2	no	—
61	C₂×C₈	—	4	2	no	—
62	C₂×C₈	—	8	2	no	—
63	C₂×C₈	—	8	2	no	—
64	C₂×C₈	#106	1	2	yes	—
65	C₂×C₈	#106	1	2	yes	—
66	C₂×C₈	#107	1	2	yes	—
67	C₂×C₈	#107	1	2	yes	—
68	C₂×C₈	#117	1	2	yes	—
69	C₂×C₈	#125	1	2	yes	—
70	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₄²	#124	1	1	yes	characteristic
72	C₄²	C₂×D₈	1	1	yes	characteristic
73	C₄²	C₂×SD₁₆	1	1	yes	characteristic
74	C₁₆	—	4	4	no	—
75	C₂³	#487	1	1	yes	characteristic
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	#382	1	1	yes	characteristic
80	C₂×C₄	C₂×#106	1	1	yes	characteristic
81	C₂×C₄	C₂×#107	1	1	yes	characteristic
82	C₂×C₄	#486	1	1	yes	characteristic
83	C₂×C₄	#486	1	1	yes	characteristic
84	C₂×C₄	#487	1	1	yes	characteristic
85	C₈	—	2	2	no	—
86	C₈	—	2	2	no	—
87	C₈	—	2	2	no	—
88	C₈	—	2	2	no	—
89	C₈	—	2	2	no	—
90	C₈	—	16	2	no	—
91	C₈	—	16	2	no	—
92	C₈	#365	1	4	yes	—
93	C₂²	—	2	1	no	—
94	C₂²	—	2	1	no	—
95	C₂²	—	2	1	no	—
96	C₂²	#1834	1	1	yes	characteristic
97	C₄	—	2	1	no	—
98	C₄	—	2	1	no	—
99	C₄	—	2	1	no	—
100	C₄	#1833	1	2	yes	—
101	C₄	C₂×#365	1	2	yes	—
102	C₄	#2161	1	2	yes	—
103	C₂	—	4	1	no	—
104	C₂	#8793	1	1	yes	characteristic
105	C₂	#12886	1	1	yes	characteristic
106	C₂	#12892	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11411

C₂

yes

maximal, characteristic

#8824

C₂

yes

maximal, characteristic

#8822

C₂

yes

maximal, characteristic

#7219

C₂

yes

maximal

#2162

C₂²

yes

—

#1848

—

#1837

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂×C₈²

C₂²

yes

characteristic

#1312

C₄

yes

—

#1311

C₄

yes

—

C₂×#108

—

#423

—

#423

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

C₂×C₄

yes

—

—

—

—

—

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

Q₈

yes

characteristic

C₈²

Q₈

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

SD₁₆

yes

—

C₂²×C₈

Q₁₆

yes

—

C₂²×C₈

#41

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

D₈

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

#41

yes

—

C₂²×C₄

#124

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

characteristic

C₂²×C₄

C₂×Q₁₆

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#106

yes

—

C₂×C₈

#106

yes

—

C₂×C₈

#107

yes

—

C₂×C₈

#107

yes

—

C₂×C₈

#117

yes

—

C₂×C₈

#125

yes

—

C₄²

C₂×C₄⋊C₄

yes

characteristic

C₄²

#124

yes

characteristic

C₄²

C₂×D₈

yes

characteristic

C₄²

C₂×SD₁₆

yes

characteristic

C₁₆

—

C₂³

#487

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

#365

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#1834

yes

characteristic

C₄

—

C₄

—

C₄

—

100

C₄

#1833

yes

—

101

C₄

C₂×#365

yes

—

102

C₄

#2161

yes

—

103

C₂

—

104

C₂

#8793

yes

characteristic

105

C₂

#12886

yes

characteristic

106

C₂

#12892

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1834
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#487
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1834

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#487

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ 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
#11411	2
#8824	2
#8822	2
#7219	2

subgroup

index

Subgroups — #135702

Conjugacy classes of subgroups

Maximal subgroups