Group #124646 — subgroups

Subgroups — #124646

Order 512.

250 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 78 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	#21734	C₂	1	1	yes	maximal, characteristic
2	#8800	C₂	1	2	yes	maximal
3	#7134	C₂	1	4	yes	maximal
4	#3160	C₄	1	1	yes	characteristic
5	#2960	C₄	1	2	yes	—
6	#1994	C₂²	1	2	yes	—
7	#1946	—	2	2	no	—
8	#1848	C₄	1	1	yes	characteristic
9	#1773	—	2	2	no	—
10	C₄×#97	C₂²	1	1	yes	characteristic
11	#1281	—	2	2	no	—
12	#1276	C₂²	1	4	yes	—
13	#1275	—	2	2	no	—
14	#533	C₂×C₄	1	2	yes	—
15	#500	—	2	1	no	—
16	#499	C₈	1	4	yes	complemented
17	#498	—	2	1	no	—
18	#446	—	4	2	no	—
19	C₈×Q₈	—	4	2	no	—
20	#431	—	2	4	no	—
21	C₂×#108	—	2	2	no	—
22	#427	—	2	2	no	—
23	#424	—	2	2	no	—
24	#423	C₂×C₄	1	2	yes	—
25	#405	—	2	2	no	—
26	C₂×#97	D₄	1	2	yes	—
27	C₂×#97	D₄	1	2	yes	—
28	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
30	C₄³	C₂³	1	1	yes	characteristic
31	#339	—	2	4	no	—
32	C₂×M₁₆	—	2	2	no	—
33	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
36	C₄⋊Q₈	—	2	2	no	—
37	#125	C₂×C₈	1	4	yes	—
38	C₂²⋊Q₈	—	4	2	no	—
39	C₄×Q₈	—	4	2	no	—
40	#117	—	2	2	no	—
41	C₂×C₄²	—	4	1	no	—
42	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄²	C₂×D₄	1	2	yes	—
44	#108	—	4	4	no	—
45	#107	—	2	4	no	—
46	#106	—	2	4	no	—
47	C₄⋊C₈	—	4	2	no	—
48	C₄⋊C₈	—	4	4	no	—
49	C₄⋊C₈	—	4	4	no	—
50	C₂²⋊C₈	—	4	4	no	—
51	#97	—	2	4	no	—
52	#97	—	4	2	no	—
53	#97	—	4	2	no	—
54	C₄×C₈	—	2	2	no	—
55	C₄×C₈	—	4	2	no	—
56	C₄×C₈	—	4	4	no	—
57	C₄×C₈	C₂×C₈	1	2	yes	—
58	C₄×C₈	M₁₆	1	1	yes	characteristic
59	C₄×C₈	M₁₆	1	1	yes	characteristic
60	C₂×Q₈	—	4	2	no	—
61	C₂²×C₄	—	2	1	no	—
62	C₂²×C₄	—	2	2	no	—
63	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	M₁₆	—	4	4	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	—	2	2	no	—
67	C₂×C₈	—	2	2	no	—
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	4	2	no	—
71	C₂×C₈	—	8	4	no	—
72	C₂×C₈	C₂²⋊C₈	1	2	yes	—
73	C₂×C₈	C₂²⋊C₈	1	2	yes	—
74	C₂×C₈	#100	1	2	yes	—
75	C₄⋊C₄	—	2	4	no	—
76	C₄⋊C₄	—	4	4	no	—
77	C₄⋊C₄	—	4	2	no	—
78	C₂²⋊C₄	—	4	2	no	—
79	C₄²	—	2	1	no	—
80	C₄²	—	2	2	no	—
81	C₄²	—	4	2	no	—
82	C₄²	—	8	2	no	—
83	C₄²	C₂²×C₈	1	1	yes	characteristic
84	C₄²	C₂×M₁₆	1	1	yes	characteristic
85	C₄²	#137	1	4	yes	—
86	C₂³	#410	1	1	yes	characteristic
87	Q₈	—	8	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	—	4	4	no	—
93	C₂×C₄	—	8	2	no	—
94	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
95	C₂×C₄	C₂×#100	1	1	yes	characteristic
96	C₂×C₄	#419	1	2	yes	—
97	C₂×C₄	#421	1	2	yes	—
98	C₈	—	2	2	no	—
99	C₈	—	2	2	no	—
100	C₈	—	2	2	no	—
101	C₈	—	4	1	no	—
102	C₈	—	8	2	no	—
103	C₈	—	16	4	no	—
104	C₂²	—	4	1	no	—
105	C₂²	#1460	1	1	yes	characteristic
106	C₂²	#1515	1	2	yes	—
107	C₄	—	2	2	no	—
108	C₄	—	4	4	no	—
109	C₄	—	4	1	no	—
110	C₄	—	8	4	no	—
111	C₄	#1479	1	2	yes	—
112	C₄	#1536	1	1	yes	characteristic
113	C₄	#1558	1	1	yes	characteristic
114	C₂	—	2	1	no	—
115	C₂	—	2	2	no	—
116	C₂	#7605	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21734

C₂

yes

maximal, characteristic

#8800

C₂

yes

maximal

#7134

C₂

yes

maximal

#3160

C₄

yes

characteristic

#2960

C₄

yes

—

#1994

C₂²

yes

—

#1946

—

#1848

C₄

yes

characteristic

#1773

—

C₄×#97

C₂²

yes

characteristic

#1281

—

#1276

C₂²

yes

—

#1275

—

#533

C₂×C₄

yes

—

#500

—

#499

C₈

yes

complemented

#498

—

#446

—

C₈×Q₈

—

—

—

—

—

C₂×C₄

yes

—

#405

—

C₂×#97

D₄

yes

—

C₂×#97

D₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₄³

C₂³

yes

characteristic

#339

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₄⋊Q₈

—

#125

C₂×C₈

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

—

#108

—

#107

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

characteristic

C₄×C₈

M₁₆

yes

characteristic

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₈

yes

—

C₂×C₈

C₂²⋊C₈

yes

—

C₂×C₈

#100

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×C₈

yes

characteristic

C₄²

C₂×M₁₆

yes

characteristic

C₄²

#137

yes

—

C₂³

#410

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂²⋊C₈

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

#421

yes

—

C₈

—

C₈

—

100

C₈

—

101

C₈

—

102

C₈

—

103

C₈

—

104

C₂²

—

105

C₂²

#1460

yes

characteristic

106

C₂²

#1515

yes

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#1479

yes

—

112

C₄

#1536

yes

characteristic

113

C₄

#1558

yes

characteristic

114

C₂

—

115

C₂

—

116

C₂

#7605

yes

characteristic

	subgroup	G/H
center	C₄	#1558
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
socle	C₂	#7605
prime radical	C₂³	#410
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₄³ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄

#1558

commutator

C₄²

C₂²×C₈

Frattini, cosocle

C₄³

C₂³

C₂

C₂³

G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G

LECS

G ▷ C₄³ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
16. #499	C₈	semidirect

H ≅ G/N

type

16. #499

C₈

semidirect

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

subgroup	index
#21734	2
#8800	2
#7134	2

subgroup

index

#21734

#8800

#7134

Subgroups — #124646

Conjugacy classes of subgroups

Complements

Maximal subgroups