Group #512517 — subgroups

Subgroups — #512517

Order 512.

526 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 218 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	#33313	C₂	1	2	yes	maximal
2	#13283	C₂	1	8	yes	maximal
3	#12889	C₂	1	4	yes	maximal
4	C₄×#346	C₂	1	1	yes	maximal, characteristic
5	C₂×#500	C₂²	1	4	yes	—
6	C₂×#498	—	2	2	no	—
7	#2862	—	2	2	no	—
8	C₂×#346	C₂²	1	4	yes	—
9	C₂×C₄×C₁₆	C₂²	1	2	yes	—
10	#1846	—	2	2	no	—
11	#1846	—	2	2	no	—
12	#1846	C₂²	1	4	yes	—
13	#1846	C₂²	1	4	yes	—
14	C₄²×C₈	C₂²	1	1	yes	characteristic
15	#1380	C₂²	1	16	yes	—
16	C₂×C₄⋊Q₈	—	2	4	no	—
17	C₂²×C₁₆	C₂³	1	4	yes	—
18	#500	—	2	16	no	—
19	#498	—	4	8	no	—
20	C₂×#107	—	2	4	no	—
21	C₂×#107	C₂³	1	8	yes	—
22	C₂×#106	—	2	4	no	—
23	C₂×#106	—	2	4	no	—
24	C₂×C₄×C₈	D₄	1	2	yes	—
25	C₂×C₄×C₈	D₄	1	1	yes	characteristic
26	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂³	1	2	yes	—
28	#391	—	4	8	no	—
29	#382	—	2	4	no	—
30	#382	—	2	4	no	—
31	C₄³	D₄	1	1	yes	characteristic
32	#346	—	2	8	no	—
33	C₄×C₁₆	Q₈	1	8	yes	complemented
34	C₂²×Q₈	—	4	4	no	—
35	C₂²×C₈	C₂×D₄	1	2	yes	—
36	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₈	C₂⁴	1	1	yes	characteristic
38	C₄⋊Q₈	—	4	8	no	—
39	C₄⋊Q₈	—	4	8	no	—
40	C₂×C₄⋊C₄	—	2	8	no	—
41	C₂×C₄⋊C₄	—	4	4	no	—
42	C₂×C₄²	D₈	1	4	yes	—
43	C₂×C₄²	C₂×D₄	1	2	yes	—
44	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
45	C₂×C₁₆	—	2	8	no	—
46	C₂×C₁₆	C₂×Q₈	1	8	yes	—
47	#107	—	2	16	no	—
48	#107	—	4	8	no	—
49	#106	—	4	8	no	—
50	#106	—	4	8	no	—
51	C₄×C₈	SD₁₆	1	8	yes	—
52	C₄×C₈	Q₁₆	1	4	yes	—
53	C₄×C₈	C₂×Q₈	1	4	yes	—
54	C₄×C₈	#41	1	8	yes	—
55	#95	—	4	8	no	—
56	#95	—	4	8	no	—
57	C₂×Q₈	—	8	16	no	—
58	C₂×Q₈	—	8	8	no	—
59	C₂²×C₄	—	4	8	no	—
60	C₂²×C₄	C₂×D₈	1	4	yes	—
61	C₂²×C₄	C₂×D₈	1	2	yes	—
62	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
63	C₂×C₈	#124	1	4	yes	—
64	C₂×C₈	C₄⋊Q₈	1	2	yes	—
65	C₂×C₈	C₂×SD₁₆	1	4	yes	—
66	C₂×C₈	C₂×SD₁₆	1	4	yes	—
67	C₂×C₈	C₂×Q₁₆	1	4	yes	—
68	C₂×C₈	C₂²×Q₈	1	2	yes	—
69	C₂×C₈	C₂×#41	1	4	yes	—
70	C₄⋊C₄	—	4	8	no	—
71	C₄⋊C₄	—	4	8	no	—
72	C₄⋊C₄	—	8	8	no	—
73	C₄²	—	4	4	no	—
74	C₄²	#124	1	4	yes	—
75	C₄²	C₄⋊Q₈	1	2	yes	—
76	C₄²	C₂×SD₁₆	1	4	yes	—
77	C₄²	C₂×Q₁₆	1	2	yes	—
78	C₁₆	—	2	8	no	—
79	C₂³	C₂²×D₈	1	1	yes	characteristic
80	Q₈	—	16	16	no	—
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	2	8	no	—
83	C₂×C₄	—	8	8	no	—
84	C₂×C₄	—	8	16	no	—
85	C₂×C₄	#486	1	8	yes	—
86	C₂×C₄	#500	1	4	yes	—
87	C₂×C₄	C₂×#124	1	2	yes	—
88	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
89	C₂×C₄	C₂²×SD₁₆	1	2	yes	—
90	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
91	C₈	#487	1	8	yes	—
92	C₈	#498	1	4	yes	—
93	C₈	#533	1	4	yes	—
94	C₂²	C₂×#486	1	2	yes	—
95	C₂²	C₂×#500	1	1	yes	characteristic
96	C₂²	#3414	1	4	yes	—
97	C₄	—	2	8	no	—
98	C₄	—	2	4	no	—
99	C₄	—	16	8	no	—
100	C₄	#2862	1	1	yes	characteristic
101	C₄	C₂×#487	1	2	yes	—
102	C₄	C₂×#498	1	1	yes	characteristic
103	C₂	#33289	1	1	yes	characteristic
104	C₂	#33706	1	4	yes	—
105	C₂	#33728	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33313

C₂

yes

maximal

#13283

C₂

yes

maximal

#12889

C₂

yes

maximal

C₄×#346

C₂

yes

maximal, characteristic

C₂×#500

C₂²

yes

—

C₂×#498

—

#2862

—

C₂×#346

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

—

#1846

—

#1846

—

#1846

C₂²

yes

—

#1846

C₂²

yes

—

C₄²×C₈

C₂²

yes

characteristic

#1380

C₂²

yes

—

C₂×C₄⋊Q₈

—

C₂²×C₁₆

C₂³

yes

—

—

—

—

C₂³

yes

—

C₂×#106

—

C₂×#106

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

#391

—

#382

—

#382

—

C₄³

D₄

yes

characteristic

#346

—

C₄×C₁₆

Q₈

yes

complemented

C₂²×Q₈

—

C₂²×C₈

C₂×D₄

yes

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂⁴

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₈

yes

—

C₂×C₄²

C₂×D₄

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×Q₈

yes

—

—

—

—

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

#41

yes

—

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₈

yes

—

C₂²×C₄

C₂×D₈

yes

—

C₂²×C₄

C₂²×D₄

yes

characteristic

C₂×C₈

#124

yes

—

C₂×C₈

C₄⋊Q₈

yes

—

C₂×C₈

C₂×SD₁₆

yes

—

C₂×C₈

C₂×SD₁₆

yes

—

C₂×C₈

C₂×Q₁₆

yes

—

C₂×C₈

C₂²×Q₈

yes

—

C₂×C₈

C₂×#41

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

#124

yes

—

C₄²

C₄⋊Q₈

yes

—

C₄²

C₂×SD₁₆

yes

—

C₄²

C₂×Q₁₆

yes

—

C₁₆

—

C₂³

C₂²×D₈

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#486

yes

—

C₂×C₄

#500

yes

—

C₂×C₄

C₂×#124

yes

—

C₂×C₄

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₄

C₂²×SD₁₆

yes

—

C₂×C₄

C₂²×Q₁₆

yes

characteristic

C₈

#487

yes

—

C₈

#498

yes

—

C₈

#533

yes

—

C₂²

C₂×#486

yes

—

C₂²

C₂×#500

yes

characteristic

C₂²

#3414

yes

—

C₄

—

C₄

—

C₄

—

100

C₄

#2862

yes

characteristic

101

C₄

C₂×#487

yes

—

102

C₄

C₂×#498

yes

characteristic

103

C₂

#33289

yes

characteristic

104

C₂

#33706

yes

—

105

C₂

#33728

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂²×D₈

commutator, Frattini, cosocle

C₂²×C₈

C₂⁴

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
33. C₄×C₁₆	Q₈	semidirect

H ≅ G/N

type

33. C₄×C₁₆

Q₈

semidirect

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

subgroup	index
#33313	2
#13283	2
#12889	2
C₄×#346	2

subgroup

index

Subgroups — #512517

Conjugacy classes of subgroups

Complements

Maximal subgroups