Group #21732 — subgroups

Subgroups — #21732

Order 256.

351 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 191 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	#3160	C₂	1	1	yes	maximal, characteristic
2	#2960	C₂	1	2	yes	maximal
3	#1994	C₂	1	2	yes	maximal
4	#1945	C₂	1	4	yes	maximal
5	#1848	C₂	1	1	yes	maximal, characteristic
6	#1772	C₂	1	4	yes	maximal
7	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
8	#533	C₂²	1	2	yes	—
9	#500	C₄	1	2	yes	complemented
10	#499	C₄	1	4	yes	complemented
11	#498	C₄	1	2	yes	complemented
12	#446	—	2	4	no	—
13	C₈×Q₈	—	2	4	no	—
14	C₂×#108	C₂²	1	4	yes	—
15	#427	C₂²	1	4	yes	—
16	#424	C₂²	1	4	yes	—
17	#423	C₂²	1	2	yes	—
18	#405	C₂²	1	4	yes	—
19	C₂×C₄×C₈	C₂²	1	2	yes	—
20	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈	C₂²	1	2	yes	—
22	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
23	C₄³	C₂²	1	1	yes	characteristic
24	#339	C₂²	1	8	yes	—
25	C₂×M₁₆	C₂³	1	4	yes	—
26	C₂²×C₈	C₂³	1	1	yes	characteristic
27	C₂²×C₈	C₂³	1	2	yes	—
28	C₂²×C₈	C₂³	1	1	yes	characteristic
29	C₄⋊Q₈	C₂×C₄	1	4	yes	—
30	#125	C₂×C₄	1	4	yes	—
31	C₂²⋊Q₈	—	2	4	no	—
32	C₄×Q₈	—	2	4	no	—
33	#117	C₂³	1	4	yes	—
34	C₂×C₄²	—	2	2	no	—
35	C₂×C₄²	C₂³	1	2	yes	—
36	C₂×C₄²	C₂³	1	1	yes	characteristic
37	#108	—	2	8	no	—
38	#107	C₂×C₄	1	8	yes	—
39	#106	C₂×C₄	1	8	yes	—
40	C₄⋊C₈	—	2	8	no	—
41	C₄⋊C₈	—	2	4	no	—
42	#97	—	2	4	no	—
43	C₄×C₈	—	2	4	no	—
44	C₄×C₈	—	2	4	no	—
45	C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈	C₂×C₄	1	2	yes	—
47	C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈	D₄	1	2	yes	—
49	C₄×C₈	D₄	1	2	yes	—
50	C₄×C₈	Q₈	1	8	yes	complemented
51	C₂×Q₈	—	2	4	no	—
52	C₂²×C₄	—	2	2	no	—
53	C₂²×C₄	#41	1	2	yes	—
54	C₂²×C₄	C₂⁴	1	1	yes	characteristic
55	M₁₆	—	2	8	no	—
56	C₂×C₈	—	2	2	no	—
57	C₂×C₈	—	2	4	no	—
58	C₂×C₈	C₂²×C₄	1	2	yes	—
59	C₂×C₈	C₂²×C₄	1	2	yes	—
60	C₂×C₈	C₂×D₄	1	2	yes	—
61	C₂×C₈	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈	C₂×Q₈	1	4	yes	—
64	C₂×C₈	C₂×Q₈	1	4	yes	—
65	C₂×C₈	#41	1	4	yes	—
66	C₄⋊C₄	—	2	8	no	—
67	C₄⋊C₄	—	2	4	no	—
68	C₄⋊C₄	C₂²×C₄	1	8	yes	—
69	C₂²⋊C₄	—	2	4	no	—
70	C₄²	—	2	8	no	—
71	C₄²	—	2	4	no	—
72	C₄²	—	2	2	no	—
73	C₄²	C₂²×C₄	1	1	yes	characteristic
74	C₄²	C₂²×C₄	1	1	yes	characteristic
75	C₄²	C₂×D₄	1	2	yes	—
76	C₄²	C₂×Q₈	1	4	yes	—
77	C₂³	C₂×#41	1	1	yes	characteristic
78	Q₈	—	4	8	no	—
79	C₂×C₄	—	2	8	no	—
80	C₂×C₄	—	2	4	no	—
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	2	8	no	—
83	C₂×C₄	C₄×D₄	1	2	yes	—
84	C₂×C₄	C₄×D₄	1	2	yes	—
85	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
86	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
87	C₂×C₄	C₂²×Q₈	1	2	yes	—
88	C₂×C₄	C₂×#41	1	2	yes	—
89	C₈	—	2	2	no	—
90	C₈	—	4	4	no	—
91	C₈	C₄×Q₈	1	4	yes	—
92	C₈	C₄×Q₈	1	4	yes	—
93	C₈	C₄⋊Q₈	1	2	yes	—
94	C₈	C₄⋊Q₈	1	2	yes	—
95	C₂²	—	2	2	no	—
96	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
97	C₂²	#533	1	2	yes	—
98	C₄	—	2	8	no	—
99	C₄	—	2	2	no	—
100	C₄	—	4	8	no	—
101	C₄	#443	1	4	yes	—
102	C₄	C₂×C₄×Q₈	1	2	yes	—
103	C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
104	C₄	#532	1	1	yes	characteristic
105	C₂	—	2	2	no	—
106	C₂	C₄×C₄⋊Q₈	1	1	yes	characteristic
107	C₂	C₂×#443	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3160

C₂

yes

maximal, characteristic

#2960

C₂

yes

maximal

#1994

C₂

yes

maximal

#1945

C₂

yes

maximal

#1848

C₂

yes

maximal, characteristic

#1772

C₂

yes

maximal

C₄²×C₈

C₂

yes

maximal, characteristic

#533

C₂²

yes

—

#500

C₄

yes

complemented

#499

C₄

yes

complemented

#498

C₄

yes

complemented

#446

—

C₈×Q₈

—

C₂×#108

C₂²

yes

—

#427

C₂²

yes

—

#424

C₂²

yes

—

#423

C₂²

yes

—

#405

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₄³

C₂²

yes

characteristic

#339

C₂²

yes

—

C₂×M₁₆

C₂³

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

C₂×C₄

yes

—

#125

C₂×C₄

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂³

yes

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

characteristic

#108

—

#107

C₂×C₄

yes

—

#106

C₂×C₄

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

Q₈

yes

complemented

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

#41

yes

—

C₂²×C₄

C₂⁴

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

#41

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×C₄

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

—

C₄²

C₂×Q₈

yes

—

C₂³

C₂×#41

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

C₂³×C₄

yes

characteristic

C₂×C₄

C₂²×D₄

yes

characteristic

C₂×C₄

C₂²×Q₈

yes

—

C₂×C₄

C₂×#41

yes

—

C₈

—

C₈

—

C₈

C₄×Q₈

yes

—

C₈

C₄×Q₈

yes

—

C₈

C₄⋊Q₈

yes

—

C₈

C₄⋊Q₈

yes

—

C₂²

—

C₂²

C₂×C₄×D₄

yes

characteristic

C₂²

#533

yes

—

C₄

—

C₄

—

100

C₄

—

101

C₄

#443

yes

—

102

C₄

C₂×C₄×Q₈

yes

—

103

C₄

C₂×C₄⋊Q₈

yes

characteristic

104

C₄

#532

yes

characteristic

105

C₂

—

106

C₂

C₄×C₄⋊Q₈

yes

characteristic

107

C₂

C₂×#443

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	C₂³	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₈

C₂×D₄

commutator

C₂×C₄

C₂³×C₄

Frattini, cosocle

C₂²×C₄

C₂⁴

socle

C₂²

C₂×C₄×D₄

prime radical

C₂³

C₂×#41

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
9. #500	C₄	semidirect
10. #499	C₄	semidirect
11. #498	C₄	semidirect
50. C₄×C₈	Q₈	semidirect

H ≅ G/N

type

9. #500

C₄

semidirect

10. #499

C₄

semidirect

11. #498

C₄

semidirect

50. C₄×C₈

Q₈

semidirect

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

subgroup	index
#3160	2
#2960	2
#1994	2
#1945	2
#1848	2
#1772	2
C₄²×C₈	2

subgroup

index

C₄²×C₈

Subgroups — #21732

Conjugacy classes of subgroups

Complements

Maximal subgroups