Group #3256 — subgroups

Subgroups — #3256

Order 128.

177 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 177 conjugacy classes of subgroups — too many to draw. Only its 84 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	#529	C₂	1	2	yes	maximal, complemented
2	#479	C₂	1	2	yes	maximal, complemented
3	#478	C₂	1	2	yes	maximal, complemented
4	#470	C₂	1	1	yes	maximal, characteristic, complemented
5	#469	C₂	1	1	yes	maximal, characteristic
6	#468	C₂	1	2	yes	maximal, complemented
7	#464	C₂	1	2	yes	maximal, complemented
8	#463	C₂	1	2	yes	maximal, complemented
9	#423	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂×SD₁₆	C₂²	1	2	yes	—
12	C₂×D₈	C₂²	1	1	yes	characteristic
13	C₂×M₁₆	C₂²	1	1	yes	characteristic
14	C₂×M₁₆	C₂²	1	1	yes	characteristic
15	#126	—	2	2	no	—
16	#125	C₂²	1	2	yes	—
17	#124	C₂²	1	2	yes	—
18	#123	—	2	2	no	—
19	C₂²⋊Q₈	C₂²	1	2	yes	—
20	#121	C₂²	1	2	yes	—
21	C₄×Q₈	C₂²	1	2	yes	—
22	C₄×D₄	—	2	2	no	—
23	C₄×D₄	C₂²	1	2	yes	—
24	#117	—	2	2	no	—
25	C₂×C₄²	C₂²	1	1	yes	characteristic
26	#107	C₂²	1	2	yes	—
27	#106	C₂²	1	1	yes	characteristic
28	#106	C₂²	1	1	yes	characteristic
29	C₄⋊C₈	C₂²	1	2	yes	—
30	C₄⋊C₈	C₂²	1	2	yes	—
31	#103	C₂²	1	2	yes	—
32	#103	C₂²	1	2	yes	—
33	#102	C₂²	1	2	yes	—
34	#102	C₂²	1	2	yes	—
35	C₂×Q₈	C₂³	1	2	yes	—
36	C₂×D₄	—	2	2	no	—
37	C₂×D₄	C₂³	1	2	yes	—
38	C₂²×C₄	—	2	2	no	—
39	C₂²×C₄	D₄	1	2	yes	—
40	C₂²×C₄	C₂³	1	1	yes	characteristic
41	Q₁₆	—	4	1	no	—
42	SD₁₆	—	4	2	no	—
43	D₈	—	4	1	no	—
44	M₁₆	—	4	1	no	—
45	M₁₆	—	4	1	no	—
46	C₂×C₈	C₂³	1	1	yes	characteristic
47	C₂×C₈	C₂³	1	2	yes	—
48	C₂×C₈	C₂³	1	1	yes	characteristic
49	C₄⋊C₄	—	2	2	no	—
50	C₄⋊C₄	—	2	2	no	—
51	C₄⋊C₄	—	2	2	no	—
52	C₄⋊C₄	—	2	2	no	—
53	C₄⋊C₄	C₂³	1	2	yes	—
54	C₄⋊C₄	C₂³	1	2	yes	—
55	C₂²⋊C₄	—	2	2	no	—
56	C₂²⋊C₄	—	2	2	no	—
57	C₂²⋊C₄	—	2	2	no	—
58	C₂²⋊C₄	—	2	2	no	—
59	C₂²⋊C₄	—	2	2	no	—
60	C₄²	—	2	2	no	—
61	C₄²	D₄	1	1	yes	characteristic
62	C₄²	D₄	1	1	yes	characteristic
63	C₄²	C₂³	1	2	yes	—
64	C₂³	—	2	2	no	—
65	C₂³	C₂×D₄	1	1	yes	characteristic
66	Q₈	—	2	2	no	—
67	Q₈	—	2	2	no	—
68	D₄	—	2	2	no	—
69	D₄	—	2	2	no	—
70	D₄	—	4	4	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	4	2	no	—
77	C₂×C₄	—	4	4	no	—
78	C₂×C₄	C₂×D₄	1	2	yes	—
79	C₂×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄	C₂×D₄	1	2	yes	—
81	C₂×C₄	C₂⁴	1	1	yes	characteristic
82	C₈	—	2	1	no	—
83	C₈	—	2	2	no	—
84	C₈	—	2	1	no	—
85	C₂²	—	2	1	no	—
86	C₂²	—	2	2	no	—
87	C₂²	—	4	2	no	—
88	C₂²	—	4	2	no	—
89	C₂²	—	4	2	no	—
90	C₂²	C₂²×D₄	1	1	yes	characteristic
91	C₄	—	2	1	no	—
92	C₄	—	2	2	no	—
93	C₄	—	2	2	no	—
94	C₄	—	4	2	no	—
95	C₄	—	4	2	no	—
96	C₄	—	4	2	no	—
97	C₄	H(5,𝔽₂)	1	1	yes	characteristic
98	C₄	2₋¹⁺⁴	1	1	yes	characteristic
99	C₂	—	4	1	no	—
100	C₂	—	8	2	no	—
101	C₂	#537	1	1	yes	characteristic
102	C₂	#575	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#529

C₂

yes

maximal, complemented

#479

C₂

yes

maximal, complemented

#478

C₂

yes

maximal, complemented

#470

C₂

yes

maximal, characteristic, complemented

#469

C₂

yes

maximal, characteristic

#468

C₂

yes

maximal, complemented

#464

C₂

yes

maximal, complemented

#463

C₂

yes

maximal, complemented

#423

C₂

yes

maximal, characteristic, complemented

C₂×Q₁₆

C₂²

yes

characteristic

C₂×SD₁₆

C₂²

yes

—

C₂×D₈

C₂²

yes

characteristic

C₂×M₁₆

C₂²

yes

characteristic

C₂×M₁₆

C₂²

yes

characteristic

#126

—

#125

C₂²

yes

—

#124

C₂²

yes

—

#123

—

C₂²⋊Q₈

C₂²

yes

—

#121

C₂²

yes

—

C₄×Q₈

C₂²

yes

—

C₄×D₄

—

C₄×D₄

C₂²

yes

—

#117

—

C₂×C₄²

C₂²

yes

characteristic

#107

C₂²

yes

—

#106

C₂²

yes

characteristic

#106

C₂²

yes

characteristic

C₄⋊C₈

C₂²

yes

—

C₄⋊C₈

C₂²

yes

—

#103

C₂²

yes

—

#103

C₂²

yes

—

#102

C₂²

yes

—

#102

C₂²

yes

—

C₂×Q₈

C₂³

yes

—

C₂×D₄

—

C₂×D₄

C₂³

yes

—

C₂²×C₄

—

C₂²×C₄

D₄

yes

—

C₂²×C₄

C₂³

yes

characteristic

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

M₁₆

—

C₂×C₈

C₂³

yes

characteristic

C₂×C₈

C₂³

yes

—

C₂×C₈

C₂³

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂³

yes

—

C₄⋊C₄

C₂³

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

D₄

yes

characteristic

C₄²

D₄

yes

characteristic

C₄²

C₂³

yes

—

C₂³

—

C₂³

C₂×D₄

yes

characteristic

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄

yes

—

C₂×C₄

C₂×D₄

yes

characteristic

C₂×C₄

C₂×D₄

yes

—

C₂×C₄

C₂⁴

yes

characteristic

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²×D₄

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

H(5,𝔽₂)

yes

characteristic

C₄

2₋¹⁺⁴

yes

characteristic

C₂

—

100

C₂

—

101

C₂

#537

yes

characteristic

102

C₂

#575

yes

—

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	#469	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴

C₂

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. #529	C₂	semidirect
2. #479	C₂	semidirect
3. #478	C₂	semidirect
4. #470	C₂	semidirect
6. #468	C₂	semidirect
7. #464	C₂	semidirect
8. #463	C₂	semidirect
9. #423	C₂	semidirect

H ≅ G/N

type

1. #529

C₂

semidirect

2. #479

C₂

semidirect

3. #478

C₂

semidirect

4. #470

C₂

semidirect

6. #468

C₂

semidirect

7. #464

C₂

semidirect

8. #463

C₂

semidirect

9. #423

C₂

semidirect

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

subgroup	index
#529	2
#479	2
#478	2
#470	2
#469	2
#468	2
#464	2
#463	2
#423	2

subgroup

index

Subgroups — #3256

Conjugacy classes of subgroups

Complements

Maximal subgroups