Group #3227 — subgroups

Subgroups — #3227

Order 128.

202 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 96 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	#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#498	C₂	1	2	yes	maximal, complemented
4	#486	C₂	1	2	yes	maximal, complemented
5	#481	C₂	1	4	yes	maximal, complemented
6	#476	C₂	1	4	yes	maximal, complemented
7	#431	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41	—	2	1	no	—
9	C₂²×Q₈	—	2	1	no	—
10	C₄⋊Q₈	—	2	1	no	—
11	C₄⋊Q₈	—	2	1	no	—
12	C₄⋊Q₈	C₂²	1	2	yes	—
13	C₄⋊Q₈	C₂²	1	2	yes	—
14	C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	#127	C₂²	1	1	yes	characteristic
16	#124	—	2	1	no	—
17	#121	—	2	1	no	—
18	#121	C₂²	1	2	yes	—
19	C₄×D₄	—	2	1	no	—
20	C₄×D₄	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄	—	2	1	no	—
22	C₂×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₄²	C₂²	1	1	yes	characteristic
24	#106	C₂²	1	8	yes	—
25	C₄⋊C₈	C₂²	1	2	yes	—
26	#102	C₂²	1	8	yes	—
27	C₂²⋊C₈	C₂²	1	2	yes	—
28	C₄×C₈	C₂²	1	2	yes	—
29	#41	—	4	4	no	—
30	C₂×Q₈	—	2	1	no	—
31	C₂×Q₈	—	2	2	no	—
32	C₂×Q₈	—	2	2	no	—
33	C₂×Q₈	—	4	4	no	—
34	C₂×D₄	—	2	2	no	—
35	C₂×D₄	—	2	1	no	—
36	C₂×D₄	—	2	1	no	—
37	C₂×D₄	C₂³	1	2	yes	—
38	C₂²×C₄	—	2	2	no	—
39	C₂²×C₄	—	2	2	no	—
40	C₂²×C₄	D₄	1	1	yes	characteristic
41	C₂²×C₄	D₄	1	1	yes	characteristic
42	C₂²×C₄	C₂³	1	1	yes	characteristic
43	C₂×C₈	C₂³	1	4	yes	—
44	C₄⋊C₄	—	2	2	no	—
45	C₄⋊C₄	—	2	2	no	—
46	C₄⋊C₄	—	2	2	no	—
47	C₄⋊C₄	—	2	1	no	—
48	C₄⋊C₄	C₂³	1	4	yes	—
49	C₄⋊C₄	C₂³	1	2	yes	—
50	C₂²⋊C₄	—	2	2	no	—
51	C₂²⋊C₄	—	2	2	no	—
52	C₄²	D₄	1	1	yes	characteristic
53	C₄²	D₄	1	1	yes	characteristic
54	C₄²	C₂³	1	1	yes	characteristic
55	C₄²	C₂³	1	1	yes	characteristic
56	C₂³	—	2	2	no	—
57	C₂³	C₂×D₄	1	1	yes	characteristic
58	Q₈	—	4	4	no	—
59	Q₈	—	4	2	no	—
60	Q₈	—	4	4	no	—
61	D₄	—	2	4	no	—
62	D₄	—	4	4	no	—
63	D₄	—	4	2	no	—
64	D₄	—	4	2	no	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	2	4	no	—
69	C₂×C₄	—	4	4	no	—
70	C₂×C₄	—	4	4	no	—
71	C₂×C₄	SD₁₆	1	4	yes	—
72	C₂×C₄	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄	C₂⁴	1	1	yes	characteristic
78	C₈	—	2	4	no	—
79	C₂²	—	2	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	—	4	2	no	—
82	C₂²	—	4	2	no	—
83	C₂²	—	4	2	no	—
84	C₂²	C₂×SD₁₆	1	2	yes	—
85	C₂²	C₂²×D₄	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	2	1	no	—
88	C₄	—	4	2	no	—
89	C₄	—	4	4	no	—
90	C₄	C₂×SD₁₆	1	2	yes	—
91	C₄	C₂×SD₁₆	1	2	yes	—
92	C₄	AGL(1,ℤ/8)	1	2	yes	—
93	C₄	2₋¹⁺⁴	1	2	yes	—
94	C₂	—	2	2	no	—
95	C₂	—	8	2	no	—
96	C₂	#536	1	1	yes	characteristic
97	C₂	C₂²×SD₁₆	1	1	yes	characteristic
98	C₂	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#532

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈

C₂

yes

maximal, characteristic, complemented

#498

C₂

yes

maximal, complemented

#486

C₂

yes

maximal, complemented

#481

C₂

yes

maximal, complemented

#476

C₂

yes

maximal, complemented

#431

C₂

yes

maximal, characteristic, complemented

C₂×#41

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂²

yes

—

C₄⋊Q₈

C₂²

yes

—

C₄⋊Q₈

C₂²

yes

characteristic

#127

C₂²

yes

—

—

C₂²

yes

—

C₄×D₄

—

C₄×D₄

C₂²

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²

yes

—

C₂×C₄²

C₂²

yes

characteristic

#106

C₂²

yes

—

C₄⋊C₈

C₂²

yes

—

#102

C₂²

yes

—

C₂²⋊C₈

C₂²

yes

—

C₄×C₈

C₂²

yes

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂³

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄

yes

characteristic

C₂²×C₄

D₄

yes

characteristic

C₂²×C₄

C₂³

yes

characteristic

C₂×C₈

C₂³

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂³

yes

—

C₄⋊C₄

C₂³

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

D₄

yes

characteristic

C₄²

D₄

yes

characteristic

C₄²

C₂³

yes

characteristic

C₄²

C₂³

yes

characteristic

C₂³

—

C₂³

C₂×D₄

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

SD₁₆

yes

—

C₂×C₄

C₂×D₄

yes

characteristic

C₂×C₄

C₂×D₄

yes

characteristic

C₂×C₄

C₂×D₄

yes

characteristic

C₂×C₄

C₂×D₄

yes

characteristic

C₂×C₄

C₂×D₄

yes

characteristic

C₂×C₄

C₂⁴

yes

characteristic

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂×SD₁₆

yes

—

C₂²

C₂²×D₄

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

C₂×SD₁₆

yes

—

C₄

C₂×SD₁₆

yes

—

C₄

AGL(1,ℤ/8)

yes

—

C₄

2₋¹⁺⁴

yes

—

C₂

—

C₂

—

C₂

#536

yes

characteristic

C₂

C₂²×SD₁₆

yes

characteristic

C₂

C₂×AGL(1,ℤ/8)

yes

characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	#532	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
2. C₂×C₄⋊Q₈	C₂	semidirect
3. #498	C₂	semidirect
4. #486	C₂	semidirect
5. #481	C₂	semidirect
6. #476	C₂	semidirect
7. #431	C₂	semidirect

H ≅ G/N

type

2. C₂×C₄⋊Q₈

C₂

semidirect

3. #498

C₂

semidirect

4. #486

C₂

semidirect

5. #481

C₂

semidirect

6. #476

C₂

semidirect

7. #431

C₂

semidirect

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

subgroup	index
#532	2
C₂×C₄⋊Q₈	2
#498	2
#486	2
#481	2
#476	2
#431	2

subgroup

index

C₂×C₄⋊Q₈

Subgroups — #3227

Conjugacy classes of subgroups

Complements

Maximal subgroups