Group #13272 — subgroups

Subgroups — #13272

Order 256.

183 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 183 conjugacy classes of subgroups — too many to draw. Only its 75 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	C₂×#498	C₂	1	1	yes	maximal, characteristic
2	C₂×#364	C₂	1	1	yes	maximal, characteristic
3	#1772	C₂	1	1	yes	maximal, characteristic
4	#1386	C₂	1	2	yes	maximal
5	#1383	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈	—	2	1	no	—
7	C₂×M₃₂	C₂²	1	2	yes	—
8	#498	—	2	2	no	—
9	#498	C₄	1	4	yes	complemented
10	C₂×#108	C₂²	1	1	yes	characteristic
11	#427	C₂²	1	1	yes	characteristic
12	C₂×#106	—	2	1	no	—
13	C₂×#106	C₂²	1	2	yes	—
14	#405	—	2	1	no	—
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	#364	—	2	2	no	—
17	#339	—	2	1	no	—
18	C₂²×Q₈	—	4	1	no	—
19	C₂×M₁₆	—	2	1	no	—
20	C₂²×C₈	D₄	1	1	yes	characteristic
21	C₂²×C₈	C₂³	1	1	yes	characteristic
22	C₄⋊Q₈	—	2	4	no	—
23	C₄⋊Q₈	—	4	2	no	—
24	#117	—	2	1	no	—
25	C₂×C₄⋊C₄	—	2	2	no	—
26	C₂×C₄⋊C₄	—	4	1	no	—
27	C₂×C₄²	D₄	1	1	yes	characteristic
28	M₃₂	—	2	2	no	—
29	M₃₂	—	2	2	no	—
30	C₂×C₁₆	—	2	2	no	—
31	#108	—	2	1	no	—
32	#108	Q₈	1	2	yes	complemented
33	#107	Q₈	1	2	yes	complemented
34	#106	—	2	4	no	—
35	#106	—	2	2	no	—
36	#106	—	2	1	no	—
37	#106	C₂×C₄	1	4	yes	—
38	#97	—	4	1	no	—
39	C₄×C₈	—	4	1	no	—
40	C₄×C₈	C₂×C₄	1	2	yes	—
41	C₄×C₈	D₄	1	2	yes	—
42	C₂×Q₈	—	4	4	no	—
43	C₂×Q₈	—	8	4	no	—
44	C₂²×C₄	—	2	1	no	—
45	C₂²×C₄	—	4	2	no	—
46	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
47	M₁₆	—	2	2	no	—
48	M₁₆	—	4	1	no	—
49	C₂×C₈	—	4	1	no	—
50	C₂×C₈	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈	SD₁₆	1	1	yes	characteristic
52	C₂×C₈	SD₁₆	1	1	yes	characteristic
53	C₂×C₈	Q₁₆	1	2	yes	—
54	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈	#41	1	1	yes	characteristic
59	C₂×C₈	#41	1	1	yes	characteristic
60	C₄⋊C₄	—	2	4	no	—
61	C₄⋊C₄	—	2	2	no	—
62	C₄⋊C₄	—	4	4	no	—
63	C₄⋊C₄	—	4	2	no	—
64	C₂²⋊C₄	—	4	1	no	—
65	C₄²	—	2	1	no	—
66	C₄²	—	4	1	no	—
67	C₄²	C₂²⋊C₄	1	2	yes	—
68	C₁₆	—	4	2	no	—
69	C₂³	AGL(1,ℤ/8)	1	1	yes	characteristic
70	Q₈	—	8	8	no	—
71	C₂×C₄	—	2	4	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	4	4	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	8	4	no	—
76	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
79	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
80	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
81	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₈	—	4	2	no	—
83	C₈	#103	1	2	yes	—
84	C₈	#103	1	2	yes	—
85	C₈	C₄×Q₈	1	1	yes	characteristic
86	C₈	C₄×Q₈	1	1	yes	characteristic
87	C₈	#124	1	1	yes	characteristic
88	C₈	C₄⋊Q₈	1	1	yes	characteristic
89	C₂²	—	2	2	no	—
90	C₂²	#418	1	1	yes	characteristic
91	C₂²	#476	1	1	yes	characteristic
92	C₂²	#477	1	1	yes	characteristic
93	C₄	—	2	4	no	—
94	C₄	—	8	4	no	—
95	C₄	—	8	2	no	—
96	C₄	#391	1	1	yes	characteristic
97	C₄	C₂×#103	1	1	yes	characteristic
98	C₄	#476	1	1	yes	characteristic
99	C₄	#477	1	1	yes	characteristic
100	C₂	—	2	1	no	—
101	C₂	—	2	1	no	—
102	C₂	#1987	1	1	yes	characteristic
103	C₂	#2230	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#498

C₂

yes

maximal, characteristic

C₂×#364

C₂

yes

maximal, characteristic

#1772

C₂

yes

maximal, characteristic

#1386

C₂

yes

maximal

#1383

C₂

yes

maximal

C₂×C₄⋊Q₈

—

C₂×M₃₂

C₂²

yes

—

#498

—

#498

C₄

yes

complemented

C₂×#108

C₂²

yes

characteristic

#427

C₂²

yes

characteristic

C₂×#106

—

C₂×#106

C₂²

yes

—

#405

—

C₂×C₄×C₈

C₂²

yes

characteristic

#364

—

#339

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#108

—

#108

Q₈

yes

complemented

#107

Q₈

yes

—

—

—

C₂×C₄

yes

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

SD₁₆

yes

characteristic

C₂×C₈

SD₁₆

yes

characteristic

C₂×C₈

Q₁₆

yes

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×Q₈

yes

characteristic

C₂×C₈

C₂×Q₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊C₄

yes

—

C₁₆

—

C₂³

AGL(1,ℤ/8)

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

C₂²⋊Q₈

yes

characteristic

C₂×C₄

C₂²⋊Q₈

yes

characteristic

C₂×C₄

C₂×SD₁₆

yes

characteristic

C₂×C₄

C₂×Q₁₆

yes

characteristic

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

C₈

—

C₈

#103

yes

—

C₈

#103

yes

—

C₈

C₄×Q₈

yes

characteristic

C₈

C₄×Q₈

yes

characteristic

C₈

#124

yes

characteristic

C₈

C₄⋊Q₈

yes

characteristic

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

C₄

yes

100

C₂

—

101

C₂

—

102

C₂

#1987

yes

characteristic

103

C₂

#2230

yes

—

	subgroup	G/H
center, socle	C₂²	#476
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#476

commutator

C₂×C₈

C₂²×C₄

Frattini, cosocle

C₂²×C₈

C₂³

prime radical

C₂³

AGL(1,ℤ/8)

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
9. #498	C₄	semidirect
32. #108	Q₈	semidirect
33. #107	Q₈	semidirect

H ≅ G/N

type

9. #498

C₄

semidirect

32. #108

Q₈

semidirect

33. #107

Q₈

semidirect

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

subgroup	index
C₂×#498	2
C₂×#364	2
#1772	2
#1386	2
#1383	2

subgroup

index

Subgroups — #13272

Conjugacy classes of subgroups

Complements

Maximal subgroups