Group #8784 — subgroups

Subgroups — #8784

Order 256.

211 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 211 conjugacy classes of subgroups — too many to draw. Only its 127 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	#1842	C₂	1	2	yes	maximal
2	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
3	#1277	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈	—	2	2	no	—
5	C₂×C₄⋊C₈	—	2	2	no	—
6	C₂×C₄⋊C₈	C₂²	1	4	yes	—
7	C₂×C₄×C₈	C₄	1	2	yes	—
8	C₂×C₄×C₈	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈	C₂²	1	2	yes	—
11	C₄³	C₂²	1	1	yes	characteristic
12	C₂²×C₈	—	2	4	no	—
13	C₂²×C₈	C₂×C₄	1	2	yes	—
14	C₂²×C₈	C₂×C₄	1	2	yes	—
15	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²	D₄	1	1	yes	characteristic
18	C₂×C₄²	D₄	1	1	yes	characteristic
19	C₂×C₄²	Q₈	1	1	yes	characteristic
20	C₂×C₄²	Q₈	1	1	yes	characteristic
21	C₂×C₄²	C₂³	1	1	yes	characteristic
22	C₄⋊C₈	—	2	8	no	—
23	C₄⋊C₈	—	4	4	no	—
24	C₄⋊C₈	—	4	4	no	—
25	C₄×C₈	—	2	4	no	—
26	C₄×C₈	—	2	2	no	—
27	C₄×C₈	—	2	4	no	—
28	C₄×C₈	C₈	1	4	yes	complemented
29	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₈	—	2	2	no	—
37	C₂×C₈	—	2	2	no	—
38	C₂×C₈	—	2	2	no	—
39	C₂×C₈	—	2	2	no	—
40	C₂×C₈	—	4	4	no	—
41	C₂×C₈	—	4	4	no	—
42	C₂×C₈	—	4	4	no	—
43	C₂×C₈	C₂×C₈	1	4	yes	—
44	C₂×C₈	M₁₆	1	4	yes	—
45	C₄²	—	2	2	no	—
46	C₄²	—	2	2	no	—
47	C₄²	—	2	2	no	—
48	C₄²	C₂×C₈	1	2	yes	—
49	C₄²	M₁₆	1	2	yes	—
50	C₄²	D₈	1	2	yes	—
51	C₄²	SD₁₆	1	4	yes	—
52	C₄²	Q₁₆	1	2	yes	—
53	C₄²	#41	1	4	yes	—
54	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	C₄⋊C₈	1	2	yes	—
62	C₂×C₄	C₄⋊C₈	1	2	yes	—
63	C₂×C₄	#106	1	2	yes	—
64	C₂×C₄	#106	1	2	yes	—
65	C₂×C₄	#107	1	2	yes	—
66	C₂×C₄	#107	1	2	yes	—
67	C₂×C₄	#108	1	2	yes	—
68	C₂×C₄	#108	1	2	yes	—
69	C₂×C₄	#117	1	2	yes	—
70	C₂×C₄	#124	1	2	yes	—
71	C₂×C₄	#125	1	2	yes	—
72	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄	C₂×M₁₆	1	2	yes	—
74	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄	C₂×SD₁₆	1	2	yes	—
77	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
78	C₈	—	2	4	no	—
79	C₈	—	2	4	no	—
80	C₈	—	8	4	no	—
81	C₂²	#382	1	1	yes	characteristic
82	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
83	C₂²	#423	1	1	yes	characteristic
84	C₂²	C₂×#106	1	1	yes	characteristic
85	C₂²	C₂×#107	1	1	yes	characteristic
86	C₂²	C₂×#108	1	1	yes	characteristic
87	C₂²	C₂×#108	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	2	2	no	—
90	C₄	—	2	2	no	—
91	C₄	#334	1	4	yes	—
92	C₄	#335	1	4	yes	—
93	C₄	#431	1	2	yes	—
94	C₄	#432	1	2	yes	—
95	C₄	#486	1	2	yes	—
96	C₄	#487	1	2	yes	—
97	C₂	C₂×#334	1	1	yes	characteristic
98	C₂	C₂×#335	1	1	yes	characteristic
99	C₂	#1566	1	1	yes	characteristic
100	C₂	#1567	1	1	yes	characteristic
101	C₂	#1831	1	1	yes	characteristic
102	C₂	#1834	1	1	yes	characteristic
103	C₂	#1837	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1842

C₂

yes

maximal

C₄²×C₈

C₂

yes

maximal, characteristic

#1277

C₂

yes

maximal

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄×C₈

C₄

yes

—

C₂×C₄×C₈

C₄

yes

characteristic

C₂×C₄×C₈

C₄

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

C₄³

C₂²

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

Q₈

yes

characteristic

C₂×C₄²

Q₈

yes

characteristic

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₈

yes

complemented

C₂²×C₄

C₄⋊C₄

yes

characteristic

C₂²×C₄

C₄⋊C₄

yes

characteristic

C₂²×C₄

C₄⋊C₄

yes

characteristic

C₂²×C₄

C₄⋊C₄

yes

characteristic

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

—

C₂×C₈

M₁₆

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×C₈

yes

—

C₄²

M₁₆

yes

—

C₄²

D₈

yes

—

C₄²

SD₁₆

yes

—

C₄²

Q₁₆

yes

—

C₄²

#41

yes

—

C₂³

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊C₈

yes

—

C₂×C₄

C₄⋊C₈

yes

—

C₂×C₄

#106

yes

—

C₂×C₄

#106

yes

—

C₂×C₄

#107

yes

—

C₂×C₄

#107

yes

—

C₂×C₄

#108

yes

—

C₂×C₄

#108

yes

—

C₂×C₄

#117

yes

—

C₂×C₄

#124

yes

—

C₂×C₄

#125

yes

—

C₂×C₄

C₂²×C₈

yes

characteristic

C₂×C₄

C₂×M₁₆

yes

—

C₂×C₄

C₂×M₁₆

yes

characteristic

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

C₂×SD₁₆

yes

—

C₂×C₄

C₂×Q₁₆

yes

characteristic

C₈

—

C₈

—

C₈

—

C₂²

#382

yes

characteristic

C₂²

C₂×C₄⋊C₈

yes

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

C₄

#334

yes

—

C₄

#335

yes

—

C₄

#431

yes

—

C₄

#432

yes

—

C₄

#486

yes

—

C₄

#487

yes

—

C₂

yes

C₂

yes

C₂

yes

100

C₂

#1567

yes

characteristic

101

C₂

#1831

yes

characteristic

102

C₂

#1834

yes

characteristic

103

C₂

#1837

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄²	C₂³
socle, prime radical	C₂³	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂²×C₄

C₂×D₄

commutator

C₂×C₄

C₂²×C₈

Frattini, cosocle

C₂×C₄²

C₂³

socle, prime radical

C₂³

C₂×C₄⋊C₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
28. C₄×C₈	C₈	semidirect

H ≅ G/N

type

28. C₄×C₈

C₈

semidirect

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

subgroup	index
#1842	2
C₄²×C₈	2
#1277	2

subgroup

index

#1842

C₄²×C₈

#1277

Subgroups — #8784

Conjugacy classes of subgroups

Complements

Maximal subgroups