Group #8617 — subgroups

Subgroups — #8617

Order 256.

199 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 199 conjugacy classes of subgroups — too many to draw. Only its 83 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×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	#1801	C₂	1	2	yes	maximal
3	#1567	C₂	1	2	yes	maximal
4	#1292	C₂	1	2	yes	maximal
5	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
6	C₈×Q₈	—	2	2	no	—
7	C₈×Q₈	—	2	2	no	—
8	#423	C₂²	1	4	yes	—
9	C₂×C₄⋊C₈	C₄	1	2	yes	—
10	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×#101	—	2	2	no	—
12	C₂×C₄×C₈	C₄	1	2	yes	—
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	#343	—	2	2	no	—
15	#335	—	2	2	no	—
16	#335	—	2	2	no	—
17	C₂²×Q₈	D₄	1	1	yes	characteristic
18	C₂×M₁₆	—	2	4	no	—
19	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
22	C₄×Q₈	—	2	2	no	—
23	C₄×Q₈	—	2	2	no	—
24	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
25	C₂×C₄⋊C₄	Q₈	1	2	yes	—
26	C₂×C₄²	C₂×C₄	1	2	yes	—
27	C₂×C₄²	C₂³	1	1	yes	characteristic
28	C₄⋊C₈	—	2	4	no	—
29	C₄⋊C₈	—	2	2	no	—
30	C₄⋊C₈	—	2	4	no	—
31	C₄⋊C₈	—	2	4	no	—
32	#101	—	4	4	no	—
33	C₄×C₈	—	2	2	no	—
34	C₄×C₈	—	2	2	no	—
35	C₄×C₈	D₄	1	2	yes	—
36	C₄×C₈	D₄	1	2	yes	—
37	C₂×Q₈	—	2	2	no	—
38	C₂×Q₈	—	2	2	no	—
39	C₂×Q₈	D₈	1	2	yes	—
40	C₂×Q₈	Q₁₆	1	2	yes	—
41	C₂²×C₄	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
46	M₁₆	—	4	4	no	—
47	M₁₆	—	4	4	no	—
48	C₂×C₈	—	2	2	no	—
49	C₂×C₈	—	2	2	no	—
50	C₂×C₈	—	4	1	no	—
51	C₂×C₈	—	4	2	no	—
52	C₂×C₈	—	4	4	no	—
53	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
56	C₄⋊C₄	—	2	2	no	—
57	C₄⋊C₄	—	2	4	no	—
58	C₄²	—	2	2	no	—
59	C₄²	—	2	2	no	—
60	C₄²	C₂×D₄	1	2	yes	—
61	C₄²	#41	1	2	yes	—
62	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
63	Q₈	—	2	4	no	—
64	Q₈	—	2	4	no	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	2	2	no	—
70	C₂×C₄	—	4	1	no	—
71	C₂×C₄	—	4	2	no	—
72	C₂×C₄	#107	1	4	yes	—
73	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄	#117	1	1	yes	characteristic
75	C₂×C₄	#121	1	2	yes	—
76	C₂×C₄	C₂²⋊Q₈	1	2	yes	—
77	C₂×C₄	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×C₄	#135	1	2	yes	—
80	C₈	—	2	1	no	—
81	C₈	—	2	2	no	—
82	C₈	—	4	2	no	—
83	C₈	—	8	4	no	—
84	C₈	#101	1	2	yes	complemented
85	C₂²	—	2	1	no	—
86	C₂²	—	2	1	no	—
87	C₂²	#380	1	1	yes	characteristic
88	C₂²	C₂×#107	1	1	yes	characteristic
89	C₂²	#427	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	2	4	no	—
92	C₄	—	2	2	no	—
93	C₄	—	4	2	no	—
94	C₄	C₂×#101	1	1	yes	characteristic
95	C₄	#413	1	1	yes	characteristic
96	C₄	#466	1	2	yes	—
97	C₄	#467	1	2	yes	—
98	C₂	—	2	1	no	—
99	C₂	—	2	1	no	—
100	C₂	#1801	1	1	yes	characteristic
101	C₂	#1808	1	1	yes	characteristic
102	C₂	#1813	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Q₈

C₂

yes

maximal, characteristic

#1801

C₂

yes

maximal

#1567

C₂

yes

maximal

#1292

C₂

yes

maximal

C₂×C₄×Q₈

C₂²

yes

characteristic

C₈×Q₈

—

C₈×Q₈

—

#423

C₂²

yes

—

C₂×C₄⋊C₈

C₄

yes

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×#101

—

C₂×C₄×C₈

C₄

yes

—

C₂×C₄×C₈

C₂²

yes

—

—

—

C₂²×Q₈

D₄

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

D₄

yes

characteristic

C₂×C₄⋊C₄

Q₈

yes

—

C₂×C₄²

C₂×C₄

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#101

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₈

yes

—

C₂×Q₈

Q₁₆

yes

—

C₂²×C₄

C₄⋊C₄

yes

—

C₂²×C₄

C₄⋊C₄

yes

—

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×D₄

yes

—

C₄²

#41

yes

—

C₂³

C₂×C₄⋊C₄

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#107

yes

—

C₂×C₄

C₂×C₂²⋊C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

C₂²⋊Q₈

yes

—

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

C₂×Q₁₆

yes

characteristic

C₂×C₄

#135

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

#101

yes

complemented

C₂²

—

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

—

C₄

#467

yes

—

C₂

—

C₂

—

100

C₂

#1801

yes

characteristic

101

C₂

#1808

yes

characteristic

102

C₂

#1813

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#380
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

C₂²

C₂²×C₄

C₂×C₄²

C₂³

prime radical

C₂³

C₂×C₄⋊C₄

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
84. C₈	#101	semidirect

H ≅ G/N

type

84. C₈

#101

semidirect

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

subgroup	index
C₂×C₈×Q₈	2
#1801	2
#1567	2
#1292	2

subgroup

index

C₂×C₈×Q₈

#1801

#1567

#1292

Subgroups — #8617

Conjugacy classes of subgroups

Complements

Maximal subgroups