Group #28701 — subgroups

Subgroups — #28701

Order 256.

299 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 135 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	#3166	C₂	1	2	yes	maximal
2	C₂×#499	C₂	1	1	yes	maximal, characteristic
3	#2960	C₂	1	1	yes	maximal, characteristic
4	#2089	C₂	1	4	yes	maximal
5	#1998	C₂	1	2	yes	maximal
6	#1951	C₂	1	4	yes	maximal
7	C₄×#108	C₂	1	1	yes	maximal, characteristic
8	#533	C₂²	1	1	yes	characteristic
9	C₂×#125	C₂²	1	2	yes	—
10	#501	—	2	4	no	—
11	#499	—	2	4	no	—
12	#499	—	2	2	no	—
13	#499	—	2	2	no	—
14	#446	—	2	2	no	—
15	C₈×Q₈	—	2	2	no	—
16	C₂×#108	C₂²	1	2	yes	—
17	#428	C₂²	1	8	yes	—
18	C₂×#107	C₂²	1	2	yes	—
19	C₂×#106	C₂²	1	2	yes	—
20	#424	C₂²	1	2	yes	—
21	#423	C₂²	1	1	yes	characteristic
22	#405	C₂²	1	2	yes	—
23	C₄×M₁₆	C₂²	1	2	yes	—
24	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
25	#343	C₂²	1	8	yes	—
26	#341	C₂²	1	2	yes	—
27	C₂×M₁₆	C₂³	1	2	yes	—
28	C₂×M₁₆	C₂³	1	4	yes	—
29	C₂²×C₈	C₂³	1	1	yes	characteristic
30	C₂²×C₈	C₂³	1	1	yes	characteristic
31	C₄⋊Q₈	D₄	1	2	yes	—
32	#125	—	2	2	no	—
33	#125	—	2	4	no	—
34	#125	—	2	2	no	—
35	#125	D₄	1	2	yes	—
36	C₂²⋊Q₈	—	2	2	no	—
37	C₄×Q₈	—	2	2	no	—
38	#117	C₂³	1	2	yes	—
39	C₂×C₄⋊C₄	—	2	4	no	—
40	C₂×C₄⋊C₄	C₂³	1	4	yes	—
41	C₂×C₄²	C₂³	1	1	yes	characteristic
42	#108	Q₈	1	8	yes	complemented
43	#107	—	2	4	no	—
44	#107	—	2	8	no	—
45	#106	—	2	2	no	—
46	#106	—	2	2	no	—
47	#106	—	2	8	no	—
48	C₄⋊C₈	—	2	4	no	—
49	C₄⋊C₈	—	2	2	no	—
50	#97	—	2	2	no	—
51	#97	—	2	2	no	—
52	C₄×C₈	—	2	2	no	—
53	C₄×C₈	—	2	2	no	—
54	C₄×C₈	D₄	1	2	yes	—
55	C₄×C₈	D₄	1	2	yes	—
56	C₂×Q₈	—	2	2	no	—
57	C₂²×C₄	—	2	4	no	—
58	C₂²×C₄	#41	1	2	yes	—
59	C₂²×C₄	C₂⁴	1	1	yes	characteristic
60	M₁₆	—	2	4	no	—
61	M₁₆	—	2	4	no	—
62	M₁₆	C₂×Q₈	1	8	yes	—
63	C₂×C₈	—	2	4	no	—
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	C₂×D₄	1	2	yes	—
67	C₂×C₈	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈	C₂×Q₈	1	4	yes	—
70	C₄⋊C₄	—	2	4	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	2	4	no	—
73	C₄⋊C₄	—	2	4	no	—
74	C₄⋊C₄	—	4	4	no	—
75	C₄⋊C₄	—	4	4	no	—
76	C₄⋊C₄	C₂×D₄	1	4	yes	—
77	C₂²⋊C₄	—	2	2	no	—
78	C₄²	—	2	2	no	—
79	C₄²	C₂×D₄	1	1	yes	characteristic
80	C₄²	C₂×D₄	1	2	yes	—
81	C₄²	C₂×D₄	1	1	yes	characteristic
82	C₂³	C₂×#41	1	1	yes	characteristic
83	Q₈	—	4	4	no	—
84	C₂×C₄	—	2	4	no	—
85	C₂×C₄	—	4	2	no	—
86	C₂×C₄	—	4	4	no	—
87	C₂×C₄	—	8	4	no	—
88	C₂×C₄	#121	1	2	yes	—
89	C₂×C₄	#121	1	2	yes	—
90	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄	C₂²×Q₈	1	2	yes	—
93	C₂×C₄	2₋¹⁺⁴	1	2	yes	—
94	C₈	—	2	2	no	—
95	C₈	—	2	8	no	—
96	C₈	—	4	2	no	—
97	C₈	C₄⋊Q₈	1	2	yes	—
98	C₈	C₄⋊Q₈	1	2	yes	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂×#121	1	1	yes	characteristic
101	C₂²	#557	1	2	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	2	no	—
104	C₄	—	4	4	no	—
105	C₄	—	8	4	no	—
106	C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
107	C₄	#536	1	1	yes	characteristic
108	C₄	D₄×Q₈	1	2	yes	—
109	C₂	—	2	2	no	—
110	C₂	#2546	1	1	yes	characteristic
111	C₂	#3067	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3166

C₂

yes

maximal

C₂×#499

C₂

yes

maximal, characteristic

#2960

C₂

yes

maximal, characteristic

#2089

C₂

yes

maximal

#1998

C₂

yes

maximal

#1951

C₂

yes

maximal

C₄×#108

C₂

yes

maximal, characteristic

#533

C₂²

yes

characteristic

C₂×#125

C₂²

yes

—

—

—

—

—

—

C₈×Q₈

—

C₂×#108

C₂²

yes

—

#428

C₂²

yes

—

C₂×#107

C₂²

yes

—

C₂×#106

C₂²

yes

—

#424

C₂²

yes

—

#423

C₂²

yes

characteristic

#405

C₂²

yes

—

C₄×M₁₆

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

#343

C₂²

yes

—

#341

C₂²

yes

—

C₂×M₁₆

C₂³

yes

—

C₂×M₁₆

C₂³

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

D₄

yes

—

—

—

—

D₄

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂³

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄²

C₂³

yes

characteristic

#108

Q₈

yes

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

#41

yes

—

C₂²×C₄

C₂⁴

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

C₂×Q₈

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×Q₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₄

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

—

C₄²

C₂×D₄

yes

characteristic

C₂³

C₂×#41

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#121

yes

—

C₂×C₄

#121

yes

—

C₂×C₄

C₂²×D₄

yes

characteristic

C₂×C₄

C₂²×D₄

yes

characteristic

C₂×C₄

C₂²×Q₈

yes

—

C₂×C₄

2₋¹⁺⁴

yes

—

C₈

—

C₈

—

C₈

—

C₈

C₄⋊Q₈

yes

—

C₈

C₄⋊Q₈

yes

—

C₂²

—

100

C₂²

C₂×#121

yes

characteristic

101

C₂²

#557

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

C₂×C₄⋊Q₈

yes

characteristic

107

C₄

#536

yes

characteristic

108

C₄

D₄×Q₈

yes

—

109

C₂

—

110

C₂

#2546

yes

characteristic

111

C₂

#3067

yes

—

	subgroup	G/H
center, socle	C₂²	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	C₂³	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂×#121

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴

prime radical

C₂³

C₂×#41

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
42. #108	Q₈	semidirect

H ≅ G/N

type

42. #108

Q₈

semidirect

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

subgroup	index
#3166	2
C₂×#499	2
#2960	2
#2089	2
#1998	2
#1951	2
C₄×#108	2

subgroup

index

Subgroups — #28701

Conjugacy classes of subgroups

Complements

Maximal subgroups