Group #28700 — subgroups

Subgroups — #28700

Order 256.

315 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 315 conjugacy classes of subgroups — too many to draw. Only its 139 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	#3165	C₂	1	2	yes	maximal
2	C₂×#498	C₂	1	1	yes	maximal, characteristic
3	#2960	C₂	1	1	yes	maximal, characteristic
4	#2090	C₂	1	4	yes	maximal
5	#1997	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×C₄⋊Q₈	C₂²	1	2	yes	—
10	#501	—	2	4	no	—
11	#500	—	2	2	no	—
12	#498	—	2	2	no	—
13	#498	—	2	2	no	—
14	#498	—	2	2	no	—
15	#446	—	2	2	no	—
16	C₈×Q₈	—	2	2	no	—
17	C₂×#108	C₂²	1	2	yes	—
18	#428	C₂²	1	8	yes	—
19	C₂×#106	C₂²	1	2	yes	—
20	C₂×#106	C₂²	1	2	yes	—
21	#424	C₂²	1	2	yes	—
22	#423	C₂²	1	1	yes	characteristic
23	#405	C₂²	1	2	yes	—
24	C₄×M₁₆	C₂²	1	2	yes	—
25	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
26	#343	C₂²	1	8	yes	—
27	#341	C₂²	1	2	yes	—
28	C₂²×Q₈	—	2	2	no	—
29	C₂×M₁₆	C₂³	1	2	yes	—
30	C₂×M₁₆	C₂³	1	4	yes	—
31	C₂²×C₈	C₂³	1	1	yes	characteristic
32	C₂²×C₈	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈	—	2	2	no	—
34	C₄⋊Q₈	—	2	2	no	—
35	C₄⋊Q₈	—	2	2	no	—
36	C₄⋊Q₈	—	2	2	no	—
37	C₄⋊Q₈	D₄	1	2	yes	—
38	#125	D₄	1	2	yes	—
39	C₂²⋊Q₈	—	2	2	no	—
40	C₄×Q₈	—	2	2	no	—
41	#117	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄	—	2	2	no	—
43	C₂×C₄⋊C₄	C₂³	1	4	yes	—
44	C₂×C₄²	C₂³	1	1	yes	characteristic
45	#108	Q₈	1	8	yes	complemented
46	#107	—	2	8	no	—
47	#106	—	2	4	no	—
48	#106	—	2	2	no	—
49	#106	—	2	2	no	—
50	#106	—	2	8	no	—
51	C₄⋊C₈	—	2	4	no	—
52	C₄⋊C₈	—	2	2	no	—
53	#97	—	2	2	no	—
54	#97	—	2	2	no	—
55	C₄×C₈	—	2	2	no	—
56	C₄×C₈	—	2	2	no	—
57	C₄×C₈	D₄	1	2	yes	—
58	C₄×C₈	D₄	1	2	yes	—
59	C₂×Q₈	—	2	2	no	—
60	C₂×Q₈	—	4	2	no	—
61	C₂×Q₈	—	4	2	no	—
62	C₂×Q₈	—	8	4	no	—
63	C₂²×C₄	—	2	4	no	—
64	C₂²×C₄	#41	1	2	yes	—
65	C₂²×C₄	C₂⁴	1	1	yes	characteristic
66	M₁₆	—	2	4	no	—
67	M₁₆	—	2	4	no	—
68	M₁₆	C₂×Q₈	1	8	yes	—
69	C₂×C₈	—	2	4	no	—
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	C₂×D₄	1	2	yes	—
73	C₂×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈	C₂×Q₈	1	4	yes	—
76	C₄⋊C₄	—	2	4	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	—	2	4	no	—
79	C₄⋊C₄	—	2	4	no	—
80	C₄⋊C₄	—	4	2	no	—
81	C₄⋊C₄	—	4	2	no	—
82	C₄⋊C₄	C₂×D₄	1	4	yes	—
83	C₂²⋊C₄	—	2	2	no	—
84	C₄²	—	2	2	no	—
85	C₄²	C₂×D₄	1	1	yes	characteristic
86	C₄²	C₂×D₄	1	2	yes	—
87	C₄²	C₂×D₄	1	1	yes	characteristic
88	C₂³	C₂×#41	1	1	yes	characteristic
89	Q₈	—	4	4	no	—
90	Q₈	—	8	4	no	—
91	Q₈	—	8	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	4	4	no	—
95	C₂×C₄	—	8	4	no	—
96	C₂×C₄	#121	1	2	yes	—
97	C₂×C₄	#121	1	2	yes	—
98	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄	C₂²×Q₈	1	2	yes	—
101	C₂×C₄	2₋¹⁺⁴	1	2	yes	—
102	C₈	—	2	2	no	—
103	C₈	—	2	8	no	—
104	C₈	—	4	2	no	—
105	C₈	C₄⋊Q₈	1	2	yes	—
106	C₈	C₄⋊Q₈	1	2	yes	—
107	C₂²	—	2	2	no	—
108	C₂²	C₂×#121	1	1	yes	characteristic
109	C₂²	#557	1	2	yes	—
110	C₄	—	2	2	no	—
111	C₄	—	4	4	no	—
112	C₄	—	8	4	no	—
113	C₄	#471	1	4	yes	—
114	C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
115	C₄	#536	1	1	yes	characteristic
116	C₄	D₄×Q₈	1	2	yes	—
117	C₂	—	2	2	no	—
118	C₂	#2546	1	1	yes	characteristic
119	C₂	C₂×#471	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3165

C₂

yes

maximal

C₂×#498

C₂

yes

maximal, characteristic

#2960

C₂

yes

maximal, characteristic

#2090

C₂

yes

maximal

#1997

C₂

yes

maximal

#1951

C₂

yes

maximal

C₄×#108

C₂

yes

maximal, characteristic

#533

C₂²

yes

characteristic

C₂×C₄⋊Q₈

C₂²

yes

—

—

—

—

—

—

—

C₈×Q₈

—

C₂×#108

C₂²

yes

—

#428

C₂²

yes

—

C₂×#106

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

—

C₂²×Q₈

—

C₂×M₁₆

C₂³

yes

—

C₂×M₁₆

C₂³

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₄

yes

—

#125

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

—

C₂×Q₈

—

C₂×Q₈

—

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

—

Q₈

—

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

100

C₂×C₄

C₂²×Q₈

yes

—

101

C₂×C₄

2₋¹⁺⁴

yes

—

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

C₄⋊Q₈

yes

—

106

C₈

C₄⋊Q₈

yes

—

107

C₂²

—

108

C₂²

C₂×#121

yes

characteristic

109

C₂²

#557

yes

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

#471

yes

—

114

C₄

C₂×C₄⋊Q₈

yes

characteristic

115

C₄

#536

yes

characteristic

116

C₄

D₄×Q₈

yes

—

117

C₂

—

118

C₂

#2546

yes

characteristic

119

C₂

C₂×#471

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
45. #108	Q₈	semidirect

H ≅ G/N

type

45. #108

Q₈

semidirect

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

subgroup	index
#3165	2
C₂×#498	2
#2960	2
#2090	2
#1997	2
#1951	2
C₄×#108	2

subgroup

index

Subgroups — #28700

Conjugacy classes of subgroups

Complements

Maximal subgroups