Group #28823 — subgroups

Subgroups — #28823

Order 256.

309 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 309 conjugacy classes of subgroups — too many to draw. Only its 143 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	#3164	C₂	1	2	yes	maximal
2	#3163	C₂	1	2	yes	maximal
3	#2061	C₂	1	4	yes	maximal
4	#1952	C₂	1	2	yes	maximal
5	#1951	C₂	1	2	yes	maximal
6	#1846	C₂	1	1	yes	maximal, characteristic
7	#1772	C₂	1	1	yes	maximal, characteristic
8	#1772	C₂	1	1	yes	maximal, characteristic
9	#544	C₂²	1	4	yes	—
10	#501	—	2	4	no	—
11	#501	—	2	4	no	—
12	#500	—	2	2	no	—
13	#499	—	2	2	no	—
14	#499	—	2	2	no	—
15	#498	—	2	2	no	—
16	C₂×#108	C₂²	1	1	yes	characteristic
17	C₂×#108	C₂²	1	1	yes	characteristic
18	#428	C₂²	1	4	yes	—
19	#428	C₂²	1	4	yes	—
20	#427	C₂²	1	1	yes	characteristic
21	#427	C₂²	1	1	yes	characteristic
22	C₂×#107	C₂²	1	2	yes	—
23	C₂×#106	C₂²	1	2	yes	—
24	#405	C₂²	1	2	yes	—
25	#405	C₂²	1	2	yes	—
26	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
27	#382	C₂²	1	2	yes	—
28	#343	C₂²	1	4	yes	—
29	#339	C₂²	1	2	yes	—
30	#339	C₂²	1	2	yes	—
31	C₂×M₁₆	C₂³	1	2	yes	—
32	C₂×M₁₆	C₂³	1	2	yes	—
33	C₂²×C₈	C₂³	1	1	yes	characteristic
34	C₂²×C₈	C₂³	1	1	yes	characteristic
35	C₄⋊Q₈	—	2	4	no	—
36	C₄⋊Q₈	—	2	4	no	—
37	#125	—	2	4	no	—
38	#125	—	2	4	no	—
39	C₂²⋊Q₈	—	4	4	no	—
40	#117	C₂³	1	2	yes	—
41	#117	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄	C₂³	1	4	yes	—
43	C₂×C₄²	C₂³	1	1	yes	characteristic
44	#108	Q₈	1	4	yes	complemented
45	#108	Q₈	1	4	yes	complemented
46	#107	—	2	2	no	—
47	#107	—	2	2	no	—
48	#107	—	2	4	no	—
49	#107	—	2	4	no	—
50	#107	D₄	1	2	yes	—
51	#107	D₄	1	2	yes	—
52	#106	—	2	2	no	—
53	#106	—	2	2	no	—
54	#106	—	2	4	no	—
55	#106	—	2	4	no	—
56	#106	D₄	1	1	yes	characteristic
57	#106	D₄	1	1	yes	characteristic
58	#106	D₄	1	2	yes	—
59	#97	—	2	2	no	—
60	#97	—	2	2	no	—
61	C₄×C₈	—	2	2	no	—
62	C₄×C₈	—	2	2	no	—
63	C₄×C₈	Q₈	1	4	yes	complemented
64	#95	—	4	2	no	—
65	C₂×Q₈	—	4	4	no	—
66	C₂²×C₄	—	2	1	no	—
67	C₂²×C₄	—	2	4	no	—
68	C₂²×C₄	C₂⁴	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	C₂×Q₈	1	4	yes	—
72	M₁₆	C₂×Q₈	1	4	yes	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈	C₂×Q₈	1	2	yes	—
80	C₂×C₈	C₂×Q₈	1	2	yes	—
81	C₂×C₈	C₂×Q₈	1	2	yes	—
82	C₂×C₈	C₂×Q₈	1	2	yes	—
83	C₄⋊C₄	—	2	4	no	—
84	C₄⋊C₄	—	2	4	no	—
85	C₄⋊C₄	—	4	4	no	—
86	C₄⋊C₄	—	4	4	no	—
87	C₄⋊C₄	—	4	4	no	—
88	C₄⋊C₄	C₂×D₄	1	2	yes	—
89	C₄⋊C₄	C₂×D₄	1	4	yes	—
90	C₄⋊C₄	C₂×D₄	1	2	yes	—
91	C₂²⋊C₄	—	2	2	no	—
92	C₂²⋊C₄	—	2	2	no	—
93	C₄²	—	2	2	no	—
94	C₄²	—	2	2	no	—
95	C₄²	—	2	1	no	—
96	C₄²	C₂×Q₈	1	2	yes	—
97	C₂³	H(5,𝔽₂)	1	1	yes	characteristic
98	Q₈	—	16	4	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	4	4	no	—
105	C₂×C₄	—	8	4	no	—
106	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
112	C₈	—	2	4	no	—
113	C₈	—	2	4	no	—
114	C₈	C₄⋊Q₈	1	2	yes	—
115	C₈	C₄⋊Q₈	1	2	yes	—
116	C₈	C₄⋊Q₈	1	2	yes	—
117	C₈	C₄⋊Q₈	1	2	yes	—
118	C₂²	—	2	2	no	—
119	C₂²	#544	1	1	yes	characteristic
120	C₂²	D₄²	1	1	yes	characteristic
121	C₂²	Q₈²	1	1	yes	characteristic
122	C₄	—	4	2	no	—
123	C₄	—	4	2	no	—
124	C₄	—	4	4	no	—
125	C₄	—	4	2	no	—
126	C₄	—	8	4	no	—
127	C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
128	C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
129	C₄	D₄×Q₈	1	1	yes	characteristic
130	C₄	D₄×Q₈	1	1	yes	characteristic
131	C₂	—	2	1	no	—
132	C₂	—	2	1	no	—
133	C₂	#2551	1	1	yes	characteristic
134	C₂	#3290	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3164

C₂

yes

maximal

#3163

C₂

yes

maximal

#2061

C₂

yes

maximal

#1952

C₂

yes

maximal

#1951

C₂

yes

maximal

#1846

C₂

yes

maximal, characteristic

#1772

C₂

yes

maximal, characteristic

#1772

C₂

yes

maximal, characteristic

#544

C₂²

yes

—

—

—

—

—

—

—

C₂²

yes

characteristic

C₂×#108

C₂²

yes

characteristic

#428

C₂²

yes

—

#428

C₂²

yes

—

#427

C₂²

yes

characteristic

#427

C₂²

yes

characteristic

C₂×#107

C₂²

yes

—

C₂×#106

C₂²

yes

—

#405

C₂²

yes

—

#405

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

#382

C₂²

yes

—

#343

C₂²

yes

—

#339

C₂²

yes

—

#339

C₂²

yes

—

C₂×M₁₆

C₂³

yes

—

C₂×M₁₆

C₂³

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₂²⋊Q₈

—

#117

C₂³

yes

—

#117

C₂³

yes

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄²

C₂³

yes

characteristic

#108

Q₈

yes

complemented

#108

Q₈

yes

—

—

—

—

D₄

yes

—

#107

D₄

yes

—

—

—

—

—

D₄

yes

characteristic

#106

D₄

yes

characteristic

#106

D₄

yes

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Q₈

yes

complemented

#95

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂⁴

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

C₂×Q₈

yes

—

M₁₆

C₂×Q₈

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

C₂×Q₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₄

yes

—

C₄⋊C₄

C₂×D₄

yes

—

C₄⋊C₄

C₂×D₄

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×Q₈

yes

—

C₂³

H(5,𝔽₂)

yes

characteristic

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂²×D₄

yes

characteristic

107

C₂×C₄

C₂²×D₄

yes

characteristic

108

C₂×C₄

C₂²×Q₈

yes

characteristic

109

C₂×C₄

C₂²×Q₈

yes

characteristic

110

C₂×C₄

C₂²×Q₈

yes

characteristic

111

C₂×C₄

2₋¹⁺⁴

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

C₄⋊Q₈

yes

—

115

C₈

C₄⋊Q₈

yes

—

116

C₈

C₄⋊Q₈

yes

—

117

C₈

C₄⋊Q₈

yes

—

118

C₂²

—

119

C₂²

#544

yes

characteristic

120

C₂²

D₄²

yes

characteristic

121

C₂²

Q₈²

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

C₂×C₄⋊Q₈

yes

characteristic

128

C₄

C₂×C₄⋊Q₈

yes

characteristic

129

C₄

D₄×Q₈

yes

characteristic

130

C₄

D₄×Q₈

yes

characteristic

131

C₂

—

132

C₂

—

133

C₂

#2551

yes

characteristic

134

C₂

#3290

yes

—

	subgroup	G/H
center, socle	C₂²	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	C₂³	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

D₄²

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴

C₂³

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ G

LECS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
44. #108	Q₈	semidirect
45. #108	Q₈	semidirect
63. C₄×C₈	Q₈	semidirect

H ≅ G/N

type

44. #108

Q₈

semidirect

45. #108

Q₈

semidirect

63. C₄×C₈

Q₈

semidirect

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

subgroup	index
#3164	2
#3163	2
#2061	2
#1952	2
#1951	2
#1846	2
#1772	2

subgroup

index

Subgroups — #28823

Conjugacy classes of subgroups

Complements

Maximal subgroups