Group #28828 — subgroups

Subgroups — #28828

Order 256.

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

normal subgroups only

This group has 303 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	#2571	C₂	1	2	yes	maximal
2	#2085	C₂	1	4	yes	maximal
3	#1948	C₂	1	2	yes	maximal
4	#1942	C₂	1	2	yes	maximal
5	#1846	C₂	1	2	yes	maximal
6	#1845	C₂	1	1	yes	maximal, characteristic
7	C₄×#107	C₂	1	1	yes	maximal, characteristic
8	C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₂×#107	C₂²	1	2	yes	—
10	C₂×#107	C₂²	1	1	yes	characteristic
11	C₂×#107	C₂²	1	1	yes	characteristic
12	C₂×#107	C₂²	1	2	yes	—
13	C₂×#106	C₂²	1	4	yes	—
14	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈	C₂²	1	2	yes	—
16	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
17	#400	—	2	4	no	—
18	#389	—	2	2	no	—
19	#389	C₂²	1	4	yes	—
20	#384	C₂²	1	2	yes	—
21	#382	—	2	2	no	—
22	#382	C₂²	1	4	yes	—
23	C₄×C₄⋊C₄	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	#340	C₂²	1	2	yes	—
26	#340	C₂²	1	4	yes	—
27	#336	C₂²	1	2	yes	—
28	C₂²×C₈	C₂³	1	1	yes	characteristic
29	C₂²×C₈	C₂³	1	2	yes	—
30	C₂²×C₈	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊C₄	—	2	2	no	—
32	C₂×C₄⋊C₄	—	2	2	no	—
33	C₂×C₄⋊C₄	—	2	4	no	—
34	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄	C₂³	1	2	yes	—
37	C₂×C₄⋊C₄	C₂³	1	2	yes	—
38	C₂×C₄⋊C₄	C₂³	1	4	yes	—
39	C₂×C₄²	—	2	2	no	—
40	C₂×C₄²	C₂³	1	2	yes	—
41	C₂×C₄²	C₂³	1	1	yes	characteristic
42	#107	—	2	2	no	—
43	#107	—	2	4	no	—
44	#107	—	2	2	no	—
45	#107	—	4	2	no	—
46	#106	—	4	4	no	—
47	C₄⋊C₈	Q₈	1	4	yes	—
48	C₄×C₈	—	2	4	no	—
49	C₄×C₈	Q₈	1	4	yes	—
50	#95	—	2	2	no	—
51	#95	—	2	4	no	—
52	#95	—	2	4	no	—
53	#95	—	2	4	no	—
54	#95	D₄	1	2	yes	—
55	C₂²×C₄	—	2	2	no	—
56	C₂²×C₄	—	2	2	no	—
57	C₂²×C₄	—	2	4	no	—
58	C₂²×C₄	C₂×D₄	1	2	yes	—
59	C₂²×C₄	C₂×D₄	1	2	yes	—
60	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄	C₂⁴	1	1	yes	characteristic
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	2	1	no	—
66	C₂×C₈	—	2	1	no	—
67	C₂×C₈	C₂×Q₈	1	2	yes	—
68	C₂×C₈	C₂×Q₈	1	4	yes	—
69	C₂×C₈	C₂×Q₈	1	2	yes	—
70	C₂×C₈	#41	1	2	yes	—
71	C₂×C₈	#41	1	2	yes	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	—	2	2	no	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	2	4	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	—	2	4	no	—
79	C₄⋊C₄	—	2	2	no	—
80	C₄⋊C₄	—	4	4	no	—
81	C₄⋊C₄	—	8	2	no	—
82	C₄⋊C₄	—	8	4	no	—
83	C₄²	—	2	2	no	—
84	C₄²	—	2	2	no	—
85	C₄²	—	4	4	no	—
86	C₄²	C₂×Q₈	1	2	yes	—
87	C₄²	C₂×Q₈	1	2	yes	—
88	C₂³	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	—	4	4	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	4	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	4	4	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	C₄⋊Q₈	1	2	yes	—
107	C₂×C₄	C₄⋊Q₈	1	2	yes	—
108	C₂×C₄	#135	1	2	yes	—
109	C₂×C₄	#135	1	2	yes	—
110	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄	C₂×#41	1	1	yes	characteristic
113	C₂×C₄	C₂×#41	1	1	yes	characteristic
114	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
115	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
116	C₈	—	2	2	no	—
117	C₈	—	2	4	no	—
118	C₈	#125	1	4	yes	—
119	C₂²	C₂×C₄⋊Q₈	1	1	yes	characteristic
120	C₂²	#547	1	1	yes	characteristic
121	C₂²	#550	1	1	yes	characteristic
122	C₂²	C₂×#135	1	1	yes	characteristic
123	C₂²	C₂×#135	1	1	yes	characteristic
124	C₂²	#576	1	1	yes	characteristic
125	C₂²	#578	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	4	2	no	—
128	C₄	—	4	2	no	—
129	C₄	—	8	2	no	—
130	C₄	—	8	2	no	—
131	C₄	—	8	4	no	—
132	C₄	#499	1	4	yes	—
133	C₄	C₂×#125	1	1	yes	characteristic
134	C₄	#544	1	1	yes	characteristic
135	C₄	#554	1	1	yes	characteristic
136	C₄	#557	1	1	yes	characteristic
137	C₂	#2552	1	1	yes	characteristic
138	C₂	C₂×#499	1	1	yes	characteristic
139	C₂	#3164	1	1	yes	characteristic
140	C₂	#3341	1	1	yes	characteristic
141	C₂	#3342	1	1	yes	characteristic
142	C₂	#3393	1	1	yes	characteristic
143	C₂	#3396	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2571

C₂

yes

maximal

#2085

C₂

yes

maximal

#1948

C₂

yes

maximal

#1942

C₂

yes

maximal

#1846

C₂

yes

maximal

#1845

C₂

yes

maximal, characteristic

C₄×#107

C₂

yes

maximal, characteristic

C₈×C₄⋊C₄

C₂

yes

maximal, characteristic

C₂×#107

C₂²

yes

—

C₂×#107

C₂²

yes

characteristic

C₂×#107

C₂²

yes

characteristic

C₂×#107

C₂²

yes

—

C₂×#106

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

—

—

C₂²

yes

—

#384

C₂²

yes

—

#382

—

#382

C₂²

yes

—

C₄×C₄⋊C₄

C₂²

yes

—

C₄×C₄⋊C₄

C₂²

yes

characteristic

#340

C₂²

yes

—

#340

C₂²

yes

—

#336

C₂²

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄

yes

characteristic

C₂×C₄⋊C₄

D₄

yes

characteristic

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

—

—

—

—

—

C₄⋊C₈

Q₈

yes

—

C₄×C₈

—

C₄×C₈

Q₈

yes

—

—

—

—

—

D₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂⁴

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

#41

yes

—

C₂×C₈

#41

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×Q₈

yes

—

C₄²

C₂×Q₈

yes

—

C₂³

C₂²×D₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₄⋊Q₈

yes

—

107

C₂×C₄

C₄⋊Q₈

yes

—

108

C₂×C₄

#135

yes

—

109

C₂×C₄

#135

yes

—

110

C₂×C₄

C₂²×Q₈

yes

characteristic

111

C₂×C₄

C₂²×Q₈

yes

characteristic

112

C₂×C₄

C₂×#41

yes

characteristic

113

C₂×C₄

C₂×#41

yes

characteristic

114

C₂×C₄

H(5,𝔽₂)

yes

characteristic

115

C₂×C₄

2₋¹⁺⁴

yes

characteristic

116

C₈

—

117

C₈

—

118

C₈

#125

yes

—

119

C₂²

C₂×C₄⋊Q₈

yes

characteristic

120

C₂²

#547

yes

characteristic

121

C₂²

#550

yes

characteristic

122

C₂²

C₂×#135

yes

characteristic

123

C₂²

C₂×#135

yes

characteristic

124

C₂²

#576

yes

characteristic

125

C₂²

#578

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

#499

yes

—

133

C₄

C₂×#125

yes

characteristic

134

C₄

#544

yes

characteristic

135

C₄

#554

yes

characteristic

136

C₄

#557

yes

characteristic

137

C₂

#2552

yes

characteristic

138

C₂

C₂×#499

yes

characteristic

139

C₂

#3164

yes

characteristic

140

C₂

#3341

yes

characteristic

141

C₂

#3342

yes

characteristic

142

C₂

#3393

yes

characteristic

143

C₂

#3396

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂²×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ G

LECS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#2571	2
#2085	2
#1948	2
#1942	2
#1846	2
#1845	2
C₄×#107	2
C₈×C₄⋊C₄	2

subgroup

index

C₈×C₄⋊C₄

Subgroups — #28828

Conjugacy classes of subgroups

Maximal subgroups