Group #28809 — subgroups

Subgroups — #28809

Order 256.

411 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 411 conjugacy classes of subgroups — too many to draw. Only its 175 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×#498	C₂	1	2	yes	maximal
2	#2551	C₂	1	2	yes	maximal
3	#2058	C₂	1	4	yes	maximal
4	#1948	C₂	1	2	yes	maximal
5	#1941	C₂	1	2	yes	maximal
6	#1844	C₂	1	1	yes	maximal, characteristic
7	C₄×#106	C₂	1	1	yes	maximal, characteristic
8	C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Q₈	—	2	2	no	—
10	C₂×C₄⋊Q₈	C₂²	1	4	yes	—
11	#498	—	2	8	no	—
12	C₂×#106	C₂²	1	4	yes	—
13	C₂×#106	C₂²	1	2	yes	—
14	C₂×#106	C₂²	1	2	yes	—
15	C₂×#106	C₂²	1	1	yes	characteristic
16	C₂×#106	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈	C₂²	1	2	yes	—
19	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
20	#395	—	2	4	no	—
21	#389	—	2	2	no	—
22	#389	C₂²	1	4	yes	—
23	#384	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	#340	C₂²	1	2	yes	—
27	#340	C₂²	1	4	yes	—
28	#336	C₂²	1	2	yes	—
29	C₂²×Q₈	—	2	4	no	—
30	C₂²×C₈	C₂³	1	1	yes	characteristic
31	C₂²×C₈	C₂³	1	2	yes	—
32	C₂²×C₈	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈	—	2	8	no	—
34	C₄⋊Q₈	—	4	4	no	—
35	C₄⋊Q₈	—	4	8	no	—
36	C₂×C₄⋊C₄	—	2	2	no	—
37	C₂×C₄⋊C₄	—	2	4	no	—
38	C₂×C₄⋊C₄	—	2	4	no	—
39	C₂×C₄⋊C₄	—	2	2	no	—
40	C₂×C₄⋊C₄	D₄	1	2	yes	—
41	C₂×C₄⋊C₄	C₂³	1	4	yes	—
42	C₂×C₄⋊C₄	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄	C₂³	1	2	yes	—
44	C₂×C₄²	—	2	2	no	—
45	C₂×C₄²	C₂³	1	2	yes	—
46	C₂×C₄²	C₂³	1	1	yes	characteristic
47	#106	—	2	4	no	—
48	#106	—	2	8	no	—
49	#106	—	4	2	no	—
50	#106	D₄	1	4	yes	—
51	#106	Q₈	1	4	yes	complemented
52	C₄⋊C₈	Q₈	1	4	yes	complemented
53	C₄×C₈	—	2	4	no	—
54	C₄×C₈	Q₈	1	4	yes	complemented
55	#95	—	2	2	no	—
56	#95	—	2	4	no	—
57	#95	D₄	1	2	yes	—
58	C₂×Q₈	—	4	8	no	—
59	C₂×Q₈	—	4	8	no	—
60	C₂×Q₈	—	8	4	no	—
61	C₂²×C₄	—	2	2	no	—
62	C₂²×C₄	—	2	2	no	—
63	C₂²×C₄	—	2	4	no	—
64	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄	C₂×D₄	1	4	yes	—
66	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄	C₂⁴	1	1	yes	characteristic
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	2	1	no	—
70	C₂×C₈	—	2	1	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	C₂×D₄	1	2	yes	—
73	C₂×C₈	C₂×Q₈	1	2	yes	—
74	C₂×C₈	C₂×Q₈	1	4	yes	—
75	C₂×C₈	C₂×Q₈	1	2	yes	—
76	C₂×C₈	C₂×Q₈	1	2	yes	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	—	2	2	no	—
79	C₄⋊C₄	—	2	4	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	4	8	no	—
82	C₄⋊C₄	—	4	8	no	—
83	C₄⋊C₄	—	4	4	no	—
84	C₄⋊C₄	—	8	2	no	—
85	C₄⋊C₄	SD₁₆	1	8	yes	—
86	C₄⋊C₄	C₂×D₄	1	4	yes	—
87	C₄⋊C₄	C₂×Q₈	1	4	yes	—
88	C₄²	—	2	2	no	—
89	C₄²	—	2	2	no	—
90	C₄²	—	4	4	no	—
91	C₄²	C₂×Q₈	1	2	yes	—
92	C₄²	C₂×Q₈	1	2	yes	—
93	C₂³	C₂²×D₄	1	1	yes	characteristic
94	Q₈	—	8	16	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	4	4	no	—
105	C₂×C₄	—	4	4	no	—
106	C₂×C₄	—	4	4	no	—
107	C₂×C₄	C₄⋊Q₈	1	2	yes	—
108	C₂×C₄	C₄⋊Q₈	1	2	yes	—
109	C₂×C₄	C₂×SD₁₆	1	8	yes	—
110	C₂×C₄	C₂×SD₁₆	1	4	yes	—
111	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
116	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
117	C₈	—	2	2	no	—
118	C₈	—	2	4	no	—
119	C₈	C₄⋊Q₈	1	4	yes	complemented
120	C₂²	C₂×C₄⋊Q₈	1	1	yes	characteristic
121	C₂²	D₄²	1	1	yes	characteristic
122	C₂²	D₄×Q₈	1	1	yes	characteristic
123	C₂²	C₂²×SD₁₆	1	2	yes	—
124	C₂²	#577	1	2	yes	—
125	C₄	—	2	2	no	—
126	C₄	—	2	8	no	—
127	C₄	—	4	4	no	—
128	C₄	—	4	4	no	—
129	C₄	—	8	4	no	—
130	C₄	#498	1	4	yes	complemented
131	C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
132	C₄	#544	1	1	yes	characteristic
133	C₄	D₄×Q₈	1	1	yes	characteristic
134	C₄	Q₈²	1	1	yes	characteristic
135	C₂	#2551	1	1	yes	characteristic
136	C₂	C₂×#498	1	1	yes	characteristic
137	C₂	#3163	1	1	yes	characteristic
138	C₂	D₄×SD₁₆	1	2	yes	—
139	C₂	Q₈×SD₁₆	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#498

C₂

yes

maximal

#2551

C₂

yes

maximal

#2058

C₂

yes

maximal

#1948

C₂

yes

maximal

#1941

C₂

yes

maximal

#1844

C₂

yes

maximal, characteristic

C₄×#106

C₂

yes

maximal, characteristic

C₈×C₄⋊C₄

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

C₂²

yes

—

#498

—

C₂×#106

C₂²

yes

—

C₂×#106

C₂²

yes

—

C₂×#106

C₂²

yes

—

C₂×#106

C₂²

yes

characteristic

C₂×#106

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

—

—

C₂²

yes

—

#384

C₂²

yes

—

C₄×C₄⋊C₄

C₂²

yes

—

C₄×C₄⋊C₄

C₂²

yes

characteristic

#340

C₂²

yes

—

#340

C₂²

yes

—

#336

C₂²

yes

—

C₂²×Q₈

—

C₂²×C₈

C₂³

yes

characteristic

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄

yes

—

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

—

—

—

D₄

yes

—

#106

Q₈

yes

complemented

C₄⋊C₈

Q₈

yes

complemented

C₄×C₈

—

C₄×C₈

Q₈

yes

—

—

D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂⁴

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

—

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

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

SD₁₆

yes

—

C₄⋊C₄

C₂×D₄

yes

—

C₄⋊C₄

C₂×Q₈

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×Q₈

yes

—

C₄²

C₂×Q₈

yes

—

C₂³

C₂²×D₄

yes

characteristic

Q₈

—

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

—

107

C₂×C₄

C₄⋊Q₈

yes

—

108

C₂×C₄

C₄⋊Q₈

yes

—

109

C₂×C₄

C₂×SD₁₆

yes

—

110

C₂×C₄

C₂×SD₁₆

yes

—

111

C₂×C₄

C₂²×D₄

yes

characteristic

112

C₂×C₄

C₂²×Q₈

yes

characteristic

113

C₂×C₄

C₂²×Q₈

yes

characteristic

114

C₂×C₄

C₂²×Q₈

yes

characteristic

115

C₂×C₄

H(5,𝔽₂)

yes

characteristic

116

C₂×C₄

2₋¹⁺⁴

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

C₄⋊Q₈

yes

complemented

120

C₂²

C₂×C₄⋊Q₈

yes

characteristic

121

C₂²

D₄²

yes

characteristic

122

C₂²

D₄×Q₈

yes

characteristic

123

C₂²

C₂²×SD₁₆

yes

—

124

C₂²

#577

yes

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#498

yes

complemented

131

C₄

C₂×C₄⋊Q₈

yes

characteristic

132

C₄

#544

yes

characteristic

133

C₄

D₄×Q₈

yes

characteristic

134

C₄

Q₈²

yes

characteristic

135

C₂

#2551

yes

characteristic

136

C₂

C₂×#498

yes

characteristic

137

C₂

#3163

yes

characteristic

138

C₂

D₄×SD₁₆

yes

—

139

C₂

Q₈×SD₁₆

yes

—

	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

N	H ≅ G/N	type
51. #106	Q₈	semidirect
52. C₄⋊C₈	Q₈	semidirect
54. C₄×C₈	Q₈	semidirect
119. C₈	C₄⋊Q₈	semidirect
130. C₄	#498	semidirect

H ≅ G/N

type

51. #106

Q₈

semidirect

52. C₄⋊C₈

Q₈

semidirect

54. C₄×C₈

Q₈

semidirect

119. C₈

C₄⋊Q₈

semidirect

130. C₄

#498

semidirect

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

subgroup	index
C₂×#498	2
#2551	2
#2058	2
#1948	2
#1941	2
#1844	2
C₄×#106	2
C₈×C₄⋊C₄	2

subgroup

index

C₈×C₄⋊C₄

Subgroups — #28809

Conjugacy classes of subgroups

Complements

Maximal subgroups