Group #9806 — subgroups

Subgroups — #9806

Order 256.

209 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 209 conjugacy classes of subgroups — too many to draw. Only its 75 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	1	yes	maximal, characteristic
2	#1997	C₂	1	1	yes	maximal, characteristic
3	#1966	C₂	1	1	yes	maximal, characteristic
4	#1567	C₂	1	1	yes	maximal, characteristic
5	#1549	C₂	1	1	yes	maximal, characteristic
6	#1546	C₂	1	1	yes	maximal, characteristic
7	#1451	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	#498	—	2	1	no	—
11	#498	—	2	1	no	—
12	#498	—	2	2	no	—
13	#431	C₂²	1	1	yes	characteristic
14	C₂×#106	C₄	1	4	yes	—
15	#423	C₂²	1	1	yes	characteristic
16	#423	C₂²	1	1	yes	characteristic
17	C₂×#101	—	2	2	no	—
18	C₄×M₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
20	#343	—	2	2	no	—
21	#335	—	2	2	no	—
22	#332	—	2	2	no	—
23	#332	—	2	2	no	—
24	#322	—	2	1	no	—
25	#322	—	2	1	no	—
26	C₂²×Q₈	—	2	1	no	—
27	C₂²×Q₈	—	2	1	no	—
28	C₂×M₁₆	—	2	1	no	—
29	C₂×M₁₆	—	2	1	no	—
30	C₂×M₁₆	D₄	1	2	yes	—
31	C₂²×C₈	C₂×C₄	1	2	yes	—
32	C₄⋊Q₈	—	2	1	no	—
33	C₄⋊Q₈	—	2	1	no	—
34	C₄⋊Q₈	—	2	1	no	—
35	C₄⋊Q₈	—	2	1	no	—
36	C₄⋊Q₈	—	2	1	no	—
37	C₄⋊Q₈	—	2	1	no	—
38	C₄⋊Q₈	—	2	1	no	—
39	C₄⋊Q₈	—	2	1	no	—
40	C₂×C₄⋊C₄	—	2	1	no	—
41	C₂×C₄⋊C₄	—	2	1	no	—
42	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
44	C₂×C₄²	C₂³	1	1	yes	characteristic
45	#106	—	2	4	no	—
46	#106	—	2	4	no	—
47	C₄⋊C₈	—	2	1	no	—
48	C₄⋊C₈	—	2	1	no	—
49	C₄⋊C₈	—	2	1	no	—
50	C₄⋊C₈	—	2	1	no	—
51	C₄⋊C₈	D₄	1	2	yes	—
52	#101	—	4	4	no	—
53	C₂²⋊C₈	—	2	1	no	—
54	#97	—	2	1	no	—
55	C₄×C₈	—	2	1	no	—
56	C₄×C₈	—	2	1	no	—
57	C₄×C₈	D₄	1	1	yes	characteristic
58	C₄×C₈	D₄	1	1	yes	characteristic
59	C₄×C₈	D₄	1	2	yes	—
60	C₂×Q₈	—	4	1	no	—
61	C₂×Q₈	—	4	1	no	—
62	C₂×Q₈	—	4	1	no	—
63	C₂×Q₈	—	4	1	no	—
64	C₂×Q₈	—	4	2	no	—
65	C₂×Q₈	—	4	2	no	—
66	C₂×Q₈	—	8	2	no	—
67	C₂²×C₄	—	2	2	no	—
68	C₂²×C₄	—	2	2	no	—
69	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄	#41	1	1	yes	characteristic
72	M₁₆	—	2	2	no	—
73	M₁₆	—	2	2	no	—
74	M₁₆	—	4	2	no	—
75	M₁₆	—	4	2	no	—
76	C₂×C₈	—	2	4	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	4	1	no	—
80	C₂×C₈	—	4	1	no	—
81	C₂×C₈	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈	C₂²⋊C₄	1	2	yes	—
83	C₄⋊C₄	—	2	2	no	—
84	C₄⋊C₄	—	2	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	4	1	no	—
88	C₄⋊C₄	—	4	1	no	—
89	C₄⋊C₄	—	4	1	no	—
90	C₄⋊C₄	—	4	1	no	—
91	C₄²	C₂×D₄	1	1	yes	characteristic
92	C₄²	C₂×D₄	1	1	yes	characteristic
93	C₄²	C₂×D₄	1	1	yes	characteristic
94	C₄²	#41	1	1	yes	characteristic
95	C₂³	C₄×D₄	1	1	yes	characteristic
96	Q₈	—	8	2	no	—
97	Q₈	—	8	2	no	—
98	Q₈	—	8	2	no	—
99	Q₈	—	8	1	no	—
100	Q₈	—	8	1	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	4	no	—
106	C₂×C₄	—	8	2	no	—
107	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄	#121	1	1	yes	characteristic
110	C₂×C₄	#121	1	1	yes	characteristic
111	C₂×C₄	#124	1	1	yes	characteristic
112	C₂×C₄	#127	1	1	yes	characteristic
113	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
114	C₂×C₄	#137	1	2	yes	—
115	C₈	—	2	2	no	—
116	C₈	—	4	2	no	—
117	C₈	—	4	2	no	—
118	C₈	—	8	1	no	—
119	C₈	—	8	1	no	—
120	C₈	#101	1	4	yes	complemented
121	C₂²	—	2	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	#390	1	1	yes	characteristic
124	C₂²	#440	1	2	yes	—
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	8	2	no	—
128	C₄	—	8	2	no	—
129	C₄	C₂×#101	1	2	yes	—
130	C₄	#468	1	2	yes	—
131	C₄	#471	1	2	yes	—
132	C₄	#496	1	1	yes	characteristic
133	C₄	#497	1	1	yes	characteristic
134	C₂	—	2	2	no	—
135	C₂	#1966	1	1	yes	characteristic
136	C₂	#1973	1	1	yes	characteristic
137	C₂	#1977	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#498

C₂

yes

maximal, characteristic

#1997

C₂

yes

maximal, characteristic

#1966

C₂

yes

maximal, characteristic

#1567

C₂

yes

maximal, characteristic

#1549

C₂

yes

maximal, characteristic

#1546

C₂

yes

maximal, characteristic

#1451

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈

C₂²

yes

characteristic

C₂×C₄⋊Q₈

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₂×#106

C₄

yes

—

#423

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

C₂×#101

—

C₄×M₁₆

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

—

—

—

—

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

D₄

yes

—

C₂²×C₈

C₂×C₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²

C₂³

yes

characteristic

#106

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₄

yes

—

#101

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

C₂²⋊C₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

#41

yes

characteristic

C₂³

C₄×D₄

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

100

Q₈

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

108

C₂×C₄

C₄×D₄

yes

characteristic

109

C₂×C₄

#121

yes

characteristic

110

C₂×C₄

#121

yes

characteristic

111

C₂×C₄

#124

yes

characteristic

112

C₂×C₄

#127

yes

characteristic

113

C₂×C₄

AGL(1,ℤ/8)

yes

—

114

C₂×C₄

#137

yes

—

115

C₈

—

116

C₈

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

#101

yes

complemented

121

C₂²

—

122

C₂²

—

123

C₂²

#390

yes

characteristic

124

C₂²

#440

yes

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

C₂×#101

yes

—

130

C₄

#468

yes

—

131

C₄

#471

yes

—

132

C₄

#496

yes

characteristic

133

C₄

#497

yes

characteristic

134

C₂

—

135

C₂

#1966

yes

characteristic

136

C₂

#1973

yes

characteristic

137

C₂

#1977

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#390
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

C₂²

C₂²×C₄

C₂×C₄²

C₂³

prime radical

C₂³

C₄×D₄

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
120. C₈	#101	semidirect

H ≅ G/N

type

120. C₈

#101

semidirect

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

subgroup	index
C₂×#498	2
#1997	2
#1966	2
#1567	2
#1549	2
#1546	2
#1451	2

subgroup

index

Subgroups — #9806

Conjugacy classes of subgroups

Complements

Maximal subgroups