Group #9898 — subgroups

Subgroups — #9898

Order 256.

269 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 269 conjugacy classes of subgroups — too many to draw. Only its 119 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	#2868	C₂	1	1	yes	maximal, characteristic
2	#1842	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#1296	C₂	1	4	yes	maximal
5	C₂×C₄⋊Q₈	—	2	1	no	—
6	C₂×C₄⋊Q₈	—	2	1	no	—
7	C₂×C₄⋊Q₈	C₄	1	2	yes	—
8	C₂×C₄⋊C₈	—	2	1	no	—
9	C₂×C₄⋊C₈	—	2	1	no	—
10	C₂×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄×C₈	—	2	1	no	—
13	#384	—	2	1	no	—
14	#384	—	2	1	no	—
15	#384	C₄	1	2	yes	—
16	#384	C₂²	1	2	yes	—
17	C₄³	C₂²	1	1	yes	characteristic
18	#336	—	2	1	no	—
19	C₂²×Q₈	—	2	2	no	—
20	C₂²×C₈	—	2	2	no	—
21	C₂²×C₈	—	2	2	no	—
22	C₄⋊Q₈	—	4	4	no	—
23	C₄⋊Q₈	—	4	2	no	—
24	C₄⋊Q₈	—	4	2	no	—
25	C₄⋊Q₈	—	4	2	no	—
26	C₄⋊Q₈	—	4	2	no	—
27	C₂×C₄⋊C₄	—	2	2	no	—
28	C₂×C₄⋊C₄	—	2	2	no	—
29	C₂×C₄⋊C₄	—	2	2	no	—
30	C₂×C₄⋊C₄	—	2	2	no	—
31	C₂×C₄⋊C₄	—	2	2	no	—
32	C₂×C₄⋊C₄	C₂×C₄	1	4	yes	—
33	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²	D₄	1	1	yes	characteristic
36	C₂×C₄²	D₄	1	1	yes	characteristic
37	C₂×C₄²	D₄	1	1	yes	characteristic
38	C₂×C₄²	D₄	1	1	yes	characteristic
39	C₂×C₄²	C₂³	1	1	yes	characteristic
40	C₄⋊C₈	—	2	2	no	—
41	C₄⋊C₈	—	2	4	no	—
42	C₄⋊C₈	—	4	2	no	—
43	C₄⋊C₈	—	4	2	no	—
44	C₄⋊C₈	Q₈	1	4	yes	complemented
45	C₄×C₈	—	4	2	no	—
46	C₂×Q₈	—	4	4	no	—
47	C₂×Q₈	—	4	4	no	—
48	C₂×Q₈	—	4	4	no	—
49	C₂²×C₄	—	2	4	no	—
50	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈	—	2	4	no	—
58	C₂×C₈	—	4	2	no	—
59	C₂×C₈	—	4	2	no	—
60	C₂×C₈	—	4	2	no	—
61	C₂×C₈	—	4	2	no	—
62	C₂×C₈	—	4	2	no	—
63	C₄⋊C₄	—	4	4	no	—
64	C₄⋊C₄	—	4	4	no	—
65	C₄⋊C₄	—	4	4	no	—
66	C₄⋊C₄	—	4	4	no	—
67	C₄⋊C₄	—	4	2	no	—
68	C₄⋊C₄	—	4	2	no	—
69	C₄⋊C₄	—	4	2	no	—
70	C₄⋊C₄	—	4	2	no	—
71	C₄²	—	2	2	no	—
72	C₄²	—	2	2	no	—
73	C₄²	—	2	2	no	—
74	C₄²	D₈	1	2	yes	—
75	C₄²	D₈	1	2	yes	—
76	C₄²	SD₁₆	1	2	yes	—
77	C₄²	SD₁₆	1	2	yes	—
78	C₄²	SD₁₆	1	2	yes	—
79	C₄²	Q₁₆	1	2	yes	—
80	C₄²	C₂×Q₈	1	2	yes	—
81	C₄²	#41	1	2	yes	—
82	C₂³	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	Q₈	—	8	8	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	4	4	no	—
91	C₂×C₄	—	4	4	no	—
92	C₂×C₄	—	4	4	no	—
93	C₂×C₄	#100	1	2	yes	—
94	C₂×C₄	#101	1	2	yes	—
95	C₂×C₄	#102	1	2	yes	—
96	C₂×C₄	#102	1	2	yes	—
97	C₂×C₄	#102	1	2	yes	—
98	C₂×C₄	#102	1	2	yes	—
99	C₂×C₄	#103	1	2	yes	—
100	C₂×C₄	#103	1	2	yes	—
101	C₂×C₄	C₄×Q₈	1	2	yes	—
102	C₂×C₄	C₂²⋊Q₈	1	2	yes	—
103	C₂×C₄	#124	1	1	yes	characteristic
104	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
105	C₂×C₄	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
111	C₂×C₄	#137	1	2	yes	—
112	C₈	—	4	4	no	—
113	C₈	—	8	2	no	—
114	C₂²	#391	1	1	yes	characteristic
115	C₂²	C₂×#100	1	1	yes	characteristic
116	C₂²	C₂×#101	1	1	yes	characteristic
117	C₂²	C₂×#102	1	1	yes	characteristic
118	C₂²	C₂×#102	1	1	yes	characteristic
119	C₂²	C₂×#103	1	1	yes	characteristic
120	C₂²	#419	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	8	4	no	—
125	C₄	#331	1	4	yes	—
126	C₄	#332	1	4	yes	—
127	C₄	#474	1	2	yes	—
128	C₄	#475	1	2	yes	—
129	C₄	#486	1	1	yes	characteristic
130	C₄	#487	1	1	yes	characteristic
131	C₄	#498	1	1	yes	characteristic
132	C₄	#500	1	1	yes	characteristic
133	C₂	C₂×#331	1	1	yes	characteristic
134	C₂	C₂×#332	1	1	yes	characteristic
135	C₂	#1546	1	1	yes	characteristic
136	C₂	#1548	1	1	yes	characteristic
137	C₂	#1988	1	1	yes	characteristic
138	C₂	#1990	1	1	yes	characteristic
139	C₂	#1997	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2868

C₂

yes

maximal, characteristic

#1842

C₂

yes

maximal, characteristic

C₄×C₄⋊C₈

C₂

yes

maximal, characteristic

#1296

C₂

yes

maximal

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

C₄

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄×C₈

—

#384

—

#384

—

#384

C₄

yes

—

#384

C₂²

yes

—

C₄³

C₂²

yes

characteristic

#336

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

Q₈

yes

complemented

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

D₈

yes

—

C₄²

D₈

yes

—

C₄²

SD₁₆

yes

—

C₄²

SD₁₆

yes

—

C₄²

SD₁₆

yes

—

C₄²

Q₁₆

yes

—

C₄²

C₂×Q₈

yes

—

C₄²

#41

yes

—

C₂³

C₂×C₂²⋊C₄

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#100

yes

—

C₂×C₄

#101

yes

—

C₂×C₄

#102

yes

—

C₂×C₄

#102

yes

—

C₂×C₄

#102

yes

—

C₂×C₄

#102

yes

—

C₂×C₄

#103

yes

—

100

C₂×C₄

#103

yes

—

101

C₂×C₄

C₄×Q₈

yes

—

102

C₂×C₄

C₂²⋊Q₈

yes

—

103

C₂×C₄

#124

yes

characteristic

104

C₂×C₄

C₄⋊Q₈

yes

characteristic

105

C₂×C₄

C₂×D₈

yes

characteristic

106

C₂×C₄

C₂×D₈

yes

characteristic

107

C₂×C₄

C₂×SD₁₆

yes

characteristic

108

C₂×C₄

C₂×SD₁₆

yes

characteristic

109

C₂×C₄

C₂×SD₁₆

yes

characteristic

110

C₂×C₄

C₂×Q₁₆

yes

characteristic

111

C₂×C₄

#137

yes

—

112

C₈

—

113

C₈

—

114

C₂²

#391

yes

characteristic

115

C₂²

C₂×#100

yes

characteristic

116

C₂²

C₂×#101

yes

characteristic

117

C₂²

C₂×#102

yes

characteristic

118

C₂²

C₂×#102

yes

characteristic

119

C₂²

C₂×#103

yes

characteristic

120

C₂²

#419

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#331

yes

—

126

C₄

#332

yes

—

127

C₄

#474

yes

—

128

C₄

#475

yes

—

129

C₄

#486

yes

characteristic

130

C₄

#487

yes

characteristic

131

C₄

#498

yes

characteristic

132

C₄

#500

yes

characteristic

133

C₂

C₂×#331

yes

characteristic

134

C₂

C₂×#332

yes

characteristic

135

C₂

#1546

yes

characteristic

136

C₂

#1548

yes

characteristic

137

C₂

#1988

yes

characteristic

138

C₂

#1990

yes

characteristic

139

C₂

#1997

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂×C₂²⋊C₄

commutator

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
44. C₄⋊C₈	Q₈	semidirect

H ≅ G/N

type

44. C₄⋊C₈

Q₈

semidirect

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

subgroup	index
#2868	2
#1842	2
C₄×C₄⋊C₈	2
#1296	2

subgroup

index

#2868

#1842

C₄×C₄⋊C₈

#1296

Subgroups — #9898

Conjugacy classes of subgroups

Complements

Maximal subgroups