Group #9604 — subgroups

Subgroups — #9604

Order 256.

194 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 97 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	#1842	C₂	1	1	yes	maximal, characteristic
2	#1767	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×#335	C₂	1	2	yes	maximal
5	#1277	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₈	—	2	1	no	—
7	C₂×C₄⋊C₈	—	2	1	no	—
8	C₂×C₄⋊C₈	C₄	1	2	yes	—
9	C₂×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×#97	—	2	1	no	—
13	C₂×C₄×C₈	C₂²	1	2	yes	—
14	C₄³	C₂²	1	1	yes	characteristic
15	#336	—	2	1	no	—
16	#336	C₄	1	1	yes	characteristic
17	#336	C₄	1	1	yes	characteristic
18	#335	—	2	4	no	—
19	C₂²×C₈	—	2	1	no	—
20	C₂²×C₈	—	2	1	no	—
21	C₂²×C₈	—	2	2	no	—
22	C₂²×C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₈	C₂×C₄	1	2	yes	—
24	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²	D₄	1	1	yes	characteristic
27	C₂×C₄²	D₄	1	1	yes	characteristic
28	C₂×C₄²	Q₈	1	1	yes	characteristic
29	C₂×C₄²	Q₈	1	1	yes	characteristic
30	C₂×C₄²	C₂³	1	1	yes	characteristic
31	C₄⋊C₈	—	2	4	no	—
32	C₄⋊C₈	—	2	2	no	—
33	C₄⋊C₈	—	2	1	no	—
34	C₄⋊C₈	—	2	1	no	—
35	C₄⋊C₈	—	2	2	no	—
36	C₄⋊C₈	—	4	2	no	—
37	C₄⋊C₈	—	4	2	no	—
38	C₄⋊C₈	D₄	1	2	yes	—
39	C₄⋊C₈	Q₈	1	2	yes	—
40	#97	—	4	2	no	—
41	C₄×C₈	—	2	2	no	—
42	C₄×C₈	—	2	2	no	—
43	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈	—	2	2	no	—
51	C₂×C₈	—	2	2	no	—
52	C₂×C₈	—	2	2	no	—
53	C₂×C₈	—	2	2	no	—
54	C₂×C₈	—	2	2	no	—
55	C₂×C₈	—	2	2	no	—
56	C₂×C₈	—	2	2	no	—
57	C₂×C₈	—	4	1	no	—
58	C₂×C₈	—	4	1	no	—
59	C₂×C₈	—	4	1	no	—
60	C₂×C₈	—	4	1	no	—
61	C₂×C₈	—	4	2	no	—
62	C₂×C₈	—	4	2	no	—
63	C₂×C₈	—	4	2	no	—
64	C₂×C₈	M₁₆	1	4	yes	—
65	C₄²	—	2	2	no	—
66	C₄²	—	2	2	no	—
67	C₄²	—	2	2	no	—
68	C₄²	—	2	1	no	—
69	C₄²	—	2	1	no	—
70	C₄²	—	4	1	no	—
71	C₄²	D₈	1	2	yes	—
72	C₄²	Q₁₆	1	2	yes	—
73	C₄²	C₂×D₄	1	1	yes	characteristic
74	C₄²	C₂×Q₈	1	1	yes	characteristic
75	C₄²	#41	1	1	yes	characteristic
76	C₄²	#41	1	1	yes	characteristic
77	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	2	1	no	—
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	1	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	#107	1	2	yes	—
90	C₂×C₄	#107	1	2	yes	—
91	C₂×C₄	#108	1	2	yes	—
92	C₂×C₄	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
94	C₂×C₄	#121	1	1	yes	characteristic
95	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄	#125	1	1	yes	characteristic
97	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
98	C₂×C₄	C₂×M₁₆	1	2	yes	—
99	C₂×C₄	#131	1	2	yes	—
100	C₂×C₄	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
102	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂×C₄	#137	1	1	yes	characteristic
104	C₈	—	2	4	no	—
105	C₈	—	4	2	no	—
106	C₈	—	4	2	no	—
107	C₈	—	4	2	no	—
108	C₈	—	8	2	no	—
109	C₂²	#389	1	1	yes	characteristic
110	C₂²	#423	1	1	yes	characteristic
111	C₂²	#424	1	1	yes	characteristic
112	C₂²	C₂×#107	1	1	yes	characteristic
113	C₂²	#428	1	1	yes	characteristic
114	C₂²	C₂×#108	1	1	yes	characteristic
115	C₂²	#430	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	2	1	no	—
120	C₄	—	2	1	no	—
121	C₄	—	4	1	no	—
122	C₄	#435	1	2	yes	—
123	C₄	#446	1	2	yes	—
124	C₄	#459	1	1	yes	characteristic
125	C₄	#462	1	1	yes	characteristic
126	C₄	#474	1	1	yes	characteristic
127	C₄	#477	1	1	yes	characteristic
128	C₂	#1567	1	1	yes	characteristic
129	C₂	#1569	1	1	yes	characteristic
130	C₂	#1571	1	1	yes	characteristic
131	C₂	#1573	1	1	yes	characteristic
132	C₂	#1940	1	1	yes	characteristic
133	C₂	#1947	1	1	yes	characteristic
134	C₂	#1950	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1842

C₂

yes

maximal, characteristic

#1767

C₂

yes

maximal, characteristic

C₄×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂×#335

C₂

yes

maximal

#1277

C₂

yes

maximal

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₄

yes

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×#97

—

C₂×C₄×C₈

C₂²

yes

—

C₄³

C₂²

yes

characteristic

#336

—

#336

C₄

yes

characteristic

#336

C₄

yes

characteristic

#335

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

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

Q₈

yes

characteristic

C₂×C₄²

Q₈

yes

characteristic

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₄

yes

—

C₄⋊C₈

Q₈

yes

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

C₄⋊C₄

yes

characteristic

C₂²×C₄

C₄⋊C₄

yes

characteristic

C₂²×C₄

C₄⋊C₄

yes

characteristic

C₂²×C₄

C₄⋊C₄

yes

characteristic

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

M₁₆

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

D₈

yes

—

C₄²

Q₁₆

yes

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×Q₈

yes

C₄²

yes

C₄²

yes

C₂³

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#107

yes

—

C₂×C₄

#107

yes

—

C₂×C₄

#108

yes

—

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×Q₈

yes

characteristic

C₂×C₄

#121

yes

characteristic

C₂×C₄

C₂²⋊Q₈

yes

characteristic

C₂×C₄

#125

yes

characteristic

C₂×C₄

C₄⋊Q₈

yes

characteristic

C₂×C₄

C₂×M₁₆

yes

—

C₂×C₄

#131

yes

—

100

C₂×C₄

C₂×D₈

yes

characteristic

101

C₂×C₄

C₂×Q₁₆

yes

characteristic

102

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

103

C₂×C₄

#137

yes

characteristic

104

C₈

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

—

109

C₂²

#389

yes

characteristic

110

C₂²

#423

yes

characteristic

111

C₂²

#424

yes

characteristic

112

C₂²

C₂×#107

yes

characteristic

113

C₂²

#428

yes

characteristic

114

C₂²

C₂×#108

yes

characteristic

115

C₂²

#430

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

#435

yes

—

123

C₄

#446

yes

—

124

C₄

#459

yes

characteristic

125

C₄

#462

yes

characteristic

126

C₄

#474

yes

characteristic

127

C₄

#477

yes

characteristic

128

C₂

#1567

yes

characteristic

129

C₂

#1569

yes

characteristic

130

C₂

#1571

yes

characteristic

131

C₂

#1573

yes

characteristic

132

C₂

#1940

yes

characteristic

133

C₂

#1947

yes

characteristic

134

C₂

#1950

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle, prime radical	C₂³	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂²×C₄

C₂×D₄

commutator

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

socle, prime radical

C₂³

C₂×C₄⋊C₄

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

subgroup	index
#1842	2
#1767	2
C₄×C₄⋊C₈	2
C₂×#335	2
#1277	2

subgroup

index

C₄×C₄⋊C₈

Subgroups — #9604

Conjugacy classes of subgroups

Maximal subgroups