Group #123400 — subgroups

Subgroups — #123400

Order 512.

158 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 48 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	#12509	C₂	1	1	yes	maximal, characteristic
2	#12448	C₂	1	1	yes	maximal, characteristic, complemented
3	#12425	C₂	1	1	yes	maximal, characteristic
4	#12054	C₂	1	1	yes	maximal, characteristic, complemented
5	#7100	C₂	1	1	yes	maximal, characteristic, complemented
6	#7098	C₂	1	1	yes	maximal, characteristic, complemented
7	#7096	C₂	1	1	yes	maximal, characteristic
8	#2240	—	2	1	no	—
9	#2240	—	2	1	no	—
10	#2227	—	2	1	no	—
11	#2207	—	2	1	no	—
12	#2183	—	2	1	no	—
13	#1713	C₂²	1	1	yes	characteristic
14	#1712	C₂²	1	1	yes	characteristic
15	C₈×Q₁₆	C₂²	1	1	yes	characteristic
16	#1370	C₂²	1	1	yes	characteristic
17	#1370	C₂²	1	1	yes	characteristic
18	#1370	C₂²	1	1	yes	characteristic
19	#1365	—	2	1	no	—
20	#1361	—	2	1	no	—
21	Q₈⋊C₁₆	—	2	1	no	—
22	C₈×C₁₆	C₂²	1	1	yes	characteristic
23	C₂×SD₃₂	—	4	1	no	—
24	#500	—	2	1	no	—
25	#500	—	2	1	no	—
26	#494	—	2	1	no	—
27	#493	—	2	1	no	—
28	#493	—	2	1	no	—
29	#493	—	2	1	no	—
30	C₈×Q₈	—	2	1	no	—
31	C₄×Q₁₆	D₄	1	1	yes	characteristic
32	#366	—	4	1	no	—
33	C₄⋊C₁₆	—	2	1	no	—
34	C₄⋊C₁₆	—	2	1	no	—
35	C₄⋊C₁₆	—	2	1	no	—
36	C₄⋊C₁₆	—	4	1	no	—
37	#358	—	4	1	no	—
38	#357	—	4	2	no	—
39	#357	—	4	1	no	—
40	#357	—	4	1	no	—
41	#357	—	4	1	no	—
42	#346	—	4	1	no	—
43	C₄×C₁₆	D₄	1	1	yes	characteristic
44	C₄×C₁₆	D₄	1	1	yes	characteristic
45	#335	D₄	1	1	yes	characteristic
46	Q₈⋊C₈	—	2	1	no	—
47	C₈²	C₂³	1	1	yes	characteristic
48	C₂×Q₁₆	—	8	1	no	—
49	C₂×Q₁₆	D₈	1	1	yes	characteristic
50	C₂×D₈	—	4	1	no	—
51	C₂×D₈	—	4	1	no	—
52	C₂×D₈	—	4	1	no	—
53	C₂×D₈	—	4	1	no	—
54	C₂×D₈	—	8	1	no	—
55	C₄⋊Q₈	—	4	1	no	—
56	#127	—	4	1	no	—
57	C₄×Q₈	—	2	1	no	—
58	SD₃₂	—	8	2	no	—
59	C₂×C₁₆	—	2	1	no	—
60	C₂×C₁₆	—	4	1	no	—
61	C₂×C₁₆	—	4	1	no	—
62	C₂×C₁₆	—	4	1	no	—
63	C₂×C₁₆	D₈	1	1	yes	characteristic
64	C₂×C₁₆	D₈	1	1	yes	characteristic
65	#107	—	4	1	no	—
66	#107	—	4	1	no	—
67	#107	—	4	1	no	—
68	#107	—	4	1	no	—
69	#107	D₈	1	1	yes	characteristic
70	C₄⋊C₈	—	2	1	no	—
71	C₄⋊C₈	—	4	1	no	—
72	#103	—	2	1	no	—
73	C₄×C₈	—	4	1	no	—
74	C₄×C₈	C₂×D₄	1	1	yes	characteristic
75	C₄×C₈	C₂×D₄	1	1	yes	characteristic
76	C₄×C₈	#41	1	1	yes	characteristic
77	C₂×Q₈	—	2	1	no	—
78	C₂×Q₈	—	8	1	no	—
79	C₂×D₄	—	8	1	no	—
80	C₂×D₄	—	8	1	no	—
81	C₂×D₄	—	8	1	no	—
82	Q₁₆	—	2	1	no	—
83	Q₁₆	—	2	1	no	—
84	Q₁₆	—	16	2	no	—
85	D₈	—	8	2	no	—
86	D₈	—	8	1	no	—
87	D₈	—	8	1	no	—
88	D₈	—	8	1	no	—
89	D₈	—	8	1	no	—
90	D₈	—	16	1	no	—
91	D₈	—	16	2	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	4	1	no	—
94	C₂×C₈	C₂×D₈	1	1	yes	characteristic
95	C₂×C₈	C₂×D₈	1	1	yes	characteristic
96	C₂×C₈	#135	1	1	yes	characteristic
97	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
98	C₄⋊C₄	—	2	1	no	—
99	C₄⋊C₄	—	4	1	no	—
100	C₄⋊C₄	—	8	1	no	—
101	C₄⋊C₄	—	8	1	no	—
102	C₄²	—	4	1	no	—
103	C₄²	#121	1	1	yes	characteristic
104	C₁₆	—	2	1	no	—
105	C₁₆	—	2	1	no	—
106	C₁₆	—	4	1	no	—
107	C₁₆	—	8	1	no	—
108	C₁₆	—	8	1	no	—
109	C₁₆	—	8	1	no	—
110	C₂³	—	16	1	no	—
111	Q₈	—	4	1	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	16	2	no	—
114	D₄	—	16	2	no	—
115	D₄	—	16	1	no	—
116	D₄	—	16	1	no	—
117	D₄	—	16	1	no	—
118	D₄	—	16	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	16	1	no	—
122	C₂×C₄	#459	1	1	yes	characteristic
123	C₂×C₄	#466	1	1	yes	characteristic
124	C₂×C₄	#472	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	2	1	no	—
127	C₈	—	2	1	no	—
128	C₈	—	8	1	no	—
129	C₈	#508	1	1	yes	characteristic
130	C₈	#509	1	1	yes	characteristic
131	C₈	#510	1	1	yes	characteristic
132	C₈	#510	1	1	yes	characteristic
133	C₂²	—	32	1	no	—
134	C₂²	—	32	1	no	—
135	C₂²	—	32	1	no	—
136	C₂²	#1667	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	8	1	no	—
139	C₄	—	8	1	no	—
140	C₄	—	32	1	no	—
141	C₄	#2211	1	1	yes	characteristic
142	C₄	#2212	1	1	yes	characteristic
143	C₄	#2220	1	1	yes	characteristic
144	C₄	#2223	1	1	yes	characteristic
145	C₂	—	64	1	no	—
146	C₂	#12429	1	1	yes	characteristic
147	C₂	#12435	1	1	yes	characteristic
148	C₂	#12436	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12509

C₂

yes

maximal, characteristic

#12448

C₂

yes

maximal, characteristic, complemented

#12425

C₂

yes

maximal, characteristic

#12054

C₂

yes

maximal, characteristic, complemented

#7100

C₂

yes

maximal, characteristic, complemented

#7098

C₂

yes

maximal, characteristic, complemented

#7096

C₂

yes

maximal, characteristic

—

—

—

—

—

C₂²

yes

characteristic

#1712

C₂²

yes

characteristic

C₈×Q₁₆

C₂²

yes

characteristic

#1370

C₂²

yes

characteristic

#1370

C₂²

yes

characteristic

#1370

C₂²

yes

characteristic

#1365

—

#1361

—

Q₈⋊C₁₆

—

C₈×C₁₆

C₂²

yes

characteristic

C₂×SD₃₂

—

—

—

—

—

—

—

C₈×Q₈

—

C₄×Q₁₆

D₄

yes

characteristic

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

—

—

—

—

—

—

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

#335

D₄

yes

characteristic

Q₈⋊C₈

—

C₈²

C₂³

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

D₈

yes

characteristic

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

C₄×Q₈

—

SD₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₈

yes

characteristic

C₂×C₁₆

D₈

yes

—

—

—

—

D₈

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

D₈

—

D₈

—

D₈

—

D₈

—

D₈

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₈

yes

characteristic

C₂×C₈

C₂×D₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄²

—

103

C₄²

#121

yes

characteristic

104

C₁₆

—

105

C₁₆

—

106

C₁₆

—

107

C₁₆

—

108

C₁₆

—

109

C₁₆

—

110

C₂³

—

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

D₄

—

115

D₄

—

116

D₄

—

117

D₄

—

118

D₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

#459

yes

characteristic

123

C₂×C₄

#466

yes

characteristic

124

C₂×C₄

#472

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

#508

yes

characteristic

130

C₈

#509

yes

characteristic

131

C₈

#510

yes

characteristic

132

C₈

#510

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

#1667

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#2211

yes

characteristic

142

C₄

#2212

yes

characteristic

143

C₄

#2220

yes

characteristic

144

C₄

#2223

yes

characteristic

145

C₂

—

146

C₂

#12429

yes

characteristic

147

C₂

#12435

yes

characteristic

148

C₂

#12436

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1667
commutator, Frattini, cosocle	C₈²	C₂³
prime radical	#1712	C₂²
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

#1667

commutator, Frattini, cosocle

C₈²

C₂³

prime radical

#1712

C₂²

LCS

G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G

LECS

G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
2. #12448	C₂	semidirect
4. #12054	C₂	semidirect
5. #7100	C₂	semidirect
6. #7098	C₂	semidirect

H ≅ G/N

type

2. #12448

C₂

semidirect

4. #12054

C₂

semidirect

5. #7100

C₂

semidirect

6. #7098

C₂

semidirect

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

subgroup	index
#12509	2
#12448	2
#12425	2
#12054	2
#7100	2
#7098	2
#7096	2

subgroup

index

Subgroups — #123400

Conjugacy classes of subgroups

Complements

Maximal subgroups