Group #123398 — subgroups

Subgroups — #123398

Order 512.

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

normal subgroups only

This group has 144 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	#12510	C₂	1	1	yes	maximal, characteristic
2	#12448	C₂	1	1	yes	maximal, characteristic
3	#12446	C₂	1	1	yes	maximal, characteristic
4	#12054	C₂	1	1	yes	maximal, characteristic
5	#7101	C₂	1	1	yes	maximal, characteristic
6	#7098	C₂	1	1	yes	maximal, characteristic
7	#7097	C₂	1	1	yes	maximal, characteristic
8	#2241	—	2	1	no	—
9	#2241	—	2	1	no	—
10	#2227	—	2	1	no	—
11	#2225	—	2	1	no	—
12	#2183	—	2	1	no	—
13	#1713	C₂²	1	1	yes	characteristic
14	#1713	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	#1364	—	2	1	no	—
21	Q₈⋊C₁₆	—	2	1	no	—
22	C₈×C₁₆	C₂²	1	1	yes	characteristic
23	#500	—	2	1	no	—
24	#500	—	2	1	no	—
25	#500	—	2	1	no	—
26	#500	—	2	1	no	—
27	#494	—	2	1	no	—
28	#494	—	2	1	no	—
29	C₈×Q₈	—	2	1	no	—
30	C₄×Q₁₆	D₄	1	1	yes	characteristic
31	#367	—	4	1	no	—
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	2	no	—
38	#358	—	4	1	no	—
39	#358	—	4	1	no	—
40	#358	—	4	1	no	—
41	#358	—	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	—	4	1	no	—
49	C₂×Q₁₆	—	4	1	no	—
50	C₂×Q₁₆	—	8	1	no	—
51	C₂×Q₁₆	D₈	1	1	yes	characteristic
52	C₄⋊Q₈	—	4	1	no	—
53	C₄⋊Q₈	—	4	1	no	—
54	C₄×Q₈	—	2	1	no	—
55	C₂×C₁₆	—	2	1	no	—
56	C₂×C₁₆	—	4	1	no	—
57	C₂×C₁₆	—	4	1	no	—
58	C₂×C₁₆	—	4	1	no	—
59	C₂×C₁₆	D₈	1	1	yes	characteristic
60	C₂×C₁₆	D₈	1	1	yes	characteristic
61	#107	—	4	1	no	—
62	#107	—	4	1	no	—
63	#107	—	4	1	no	—
64	#107	—	4	1	no	—
65	#107	—	4	1	no	—
66	#107	—	4	1	no	—
67	#107	—	8	1	no	—
68	#107	D₈	1	1	yes	characteristic
69	C₄⋊C₈	—	2	1	no	—
70	C₄⋊C₈	—	4	1	no	—
71	#103	—	2	1	no	—
72	C₄×C₈	—	4	1	no	—
73	C₄×C₈	C₂×D₄	1	1	yes	characteristic
74	C₄×C₈	C₂×D₄	1	1	yes	characteristic
75	C₄×C₈	#41	1	1	yes	characteristic
76	C₂×Q₈	—	2	1	no	—
77	C₂×Q₈	—	8	1	no	—
78	C₂×Q₈	—	8	1	no	—
79	Q₁₆	—	2	1	no	—
80	Q₁₆	—	2	1	no	—
81	Q₁₆	—	8	2	no	—
82	Q₁₆	—	8	1	no	—
83	Q₁₆	—	8	1	no	—
84	Q₁₆	—	16	2	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	4	1	no	—
87	C₂×C₈	C₂×D₈	1	1	yes	characteristic
88	C₂×C₈	C₂×D₈	1	1	yes	characteristic
89	C₂×C₈	#135	1	1	yes	characteristic
90	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
91	C₄⋊C₄	—	2	1	no	—
92	C₄⋊C₄	—	4	1	no	—
93	C₄⋊C₄	—	8	1	no	—
94	C₄⋊C₄	—	8	1	no	—
95	C₄⋊C₄	—	8	1	no	—
96	C₄⋊C₄	—	8	1	no	—
97	C₄²	—	4	1	no	—
98	C₄²	#121	1	1	yes	characteristic
99	C₁₆	—	2	1	no	—
100	C₁₆	—	2	1	no	—
101	C₁₆	—	4	1	no	—
102	C₁₆	—	8	1	no	—
103	C₁₆	—	8	1	no	—
104	C₁₆	—	8	1	no	—
105	Q₈	—	4	1	no	—
106	Q₈	—	4	1	no	—
107	Q₈	—	16	2	no	—
108	Q₈	—	16	1	no	—
109	Q₈	—	16	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	16	1	no	—
113	C₂×C₄	—	16	1	no	—
114	C₂×C₄	#459	1	1	yes	characteristic
115	C₂×C₄	#466	1	1	yes	characteristic
116	C₂×C₄	#472	1	1	yes	characteristic
117	C₈	—	2	2	no	—
118	C₈	—	2	1	no	—
119	C₈	—	2	1	no	—
120	C₈	—	8	1	no	—
121	C₈	#508	1	1	yes	characteristic
122	C₈	#510	1	1	yes	characteristic
123	C₈	#510	1	1	yes	characteristic
124	C₈	#510	1	1	yes	characteristic
125	C₂²	#1667	1	1	yes	characteristic
126	C₄	—	2	1	no	—
127	C₄	—	8	1	no	—
128	C₄	—	8	1	no	—
129	C₄	—	32	1	no	—
130	C₄	—	32	1	no	—
131	C₄	#2211	1	1	yes	characteristic
132	C₄	#2212	1	1	yes	characteristic
133	C₄	#2219	1	1	yes	characteristic
134	C₄	#2222	1	1	yes	characteristic
135	C₂	#12429	1	1	yes	characteristic
136	C₂	#12434	1	1	yes	characteristic
137	C₂	#12438	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12510

C₂

yes

maximal, characteristic

#12448

C₂

yes

maximal, characteristic

#12446

C₂

yes

maximal, characteristic

#12054

C₂

yes

maximal, characteristic

#7101

C₂

yes

maximal, characteristic

#7098

C₂

yes

maximal, characteristic

#7097

C₂

yes

maximal, characteristic

—

—

—

—

—

C₂²

yes

characteristic

#1713

C₂²

yes

characteristic

C₈×Q₁₆

C₂²

yes

characteristic

#1370

C₂²

yes

characteristic

#1370

C₂²

yes

characteristic

#1370

C₂²

yes

characteristic

#1365

—

#1364

—

Q₈⋊C₁₆

—

C₈×C₁₆

C₂²

yes

—

—

—

—

—

—

C₈×Q₈

—

C₄×Q₁₆

D₄

yes

characteristic

#367

—

#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

—

C₂×Q₁₆

—

C₂×Q₁₆

D₈

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

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×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

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

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

#121

yes

characteristic

C₁₆

—

100

C₁₆

—

101

C₁₆

—

102

C₁₆

—

103

C₁₆

—

104

C₁₆

—

105

Q₈

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

109

Q₈

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

#459

yes

characteristic

115

C₂×C₄

#466

yes

characteristic

116

C₂×C₄

#472

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

#508

yes

characteristic

122

C₈

#510

yes

characteristic

123

C₈

#510

yes

characteristic

124

C₈

#510

yes

characteristic

125

C₂²

#1667

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#2211

yes

characteristic

132

C₄

#2212

yes

characteristic

133

C₄

#2219

yes

characteristic

134

C₄

#2222

yes

characteristic

135

C₂

#12429

yes

characteristic

136

C₂

#12434

yes

characteristic

137

C₂

#12438

yes

characteristic

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

subgroup

G/H

center, socle, prime radical

C₂²

#1667

commutator, Frattini, cosocle

C₈²

C₂³

LCS

G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G

LECS

G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

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

subgroup	index
#12510	2
#12448	2
#12446	2
#12054	2
#7101	2
#7098	2
#7097	2

subgroup

index

Subgroups — #123398

Conjugacy classes of subgroups

Maximal subgroups