Group #19664 — subgroups

Subgroups — #19664

Order 256.

345 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 139 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	#3230	C₂	1	2	yes	maximal
2	#3209	C₂	1	2	yes	maximal
3	#3160	C₂	1	1	yes	maximal, characteristic
4	C₂×#500	C₂	1	1	yes	maximal, characteristic
5	#1723	C₂	1	4	yes	maximal, complemented
6	#1713	C₂	1	2	yes	maximal, complemented
7	#1710	C₂	1	2	yes	maximal, complemented
8	#1450	C₂	1	1	yes	maximal, characteristic
9	#533	C₂²	1	2	yes	—
10	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
11	#500	—	2	2	no	—
12	#500	C₂²	1	2	yes	—
13	#500	C₂²	1	2	yes	—
14	#500	C₂²	1	1	yes	characteristic
15	#500	C₂²	1	1	yes	characteristic
16	#500	C₂²	1	4	yes	—
17	#499	—	2	2	no	—
18	#498	C₂²	1	2	yes	—
19	#494	C₂²	1	4	yes	—
20	#487	C₂²	1	4	yes	—
21	#487	C₂²	1	4	yes	—
22	#484	—	2	4	no	—
23	#477	—	2	4	no	—
24	#462	—	2	4	no	—
25	#451	—	2	4	no	—
26	#431	C₂²	1	1	yes	characteristic
27	#431	C₂²	1	1	yes	characteristic
28	#427	—	2	2	no	—
29	C₂×#107	—	2	2	no	—
30	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
31	#322	C₂²	1	2	yes	—
32	C₈²	C₂²	1	2	yes	—
33	C₂²×Q₈	—	2	2	no	—
34	C₂×Q₁₆	—	2	8	no	—
35	C₂²×C₈	D₄	1	2	yes	—
36	C₄⋊Q₈	—	2	2	no	—
37	C₄⋊Q₈	—	2	2	no	—
38	C₄⋊Q₈	—	2	2	no	—
39	C₄⋊Q₈	C₂³	1	4	yes	—
40	C₄⋊Q₈	C₂³	1	2	yes	—
41	C₄⋊Q₈	C₂³	1	2	yes	—
42	#125	—	2	2	no	—
43	C₂²⋊Q₈	—	2	2	no	—
44	C₂²⋊Q₈	—	2	2	no	—
45	C₄×Q₈	—	2	2	no	—
46	C₄×Q₈	—	2	2	no	—
47	#117	—	2	2	no	—
48	C₂×C₄⋊C₄	—	2	2	no	—
49	C₂×C₄⋊C₄	—	2	2	no	—
50	C₂×C₄²	C₂³	1	1	yes	characteristic
51	#107	—	2	4	no	—
52	#107	—	2	4	no	—
53	#107	—	2	2	no	—
54	#107	—	2	2	no	—
55	#107	—	2	8	no	—
56	#106	—	2	4	no	—
57	C₄⋊C₈	D₄	1	2	yes	—
58	C₄⋊C₈	D₄	1	2	yes	—
59	#103	—	2	8	no	—
60	#103	—	2	8	no	—
61	C₂²⋊C₈	D₄	1	2	yes	—
62	C₂²⋊C₈	D₄	1	2	yes	—
63	C₄×C₈	D₄	1	2	yes	—
64	C₄×C₈	C₂³	1	1	yes	characteristic
65	C₄×C₈	C₂³	1	2	yes	—
66	C₄×C₈	C₂³	1	2	yes	—
67	C₄×C₈	C₂³	1	1	yes	characteristic
68	C₂×Q₈	—	2	2	no	—
69	C₂×Q₈	—	2	4	no	—
70	C₂×Q₈	—	2	2	no	—
71	C₂×Q₈	—	4	2	no	—
72	C₂×Q₈	—	8	4	no	—
73	C₂²×C₄	—	4	2	no	—
74	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
77	Q₁₆	—	8	8	no	—
78	C₂×C₈	Q₁₆	1	8	yes	complemented
79	C₂×C₈	C₂×D₄	1	4	yes	—
80	C₂×C₈	C₂×D₄	1	4	yes	—
81	C₂×C₈	C₂×D₄	1	2	yes	—
82	C₂×C₈	C₂×D₄	1	2	yes	—
83	C₄⋊C₄	—	2	2	no	—
84	C₄⋊C₄	—	2	4	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	2	4	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	4	2	no	—
90	C₄⋊C₄	—	4	2	no	—
91	C₄⋊C₄	—	4	2	no	—
92	C₄⋊C₄	—	4	2	no	—
93	C₂²⋊C₄	—	4	2	no	—
94	C₄²	—	4	2	no	—
95	C₄²	C₂×D₄	1	1	yes	characteristic
96	C₄²	C₂×D₄	1	1	yes	characteristic
97	C₄²	C₂×D₄	1	1	yes	characteristic
98	C₄²	C₂⁴	1	1	yes	characteristic
99	C₂³	#127	1	1	yes	characteristic
100	Q₈	—	4	4	no	—
101	Q₈	—	4	8	no	—
102	Q₈	—	4	4	no	—
103	Q₈	—	8	4	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	4	no	—
109	C₂×C₄	—	8	4	no	—
110	C₂×C₄	#127	1	1	yes	characteristic
111	C₂×C₄	#127	1	1	yes	characteristic
112	C₂×C₄	#127	1	1	yes	characteristic
113	C₂×C₄	C₂×Q₁₆	1	4	yes	—
114	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
117	C₈	—	2	4	no	—
118	C₈	—	2	4	no	—
119	C₈	C₂×Q₁₆	1	4	yes	—
120	C₈	C₂×Q₁₆	1	4	yes	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	#494	1	2	yes	—
124	C₂²	C₂×#127	1	1	yes	characteristic
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	8	4	no	—
128	C₄	—	8	2	no	—
129	C₄	—	8	2	no	—
130	C₄	#494	1	2	yes	—
131	C₄	C₂²×Q₁₆	1	2	yes	—
132	C₄	#576	1	2	yes	—
133	C₄	#578	1	2	yes	—
134	C₂	—	2	2	no	—
135	C₂	C₂×#494	1	1	yes	characteristic
136	C₂	#3155	1	1	yes	characteristic
137	C₂	#3156	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3230

C₂

yes

maximal

#3209

C₂

yes

maximal

#3160

C₂

yes

maximal, characteristic

C₂×#500

C₂

yes

maximal, characteristic

#1723

C₂

yes

maximal, complemented

#1713

C₂

yes

maximal, complemented

#1710

C₂

yes

maximal, complemented

#1450

C₂

yes

maximal, characteristic

#533

C₂²

yes

—

C₂×C₄⋊Q₈

C₂²

yes

—

#500

—

#500

C₂²

yes

—

#500

C₂²

yes

—

#500

C₂²

yes

characteristic

#500

C₂²

yes

characteristic

#500

C₂²

yes

—

#499

—

#498

C₂²

yes

—

#494

C₂²

yes

—

#487

C₂²

yes

—

#487

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

#431

C₂²

yes

characteristic

#427

—

C₂×#107

—

C₂×C₄×C₈

C₂²

yes

characteristic

#322

C₂²

yes

—

C₈²

C₂²

yes

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂³

yes

—

C₄⋊Q₈

C₂³

yes

—

C₄⋊Q₈

C₂³

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³

yes

—

—

—

—

—

—

C₄⋊C₈

D₄

yes

—

C₄⋊C₈

D₄

yes

—

#103

—

#103

—

C₂²⋊C₈

D₄

yes

—

C₂²⋊C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

C₂×C₈

Q₁₆

yes

complemented

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂⁴

yes

characteristic

C₂³

#127

yes

characteristic

100

Q₈

—

101

Q₈

—

102

Q₈

—

103

Q₈

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

#127

yes

characteristic

111

C₂×C₄

#127

yes

characteristic

112

C₂×C₄

#127

yes

characteristic

113

C₂×C₄

C₂×Q₁₆

yes

—

114

C₂×C₄

C₂²×D₄

yes

characteristic

115

C₂×C₄

C₂²×D₄

yes

characteristic

116

C₂×C₄

C₂²×D₄

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

C₂×Q₁₆

yes

—

120

C₈

C₂×Q₁₆

yes

—

121

C₂²

—

122

C₂²

—

123

C₂²

#494

yes

—

124

C₂²

C₂×#127

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#494

yes

—

131

C₄

C₂²×Q₁₆

yes

—

132

C₄

#576

yes

—

133

C₄

#578

yes

—

134

C₂

—

135

C₂

C₂×#494

yes

characteristic

136

C₂

#3155

yes

characteristic

137

C₂

#3156

yes

characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂³	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂×#127

commutator, Frattini, cosocle

C₄²

C₂⁴

C₂³

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
5. #1723	C₂	semidirect
6. #1713	C₂	semidirect
7. #1710	C₂	semidirect
78. C₂×C₈	Q₁₆	semidirect

H ≅ G/N

type

5. #1723

C₂

semidirect

6. #1713

C₂

semidirect

7. #1710

C₂

semidirect

78. C₂×C₈

Q₁₆

semidirect

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

subgroup	index
#3230	2
#3209	2
#3160	2
C₂×#500	2
#1723	2
#1713	2
#1710	2
#1450	2

subgroup

index

Subgroups — #19664

Conjugacy classes of subgroups

Complements

Maximal subgroups