Group #20006 — subgroups

Subgroups — #20006

Order 256.

293 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 293 conjugacy classes of subgroups — too many to draw. Only its 111 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	#3380	C₂	1	2	yes	maximal
2	#3338	C₂	1	2	yes	maximal, complemented
3	#1713	C₂	1	1	yes	maximal, characteristic, complemented
4	#1676	C₂	1	2	yes	maximal, complemented
5	#1675	C₂	1	2	yes	maximal, complemented
6	#1632	C₂	1	4	yes	maximal, complemented
7	C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₈	C₂	1	1	yes	maximal, characteristic
9	#554	C₂²	1	4	yes	—
10	#500	C₂²	1	2	yes	—
11	#500	C₂²	1	2	yes	—
12	#499	—	2	2	no	—
13	#494	C₂²	1	2	yes	—
14	#484	—	2	4	no	—
15	#478	—	2	4	no	—
16	#477	C₂²	1	4	yes	—
17	#476	C₂²	1	4	yes	—
18	#467	—	2	2	no	—
19	C₈×Q₈	C₂²	1	2	yes	—
20	C₄×Q₁₆	C₂²	1	1	yes	characteristic
21	C₄×D₈	—	2	2	no	—
22	C₄×D₈	C₂²	1	1	yes	characteristic
23	C₈×D₄	C₂²	1	2	yes	—
24	C₂×#107	—	2	2	no	—
25	#335	C₂²	1	1	yes	characteristic
26	#335	C₂²	1	1	yes	characteristic
27	#333	C₂²	1	4	yes	—
28	Q₈⋊C₈	C₂²	1	2	yes	—
29	D₄⋊C₈	C₂²	1	2	yes	—
30	C₈²	C₂²	1	1	yes	characteristic
31	C₂×Q₁₆	—	4	2	no	—
32	C₂×Q₁₆	D₄	1	1	yes	characteristic
33	C₂×D₈	D₄	1	1	yes	characteristic
34	C₂²×C₈	—	2	2	no	—
35	C₄⋊Q₈	C₂³	1	4	yes	—
36	#125	—	2	4	no	—
37	C₂²⋊Q₈	—	2	4	no	—
38	C₂²⋊Q₈	—	4	4	no	—
39	C₄×Q₈	C₂³	1	2	yes	—
40	C₄×D₄	—	2	4	no	—
41	C₄×D₄	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄	—	2	4	no	—
43	#107	—	2	2	no	—
44	#107	—	2	2	no	—
45	#107	—	2	2	no	—
46	#107	—	2	2	no	—
47	#107	—	2	4	no	—
48	#107	D₄	1	1	yes	characteristic
49	#107	D₄	1	1	yes	characteristic
50	#106	—	2	4	no	—
51	C₄⋊C₈	—	2	2	no	—
52	C₄⋊C₈	C₂³	1	2	yes	—
53	C₄⋊C₈	C₂³	1	2	yes	—
54	#103	—	2	4	no	—
55	#103	D₄	1	2	yes	—
56	#102	—	2	4	no	—
57	#102	D₄	1	2	yes	—
58	C₂²⋊C₈	—	2	2	no	—
59	C₄×C₈	—	2	2	no	—
60	C₄×C₈	C₂³	1	1	yes	characteristic
61	C₄×C₈	C₂³	1	1	yes	characteristic
62	C₄×C₈	C₂³	1	1	yes	characteristic
63	C₂×Q₈	—	2	4	no	—
64	C₂×Q₈	C₂×D₄	1	2	yes	—
65	C₂×D₄	C₂×D₄	1	2	yes	—
66	C₂²×C₄	—	2	2	no	—
67	C₂²×C₄	—	4	4	no	—
68	Q₁₆	—	2	2	no	—
69	Q₁₆	—	8	4	no	—
70	D₈	Q₁₆	1	4	yes	complemented
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	4	4	no	—
75	C₂×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈	C₂×D₄	1	1	yes	characteristic
79	C₄⋊C₄	—	2	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	4	no	—
82	C₄⋊C₄	—	4	4	no	—
83	C₄⋊C₄	—	4	4	no	—
84	C₄⋊C₄	C₂×D₄	1	2	yes	—
85	C₄⋊C₄	C₂×D₄	1	2	yes	—
86	C₂²⋊C₄	—	2	2	no	—
87	C₂²⋊C₄	—	4	4	no	—
88	C₄²	—	2	2	no	—
89	C₄²	C₂⁴	1	1	yes	characteristic
90	C₂³	—	2	2	no	—
91	Q₈	—	2	2	no	—
92	Q₈	—	2	2	no	—
93	Q₈	—	8	4	no	—
94	D₄	—	2	2	no	—
95	D₄	C₂×Q₁₆	1	4	yes	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	4	no	—
102	C₂×C₄	—	16	4	no	—
103	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
106	C₈	—	2	2	no	—
107	C₈	—	2	1	no	—
108	C₈	—	2	1	no	—
109	C₈	—	4	2	no	—
110	C₈	—	4	2	no	—
111	C₈	C₂×Q₁₆	1	2	yes	—
112	C₈	#135	1	2	yes	—
113	C₂²	—	2	4	no	—
114	C₂²	—	4	2	no	—
115	C₂²	—	4	2	no	—
116	C₂²	D₄²	1	1	yes	characteristic
117	C₄	—	2	1	no	—
118	C₄	—	4	2	no	—
119	C₄	—	4	2	no	—
120	C₄	—	4	2	no	—
121	C₄	—	8	4	no	—
122	C₄	C₂²×Q₁₆	1	1	yes	characteristic
123	C₄	C₂×#135	1	1	yes	characteristic
124	C₄	#578	1	1	yes	characteristic
125	C₄	#578	1	1	yes	characteristic
126	C₂	—	4	4	no	—
127	C₂	#3283	1	1	yes	characteristic
128	C₂	D₄×Q₁₆	1	1	yes	characteristic
129	C₂	#3292	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3380

C₂

yes

maximal

#3338

C₂

yes

maximal, complemented

#1713

C₂

yes

maximal, characteristic, complemented

#1676

C₂

yes

maximal, complemented

#1675

C₂

yes

maximal, complemented

#1632

C₂

yes

maximal, complemented

C₈×Q₁₆

C₂

yes

maximal, characteristic, complemented

C₈×D₈

C₂

yes

maximal, characteristic

#554

C₂²

yes

—

#500

C₂²

yes

—

#500

C₂²

yes

—

#499

—

#494

C₂²

yes

—

#484

—

#478

—

#477

C₂²

yes

—

#476

C₂²

yes

—

#467

—

C₈×Q₈

C₂²

yes

—

C₄×Q₁₆

C₂²

yes

characteristic

C₄×D₈

—

C₄×D₈

C₂²

yes

characteristic

C₈×D₄

C₂²

yes

—

C₂×#107

—

#335

C₂²

yes

characteristic

#335

C₂²

yes

characteristic

#333

C₂²

yes

—

Q₈⋊C₈

C₂²

yes

—

D₄⋊C₈

C₂²

yes

—

C₈²

C₂²

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

D₄

yes

characteristic

C₂×D₈

D₄

yes

characteristic

C₂²×C₈

—

C₄⋊Q₈

C₂³

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

C₂³

yes

—

C₄×D₄

—

C₄×D₄

C₂³

yes

—

C₂×C₄⋊C₄

—

—

—

—

—

—

D₄

yes

characteristic

#107

D₄

yes

characteristic

#106

—

C₄⋊C₈

—

C₄⋊C₈

C₂³

yes

—

C₄⋊C₈

C₂³

yes

—

#103

—

#103

D₄

yes

—

#102

—

#102

D₄

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₂×D₄

yes

—

C₂×D₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

Q₁₆

—

Q₁₆

—

D₈

Q₁₆

yes

complemented

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₄

yes

—

C₄⋊C₄

C₂×D₄

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂⁴

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

C₂×Q₁₆

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂²×D₄

yes

characteristic

104

C₂×C₄

C₂²×D₄

yes

characteristic

105

C₂×C₄

H(5,𝔽₂)

yes

characteristic

106

C₈

—

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

C₂×Q₁₆

yes

—

112

C₈

#135

yes

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

D₄²

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

C₂²×Q₁₆

yes

characteristic

123

C₄

C₂×#135

yes

characteristic

124

C₄

#578

yes

characteristic

125

C₄

#578

yes

characteristic

126

C₂

—

127

C₂

#3283

yes

characteristic

128

C₂

D₄×Q₁₆

yes

characteristic

129

C₂

#3292

yes

characteristic

	subgroup	G/H
center, socle	C₂²	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂×D₈	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

D₄²

commutator, Frattini, cosocle

C₄²

C₂⁴

prime radical

C₂×D₈

D₄

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
2. #3338	C₂	semidirect
3. #1713	C₂	semidirect
4. #1676	C₂	semidirect
5. #1675	C₂	semidirect
6. #1632	C₂	semidirect
7. C₈×Q₁₆	C₂	semidirect
70. D₈	Q₁₆	semidirect

H ≅ G/N

type

2. #3338

C₂

semidirect

3. #1713

C₂

semidirect

4. #1676

C₂

semidirect

5. #1675

C₂

semidirect

6. #1632

C₂

semidirect

7. C₈×Q₁₆

C₂

semidirect

70. D₈

Q₁₆

semidirect

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

subgroup	index
#3380	2
#3338	2
#1713	2
#1676	2
#1675	2
#1632	2
C₈×Q₁₆	2
C₈×D₈	2

subgroup

index

C₈×Q₁₆

C₈×D₈

Subgroups — #20006

Conjugacy classes of subgroups

Complements

Maximal subgroups