Group #19975 — subgroups

Subgroups — #19975

Order 256.

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

normal subgroups only

This group has 379 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	#3391	C₂	1	2	yes	maximal, complemented
2	#3287	C₂	1	2	yes	maximal, complemented
3	#1712	C₂	1	1	yes	maximal, characteristic, complemented
4	#1668	C₂	1	2	yes	maximal, complemented
5	#1667	C₂	1	2	yes	maximal, complemented
6	#1621	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, complemented
9	C₂×#135	—	2	2	no	—
10	#550	C₂²	1	4	yes	—
11	#493	—	2	2	no	—
12	#493	C₂²	1	2	yes	—
13	#493	C₂²	1	2	yes	—
14	#493	C₂²	1	2	yes	—
15	#466	—	2	2	no	—
16	#460	C₂²	1	4	yes	—
17	#459	—	2	4	no	—
18	#459	C₂²	1	4	yes	—
19	#449	—	2	4	no	—
20	C₈×Q₈	C₂²	1	2	yes	—
21	C₄×Q₁₆	C₂²	1	1	yes	characteristic
22	C₄×D₈	—	2	2	no	—
23	C₄×D₈	C₂²	1	1	yes	characteristic
24	C₈×D₄	C₂²	1	2	yes	—
25	#335	C₂²	1	1	yes	characteristic
26	#335	C₂²	1	1	yes	characteristic
27	#331	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×#41	—	2	4	no	—
32	#135	—	4	8	no	—
33	C₂×Q₁₆	D₄	1	1	yes	characteristic
34	C₂×SD₁₆	—	2	4	no	—
35	C₂×D₈	—	2	4	no	—
36	C₂×D₈	—	2	2	no	—
37	C₂×D₈	—	2	2	no	—
38	C₂×D₈	—	2	2	no	—
39	C₂×D₈	—	2	2	no	—
40	C₂×D₈	—	4	2	no	—
41	C₂×D₈	D₄	1	1	yes	characteristic
42	C₂²×C₈	—	2	2	no	—
43	#127	—	2	4	no	—
44	#127	C₂³	1	4	yes	—
45	#121	—	2	4	no	—
46	#121	—	4	4	no	—
47	C₄×Q₈	C₂³	1	2	yes	—
48	C₄×D₄	—	2	4	no	—
49	C₄×D₄	C₂³	1	2	yes	—
50	#107	D₄	1	1	yes	characteristic
51	#107	D₄	1	1	yes	characteristic
52	C₄⋊C₈	—	2	2	no	—
53	C₄⋊C₈	C₂³	1	2	yes	—
54	C₄⋊C₈	C₂³	1	2	yes	—
55	#103	D₄	1	2	yes	—
56	#102	—	2	4	no	—
57	#102	—	2	4	no	—
58	#102	D₄	1	2	yes	—
59	C₂²⋊C₈	—	2	2	no	—
60	C₄×C₈	—	2	2	no	—
61	C₄×C₈	C₂³	1	1	yes	characteristic
62	C₄×C₈	C₂³	1	1	yes	characteristic
63	C₄×C₈	C₂³	1	1	yes	characteristic
64	#41	—	4	8	no	—
65	#41	—	8	4	no	—
66	C₂×Q₈	C₂×D₄	1	2	yes	—
67	C₂×D₄	—	2	4	no	—
68	C₂×D₄	—	2	4	no	—
69	C₂×D₄	—	2	4	no	—
70	C₂×D₄	—	4	4	no	—
71	C₂×D₄	—	4	4	no	—
72	C₂×D₄	C₂×D₄	1	2	yes	—
73	C₂²×C₄	—	2	2	no	—
74	C₂²×C₄	—	4	4	no	—
75	Q₁₆	D₈	1	4	yes	complemented
76	SD₁₆	—	4	8	no	—
77	D₈	—	2	2	no	—
78	D₈	—	4	4	no	—
79	D₈	—	4	4	no	—
80	D₈	—	8	2	no	—
81	D₈	—	8	2	no	—
82	D₈	—	8	4	no	—
83	D₈	—	8	4	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	4	4	no	—
88	C₂×C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈	C₂×D₄	1	1	yes	characteristic
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	C₂×D₄	1	2	yes	—
94	C₄⋊C₄	C₂×D₄	1	2	yes	—
95	C₂²⋊C₄	—	2	2	no	—
96	C₂²⋊C₄	—	4	4	no	—
97	C₄²	—	2	2	no	—
98	C₄²	C₂⁴	1	1	yes	characteristic
99	C₂³	—	2	2	no	—
100	C₂³	—	4	4	no	—
101	Q₈	—	2	2	no	—
102	Q₈	C₂×D₈	1	4	yes	—
103	D₄	—	2	2	no	—
104	D₄	—	2	2	no	—
105	D₄	—	4	4	no	—
106	D₄	—	4	4	no	—
107	D₄	—	4	4	no	—
108	D₄	—	4	4	no	—
109	D₄	—	8	8	no	—
110	D₄	—	8	4	no	—
111	D₄	—	16	4	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	8	8	no	—
118	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	—	2	1	no	—
123	C₈	—	2	1	no	—
124	C₈	—	4	2	no	—
125	C₈	—	4	2	no	—
126	C₈	C₂×D₈	1	2	yes	—
127	C₈	#135	1	2	yes	complemented
128	C₂²	—	4	2	no	—
129	C₂²	—	4	2	no	—
130	C₂²	—	4	2	no	—
131	C₂²	—	8	4	no	—
132	C₂²	—	8	4	no	—
133	C₂²	—	8	4	no	—
134	C₂²	D₄²	1	1	yes	characteristic
135	C₄	—	2	4	no	—
136	C₄	—	2	1	no	—
137	C₄	—	4	2	no	—
138	C₄	—	4	2	no	—
139	C₄	C₂²×D₈	1	1	yes	characteristic
140	C₄	C₂×#135	1	1	yes	characteristic
141	C₄	#576	1	1	yes	characteristic
142	C₄	#576	1	1	yes	characteristic
143	C₂	—	8	2	no	—
144	C₂	—	16	4	no	—
145	C₂	D₄×D₈	1	1	yes	characteristic
146	C₂	#3287	1	1	yes	characteristic
147	C₂	#3292	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3391

C₂

yes

maximal, complemented

#3287

C₂

yes

maximal, complemented

#1712

C₂

yes

maximal, characteristic, complemented

#1668

C₂

yes

maximal, complemented

#1667

C₂

yes

maximal, complemented

#1621

C₂

yes

maximal, complemented

C₈×Q₁₆

C₂

yes

maximal, characteristic, complemented

C₈×D₈

C₂

yes

maximal, characteristic, complemented

C₂×#135

—

#550

C₂²

yes

—

#493

—

#493

C₂²

yes

—

#493

C₂²

yes

—

#493

C₂²

yes

—

#466

—

#460

C₂²

yes

—

#459

—

#459

C₂²

yes

—

#449

—

C₈×Q₈

C₂²

yes

—

C₄×Q₁₆

C₂²

yes

characteristic

C₄×D₈

—

C₄×D₈

C₂²

yes

characteristic

C₈×D₄

C₂²

yes

—

#335

C₂²

yes

characteristic

#335

C₂²

yes

characteristic

#331

C₂²

yes

—

Q₈⋊C₈

C₂²

yes

—

D₄⋊C₈

C₂²

yes

—

C₈²

C₂²

yes

characteristic

C₂×#41

—

#135

—

C₂×Q₁₆

D₄

yes

characteristic

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

D₄

yes

characteristic

C₂²×C₈

—

#127

—

#127

C₂³

yes

—

#121

—

#121

—

C₄×Q₈

C₂³

yes

—

C₄×D₄

—

C₄×D₄

C₂³

yes

—

#107

D₄

yes

characteristic

#107

D₄

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂³

yes

—

C₄⋊C₈

C₂³

yes

—

#103

D₄

yes

—

#102

—

#102

—

#102

D₄

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

#41

—

#41

—

C₂×Q₈

C₂×D₄

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

Q₁₆

D₈

yes

complemented

SD₁₆

—

D₈

—

D₈

—

D₈

—

D₈

—

D₈

—

D₈

—

D₈

—

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

C₂×D₄

yes

—

C₄⋊C₄

C₂×D₄

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂⁴

yes

characteristic

C₂³

—

100

C₂³

—

101

Q₈

—

102

Q₈

C₂×D₈

yes

—

103

D₄

—

104

D₄

—

105

D₄

—

106

D₄

—

107

D₄

—

108

D₄

—

109

D₄

—

110

D₄

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₂²×D₄

yes

characteristic

119

C₂×C₄

C₂²×D₄

yes

characteristic

120

C₂×C₄

H(5,𝔽₂)

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

C₂×D₈

yes

—

127

C₈

#135

yes

complemented

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

D₄²

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

C₂²×D₈

yes

characteristic

140

C₄

C₂×#135

yes

characteristic

141

C₄

#576

yes

characteristic

142

C₄

#576

yes

characteristic

143

C₂

—

144

C₂

—

145

C₂

D₄×D₈

yes

characteristic

146

C₂

#3287

yes

characteristic

147

C₂

#3292

yes

characteristic

	subgroup	G/H
center, socle	C₂²	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴
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

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
1. #3391	C₂	semidirect
2. #3287	C₂	semidirect
3. #1712	C₂	semidirect
4. #1668	C₂	semidirect
5. #1667	C₂	semidirect
6. #1621	C₂	semidirect
7. C₈×Q₁₆	C₂	semidirect
8. C₈×D₈	C₂	semidirect
75. Q₁₆	D₈	semidirect
127. C₈	#135	semidirect

H ≅ G/N

type

1. #3391

C₂

semidirect

2. #3287

C₂

semidirect

3. #1712

C₂

semidirect

4. #1668

C₂

semidirect

5. #1667

C₂

semidirect

6. #1621

C₂

semidirect

7. C₈×Q₁₆

C₂

semidirect

8. C₈×D₈

C₂

semidirect

75. Q₁₆

D₈

semidirect

127. C₈

#135

semidirect

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

subgroup	index
#3391	2
#3287	2
#1712	2
#1668	2
#1667	2
#1621	2
C₈×Q₁₆	2
C₈×D₈	2

subgroup

index

C₈×Q₁₆

C₈×D₈

Subgroups — #19975

Conjugacy classes of subgroups

Complements

Maximal subgroups