Group #3276 — subgroups

Subgroups — #3276

Order 128.

236 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 96 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	C₂×#137	C₂	1	2	yes	maximal, complemented
2	D₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#547	C₂	1	1	yes	maximal, characteristic
4	#494	C₂	1	1	yes	maximal, characteristic, complemented
5	#470	C₂	1	2	yes	maximal, complemented
6	#462	C₂	1	1	yes	maximal, characteristic, complemented
7	#461	C₂	1	1	yes	maximal, characteristic, complemented
8	#452	C₂	1	2	yes	maximal, complemented
9	#451	C₂	1	2	yes	maximal, complemented
10	C₄×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41	C₂²	1	2	yes	—
13	C₂²×Q₈	C₂²	1	2	yes	—
14	#137	—	2	8	no	—
15	C₂×Q₁₆	C₂²	1	2	yes	—
16	C₂×Q₁₆	C₂²	1	2	yes	—
17	C₂×Q₁₆	C₂²	1	2	yes	—
18	C₂×SD₁₆	C₂²	1	2	yes	—
19	C₂×SD₁₆	C₂²	1	1	yes	characteristic
20	C₂×M₁₆	C₂²	1	2	yes	—
21	C₄⋊Q₈	—	2	1	no	—
22	C₄⋊Q₈	C₂²	1	1	yes	characteristic
23	C₄⋊Q₈	C₂²	1	1	yes	characteristic
24	#123	—	2	2	no	—
25	C₂²⋊Q₈	—	2	2	no	—
26	C₂²⋊Q₈	C₂²	1	2	yes	—
27	C₂²⋊Q₈	C₂²	1	2	yes	—
28	#121	—	2	1	no	—
29	C₄×Q₈	C₂²	1	1	yes	characteristic
30	C₄×D₄	—	2	1	no	—
31	C₄×D₄	C₂²	1	1	yes	characteristic
32	C₄×D₄	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₄	—	2	1	no	—
34	#106	C₂²	1	1	yes	characteristic
35	C₄⋊C₈	C₂²	1	1	yes	characteristic
36	#103	C₂²	1	1	yes	characteristic
37	#103	C₂²	1	2	yes	—
38	#103	C₂²	1	2	yes	—
39	#102	C₂²	1	1	yes	characteristic
40	C₂²⋊C₈	C₂²	1	2	yes	—
41	C₄×C₈	C₂²	1	1	yes	characteristic
42	#41	—	2	4	no	—
43	#41	—	4	2	no	—
44	C₂×Q₈	—	2	2	no	—
45	C₂×Q₈	—	2	4	no	—
46	C₂×Q₈	—	4	2	no	—
47	C₂×Q₈	C₂³	1	2	yes	—
48	C₂×Q₈	C₂³	1	2	yes	—
49	C₂×Q₈	C₂³	1	1	yes	characteristic
50	C₂×D₄	—	2	2	no	—
51	C₂×D₄	D₄	1	1	yes	characteristic
52	C₂×D₄	C₂³	1	1	yes	characteristic
53	C₂²×C₄	—	2	2	no	—
54	C₂²×C₄	—	2	1	no	—
55	C₂²×C₄	—	2	2	no	—
56	C₂²×C₄	C₂³	1	2	yes	—
57	Q₁₆	—	2	4	no	—
58	Q₁₆	—	2	4	no	—
59	Q₁₆	—	4	2	no	—
60	SD₁₆	—	2	4	no	—
61	SD₁₆	D₄	1	4	yes	complemented
62	M₁₆	—	2	4	no	—
63	C₂×C₈	C₂³	1	1	yes	characteristic
64	C₂×C₈	C₂³	1	2	yes	—
65	C₂×C₈	C₂³	1	1	yes	characteristic
66	C₄⋊C₄	—	2	1	no	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	2	1	no	—
69	C₄⋊C₄	—	2	1	no	—
70	C₄⋊C₄	—	2	1	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	2	1	no	—
73	C₄⋊C₄	D₄	1	1	yes	characteristic
74	C₄⋊C₄	C₂³	1	1	yes	characteristic
75	C₄⋊C₄	C₂³	1	1	yes	characteristic
76	C₂²⋊C₄	—	2	2	no	—
77	C₂²⋊C₄	—	2	2	no	—
78	C₂²⋊C₄	—	2	1	no	—
79	C₂²⋊C₄	D₄	1	2	yes	—
80	C₄²	—	2	1	no	—
81	C₄²	C₂³	1	1	yes	characteristic
82	C₂³	—	2	1	no	—
83	C₂³	C₂×D₄	1	2	yes	—
84	Q₈	—	2	1	no	—
85	Q₈	—	2	2	no	—
86	Q₈	—	2	2	no	—
87	Q₈	—	2	2	no	—
88	Q₈	—	2	2	no	—
89	Q₈	—	4	4	no	—
90	Q₈	C₂×D₄	1	2	yes	—
91	D₄	—	2	2	no	—
92	D₄	—	2	1	no	—
93	D₄	—	4	4	no	—
94	D₄	C₂×D₄	1	2	yes	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	4	4	no	—
103	C₂×C₄	—	4	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄	C₂×D₄	1	2	yes	—
107	C₂×C₄	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄	C₂⁴	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	2	1	no	—
111	C₈	C₂×D₄	1	2	yes	—
112	C₂²	—	2	2	no	—
113	C₂²	—	2	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	—	4	1	no	—
117	C₂²	—	4	1	no	—
118	C₂²	C₂²×D₄	1	1	yes	characteristic
119	C₄	—	2	2	no	—
120	C₄	—	2	2	no	—
121	C₄	—	2	1	no	—
122	C₄	—	4	2	no	—
123	C₄	—	4	2	no	—
124	C₄	—	4	1	no	—
125	C₄	—	4	1	no	—
126	C₄	#137	1	2	yes	complemented
127	C₄	C₂²×D₄	1	1	yes	characteristic
128	C₄	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂	—	4	2	no	—
130	C₂	—	4	2	no	—
131	C₂	D₄²	1	1	yes	characteristic
132	C₂	C₂×#137	1	1	yes	characteristic
133	C₂	#578	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#137

C₂

yes

maximal, complemented

D₄×Q₈

C₂

yes

maximal, characteristic, complemented

#547

C₂

yes

maximal, characteristic

#494

C₂

yes

maximal, characteristic, complemented

#470

C₂

yes

maximal, complemented

#462

C₂

yes

maximal, characteristic, complemented

#461

C₂

yes

maximal, characteristic, complemented

#452

C₂

yes

maximal, complemented

#451

C₂

yes

maximal, complemented

C₄×SD₁₆

C₂

yes

maximal, characteristic, complemented

#436

C₂

yes

maximal, characteristic, complemented

C₂×#41

C₂²

yes

—

C₂²×Q₈

C₂²

yes

—

#137

—

C₂×Q₁₆

C₂²

yes

—

C₂×Q₁₆

C₂²

yes

—

C₂×Q₁₆

C₂²

yes

—

C₂×SD₁₆

C₂²

yes

—

C₂×SD₁₆

C₂²

yes

characteristic

C₂×M₁₆

C₂²

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂²

yes

characteristic

C₄⋊Q₈

C₂²

yes

characteristic

#123

—

C₂²⋊Q₈

—

C₂²⋊Q₈

C₂²

yes

—

C₂²⋊Q₈

C₂²

yes

—

#121

—

C₄×Q₈

C₂²

yes

characteristic

C₄×D₄

—

C₄×D₄

C₂²

yes

characteristic

C₄×D₄

C₂²

yes

characteristic

C₂×C₄⋊C₄

—

#106

C₂²

yes

characteristic

C₄⋊C₈

C₂²

yes

characteristic

#103

C₂²

yes

characteristic

#103

C₂²

yes

—

#103

C₂²

yes

—

#102

C₂²

yes

characteristic

C₂²⋊C₈

C₂²

yes

—

C₄×C₈

C₂²

yes

characteristic

#41

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂³

yes

—

C₂×Q₈

C₂³

yes

—

C₂×Q₈

C₂³

yes

characteristic

C₂×D₄

—

C₂×D₄

D₄

yes

characteristic

C₂×D₄

C₂³

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂³

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

SD₁₆

—

SD₁₆

D₄

yes

complemented

M₁₆

—

C₂×C₈

C₂³

yes

characteristic

C₂×C₈

C₂³

yes

—

C₂×C₈

C₂³

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄

yes

characteristic

C₄⋊C₄

C₂³

yes

characteristic

C₄⋊C₄

C₂³

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

D₄

yes

—

C₄²

—

C₄²

C₂³

yes

characteristic

C₂³

—

C₂³

C₂×D₄

yes

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

C₂×D₄

yes

—

D₄

—

D₄

—

D₄

—

D₄

C₂×D₄

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂×D₄

yes

characteristic

106

C₂×C₄

C₂×D₄

yes

—

107

C₂×C₄

C₂×D₄

yes

characteristic

108

C₂×C₄

C₂⁴

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

C₂×D₄

yes

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

C₂²×D₄

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#137

yes

complemented

127

C₄

C₂²×D₄

yes

characteristic

128

C₄

H(5,𝔽₂)

yes

characteristic

129

C₂

—

130

C₂

—

131

C₂

D₄²

yes

characteristic

132

C₂

C₂×#137

yes

characteristic

133

C₂

#578

yes

characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	#547	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴

C₂

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂² ◁ C₄×D₄ ◁ G

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. C₂×#137	C₂	semidirect
2. D₄×Q₈	C₂	semidirect
4. #494	C₂	semidirect
5. #470	C₂	semidirect
6. #462	C₂	semidirect
7. #461	C₂	semidirect
8. #452	C₂	semidirect
9. #451	C₂	semidirect
10. C₄×SD₁₆	C₂	semidirect
11. #436	C₂	semidirect
61. SD₁₆	D₄	semidirect
126. C₄	#137	semidirect

H ≅ G/N

type

1. C₂×#137

C₂

semidirect

2. D₄×Q₈

C₂

semidirect

4. #494

C₂

semidirect

5. #470

C₂

semidirect

6. #462

C₂

semidirect

7. #461

C₂

semidirect

8. #452

C₂

semidirect

9. #451

C₂

semidirect

10. C₄×SD₁₆

C₂

semidirect

11. #436

C₂

semidirect

61. SD₁₆

D₄

semidirect

126. C₄

#137

semidirect

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

subgroup	index
C₂×#137	2
D₄×Q₈	2
#547	2
#494	2
#470	2
#462	2
#461	2
#452	2
#451	2
C₄×SD₁₆	2
#436	2

subgroup

index

D₄×Q₈

C₄×SD₁₆

Subgroups — #3276

Conjugacy classes of subgroups

Complements

Maximal subgroups