Group #12648 — subgroups

Subgroups — #12648

Order 256.

211 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 211 conjugacy classes of subgroups — too many to draw. Only its 87 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×#441	C₂	1	1	yes	maximal, characteristic
2	C₂×#362	C₂	1	1	yes	maximal, characteristic
3	#2141	C₂	1	1	yes	maximal, characteristic
4	C₂×#347	C₂	1	1	yes	maximal, characteristic
5	#1773	C₂	1	1	yes	maximal, characteristic
6	#1386	C₂	1	1	yes	maximal, characteristic
7	#1383	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₄×Q₈	—	2	1	no	—
10	C₂×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×M₃₂	C₂²	1	1	yes	characteristic
12	#441	—	2	2	no	—
13	#441	C₄	1	4	yes	complemented
14	C₂×#108	C₂²	1	1	yes	characteristic
15	#427	C₂²	1	1	yes	characteristic
16	C₂×#106	C₂²	1	1	yes	characteristic
17	C₂×#103	—	2	1	no	—
18	#405	—	2	1	no	—
19	C₂×#97	C₂²	1	1	yes	characteristic
20	#362	C₄	1	4	yes	complemented
21	#358	C₄	1	4	yes	complemented
22	#347	—	2	2	no	—
23	#339	—	2	1	no	—
24	C₂²×Q₈	—	2	1	no	—
25	C₂×Q₁₆	—	2	1	no	—
26	C₂×Q₁₆	—	2	1	no	—
27	C₂×Q₁₆	—	2	1	no	—
28	C₂×Q₁₆	C₂×C₄	1	2	yes	—
29	C₂×Q₁₆	C₂×C₄	1	2	yes	—
30	C₂×Q₁₆	C₂×C₄	1	2	yes	—
31	C₂×M₁₆	—	2	1	no	—
32	C₂²×C₈	D₄	1	1	yes	characteristic
33	C₂²×C₈	C₂³	1	1	yes	characteristic
34	C₄×Q₈	—	2	4	no	—
35	C₄×Q₈	—	4	2	no	—
36	#117	—	2	1	no	—
37	C₂×C₄⋊C₄	—	2	1	no	—
38	C₂×C₄⋊C₄	—	4	1	no	—
39	C₂×C₄²	—	4	1	no	—
40	C₂×C₄²	D₄	1	1	yes	characteristic
41	M₃₂	—	2	1	no	—
42	M₃₂	—	2	1	no	—
43	M₃₂	—	2	1	no	—
44	M₃₂	C₂×C₄	1	2	yes	—
45	C₂×C₁₆	—	2	1	no	—
46	C₂×C₁₆	C₂×C₄	1	2	yes	—
47	#108	—	2	1	no	—
48	#108	C₂×C₄	1	2	yes	—
49	#107	C₂×C₄	1	2	yes	—
50	#106	—	2	1	no	—
51	#106	—	2	1	no	—
52	#106	C₂×C₄	1	2	yes	—
53	#103	—	2	4	no	—
54	#97	—	4	1	no	—
55	#97	C₂×C₄	1	2	yes	—
56	#97	D₄	1	2	yes	—
57	C₄×C₈	—	4	1	no	—
58	C₂×Q₈	—	2	2	no	—
59	C₂×Q₈	—	2	2	no	—
60	C₂×Q₈	—	2	2	no	—
61	C₂×Q₈	—	4	1	no	—
62	C₂×Q₈	—	4	1	no	—
63	C₂×Q₈	—	4	1	no	—
64	C₂²×C₄	—	2	1	no	—
65	C₂²×C₄	—	4	1	no	—
66	C₂²×C₄	—	4	1	no	—
67	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
68	Q₁₆	—	2	2	no	—
69	Q₁₆	—	2	2	no	—
70	Q₁₆	—	2	2	no	—
71	Q₁₆	C₄²	1	4	yes	—
72	M₁₆	—	2	2	no	—
73	M₁₆	—	4	1	no	—
74	C₂×C₈	—	2	1	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	4	1	no	—
77	C₂×C₈	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈	#41	1	1	yes	characteristic
83	C₂×C₈	#41	1	1	yes	characteristic
84	C₄⋊C₄	—	2	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	4	4	no	—
87	C₄⋊C₄	—	4	1	no	—
88	C₂²⋊C₄	—	4	1	no	—
89	C₄²	—	2	1	no	—
90	C₄²	—	4	4	no	—
91	C₄²	—	4	1	no	—
92	C₄²	C₂²⋊C₄	1	2	yes	—
93	C₁₆	—	2	2	no	—
94	C₁₆	—	2	2	no	—
95	C₂³	AGL(1,ℤ/8)	1	1	yes	characteristic
96	Q₈	—	2	4	no	—
97	Q₈	—	4	2	no	—
98	Q₈	—	4	2	no	—
99	Q₈	—	4	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	8	2	no	—
108	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₂×C₄	#137	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	4	2	no	—
116	C₈	C₂×C₄²	1	1	yes	characteristic
117	C₈	#117	1	1	yes	characteristic
118	C₈	C₄×D₄	1	1	yes	characteristic
119	C₈	C₄×D₄	1	1	yes	characteristic
120	C₂²	—	2	2	no	—
121	C₂²	#418	1	1	yes	characteristic
122	C₂²	#440	1	1	yes	characteristic
123	C₂²	#442	1	1	yes	characteristic
124	C₄	—	4	2	no	—
125	C₄	—	4	4	no	—
126	C₄	—	8	2	no	—
127	C₄	—	8	2	no	—
128	C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
129	C₄	#417	1	1	yes	characteristic
130	C₄	#440	1	1	yes	characteristic
131	C₄	#442	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	2	1	no	—
134	C₂	#1762	1	1	yes	characteristic
135	C₂	#2180	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#441

C₂

yes

maximal, characteristic

C₂×#362

C₂

yes

maximal, characteristic

#2141

C₂

yes

maximal, characteristic

C₂×#347

C₂

yes

maximal, characteristic

#1773

C₂

yes

maximal, characteristic

#1386

C₂

yes

maximal, characteristic

#1383

C₂

yes

maximal, characteristic

C₂²×Q₁₆

C₂²

yes

characteristic

C₂×C₄×Q₈

—

C₂×M₃₂

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

characteristic

#441

—

#441

C₄

yes

complemented

C₂×#108

C₂²

yes

characteristic

#427

C₂²

yes

characteristic

C₂×#106

C₂²

yes

characteristic

C₂×#103

—

#405

—

C₂×#97

C₂²

yes

characteristic

#362

C₄

yes

complemented

#358

C₄

yes

complemented

#347

—

#339

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×C₄

yes

—

C₂×Q₁₆

C₂×C₄

yes

—

C₂×Q₁₆

C₂×C₄

yes

—

C₂×M₁₆

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

D₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

C₂×C₄

yes

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄

yes

—

#108

—

#108

C₂×C₄

yes

—

#107

C₂×C₄

yes

—

#106

—

#106

—

#106

C₂×C₄

yes

—

#103

—

#97

—

#97

C₂×C₄

yes

—

#97

D₄

yes

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

C₄²

yes

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²⋊C₄

yes

—

C₁₆

—

C₁₆

—

C₂³

AGL(1,ℤ/8)

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

109

C₂×C₄

C₄×D₄

yes

characteristic

110

C₂×C₄

C₄×D₄

yes

characteristic

111

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

112

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

113

C₂×C₄

#137

yes

characteristic

114

C₈

—

115

C₈

—

116

C₈

C₂×C₄²

yes

characteristic

117

C₈

#117

yes

characteristic

118

C₈

C₄×D₄

yes

characteristic

119

C₈

C₄×D₄

yes

characteristic

120

C₂²

—

121

C₂²

#418

yes

characteristic

122

C₂²

#440

yes

characteristic

123

C₂²

#442

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

C₄×C₂²⋊C₄

yes

characteristic

129

C₄

#417

yes

characteristic

130

C₄

#440

yes

characteristic

131

C₄

#442

yes

characteristic

132

C₂

—

133

C₂

—

134

C₂

#1762

yes

characteristic

135

C₂

#2180

yes

—

	subgroup	G/H
center, socle	C₂²	#442
commutator	C₈	C₂×C₄²
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	AGL(1,ℤ/8)
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#442

commutator

C₈

C₂×C₄²

Frattini, cosocle

C₂²×C₈

C₂³

prime radical

C₂³

AGL(1,ℤ/8)

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
13. #441	C₄	semidirect
20. #362	C₄	semidirect
21. #358	C₄	semidirect

H ≅ G/N

type

13. #441

C₄

semidirect

20. #362

C₄

semidirect

21. #358

C₄

semidirect

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

subgroup	index
C₂×#441	2
C₂×#362	2
#2141	2
C₂×#347	2
#1773	2
#1386	2
#1383	2

subgroup

index

Subgroups — #12648

Conjugacy classes of subgroups

Complements

Maximal subgroups