Group #12784 — subgroups

Subgroups — #12784

Order 256.

239 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 239 conjugacy classes of subgroups — too many to draw. Only its 103 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×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊C₁₆	C₂	1	2	yes	maximal
4	#1374	C₂	1	2	yes	maximal, complemented
5	C₂²×M₁₆	C₄	1	1	yes	characteristic
6	C₂³×C₈	C₂²	1	1	yes	characteristic
7	C₂×#117	C₄	1	1	yes	characteristic
8	C₂²×C₁₆	C₂²	1	4	yes	—
9	#405	—	2	4	no	—
10	C₂×#97	C₄	1	1	yes	characteristic
11	C₂×#97	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆	—	2	4	no	—
15	C₂²⋊C₁₆	—	2	4	no	—
16	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
17	C₂×M₁₆	—	2	2	no	—
18	C₂×M₁₆	—	2	2	no	—
19	C₂×M₁₆	—	2	2	no	—
20	C₂²×C₈	—	2	2	no	—
21	C₂²×C₈	—	2	2	no	—
22	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈	D₄	1	2	yes	—
26	C₂²×C₈	Q₈	1	2	yes	—
27	C₂²×C₈	C₂³	1	1	yes	characteristic
28	#117	—	2	2	no	—
29	#117	—	2	2	no	—
30	C₂×C₄⋊C₄	C₈	1	2	yes	—
31	C₂×C₂²⋊C₄	C₈	1	2	yes	—
32	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆	—	2	4	no	—
35	C₂×C₁₆	—	2	4	no	—
36	C₂×C₁₆	—	2	4	no	—
37	#97	—	2	2	no	—
38	#97	—	2	2	no	—
39	C₄×C₈	—	2	2	no	—
40	C₄×C₈	D₄	1	4	yes	—
41	C₂⁴	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄	—	2	1	no	—
43	C₂²×C₄	—	2	1	no	—
44	C₂²×C₄	—	2	1	no	—
45	C₂²×C₄	—	2	1	no	—
46	C₂²×C₄	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄	C₂×C₈	1	2	yes	—
48	C₂²×C₄	C₂×C₈	1	2	yes	—
49	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄	M₁₆	1	2	yes	—
51	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
52	M₁₆	—	2	8	no	—
53	C₂×C₈	—	2	1	no	—
54	C₂×C₈	—	2	1	no	—
55	C₂×C₈	—	2	1	no	—
56	C₂×C₈	—	2	1	no	—
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	—	2	4	no	—
60	C₂×C₈	—	2	4	no	—
61	C₂×C₈	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈	C₄⋊C₄	1	2	yes	—
63	C₂×C₈	M₁₆	1	2	yes	—
64	C₂×C₈	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈	C₂×D₄	1	2	yes	—
66	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈	#41	1	2	yes	—
68	C₄⋊C₄	—	2	4	no	—
69	C₂²⋊C₄	—	2	4	no	—
70	C₄²	—	2	1	no	—
71	C₄²	—	2	1	no	—
72	C₄²	C₂²⋊C₄	1	2	yes	—
73	C₄²	M₁₆	1	2	yes	—
74	C₁₆	—	4	4	no	—
75	C₂³	—	2	1	no	—
76	C₂³	—	2	1	no	—
77	C₂³	—	2	4	no	—
78	C₂³	C₄⋊C₈	1	2	yes	—
79	C₂³	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	C₂²⋊C₈	1	4	yes	—
90	C₂×C₄	C₄⋊C₈	1	2	yes	—
91	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄	#117	1	1	yes	characteristic
94	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
95	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
96	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
97	C₈	—	2	2	no	—
98	C₈	—	2	2	no	—
99	C₈	—	2	2	no	—
100	C₈	#121	1	2	yes	—
101	C₈	C₂²⋊Q₈	1	2	yes	—
102	C₂²	—	2	2	no	—
103	C₂²	—	2	4	no	—
104	C₂²	—	4	2	no	—
105	C₂²	—	4	2	no	—
106	C₂²	C₂×C₂²⋊C₈	1	1	yes	characteristic
107	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
108	C₂²	#431	1	1	yes	characteristic
109	C₂²	#504	1	4	yes	—
110	C₄	—	2	2	no	—
111	C₄	—	2	4	no	—
112	C₄	—	4	2	no	—
113	C₄	#380	1	1	yes	characteristic
114	C₄	#407	1	1	yes	characteristic
115	C₄	#423	1	1	yes	characteristic
116	C₄	#432	1	1	yes	characteristic
117	C₂	—	4	2	no	—
118	C₂	#1797	1	1	yes	characteristic
119	C₂	#2113	1	2	yes	—
120	C₂	#2151	1	2	yes	—
121	C₂	#2164	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#405

C₂

yes

maximal, characteristic

C₂×C₄⋊C₁₆

C₂

yes

maximal, complemented

C₂×C₂²⋊C₁₆

C₂

yes

maximal

#1374

C₂

yes

maximal, complemented

C₂²×M₁₆

C₄

yes

characteristic

C₂³×C₈

C₂²

yes

characteristic

C₂×#117

C₄

yes

characteristic

C₂²×C₁₆

C₂²

yes

—

#405

—

C₂×#97

C₄

yes

characteristic

C₂×#97

C₄

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₂³×C₄

C₂×C₄

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

D₄

yes

—

C₂²×C₈

Q₈

yes

—

C₂²×C₈

C₂³

yes

characteristic

#117

—

#117

—

C₂×C₄⋊C₄

C₈

yes

—

C₂×C₂²⋊C₄

C₈

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

D₄

yes

—

C₂⁴

C₂×C₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄⋊C₄

yes

—

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

M₁₆

yes

—

C₂²×C₄

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

M₁₆

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×Q₈

yes

characteristic

C₂×C₈

#41

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊C₄

yes

—

C₄²

M₁₆

yes

—

C₁₆

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₄⋊C₈

yes

—

C₂³

C₂²×C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₈

yes

—

C₂×C₄

C₄⋊C₈

yes

—

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

#117

yes

characteristic

C₂×C₄

C₂×M₁₆

yes

characteristic

C₂×C₄

C₂×M₁₆

yes

characteristic

C₂×C₄

C₂×M₁₆

yes

characteristic

C₈

—

C₈

—

C₈

—

100

C₈

#121

yes

—

101

C₈

C₂²⋊Q₈

yes

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

C₂×C₂²⋊C₈

yes

characteristic

107

C₂²

C₂×C₄⋊C₈

yes

characteristic

108

C₂²

#431

yes

characteristic

109

C₂²

#504

yes

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

#380

yes

characteristic

114

C₄

#407

yes

characteristic

115

C₄

#423

yes

characteristic

116

C₄

#432

yes

characteristic

117

C₂

—

118

C₂

#1797

yes

characteristic

119

C₂

#2113

yes

—

120

C₂

#2151

yes

—

121

C₂

#2164

yes

—

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂³
commutator, socle	C₂³	C₂²×C₈
prime radical	C₂⁴	C₂×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, Frattini, cosocle

C₂²×C₈

C₂³

commutator, socle

C₂³

C₂²×C₈

prime radical

C₂⁴

C₂×C₈

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂²×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. C₂×C₄⋊C₁₆	C₂	semidirect
4. #1374	C₂	semidirect

H ≅ G/N

type

2. C₂×C₄⋊C₁₆

C₂

semidirect

4. #1374

C₂

semidirect

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

subgroup	index
C₂×#405	2
C₂×C₄⋊C₁₆	2
C₂×C₂²⋊C₁₆	2
#1374	2

subgroup

index

C₂×#405

C₂×C₄⋊C₁₆

C₂×C₂²⋊C₁₆

#1374

Subgroups — #12784

Conjugacy classes of subgroups

Complements

Maximal subgroups