Group #12912 — subgroups

Subgroups — #12912

Order 256.

231 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 107 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	C₂³×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₈×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
5	#1374	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆	—	2	2	no	—
8	C₂²×C₁₆	—	2	4	no	—
9	C₂²×C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆	C₂²	1	2	yes	—
12	C₂×C₂²⋊C₈	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈	C₂²	1	2	yes	—
14	C₄×C₂²⋊C₄	C₄	1	1	yes	characteristic
15	C₄⋊C₁₆	—	2	4	no	—
16	C₂²⋊C₁₆	—	2	2	no	—
17	#336	C₄	1	2	yes	—
18	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈	—	2	2	no	—
20	C₂²×C₈	—	2	2	no	—
21	C₂²×C₈	C₂×C₄	1	2	yes	—
22	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈	D₄	1	2	yes	—
24	C₂²×C₈	Q₈	1	2	yes	—
25	C₂²×C₈	C₂³	1	1	yes	characteristic
26	C₂×C₂²⋊C₄	C₈	1	1	yes	characteristic
27	C₂×C₂²⋊C₄	C₈	1	1	yes	characteristic
28	C₂×C₄²	C₂×C₄	1	2	yes	—
29	C₂×C₁₆	—	2	8	no	—
30	C₂×C₁₆	—	2	2	no	—
31	C₂×C₁₆	—	2	2	no	—
32	C₂×C₁₆	—	2	1	no	—
33	C₂×C₁₆	—	2	1	no	—
34	C₂×C₁₆	—	2	2	no	—
35	C₂×C₁₆	—	2	2	no	—
36	C₂×C₁₆	D₄	1	4	yes	—
37	C₂²⋊C₈	—	2	2	no	—
38	C₄×C₈	—	2	2	no	—
39	C₄×C₈	—	2	2	no	—
40	#95	C₈	1	2	yes	—
41	C₂⁴	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄	—	2	2	no	—
43	C₂²×C₄	—	2	2	no	—
44	C₂²×C₄	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄	C₂×C₈	1	2	yes	—
46	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄	C₂×C₈	1	2	yes	—
48	C₂²×C₄	M₁₆	1	2	yes	—
49	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈	—	2	2	no	—
51	C₂×C₈	—	2	2	no	—
52	C₂×C₈	—	2	4	no	—
53	C₂×C₈	—	2	2	no	—
54	C₂×C₈	—	2	2	no	—
55	C₂×C₈	—	2	2	no	—
56	C₂×C₈	—	2	2	no	—
57	C₂×C₈	C₄⋊C₄	1	2	yes	—
58	C₂×C₈	C₂×D₄	1	2	yes	—
59	C₂×C₈	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈	#41	1	2	yes	—
62	C₂²⋊C₄	—	2	2	no	—
63	C₂²⋊C₄	C₁₆	1	4	yes	complemented
64	C₄²	—	2	2	no	—
65	C₄²	—	2	2	no	—
66	C₁₆	—	2	4	no	—
67	C₁₆	—	4	1	no	—
68	C₁₆	—	4	1	no	—
69	C₂³	—	2	2	no	—
70	C₂³	—	2	2	no	—
71	C₂³	C₄⋊C₈	1	2	yes	—
72	C₂³	C₂×C₁₆	1	2	yes	—
73	C₂³	M₃₂	1	2	yes	—
74	C₂³	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	4	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	C₄⋊C₈	1	2	yes	—
85	C₂×C₄	C₂×C₁₆	1	4	yes	complemented
86	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄	C₄×D₄	1	2	yes	—
88	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₄	#131	1	2	yes	—
90	C₈	—	2	2	no	—
91	C₈	—	4	2	no	—
92	C₈	#120	1	1	yes	characteristic
93	C₈	C₂²⋊Q₈	1	2	yes	—
94	C₈	#123	1	1	yes	characteristic
95	C₂²	—	2	2	no	—
96	C₂²	—	2	2	no	—
97	C₂²	—	2	4	no	—
98	C₂²	—	2	4	no	—
99	C₂²	C₄⋊C₁₆	1	4	yes	complemented
100	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
101	C₂²	C₈×D₄	1	2	yes	—
102	C₂²	C₂²×C₁₆	1	1	yes	characteristic
103	C₂²	C₂×M₃₂	1	1	yes	characteristic
104	C₂²	#504	1	2	yes	—
105	C₄	—	2	2	no	—
106	C₄	—	2	4	no	—
107	C₄	—	4	2	no	—
108	C₄	#385	1	1	yes	characteristic
109	C₄	#424	1	1	yes	characteristic
110	C₄	#435	1	2	yes	—
111	C₂	—	2	4	no	—
112	C₂	#1850	1	1	yes	characteristic
113	C₂	C₂×C₄⋊C₁₆	1	1	yes	characteristic
114	C₂	#2151	1	1	yes	characteristic
115	C₂	D₄×C₁₆	1	2	yes	—
116	C₂	#2168	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₁₆

C₂

yes

maximal, characteristic

C₂×C₄⋊C₁₆

C₂

yes

maximal, complemented

C₂×C₂²⋊C₁₆

C₂

yes

maximal, characteristic

C₈×C₂²⋊C₄

C₂

yes

maximal, characteristic

#1374

C₂

yes

maximal, complemented

C₂³×C₈

C₂²

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂²

yes

characteristic

C₂²×C₁₆

C₂²

yes

characteristic

C₂²×C₁₆

C₂²

yes

—

C₂×C₂²⋊C₈

C₄

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

C₄×C₂²⋊C₄

C₄

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#336

C₄

yes

—

C₂³×C₄

C₂×C₄

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

D₄

yes

—

C₂²×C₈

Q₈

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₂×C₂²⋊C₄

C₈

yes

characteristic

C₂×C₂²⋊C₄

C₈

yes

characteristic

C₂×C₄²

C₂×C₄

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

#95

C₈

yes

—

C₂⁴

C₂×C₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄⋊C₄

yes

—

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

M₁₆

yes

—

C₂²×C₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×Q₈

yes

characteristic

C₂×C₈

#41

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

C₁₆

yes

complemented

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

—

C₂³

—

C₂³

C₄⋊C₈

yes

—

C₂³

C₂×C₁₆

yes

—

C₂³

M₃₂

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

C₄⋊C₈

yes

—

C₂×C₄

C₂×C₁₆

yes

complemented

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

C₂×M₁₆

yes

characteristic

C₂×C₄

#131

yes

—

C₈

—

C₈

—

C₈

#120

yes

characteristic

C₈

C₂²⋊Q₈

yes

—

C₈

#123

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₄⋊C₁₆

yes

complemented

100

C₂²

C₂×C₄⋊C₈

yes

characteristic

101

C₂²

C₈×D₄

yes

—

102

C₂²

C₂²×C₁₆

yes

characteristic

103

C₂²

C₂×M₃₂

yes

characteristic

104

C₂²

#504

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

#385

yes

characteristic

109

C₄

#424

yes

characteristic

110

C₄

#435

yes

—

111

C₂

—

112

C₂

#1850

yes

characteristic

113

C₂

C₂×C₄⋊C₁₆

yes

characteristic

114

C₂

#2151

yes

characteristic

115

C₂

D₄×C₁₆

yes

—

116

C₂

#2168

yes

—

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂³
commutator	C₂²	C₂²×C₁₆
socle	C₂³	C₂²×C₈
prime radical	C₂⁴	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, Frattini, cosocle

C₂²×C₈

C₂³

commutator

C₂²

C₂²×C₁₆

socle

C₂³

C₂²×C₈

prime radical

C₂⁴

C₂×C₈

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. C₂×C₄⋊C₁₆	C₂	semidirect
5. #1374	C₂	semidirect
63. C₂²⋊C₄	C₁₆	semidirect
85. C₂×C₄	C₂×C₁₆	semidirect
99. C₂²	C₄⋊C₁₆	semidirect

H ≅ G/N

type

2. C₂×C₄⋊C₁₆

C₂

semidirect

5. #1374

C₂

semidirect

63. C₂²⋊C₄

C₁₆

semidirect

85. C₂×C₄

C₂×C₁₆

semidirect

99. C₂²

C₄⋊C₁₆

semidirect

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

subgroup	index
C₂³×C₁₆	2
C₂×C₄⋊C₁₆	2
C₂×C₂²⋊C₁₆	2
C₈×C₂²⋊C₄	2
#1374	2

subgroup

index

C₂³×C₁₆

C₂×C₄⋊C₁₆

C₂×C₂²⋊C₁₆

C₈×C₂²⋊C₄

#1374

Subgroups — #12912

Conjugacy classes of subgroups

Complements

Maximal subgroups