Group #12551 — subgroups

Subgroups — #12551

Order 256.

293 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 293 conjugacy classes of subgroups — too many to draw. Only its 175 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	C₂²×M₃₂	C₂	1	2	yes	maximal
2	C₂×#405	C₂	1	1	yes	maximal, characteristic
3	#1377	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆	C₄	1	1	yes	characteristic
5	C₂³×C₈	C₂²	1	1	yes	characteristic
6	C₂×#117	C₄	1	1	yes	characteristic
7	C₂×M₃₂	—	2	4	no	—
8	C₂×M₃₂	—	2	2	no	—
9	C₂×M₃₂	C₄	1	8	yes	complemented
10	C₂×M₃₂	C₂²	1	4	yes	—
11	C₂²×C₁₆	—	2	2	no	—
12	#405	—	2	2	no	—
13	#405	—	2	2	no	—
14	C₂×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₈	C₄	1	2	yes	—
16	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
17	C₂×M₁₆	—	2	2	no	—
18	C₂×M₁₆	—	2	2	no	—
19	C₂×M₁₆	C₈	1	4	yes	—
20	C₂²×C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈	C₂×C₄	1	4	yes	—
23	C₂²×C₈	D₄	1	2	yes	—
24	C₂²×C₈	D₄	1	2	yes	—
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	2	yes	—
33	M₃₂	—	2	8	no	—
34	M₃₂	—	2	4	no	—
35	M₃₂	C₂×C₄	1	8	yes	—
36	C₂×C₁₆	—	2	8	no	—
37	C₂×C₁₆	—	2	4	no	—
38	#97	—	2	4	no	—
39	C₄×C₈	—	2	2	no	—
40	C₄×C₈	—	2	2	no	—
41	C₂⁴	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄	C₄²	1	2	yes	—
43	C₂²×C₄	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄	C₂×C₈	1	4	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	M₁₆	1	2	yes	—
50	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
51	M₁₆	—	2	8	no	—
52	C₂×C₈	—	2	2	no	—
53	C₂×C₈	—	2	2	no	—
54	C₂×C₈	C₄²	1	2	yes	—
55	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈	C₂²⋊C₄	1	4	yes	—
57	C₂×C₈	C₂²⋊C₄	1	4	yes	—
58	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈	C₄⋊C₄	1	2	yes	—
60	C₂×C₈	C₄⋊C₄	1	4	yes	—
61	C₂×C₈	C₄⋊C₄	1	4	yes	—
62	C₂×C₈	C₂×C₈	1	4	yes	—
63	C₂×C₈	C₂²×C₄	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	C₄⋊C₄	—	2	4	no	—
68	C₂²⋊C₄	—	2	4	no	—
69	C₄²	—	2	2	no	—
70	C₄²	—	2	2	no	—
71	C₁₆	—	2	8	no	—
72	C₂³	—	4	2	no	—
73	C₂³	C₄×C₈	1	2	yes	—
74	C₂³	C₂²⋊C₈	1	2	yes	—
75	C₂³	C₄⋊C₈	1	2	yes	—
76	C₂³	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄	—	2	4	no	—
78	C₂×C₄	—	4	4	no	—
79	C₂×C₄	#95	1	4	yes	—
80	C₂×C₄	C₄×C₈	1	2	yes	—
81	C₂×C₄	#97	1	4	yes	—
82	C₂×C₄	C₂²⋊C₈	1	2	yes	—
83	C₂×C₄	C₂²⋊C₈	1	4	yes	—
84	C₂×C₄	C₄⋊C₈	1	2	yes	—
85	C₂×C₄	C₄⋊C₈	1	4	yes	—
86	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
87	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
92	C₈	—	2	4	no	—
93	C₈	#95	1	2	yes	—
94	C₈	#95	1	2	yes	—
95	C₈	C₂×C₂²⋊C₄	1	2	yes	—
96	C₈	C₂×C₄⋊C₄	1	2	yes	—
97	C₂²	—	2	2	no	—
98	C₂²	—	2	4	no	—
99	C₂²	—	4	2	no	—
100	C₂²	—	4	2	no	—
101	C₂²	#336	1	4	yes	—
102	C₂²	C₂×C₄×C₈	1	1	yes	characteristic
103	C₂²	C₂×C₂²⋊C₈	1	1	yes	characteristic
104	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
105	C₄	—	4	4	no	—
106	C₄	#336	1	4	yes	—
107	C₄	C₂×#95	1	1	yes	characteristic
108	C₄	C₂×#97	1	1	yes	characteristic
109	C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
110	C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
111	C₂	—	2	4	no	—
112	C₂	—	2	1	no	—
113	C₂	—	2	1	no	—
114	C₂	C₂×#336	1	1	yes	characteristic
115	C₂	#2115	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×M₃₂

C₂

yes

maximal

C₂×#405

C₂

yes

maximal, characteristic

#1377

C₂

yes

maximal, complemented

C₂²×M₁₆

C₄

yes

characteristic

C₂³×C₈

C₂²

yes

characteristic

C₂×#117

C₄

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

—

C₂×M₃₂

C₄

yes

complemented

C₂×M₃₂

C₂²

yes

—

C₂²×C₁₆

—

#405

—

#405

—

C₂×#97

C₂²

yes

—

C₂×C₄×C₈

C₄

yes

—

C₂³×C₄

C₂×C₄

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₈

yes

—

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

D₄

yes

—

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

—

M₃₂

—

M₃₂

—

M₃₂

C₂×C₄

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₂⁴

C₂×C₈

yes

characteristic

C₂²×C₄

C₄²

yes

—

C₂²×C₄

C₂²⋊C₄

yes

—

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

M₁₆

yes

—

C₂²×C₄

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²

yes

—

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×Q₈

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₁₆

—

C₂³

—

C₂³

C₄×C₈

yes

—

C₂³

C₂²⋊C₈

yes

—

C₂³

C₄⋊C₈

yes

—

C₂³

C₂²×C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#95

yes

—

C₂×C₄

C₄×C₈

yes

—

C₂×C₄

#97

yes

—

C₂×C₄

C₂²⋊C₈

yes

—

C₂×C₄

C₂²⋊C₈

yes

—

C₂×C₄

C₄⋊C₈

yes

—

C₂×C₄

C₄⋊C₈

yes

—

C₂×C₄

C₂×C₄²

yes

characteristic

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

C₂²×C₈

yes

characteristic

C₂×C₄

C₂×M₁₆

yes

characteristic

C₂×C₄

C₂×M₁₆

yes

characteristic

C₈

—

C₈

#95

yes

—

C₈

#95

yes

—

C₈

C₂×C₂²⋊C₄

yes

—

C₈

C₂×C₄⋊C₄

yes

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

#336

yes

—

102

C₂²

C₂×C₄×C₈

yes

characteristic

103

C₂²

C₂×C₂²⋊C₈

yes

characteristic

104

C₂²

C₂×C₄⋊C₈

yes

characteristic

105

C₄

—

106

C₄

#336

yes

—

107

C₄

C₂×#95

yes

characteristic

108

C₄

C₂×#97

yes

characteristic

109

C₄

C₂×C₂²⋊C₈

yes

characteristic

110

C₄

C₂×C₄⋊C₈

yes

characteristic

111

C₂

—

112

C₂

—

113

C₂

—

114

C₂

C₂×#336

yes

characteristic

115

C₂

#2115

yes

—

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

subgroup

G/H

center

C₂×C₈

C₂²⋊C₄

commutator

C₂²

C₂×C₄×C₈

Frattini, cosocle

C₂²×C₈

C₂³

socle

C₂²

C₂×C₂²⋊C₈

prime radical

C₂⁴

C₂×C₈

LCS

G ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
3. #1377	C₂	semidirect
9. C₂×M₃₂	C₄	semidirect

H ≅ G/N

type

3. #1377

C₂

semidirect

9. C₂×M₃₂

C₄

semidirect

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

subgroup	index
C₂²×M₃₂	2
C₂×#405	2
#1377	2

subgroup

index

C₂²×M₃₂

C₂×#405

#1377

Subgroups — #12551

Conjugacy classes of subgroups

Complements

Maximal subgroups