Group #12549 — subgroups

Subgroups — #12549

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	#1376	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	4	no	—
13	C₂×#97	C₄	1	2	yes	—
14	C₂×C₄×C₈	C₂²	1	2	yes	—
15	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
16	C₂×M₁₆	—	2	2	no	—
17	C₂×M₁₆	—	2	2	no	—
18	C₂×M₁₆	C₈	1	4	yes	—
19	C₂²×C₈	C₂×C₄	1	2	yes	—
20	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈	C₂×C₄	1	4	yes	—
22	C₂²×C₈	D₄	1	2	yes	—
23	C₂²×C₈	D₄	1	4	yes	—
24	C₂²×C₈	Q₈	1	2	yes	—
25	C₂²×C₈	C₂³	1	1	yes	characteristic
26	#117	—	2	2	no	—
27	#117	C₈	1	4	yes	—
28	C₂×C₄⋊C₄	—	2	1	no	—
29	C₂×C₂²⋊C₄	—	2	1	no	—
30	C₂×C₄²	C₂×C₄	1	2	yes	—
31	M₃₂	—	2	8	no	—
32	M₃₂	—	2	4	no	—
33	M₃₂	C₂×C₄	1	8	yes	—
34	C₂×C₁₆	—	2	8	no	—
35	C₂×C₁₆	—	2	4	no	—
36	#97	—	2	4	no	—
37	C₄×C₈	—	2	4	no	—
38	C₂⁴	M₁₆	1	1	yes	characteristic
39	C₂²×C₄	—	2	2	no	—
40	C₂²×C₄	C₄²	1	2	yes	—
41	C₂²×C₄	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄	C₂×C₈	1	2	yes	—
44	C₂²×C₄	C₂×C₈	1	2	yes	—
45	C₂²×C₄	M₁₆	1	1	yes	characteristic
46	C₂²×C₄	M₁₆	1	2	yes	—
47	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
48	M₁₆	—	2	8	no	—
49	C₂×C₈	—	2	2	no	—
50	C₂×C₈	—	2	2	no	—
51	C₂×C₈	C₄²	1	2	yes	—
52	C₂×C₈	C₂²⋊C₄	1	2	yes	—
53	C₂×C₈	C₂²⋊C₄	1	4	yes	—
54	C₂×C₈	C₂²⋊C₄	1	4	yes	—
55	C₂×C₈	C₄⋊C₄	1	2	yes	—
56	C₂×C₈	C₄⋊C₄	1	8	yes	—
57	C₂×C₈	C₂×C₈	1	4	yes	—
58	C₂×C₈	C₂²×C₄	1	2	yes	—
59	C₂×C₈	C₂×D₄	1	2	yes	—
60	C₂×C₈	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
62	C₄⋊C₄	—	2	4	no	—
63	C₂²⋊C₄	—	2	4	no	—
64	C₄²	—	2	2	no	—
65	C₄²	C₂×C₈	1	4	yes	—
66	C₁₆	—	2	8	no	—
67	C₂³	—	4	2	no	—
68	C₂³	#97	1	2	yes	—
69	C₂³	C₂²⋊C₈	1	2	yes	—
70	C₂³	C₄⋊C₈	1	2	yes	—
71	C₂³	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₄	—	2	4	no	—
73	C₂×C₄	—	4	4	no	—
74	C₂×C₄	#95	1	4	yes	—
75	C₂×C₄	C₄×C₈	1	4	yes	—
76	C₂×C₄	#97	1	2	yes	—
77	C₂×C₄	C₂²⋊C₈	1	2	yes	—
78	C₂×C₄	C₂²⋊C₈	1	4	yes	—
79	C₂×C₄	C₄⋊C₈	1	2	yes	—
80	C₂×C₄	C₄⋊C₈	1	4	yes	—
81	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
87	C₈	—	2	4	no	—
88	C₈	#95	1	4	yes	—
89	C₈	C₂×C₂²⋊C₄	1	2	yes	—
90	C₈	C₂×C₄⋊C₄	1	2	yes	—
91	C₂²	—	2	2	no	—
92	C₂²	—	2	4	no	—
93	C₂²	—	4	2	no	—
94	C₂²	—	4	2	no	—
95	C₂²	#336	1	4	yes	—
96	C₂²	C₂×#97	1	1	yes	characteristic
97	C₂²	C₂×C₂²⋊C₈	1	1	yes	characteristic
98	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
99	C₄	—	4	4	no	—
100	C₄	#336	1	4	yes	—
101	C₄	C₂×#95	1	1	yes	characteristic
102	C₄	C₂×C₄×C₈	1	1	yes	characteristic
103	C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
104	C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
105	C₂	—	2	1	no	—
106	C₂	—	2	4	no	—
107	C₂	—	2	1	no	—
108	C₂	C₂×#336	1	1	yes	characteristic
109	C₂	#2153	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×M₃₂

C₂

yes

maximal

C₂×#405

C₂

yes

maximal, characteristic

#1376

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

—

C₂×#97

C₄

yes

—

C₂×C₄×C₈

C₂²

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

Q₈

yes

—

C₂²×C₈

C₂³

yes

characteristic

#117

—

#117

C₈

yes

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×C₄

yes

—

M₃₂

—

M₃₂

—

M₃₂

C₂×C₄

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

C₄×C₈

—

C₂⁴

M₁₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄²

yes

—

C₂²×C₄

C₂²⋊C₄

yes

—

C₂²×C₄

C₄⋊C₄

yes

—

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

M₁₆

yes

characteristic

C₂²×C₄

M₁₆

yes

—

C₂²×C₄

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²

yes

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

C₂²⋊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

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×Q₈

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×C₈

yes

—

C₁₆

—

C₂³

—

C₂³

#97

yes

—

C₂³

C₂²⋊C₈

yes

—

C₂³

C₄⋊C₈

yes

—

C₂³

C₂×M₁₆

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

C₂²×C₈

yes

characteristic

C₂×C₄

C₂×M₁₆

yes

characteristic

C₈

—

C₈

#95

yes

—

C₈

C₂×C₂²⋊C₄

yes

—

C₈

C₂×C₄⋊C₄

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#336

yes

—

C₂²

C₂×#97

yes

characteristic

C₂²

C₂×C₂²⋊C₈

yes

characteristic

C₂²

C₂×C₄⋊C₈

yes

characteristic

C₄

—

100

C₄

#336

yes

—

101

C₄

C₂×#95

yes

characteristic

102

C₄

C₂×C₄×C₈

yes

characteristic

103

C₄

C₂×C₂²⋊C₈

yes

characteristic

104

C₄

C₂×C₄⋊C₈

yes

characteristic

105

C₂

—

106

C₂

—

107

C₂

—

108

C₂

C₂×#336

yes

characteristic

109

C₂

#2153

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₄
commutator	C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	C₂×C₄⋊C₈
prime radical	C₂⁴	M₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₈

C₂×D₄

commutator

C₄

C₂×C₄×C₈

Frattini, cosocle

C₂²×C₈

C₂³

socle

C₂²

C₂×C₄⋊C₈

prime radical

C₂⁴

M₁₆

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
3. #1376	C₂	semidirect
9. C₂×M₃₂	C₄	semidirect

H ≅ G/N

type

3. #1376

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
#1376	2

subgroup

index

C₂²×M₃₂

C₂×#405

#1376

Subgroups — #12549

Conjugacy classes of subgroups

Complements

Maximal subgroups