Group #12544 — subgroups

Subgroups — #12544

Order 256.

339 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 339 conjugacy classes of subgroups — too many to draw. Only its 191 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	C₂²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	#1374	C₂	1	4	yes	maximal, complemented
4	C₂³×C₈	C₄	1	1	yes	characteristic
5	C₂³×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₄²	C₄	1	1	yes	characteristic
7	C₂×M₃₂	—	2	4	no	—
8	C₂×M₃₂	—	2	4	no	—
9	C₂×M₃₂	C₄	1	8	yes	complemented
10	C₂²×C₁₆	C₂²	1	4	yes	—
11	C₂×C₄×C₈	—	2	4	no	—
12	C₂×C₄×C₈	C₄	1	2	yes	—
13	C₂×C₄×C₈	C₂²	1	2	yes	—
14	C₂³×C₄	C₈	1	2	yes	—
15	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈	—	2	2	no	—
17	C₂²×C₈	—	2	2	no	—
18	C₂²×C₈	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	C₂×C₄²	—	2	2	no	—
27	C₂×C₄²	—	2	2	no	—
28	C₂×C₄²	C₈	1	2	yes	—
29	C₂×C₄²	C₂×C₄	1	2	yes	—
30	M₃₂	—	2	16	no	—
31	C₂×C₁₆	—	2	4	no	—
32	C₂×C₁₆	—	2	4	no	—
33	C₂×C₁₆	C₂×C₄	1	8	yes	complemented
34	C₄×C₈	—	2	4	no	—
35	C₄×C₈	—	2	4	no	—
36	C₂⁴	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄	—	2	4	no	—
38	C₂²×C₄	—	2	8	no	—
39	C₂²×C₄	C₄²	1	2	yes	—
40	C₂²×C₄	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄	C₂×C₈	1	4	yes	—
43	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄	C₂×C₈	1	2	yes	—
45	C₂²×C₄	M₁₆	1	2	yes	—
46	C₂²×C₄	M₁₆	1	2	yes	—
47	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈	—	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	8	yes	—
54	C₂×C₈	C₄⋊C₄	1	2	yes	—
55	C₂×C₈	C₄⋊C₄	1	8	yes	—
56	C₂×C₈	C₂×C₈	1	4	yes	—
57	C₂×C₈	C₂²×C₄	1	2	yes	—
58	C₂×C₈	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈	C₂×D₄	1	2	yes	—
60	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
61	C₄²	—	2	4	no	—
62	C₄²	—	2	2	no	—
63	C₄²	—	2	2	no	—
64	C₁₆	—	2	8	no	—
65	C₂³	—	2	4	no	—
66	C₂³	C₄×C₈	1	2	yes	—
67	C₂³	C₂²⋊C₈	1	2	yes	—
68	C₂³	C₄⋊C₈	1	2	yes	—
69	C₂³	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄	—	2	16	no	—
71	C₂×C₄	—	2	4	no	—
72	C₂×C₄	—	2	4	no	—
73	C₂×C₄	#95	1	4	yes	—
74	C₂×C₄	C₄×C₈	1	2	yes	—
75	C₂×C₄	#97	1	4	yes	—
76	C₂×C₄	C₂²⋊C₈	1	2	yes	—
77	C₂×C₄	C₂²⋊C₈	1	4	yes	—
78	C₂×C₄	C₄⋊C₈	1	2	yes	—
79	C₂×C₄	C₄⋊C₈	1	4	yes	—
80	C₂×C₄	M₃₂	1	8	yes	complemented
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	C₂×M₁₆	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	4	no	—
92	C₂²	—	2	4	no	—
93	C₂²	—	2	4	no	—
94	C₂²	#336	1	4	yes	—
95	C₂²	C₂×C₄×C₈	1	1	yes	characteristic
96	C₂²	C₂×C₂²⋊C₈	1	1	yes	characteristic
97	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
98	C₂²	C₂×M₃₂	1	4	yes	—
99	C₄	—	2	8	no	—
100	C₄	#336	1	4	yes	—
101	C₄	C₂×#95	1	1	yes	characteristic
102	C₄	C₂×#97	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	4	no	—
106	C₂	C₂×#336	1	1	yes	characteristic
107	C₂	C₄×M₃₂	1	2	yes	—
108	C₂	#2112	1	2	yes	—
109	C₂	#2150	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×M₃₂

C₂

yes

maximal

C₂²×C₄×C₈

C₂

yes

maximal, characteristic

#1374

C₂

yes

maximal, complemented

C₂³×C₈

C₄

yes

characteristic

C₂³×C₈

C₂²

yes

characteristic

C₂²×C₄²

C₄

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

—

C₂×M₃₂

C₄

yes

complemented

C₂²×C₁₆

C₂²

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₄

yes

—

C₂×C₄×C₈

C₂²

yes

—

C₂³×C₄

C₈

yes

—

C₂³×C₄

C₂×C₄

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

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

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₈

yes

—

C₂×C₄²

C₂×C₄

yes

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄

yes

complemented

C₄×C₈

—

C₄×C₈

—

C₂⁴

C₂×C₈

yes

characteristic

C₂²×C₄

—

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

characteristic

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

M₁₆

yes

—

C₂²×C₄

M₁₆

yes

—

C₂²×C₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²

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

characteristic

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×Q₈

yes

characteristic

C₄²

—

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

—

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

M₃₂

yes

complemented

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

C₂×C₂²⋊C₄

yes

—

C₈

C₂×C₄⋊C₄

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#336

yes

—

C₂²

C₂×C₄×C₈

yes

characteristic

C₂²

C₂×C₂²⋊C₈

yes

characteristic

C₂²

C₂×C₄⋊C₈

yes

characteristic

C₂²

C₂×M₃₂

yes

—

C₄

—

100

C₄

#336

yes

—

101

C₄

C₂×#95

yes

characteristic

102

C₄

C₂×#97

yes

characteristic

103

C₄

C₂×C₂²⋊C₈

yes

characteristic

104

C₄

C₂×C₄⋊C₈

yes

characteristic

105

C₂

—

106

C₂

C₂×#336

yes

characteristic

107

C₂

C₄×M₃₂

yes

—

108

C₂

#2112

yes

—

109

C₂

#2150

yes

—

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂³
commutator	C₂²	C₂×C₄×C₈
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

C₂×C₄×C₈

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
3. #1374	C₂	semidirect
9. C₂×M₃₂	C₄	semidirect
33. C₂×C₁₆	C₂×C₄	semidirect
80. C₂×C₄	M₃₂	semidirect

H ≅ G/N

type

3. #1374

C₂

semidirect

9. C₂×M₃₂

C₄

semidirect

33. C₂×C₁₆

C₂×C₄

semidirect

80. C₂×C₄

M₃₂

semidirect

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

subgroup	index
C₂²×M₃₂	2
C₂²×C₄×C₈	2
#1374	2

subgroup

index

C₂²×M₃₂

C₂²×C₄×C₈

#1374

Subgroups — #12544

Conjugacy classes of subgroups

Complements

Maximal subgroups