Group #12545 — subgroups

Subgroups — #12545

Order 256.

307 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 307 conjugacy classes of subgroups — too many to draw. Only its 183 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	1	yes	maximal, characteristic
2	C₂³×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#405	C₂	1	1	yes	maximal, characteristic
4	#1374	C₂	1	4	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	C₂×M₃₂	—	2	2	no	—
9	C₂×M₃₂	—	2	2	no	—
10	C₂×M₃₂	C₄	1	4	yes	complemented
11	C₂²×C₁₆	—	2	2	no	—
12	C₂²×C₁₆	—	2	2	no	—
13	C₂²×C₁₆	C₄	1	4	yes	complemented
14	C₂²×C₁₆	C₂²	1	2	yes	—
15	C₂²×C₁₆	C₂²	1	2	yes	—
16	#405	—	2	2	no	—
17	#405	—	2	2	no	—
18	C₂×#97	C₄	1	2	yes	—
19	C₂×C₄×C₈	C₂²	1	2	yes	—
20	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
21	C₂×M₁₆	—	2	2	no	—
22	C₂×M₁₆	—	2	2	no	—
23	C₂×M₁₆	C₈	1	4	yes	—
24	C₂²×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈	C₂×C₄	1	2	yes	—
27	C₂²×C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₈	D₄	1	2	yes	—
29	C₂²×C₈	D₄	1	2	yes	—
30	C₂²×C₈	D₄	1	2	yes	—
31	C₂²×C₈	Q₈	1	2	yes	—
32	C₂²×C₈	C₂³	1	1	yes	characteristic
33	#117	—	2	2	no	—
34	#117	—	2	2	no	—
35	C₂×C₄⋊C₄	C₈	1	2	yes	—
36	C₂×C₂²⋊C₄	C₈	1	2	yes	—
37	C₂×C₄²	C₂×C₄	1	2	yes	—
38	M₃₂	—	2	8	no	—
39	C₂×C₁₆	—	2	8	no	—
40	C₂×C₁₆	—	2	2	no	—
41	C₂×C₁₆	—	2	2	no	—
42	C₂×C₁₆	—	2	2	no	—
43	C₂×C₁₆	—	2	2	no	—
44	C₂×C₁₆	C₂×C₄	1	4	yes	—
45	C₂×C₁₆	C₂×C₄	1	4	yes	—
46	#97	—	2	4	no	—
47	C₄×C₈	—	2	2	no	—
48	C₄×C₈	—	2	2	no	—
49	C₂⁴	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄	C₄²	1	2	yes	—
51	C₂²×C₄	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄	C₂×C₈	1	4	yes	—
54	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄	C₂×C₈	1	2	yes	—
56	C₂²×C₄	M₁₆	1	2	yes	—
57	C₂²×C₄	M₁₆	1	2	yes	—
58	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
59	M₁₆	—	2	8	no	—
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	C₄²	1	2	yes	—
63	C₂×C₈	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈	C₂²⋊C₄	1	4	yes	—
65	C₂×C₈	C₂²⋊C₄	1	4	yes	—
66	C₂×C₈	C₄⋊C₄	1	2	yes	—
67	C₂×C₈	C₄⋊C₄	1	4	yes	—
68	C₂×C₈	C₄⋊C₄	1	4	yes	—
69	C₂×C₈	C₂×C₈	1	4	yes	—
70	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
76	C₄⋊C₄	—	2	4	no	—
77	C₂²⋊C₄	—	2	4	no	—
78	C₄²	—	2	2	no	—
79	C₄²	—	2	2	no	—
80	C₁₆	—	2	4	no	—
81	C₁₆	—	2	4	no	—
82	C₂³	—	2	4	no	—
83	C₂³	C₄×C₈	1	2	yes	—
84	C₂³	C₂²⋊C₈	1	2	yes	—
85	C₂³	C₄⋊C₈	1	2	yes	—
86	C₂³	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	2	8	no	—
89	C₂×C₄	#95	1	4	yes	—
90	C₂×C₄	C₄×C₈	1	2	yes	—
91	C₂×C₄	#97	1	4	yes	—
92	C₂×C₄	C₂²⋊C₈	1	2	yes	—
93	C₂×C₄	C₂²⋊C₈	1	4	yes	—
94	C₂×C₄	C₄⋊C₈	1	2	yes	—
95	C₂×C₄	C₄⋊C₈	1	4	yes	—
96	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
102	C₈	—	2	4	no	—
103	C₈	#95	1	4	yes	—
104	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂²	—	2	4	no	—
109	C₂²	—	2	4	no	—
110	C₂²	—	2	4	no	—
111	C₂²	#336	1	4	yes	—
112	C₂²	C₂×C₄×C₈	1	1	yes	characteristic
113	C₂²	C₂×C₂²⋊C₈	1	1	yes	characteristic
114	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
115	C₂²	#504	1	4	yes	—
116	C₄	—	4	4	no	—
117	C₄	#336	1	4	yes	—
118	C₄	C₂×#95	1	1	yes	characteristic
119	C₄	C₂×#97	1	1	yes	characteristic
120	C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
121	C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
122	C₂	—	2	4	no	—
123	C₂	C₂×#336	1	1	yes	characteristic
124	C₂	#2108	1	2	yes	—
125	C₂	#2113	1	2	yes	—
126	C₂	#2151	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×M₃₂

C₂

yes

maximal, characteristic

C₂³×C₁₆

C₂

yes

maximal, characteristic

C₂×#405

C₂

yes

maximal, characteristic

#1374

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

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

C₄

yes

complemented

C₂²×C₁₆

C₂²

yes

—

C₂²×C₁₆

C₂²

yes

—

#405

—

#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

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

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄

yes

—

C₂×C₁₆

C₂×C₄

yes

—

#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

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

yes

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×Q₈

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₁₆

—

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

100

C₂×C₄

C₂×M₁₆

yes

characteristic

101

C₂×C₄

C₂×M₁₆

yes

characteristic

102

C₈

—

103

C₈

#95

yes

—

104

C₈

C₂×C₂²⋊C₄

yes

characteristic

105

C₈

C₂×C₂²⋊C₄

yes

characteristic

106

C₈

C₂×C₄⋊C₄

yes

characteristic

107

C₈

C₂×C₄⋊C₄

yes

characteristic

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

#336

yes

—

112

C₂²

C₂×C₄×C₈

yes

characteristic

113

C₂²

C₂×C₂²⋊C₈

yes

characteristic

114

C₂²

C₂×C₄⋊C₈

yes

characteristic

115

C₂²

#504

yes

—

116

C₄

—

117

C₄

#336

yes

—

118

C₄

C₂×#95

yes

characteristic

119

C₄

C₂×#97

yes

characteristic

120

C₄

C₂×C₂²⋊C₈

yes

characteristic

121

C₄

C₂×C₄⋊C₈

yes

characteristic

122

C₂

—

123

C₂

C₂×#336

yes

characteristic

124

C₂

#2108

yes

—

125

C₂

#2113

yes

—

126

C₂

#2151

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
4. #1374	C₂	semidirect
10. C₂×M₃₂	C₄	semidirect
13. C₂²×C₁₆	C₄	semidirect

H ≅ G/N

type

4. #1374

C₂

semidirect

10. C₂×M₃₂

C₄

semidirect

13. C₂²×C₁₆

C₄

semidirect

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

subgroup	index
C₂²×M₃₂	2
C₂³×C₁₆	2
C₂×#405	2
#1374	2

subgroup

index

C₂²×M₃₂

C₂³×C₁₆

C₂×#405

#1374

Subgroups — #12545

Conjugacy classes of subgroups

Complements

Maximal subgroups