Group #12914 — subgroups

Subgroups — #12914

Order 256.

210 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₂×#364	C₂	1	2	yes	maximal, complemented
3	#2112	C₂	1	1	yes	maximal, characteristic
4	C₈×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
5	#1376	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈	C₂²	1	1	yes	characteristic
7	C₂×M₃₂	—	2	1	no	—
8	C₂×M₃₂	—	2	4	no	—
9	C₂×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×M₃₂	C₂²	1	2	yes	—
12	C₂²×C₁₆	—	2	1	no	—
13	C₂×C₂²⋊C₈	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈	C₂²	1	2	yes	—
15	C₄×C₂²⋊C₄	C₄	1	1	yes	characteristic
16	#364	—	2	4	no	—
17	C₂²⋊C₁₆	—	2	1	no	—
18	C₂²⋊C₁₆	—	2	1	no	—
19	#336	C₄	1	2	yes	—
20	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈	—	2	2	no	—
22	C₂²×C₈	—	2	2	no	—
23	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₈	D₄	1	1	yes	characteristic
26	C₂²×C₈	D₄	1	1	yes	characteristic
27	C₂²×C₈	Q₈	1	1	yes	characteristic
28	C₂²×C₈	Q₈	1	1	yes	characteristic
29	C₂²×C₈	C₂³	1	1	yes	characteristic
30	C₂×C₂²⋊C₄	—	2	1	no	—
31	C₂×C₄²	C₂×C₄	1	2	yes	—
32	M₃₂	—	2	4	no	—
33	M₃₂	—	2	2	no	—
34	M₃₂	—	2	2	no	—
35	M₃₂	—	2	2	no	—
36	M₃₂	D₄	1	4	yes	—
37	C₂×C₁₆	—	2	4	no	—
38	C₂×C₁₆	—	2	2	no	—
39	C₂×C₁₆	—	2	1	no	—
40	C₂×C₁₆	—	2	1	no	—
41	C₂²⋊C₈	C₈	1	2	yes	—
42	C₂²⋊C₈	C₈	1	2	yes	—
43	C₄×C₈	—	2	2	no	—
44	C₄×C₈	—	2	2	no	—
45	#95	—	2	1	no	—
46	C₂⁴	M₁₆	1	1	yes	characteristic
47	C₂²×C₄	—	2	2	no	—
48	C₂²×C₄	—	2	2	no	—
49	C₂²×C₄	—	2	2	no	—
50	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄	M₁₆	1	1	yes	characteristic
55	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈	—	2	2	no	—
57	C₂×C₈	—	2	2	no	—
58	C₂×C₈	—	2	2	no	—
59	C₂×C₈	—	2	1	no	—
60	C₂×C₈	—	2	1	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	1	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₈	C₂×C₈	1	4	yes	—
68	C₂×C₈	C₂×D₄	1	2	yes	—
69	C₂×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₈	#41	1	2	yes	—
72	C₂²⋊C₄	—	2	4	no	—
73	C₄²	—	2	2	no	—
74	C₄²	—	2	2	no	—
75	C₁₆	—	2	4	no	—
76	C₁₆	—	4	1	no	—
77	C₁₆	—	4	1	no	—
78	C₂³	—	2	2	no	—
79	C₂³	—	2	2	no	—
80	C₂³	—	4	1	no	—
81	C₂³	C₄⋊C₈	1	1	yes	characteristic
82	C₂³	C₄⋊C₈	1	1	yes	characteristic
83	C₂³	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	4	2	no	—
94	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
95	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
96	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄	C₄×D₄	1	2	yes	—
98	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄	#131	1	2	yes	—
100	C₈	—	2	1	no	—
101	C₈	—	2	1	no	—
102	C₈	—	2	4	no	—
103	C₈	#120	1	1	yes	characteristic
104	C₈	C₂²⋊Q₈	1	2	yes	—
105	C₈	#123	1	1	yes	characteristic
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	—	2	1	no	—
110	C₂²	—	2	1	no	—
111	C₂²	—	2	1	no	—
112	C₂²	—	2	1	no	—
113	C₂²	—	4	4	no	—
114	C₂²	#364	1	2	yes	complemented
115	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
116	C₂²	#435	1	2	yes	—
117	C₄	—	2	1	no	—
118	C₄	—	2	1	no	—
119	C₄	—	4	4	no	—
120	C₄	#385	1	1	yes	characteristic
121	C₄	#424	1	1	yes	characteristic
122	C₄	C₈×D₄	1	2	yes	—
123	C₂	—	2	2	no	—
124	C₂	—	2	2	no	—
125	C₂	—	4	1	no	—
126	C₂	#1850	1	1	yes	characteristic
127	C₂	C₂×#364	1	1	yes	characteristic
128	C₂	#2153	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×M₃₂

C₂

yes

maximal, characteristic

C₂×#364

C₂

yes

maximal, complemented

#2112

C₂

yes

maximal, characteristic

C₈×C₂²⋊C₄

C₂

yes

maximal, characteristic

#1376

C₂

yes

maximal, complemented

C₂³×C₈

C₂²

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

—

C₂×M₃₂

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

—

C₂²×C₁₆

—

C₂×C₂²⋊C₈

C₄

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

C₄×C₂²⋊C₄

C₄

yes

characteristic

#364

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

#336

C₄

yes

—

C₂³×C₄

C₂×C₄

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

Q₈

yes

characteristic

C₂²×C₈

Q₈

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×C₄

yes

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

D₄

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂²⋊C₈

C₈

yes

—

C₂²⋊C₈

C₈

yes

—

C₄×C₈

—

C₄×C₈

—

#95

—

C₂⁴

M₁₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄⋊C₄

yes

characteristic

C₂²×C₄

C₄⋊C₄

yes

characteristic

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

M₁₆

yes

characteristic

C₂²×C₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄

yes

characteristic

C₂×C₈

C₄⋊C₄

yes

characteristic

C₂×C₈

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×Q₈

yes

characteristic

C₂×C₈

#41

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₄⋊C₈

yes

characteristic

C₂³

C₄⋊C₈

yes

characteristic

C₂³

C₂×M₁₆

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊C₈

yes

characteristic

C₂×C₄

C₄⋊C₈

yes

characteristic

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

C₂²×C₈

yes

characteristic

C₂×C₄

#131

yes

—

100

C₈

—

101

C₈

—

102

C₈

—

103

C₈

#120

yes

characteristic

104

C₈

C₂²⋊Q₈

yes

—

105

C₈

#123

yes

characteristic

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

#364

yes

complemented

115

C₂²

C₂×C₄⋊C₈

yes

characteristic

116

C₂²

#435

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

#385

yes

characteristic

121

C₄

#424

yes

characteristic

122

C₄

C₈×D₄

yes

—

123

C₂

—

124

C₂

—

125

C₂

—

126

C₂

#1850

yes

characteristic

127

C₂

C₂×#364

yes

characteristic

128

C₂

#2153

yes

characteristic

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

C₂×C₄

C₂²×C₈

Frattini, cosocle

C₂²×C₈

C₂³

socle

C₂²

C₂×C₄⋊C₈

prime radical

C₂⁴

M₁₆

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. C₂×#364	C₂	semidirect
5. #1376	C₂	semidirect
114. C₂²	#364	semidirect

H ≅ G/N

type

2. C₂×#364

C₂

semidirect

5. #1376

C₂

semidirect

114. C₂²

#364

semidirect

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

subgroup	index
C₂²×M₃₂	2
C₂×#364	2
#2112	2
C₈×C₂²⋊C₄	2
#1376	2

subgroup

index

C₂²×M₃₂

C₂×#364

#2112

C₈×C₂²⋊C₄

#1376

Subgroups — #12914

Conjugacy classes of subgroups

Complements

Maximal subgroups