Group #10867 — subgroups

Subgroups — #10867

Order 256.

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

normal subgroups only

This group has 357 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×#108	C₂	1	2	yes	maximal
2	C₂×#405	C₂	1	1	yes	maximal, characteristic
3	#1742	C₂	1	2	yes	maximal
4	C₂×#338	C₂	1	2	yes	maximal
5	C₂²×M₁₆	C₂²	1	4	yes	—
6	C₂²×M₁₆	C₂²	1	1	yes	characteristic
7	C₂³×C₈	C₂²	1	1	yes	characteristic
8	C₂×#117	C₂²	1	1	yes	characteristic
9	C₂×#108	—	2	4	no	—
10	C₂×#108	—	2	4	no	—
11	C₂×#108	C₄	1	8	yes	complemented
12	#405	—	2	4	no	—
13	C₂×#97	C₄	1	2	yes	—
14	C₂×C₄×C₈	C₄	1	2	yes	—
15	#343	—	2	4	no	—
16	#338	—	2	4	no	—
17	C₂³×C₄	C₂³	1	1	yes	characteristic
18	C₂×M₁₆	—	2	2	no	—
19	C₂×M₁₆	—	2	2	no	—
20	C₂×M₁₆	—	2	4	no	—
21	C₂×M₁₆	—	2	4	no	—
22	C₂×M₁₆	—	2	4	no	—
23	C₂×M₁₆	—	2	4	no	—
24	C₂×M₁₆	—	2	4	no	—
25	C₂×M₁₆	C₂×C₄	1	8	yes	—
26	C₂×M₁₆	D₄	1	2	yes	—
27	C₂×M₁₆	Q₈	1	2	yes	—
28	C₂²×C₈	—	2	4	no	—
29	C₂²×C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈	C₂×C₄	1	4	yes	—
33	C₂²×C₈	D₄	1	4	yes	—
34	C₂²×C₈	D₄	1	2	yes	—
35	C₂²×C₈	Q₈	1	2	yes	—
36	#117	—	2	2	no	—
37	#117	—	2	2	no	—
38	C₂×C₄⋊C₄	D₄	1	2	yes	—
39	C₂×C₂²⋊C₄	Q₈	1	2	yes	—
40	C₂×C₄²	C₂×C₄	1	2	yes	—
41	#108	—	2	16	no	—
42	#97	—	2	4	no	—
43	C₄×C₈	—	2	4	no	—
44	C₂⁴	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄	C₂²×C₄	1	2	yes	—
48	C₂²×C₄	C₂×D₄	1	2	yes	—
49	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄	#41	1	2	yes	—
55	C₂²×C₄	#41	1	2	yes	—
56	M₁₆	—	2	4	no	—
57	M₁₆	—	2	4	no	—
58	M₁₆	—	2	8	no	—
59	M₁₆	—	2	8	no	—
60	M₁₆	—	4	4	no	—
61	M₁₆	—	4	4	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	2	8	no	—
65	C₂×C₈	—	4	4	no	—
66	C₂×C₈	—	4	4	no	—
67	C₂×C₈	C₄²	1	2	yes	—
68	C₂×C₈	C₄²	1	2	yes	—
69	C₂×C₈	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈	C₂²⋊C₄	1	2	yes	—
71	C₂×C₈	C₂²⋊C₄	1	8	yes	—
72	C₂×C₈	C₄⋊C₄	1	4	yes	—
73	C₂×C₈	C₄⋊C₄	1	2	yes	—
74	C₂×C₈	C₄⋊C₄	1	2	yes	—
75	C₂×C₈	C₄⋊C₄	1	8	yes	—
76	C₄⋊C₄	—	2	4	no	—
77	C₂²⋊C₄	—	2	4	no	—
78	C₄²	—	2	2	no	—
79	C₄²	—	2	2	no	—
80	C₂³	—	4	2	no	—
81	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂³	C₄×Q₈	1	2	yes	—
83	C₂³	C₂²⋊Q₈	1	2	yes	—
84	C₂³	C₄⋊Q₈	1	1	yes	characteristic
85	C₂³	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	4	4	no	—
88	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
89	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄	C₂×C₂²⋊C₄	1	2	yes	—
91	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄	#117	1	1	yes	characteristic
95	C₂×C₄	#117	1	1	yes	characteristic
96	C₂×C₄	C₄×D₄	1	2	yes	—
97	C₂×C₄	C₄×D₄	1	2	yes	—
98	C₂×C₄	C₄×Q₈	1	2	yes	—
99	C₂×C₄	#121	1	2	yes	—
100	C₂×C₄	#121	1	2	yes	—
101	C₂×C₄	C₂²⋊Q₈	1	2	yes	—
102	C₂×C₄	#124	1	2	yes	—
103	C₂×C₄	#125	1	2	yes	—
104	C₂×C₄	#127	1	1	yes	characteristic
105	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
106	C₈	—	2	4	no	—
107	C₈	—	4	8	no	—
108	C₈	#95	1	4	yes	—
109	C₈	#95	1	4	yes	—
110	C₂²	—	2	2	no	—
111	C₂²	—	2	4	no	—
112	C₂²	—	4	2	no	—
113	C₂²	—	4	1	no	—
114	C₂²	—	4	1	no	—
115	C₂²	C₄×C₄⋊C₄	1	1	yes	characteristic
116	C₂²	#380	1	1	yes	characteristic
117	C₂²	#384	1	1	yes	characteristic
118	C₂²	#389	1	2	yes	—
119	C₂²	#391	1	2	yes	—
120	C₄	—	4	4	no	—
121	C₄	C₂×#95	1	1	yes	characteristic
122	C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
123	C₄	#380	1	1	yes	characteristic
124	C₄	#382	1	1	yes	characteristic
125	C₄	#389	1	2	yes	—
126	C₄	#390	1	2	yes	—
127	C₂	—	2	1	no	—
128	C₂	—	2	1	no	—
129	C₂	—	2	2	no	—
130	C₂	—	2	2	no	—
131	C₂	C₄⋊#95	1	1	yes	characteristic
132	C₂	#1813	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#108

C₂

yes

maximal

C₂×#405

C₂

yes

maximal, characteristic

#1742

C₂

yes

maximal

C₂×#338

C₂

yes

maximal

C₂²×M₁₆

C₂²

yes

—

C₂²×M₁₆

C₂²

yes

characteristic

C₂³×C₈

C₂²

yes

characteristic

C₂×#117

C₂²

yes

—

—

C₄

yes

complemented

#405

—

C₂×#97

C₄

yes

—

C₂×C₄×C₈

C₄

yes

—

#343

—

#338

—

C₂³×C₄

C₂³

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×C₄

yes

—

C₂×M₁₆

D₄

yes

—

C₂×M₁₆

Q₈

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

Q₈

yes

—

#117

—

#117

—

C₂×C₄⋊C₄

D₄

yes

—

C₂×C₂²⋊C₄

Q₈

yes

—

C₂×C₄²

C₂×C₄

yes

—

#108

—

#97

—

C₄×C₈

—

C₂⁴

C₂×Q₈

yes

characteristic

C₂²×C₄

C₄⋊C₄

yes

—

C₂²×C₄

yes

characteristic

C₂²×C₄

yes

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

characteristic

C₂²×C₄

#41

yes

—

C₂²×C₄

#41

yes

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²

yes

—

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

C₄⋊C₄

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₂³

—

C₂³

C₂×C₄⋊C₄

yes

characteristic

C₂³

C₄×Q₈

yes

—

C₂³

C₂²⋊Q₈

yes

—

C₂³

C₄⋊Q₈

yes

characteristic

C₂³

C₄⋊Q₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄²

yes

characteristic

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

C₂×C₂²⋊C₄

yes

—

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

C₂×C₄⋊C₄

yes

—

C₂×C₄

C₂×C₄⋊C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

C₄×Q₈

yes

—

C₂×C₄

#121

yes

—

100

C₂×C₄

#121

yes

—

101

C₂×C₄

C₂²⋊Q₈

yes

—

102

C₂×C₄

#124

yes

—

103

C₂×C₄

#125

yes

—

104

C₂×C₄

#127

yes

characteristic

105

C₂×C₄

C₄⋊Q₈

yes

characteristic

106

C₈

—

107

C₈

—

108

C₈

#95

yes

—

109

C₈

#95

yes

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₄×C₄⋊C₄

yes

characteristic

116

C₂²

#380

yes

characteristic

117

C₂²

#384

yes

characteristic

118

C₂²

#389

yes

—

119

C₂²

#391

yes

—

120

C₄

—

121

C₄

C₂×#95

yes

characteristic

122

C₄

C₄×C₄⋊C₄

yes

characteristic

123

C₄

#380

yes

characteristic

124

C₄

#382

yes

characteristic

125

C₄

#389

yes

—

126

C₄

#390

yes

—

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

—

131

C₂

C₄⋊#95

yes

characteristic

132

C₂

#1813

yes

—

	subgroup	G/H
center	C₂×C₄	#117
commutator	C₂×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₄	C₂³
socle	C₂²	#380
prime radical	C₂⁴	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#117

commutator

C₂×C₄

C₂×C₄²

Frattini, cosocle

C₂³×C₄

C₂³

socle

C₂²

#380

prime radical

C₂⁴

C₂×Q₈

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ G

LECS

G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
11. C₂×#108	C₄	semidirect

H ≅ G/N

type

11. C₂×#108

C₄

semidirect

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

subgroup	index
C₂²×#108	2
C₂×#405	2
#1742	2
C₂×#338	2

subgroup

index

C₂²×#108

C₂×#405

#1742

C₂×#338

Subgroups — #10867

Conjugacy classes of subgroups

Complements

Maximal subgroups