Group #10664 — subgroups

Subgroups — #10664

Order 256.

152 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 53 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	#1985	C₂	1	1	yes	maximal, characteristic
2	#1838	C₂	1	1	yes	maximal, characteristic
3	#1571	C₂	1	1	yes	maximal, characteristic
4	#1570	C₂	1	1	yes	maximal, characteristic
5	#1487	C₂	1	1	yes	maximal, characteristic
6	#1487	C₂	1	1	yes	maximal, characteristic
7	#1275	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
9	#431	C₂²	1	1	yes	characteristic
10	#431	C₂²	1	1	yes	characteristic
11	#423	C₂²	1	1	yes	characteristic
12	#423	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×#97	C₂²	1	1	yes	characteristic
15	#341	—	2	2	no	—
16	#336	—	2	2	no	—
17	#335	—	2	2	no	—
18	#334	—	2	2	no	—
19	Q₈⋊C₈	—	2	2	no	—
20	Q₈⋊C₈	—	2	2	no	—
21	C₂²×Q₈	Q₈	1	1	yes	characteristic
22	C₂×M₁₆	—	2	1	no	—
23	C₂×M₁₆	—	2	1	no	—
24	C₂²×C₈	—	2	1	no	—
25	C₂²×C₈	D₄	1	2	yes	—
26	C₄×Q₈	—	2	1	no	—
27	C₄×Q₈	—	2	1	no	—
28	C₄×Q₈	—	2	1	no	—
29	C₄×Q₈	—	2	1	no	—
30	C₂×C₄⋊C₄	—	2	1	no	—
31	C₂×C₄⋊C₄	Q₈	1	1	yes	characteristic
32	C₂×C₄²	—	2	1	no	—
33	C₂×C₄²	C₂³	1	1	yes	characteristic
34	C₄⋊C₈	—	2	1	no	—
35	C₄⋊C₈	—	2	1	no	—
36	C₄⋊C₈	—	2	1	no	—
37	C₄⋊C₈	—	2	1	no	—
38	C₄⋊C₈	—	2	1	no	—
39	C₄⋊C₈	—	2	1	no	—
40	C₄⋊C₈	—	2	1	no	—
41	C₄⋊C₈	—	2	1	no	—
42	C₂²⋊C₈	—	2	1	no	—
43	C₂²⋊C₈	—	2	1	no	—
44	#97	—	2	1	no	—
45	#97	—	2	1	no	—
46	C₄×C₈	D₄	1	2	yes	—
47	C₄×C₈	D₄	1	2	yes	—
48	C₂×Q₈	—	2	1	no	—
49	C₂×Q₈	—	2	1	no	—
50	C₂×Q₈	—	4	2	no	—
51	C₂²×C₄	—	2	1	no	—
52	C₂²×C₄	—	2	1	no	—
53	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄	#41	1	1	yes	characteristic
56	M₁₆	—	4	2	no	—
57	M₁₆	—	4	2	no	—
58	C₂×C₈	—	2	2	no	—
59	C₂×C₈	—	2	2	no	—
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	4	1	no	—
64	C₂×C₈	—	4	1	no	—
65	C₂×C₈	—	4	1	no	—
66	C₂×C₈	—	4	2	no	—
67	C₄⋊C₄	—	2	1	no	—
68	C₄⋊C₄	—	2	1	no	—
69	C₄⋊C₄	—	4	1	no	—
70	C₄⋊C₄	—	4	1	no	—
71	C₄²	—	4	1	no	—
72	C₄²	—	4	1	no	—
73	C₄²	C₂×D₄	1	1	yes	characteristic
74	C₄²	C₂×D₄	1	1	yes	characteristic
75	C₄²	#41	1	1	yes	characteristic
76	C₄²	#41	1	1	yes	characteristic
77	C₂³	C₂²⋊Q₈	1	1	yes	characteristic
78	Q₈	—	2	2	no	—
79	Q₈	—	2	2	no	—
80	Q₈	—	4	1	no	—
81	Q₈	—	4	1	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	4	1	no	—
84	C₂×C₄	—	4	1	no	—
85	C₂×C₄	—	4	1	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	—	4	1	no	—
89	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
90	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂×C₄	#123	1	1	yes	characteristic
92	C₂×C₄	#123	1	1	yes	characteristic
93	C₂×C₄	#123	1	1	yes	characteristic
94	C₂×C₄	#127	1	1	yes	characteristic
95	C₂×C₄	#135	1	2	yes	—
96	C₂×C₄	#137	1	2	yes	—
97	C₈	—	4	2	no	—
98	C₈	—	4	2	no	—
99	C₈	—	4	2	no	—
100	C₈	—	8	1	no	—
101	C₈	—	8	1	no	—
102	C₈	—	8	1	no	—
103	C₂²	—	2	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	#399	1	1	yes	characteristic
106	C₂²	#479	1	2	yes	—
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	4	2	no	—
112	C₄	—	8	1	no	—
113	C₄	#485	1	2	yes	—
114	C₄	#495	1	1	yes	characteristic
115	C₄	#497	1	1	yes	characteristic
116	C₂	—	2	2	no	—
117	C₂	#2091	1	1	yes	characteristic
118	C₂	#2097	1	1	yes	characteristic
119	C₂	#2098	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1985

C₂

yes

maximal, characteristic

#1838

C₂

yes

maximal, characteristic

#1571

C₂

yes

maximal, characteristic

#1570

C₂

yes

maximal, characteristic

#1487

C₂

yes

maximal, characteristic

#1487

C₂

yes

maximal, characteristic

#1275

C₂

yes

maximal, characteristic

C₂×C₄×Q₈

C₂²

yes

characteristic

#431

C₂²

yes

characteristic

#431

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×#97

C₂²

yes

—

—

—

—

Q₈⋊C₈

—

Q₈⋊C₈

—

C₂²×Q₈

Q₈

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Q₈

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

characteristic

C₂²×C₄

#41

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

C₄²

yes

C₄²

yes

C₂³

C₂²⋊Q₈

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Q₈

yes

characteristic

C₂×C₄

C₂²⋊Q₈

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

#137

yes

—

C₈

—

C₈

—

C₈

—

100

C₈

—

101

C₈

—

102

C₈

—

103

C₂²

—

104

C₂²

—

105

C₂²

#399

yes

characteristic

106

C₂²

#479

yes

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

#485

yes

—

114

C₄

#495

yes

characteristic

115

C₄

#497

yes

characteristic

116

C₂

—

117

C₂

#2091

yes

characteristic

118

C₂

#2097

yes

characteristic

119

C₂

#2098

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#399
prime radical	C₂³	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₄

#127

commutator, Frattini, cosocle

C₂×C₄²

C₂³

socle

C₂²

#399

prime radical

C₂³

C₂²⋊Q₈

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

subgroup	index
#1985	2
#1838	2
#1571	2
#1570	2
#1487	2
#1275	2

subgroup

index

Subgroups — #10664

Conjugacy classes of subgroups

Maximal subgroups