Group #135646 — subgroups

Subgroups — #135646

Order 512.

256 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 134 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	#12900	C₂	1	2	yes	maximal
2	#7841	C₂	1	1	yes	maximal, characteristic
3	#7152	C₂	1	4	yes	maximal
4	C₂×#364	—	2	2	no	—
5	C₂×#364	—	2	2	no	—
6	#2150	C₂²	1	4	yes	—
7	#1744	C₄	1	2	yes	—
8	C₄²×C₈	C₂²	1	1	yes	characteristic
9	C₂×#322	C₂²	1	2	yes	—
10	C₂×C₈²	C₄	1	1	yes	characteristic
11	C₂×C₈²	C₄	1	1	yes	characteristic
12	C₂×M₃₂	—	2	4	no	—
13	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
15	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₈	D₄	1	1	yes	characteristic
17	C₂×C₄×C₈	D₄	1	1	yes	characteristic
18	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
19	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
21	C₄³	C₂×C₄	1	1	yes	characteristic
22	#364	—	4	4	no	—
23	#364	—	4	4	no	—
24	C₄⋊C₁₆	—	2	8	no	—
25	#336	—	2	4	no	—
26	#322	—	2	4	no	—
27	C₈²	—	2	2	no	—
28	C₈²	C₈	1	4	yes	—
29	C₂²×C₈	—	2	2	no	—
30	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₈	M₁₆	1	4	yes	—
33	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄²	—	2	2	no	—
36	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
39	M₃₂	—	4	4	no	—
40	M₃₂	—	4	4	no	—
41	C₂×C₁₆	—	4	4	no	—
42	C₄×C₈	—	2	2	no	—
43	C₄×C₈	—	2	2	no	—
44	C₄×C₈	—	2	2	no	—
45	C₄×C₈	—	2	2	no	—
46	C₄×C₈	—	2	2	no	—
47	C₄×C₈	C₂×C₈	1	4	yes	—
48	C₄×C₈	C₂×C₈	1	2	yes	—
49	C₄×C₈	M₁₆	1	2	yes	—
50	C₄×C₈	D₈	1	2	yes	—
51	C₄×C₈	SD₁₆	1	4	yes	—
52	C₄×C₈	Q₁₆	1	2	yes	—
53	C₄×C₈	#41	1	4	yes	—
54	C₂²×C₄	—	2	1	no	—
55	C₂²×C₄	—	2	1	no	—
56	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄	C₂×M₁₆	1	2	yes	—
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	—	2	1	no	—
60	C₂×C₈	—	2	4	no	—
61	C₂×C₈	—	2	4	no	—
62	C₂×C₈	—	4	4	no	—
63	C₂×C₈	—	4	4	no	—
64	C₂×C₈	C₄⋊C₈	1	2	yes	—
65	C₂×C₈	C₄⋊C₈	1	2	yes	—
66	C₂×C₈	#106	1	2	yes	—
67	C₂×C₈	#107	1	2	yes	—
68	C₂×C₈	#108	1	2	yes	—
69	C₂×C₈	#108	1	2	yes	—
70	C₂×C₈	#124	1	2	yes	—
71	C₂×C₈	#125	1	2	yes	—
72	C₂×C₈	C₂×D₈	1	1	yes	characteristic
73	C₂×C₈	C₂×SD₁₆	1	2	yes	—
74	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
75	C₄²	—	2	1	no	—
76	C₄²	—	2	1	no	—
77	C₄²	—	4	4	no	—
78	C₄²	#106	1	2	yes	—
79	C₄²	#107	1	2	yes	—
80	C₄²	#117	1	2	yes	—
81	C₄²	C₂²×C₈	1	1	yes	characteristic
82	C₄²	C₂×M₁₆	1	1	yes	characteristic
83	C₁₆	—	8	4	no	—
84	C₂³	#423	1	1	yes	characteristic
85	C₂×C₄	—	2	4	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	#382	1	1	yes	characteristic
91	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
92	C₂×C₄	C₂×#106	1	1	yes	characteristic
93	C₂×C₄	C₂×#107	1	1	yes	characteristic
94	C₂×C₄	C₂×#108	1	1	yes	characteristic
95	C₂×C₄	C₂×#108	1	1	yes	characteristic
96	C₂×C₄	#431	1	2	yes	—
97	C₂×C₄	#432	1	2	yes	—
98	C₈	—	2	2	no	—
99	C₈	—	4	4	no	—
100	C₈	—	4	4	no	—
101	C₈	#334	1	4	yes	—
102	C₈	#335	1	4	yes	—
103	C₈	#486	1	2	yes	—
104	C₈	#487	1	2	yes	—
105	C₂²	—	2	2	no	—
106	C₂²	#1566	1	1	yes	characteristic
107	C₂²	#1567	1	1	yes	characteristic
108	C₂²	#1831	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	—	4	2	no	—
111	C₄	—	4	2	no	—
112	C₄	C₂×#334	1	1	yes	characteristic
113	C₄	C₂×#335	1	1	yes	characteristic
114	C₄	#1834	1	1	yes	characteristic
115	C₄	#1837	1	1	yes	characteristic
116	C₄	#2166	1	4	yes	—
117	C₂	—	2	1	no	—
118	C₂	—	2	1	no	—
119	C₂	#8784	1	1	yes	characteristic
120	C₂	#12881	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#12900

C₂

yes

maximal

#7841

C₂

yes

maximal, characteristic

#7152

C₂

yes

—

—

C₂²

yes

—

#1744

C₄

yes

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#322

C₂²

yes

—

C₂×C₈²

C₄

yes

characteristic

C₂×C₈²

C₄

yes

characteristic

C₂×M₃₂

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

Q₈

yes

characteristic

C₂×C₄×C₈

Q₈

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

C₂×C₄

yes

characteristic

#364

—

#364

—

C₄⋊C₁₆

—

#336

—

#322

—

C₈²

—

C₈²

C₈

yes

—

C₂²×C₈

—

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

M₁₆

yes

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₂²×C₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

D₈

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

#41

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

C₂×M₁₆

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

#106

yes

—

C₂×C₈

#107

yes

—

C₂×C₈

#108

yes

—

C₂×C₈

#108

yes

—

C₂×C₈

#124

yes

—

C₂×C₈

#125

yes

—

C₂×C₈

C₂×D₈

yes

characteristic

C₂×C₈

C₂×SD₁₆

yes

—

C₂×C₈

C₂×Q₁₆

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₄²

#106

yes

—

C₄²

#107

yes

—

C₄²

#117

yes

—

C₄²

C₂²×C₈

yes

characteristic

C₄²

C₂×M₁₆

yes

characteristic

C₁₆

—

C₂³

#423

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#382

yes

characteristic

C₂×C₄

C₂×C₄⋊C₈

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

#432

yes

—

C₈

—

C₈

—

100

C₈

—

101

C₈

#334

yes

—

102

C₈

#335

yes

—

103

C₈

#486

yes

—

104

C₈

#487

yes

—

105

C₂²

—

106

C₂²

#1566

yes

characteristic

107

C₂²

#1567

yes

characteristic

108

C₂²

#1831

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

C₂×#334

yes

characteristic

113

C₄

C₂×#335

yes

characteristic

114

C₄

#1834

yes

characteristic

115

C₄

#1837

yes

characteristic

116

C₄

#2166

yes

—

117

C₂

—

118

C₂

—

119

C₂

#8784

yes

characteristic

120

C₂

#12881

yes

—

	subgroup	G/H
center	C₂×C₄	#382
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1831
prime radical	C₂³	#423
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#382

commutator

C₄²

C₂²×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#12900	2
#7841	2
#7152	2

subgroup

index

#12900

#7841

#7152

Subgroups — #135646

Conjugacy classes of subgroups

Maximal subgroups