Group #93253 — subgroups

Subgroups — #93253

Order 512.

191 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 191 conjugacy classes of subgroups — too many to draw. Only its 88 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	#12593	C₂	1	1	yes	maximal, characteristic
2	#8137	C₂	1	1	yes	maximal, characteristic
3	#8137	C₂	1	1	yes	maximal, characteristic
4	C₄×M₃₂	C₄	1	2	yes	—
5	#1726	C₂²	1	1	yes	characteristic
6	#1450	C₄	1	2	yes	—
7	#1450	C₄	1	2	yes	—
8	#1375	—	2	1	no	—
9	#1375	—	2	1	no	—
10	#1275	—	2	1	no	—
11	#1275	—	2	1	no	—
12	#1274	—	2	1	no	—
13	#1274	—	2	1	no	—
14	C₂×M₃₂	—	2	2	no	—
15	#431	—	2	2	no	—
16	#431	—	2	2	no	—
17	C₂×#97	D₄	1	1	yes	characteristic
18	C₂×#97	D₄	1	1	yes	characteristic
19	C₂×#97	Q₈	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
23	#376	D₄	1	1	yes	characteristic
24	#346	C₈	1	4	yes	complemented
25	C₄×C₁₆	—	2	2	no	—
26	#322	—	2	2	no	—
27	#322	C₈	1	4	yes	complemented
28	C₈²	—	2	2	no	—
29	C₈²	C₈	1	4	yes	complemented
30	C₂²×C₈	C₄²	1	1	yes	characteristic
31	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
37	M₃₂	—	4	2	no	—
38	M₃₂	—	4	2	no	—
39	C₂×C₁₆	—	2	4	no	—
40	C₄⋊C₈	—	4	2	no	—
41	C₄⋊C₈	—	4	2	no	—
42	C₂²⋊C₈	—	4	2	no	—
43	C₂²⋊C₈	—	4	2	no	—
44	#97	—	4	1	no	—
45	#97	—	4	1	no	—
46	#97	—	4	1	no	—
47	C₄×C₈	—	2	4	no	—
48	C₄×C₈	—	2	4	no	—
49	C₄×C₈	C₂×C₈	1	2	yes	—
50	C₄×C₈	C₂×C₈	1	2	yes	—
51	C₄×C₈	C₂×C₈	1	2	yes	—
52	C₄×C₈	M₁₆	1	2	yes	—
53	C₄×C₈	M₁₆	1	2	yes	—
54	C₄×C₈	M₁₆	1	2	yes	—
55	#95	—	4	1	no	—
56	C₂²×C₄	—	2	1	no	—
57	C₂²×C₄	—	2	1	no	—
58	C₂²×C₄	—	2	1	no	—
59	C₂²×C₄	#95	1	1	yes	characteristic
60	C₂×C₈	—	2	1	no	—
61	C₂×C₈	—	2	1	no	—
62	C₂×C₈	—	2	1	no	—
63	C₂×C₈	—	2	1	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	—	2	1	no	—
66	C₂×C₈	—	4	4	no	—
67	C₂×C₈	—	4	4	no	—
68	C₂×C₈	C₄×C₈	1	1	yes	characteristic
69	C₂×C₈	C₄×C₈	1	1	yes	characteristic
70	C₂×C₈	C₄×C₈	1	1	yes	characteristic
71	C₂×C₈	#97	1	1	yes	characteristic
72	C₂×C₈	#97	1	1	yes	characteristic
73	C₂×C₈	#97	1	1	yes	characteristic
74	C₂×C₈	C₂²⋊C₈	1	2	yes	—
75	C₂×C₈	C₄⋊C₈	1	2	yes	—
76	C₂×C₈	C₄⋊C₈	1	2	yes	—
77	C₄²	—	2	1	no	—
78	C₄²	—	2	1	no	—
79	C₄²	—	2	1	no	—
80	C₄²	C₂²⋊C₈	1	2	yes	—
81	C₄²	C₂²⋊C₈	1	2	yes	—
82	C₄²	C₄⋊C₈	1	2	yes	—
83	C₁₆	—	4	4	no	—
84	C₂³	H(3,ℤ/4)	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	8	1	no	—
89	C₂×C₄	—	8	1	no	—
90	C₂×C₄	—	8	1	no	—
91	C₂×C₄	#336	1	1	yes	characteristic
92	C₂×C₄	#336	1	1	yes	characteristic
93	C₂×C₄	#336	1	1	yes	characteristic
94	C₂×C₄	H(3,ℤ/4)	1	1	yes	characteristic
95	C₂×C₄	H(3,ℤ/4)	1	1	yes	characteristic
96	C₂×C₄	#338	1	1	yes	characteristic
97	C₈	—	2	2	no	—
98	C₈	—	2	2	no	—
99	C₈	—	2	2	no	—
100	C₈	—	8	4	no	—
101	C₈	—	8	4	no	—
102	C₈	C₈²	1	1	yes	characteristic
103	C₈	#322	1	1	yes	characteristic
104	C₈	#322	1	1	yes	characteristic
105	C₈	#322	1	1	yes	characteristic
106	C₂²	—	4	1	no	—
107	C₂²	#1274	1	1	yes	characteristic
108	C₂²	#1274	1	1	yes	characteristic
109	C₂²	#1275	1	1	yes	characteristic
110	C₄	—	4	2	no	—
111	C₄	—	4	2	no	—
112	C₄	—	4	2	no	—
113	C₄	#1273	1	1	yes	characteristic
114	C₄	#1274	1	1	yes	characteristic
115	C₄	#1275	1	1	yes	characteristic
116	C₄	#1275	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	1	no	—
119	C₂	—	2	1	no	—
120	C₂	#7014	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12593

C₂

yes

maximal, characteristic

#8137

C₂

yes

maximal, characteristic

#8137

C₂

yes

maximal, characteristic

C₄×M₃₂

C₄

yes

—

#1726

C₂²

yes

characteristic

#1450

C₄

yes

—

#1450

C₄

yes

—

—

—

—

—

—

—

C₂×M₃₂

—

#431

—

#431

—

C₂×#97

D₄

yes

characteristic

C₂×#97

D₄

yes

characteristic

C₂×#97

Q₈

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

#376

D₄

yes

characteristic

#346

C₈

yes

complemented

C₄×C₁₆

—

#322

—

#322

C₈

yes

complemented

C₈²

—

C₈²

C₈

yes

complemented

C₂²×C₈

C₄²

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

M₁₆

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#95

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×C₈

yes

characteristic

C₂×C₈

C₄×C₈

yes

characteristic

C₂×C₈

C₄×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₂²⋊C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²⋊C₈

yes

—

C₄²

C₂²⋊C₈

yes

—

C₄²

C₄⋊C₈

yes

—

C₁₆

—

C₂³

H(3,ℤ/4)

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

100

C₈

—

101

C₈

—

102

C₈

C₈²

yes

characteristic

103

C₈

#322

yes

characteristic

104

C₈

#322

yes

characteristic

105

C₈

#322

yes

characteristic

106

C₂²

—

107

C₂²

#1274

yes

characteristic

108

C₂²

#1274

yes

characteristic

109

C₂²

#1275

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

#1273

yes

characteristic

114

C₄

#1274

yes

characteristic

115

C₄

#1275

yes

characteristic

116

C₄

#1275

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

120

C₂

#7014

yes

characteristic

	subgroup	G/H
center	C₈	#322
commutator	C₈	C₈²
Frattini, cosocle	#1726	C₂²
socle	C₂	#7014
prime radical	C₂³	H(3,ℤ/4)
LCS	G ▷ C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ #1726 ◁ G
LECS	G ▷ #1726 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₈

#322

commutator

C₈

C₈²

C₂²

C₂

C₂³

G ▷ C₈ ▷ C₂ ▷ C₁

C₁ ◁ C₈ ◁ #1726 ◁ G

G ▷ #1726 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
24. #346	C₈	semidirect
27. #322	C₈	semidirect
29. C₈²	C₈	semidirect

H ≅ G/N

type

24. #346

C₈

semidirect

27. #322

C₈

semidirect

29. C₈²

C₈

semidirect

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

subgroup	index
#12593	2
#8137	2

subgroup

index

#12593

#8137

Subgroups — #93253

Conjugacy classes of subgroups

Complements

Maximal subgroups