Group #93016 — subgroups

Subgroups — #93016

Order 512.

176 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 54 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	#13263	C₂	1	1	yes	maximal, characteristic
2	#8623	C₂	1	2	yes	maximal
3	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
4	C₂×C₄⋊C₁₆	C₄	1	1	yes	characteristic
5	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
6	#1568	—	2	2	no	—
7	#1467	—	2	2	no	—
8	#1374	—	2	2	no	—
9	#1290	C₄	1	4	yes	—
10	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
11	C₂²×C₁₆	—	2	1	no	—
12	C₂²×C₁₆	—	2	1	no	—
13	C₈×Q₈	—	4	1	no	—
14	C₈×Q₈	—	4	1	no	—
15	#432	—	2	4	no	—
16	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
17	C₂×C₄⋊C₈	Q₈	1	1	yes	characteristic
18	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₈	D₄	1	2	yes	—
20	C₄⋊C₁₆	—	4	1	no	—
21	C₄×C₁₆	—	4	1	no	—
22	#335	—	4	2	no	—
23	#334	—	4	2	no	—
24	#324	—	4	2	no	—
25	#324	—	4	2	no	—
26	C₂²×Q₈	Q₁₆	1	1	yes	characteristic
27	C₂²×C₈	C₄²	1	1	yes	characteristic
28	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₈	C₄⋊C₄	1	2	yes	—
30	C₄×Q₈	—	4	1	no	—
31	C₄×Q₈	—	4	1	no	—
32	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
33	C₂×C₄⋊C₄	SD₁₆	1	2	yes	—
34	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
35	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₁₆	—	2	2	no	—
37	C₂×C₁₆	—	4	1	no	—
38	C₂×C₁₆	—	4	1	no	—
39	C₂×C₁₆	—	4	2	no	—
40	C₂×C₁₆	—	4	1	no	—
41	C₄⋊C₈	—	4	2	no	—
42	C₄⋊C₈	—	4	1	no	—
43	C₄⋊C₈	—	4	4	no	—
44	C₂²⋊C₈	—	4	4	no	—
45	#97	—	4	4	no	—
46	C₄×C₈	—	2	2	no	—
47	C₄×C₈	—	2	2	no	—
48	C₄×C₈	—	2	1	no	—
49	C₄×C₈	—	2	1	no	—
50	C₂×Q₈	—	2	2	no	—
51	C₂×Q₈	—	4	1	no	—
52	C₂×Q₈	—	4	1	no	—
53	C₂²×C₄	#95	1	1	yes	characteristic
54	C₂²×C₄	#102	1	2	yes	—
55	C₂²×C₄	#103	1	2	yes	—
56	C₂²×C₄	#106	1	1	yes	characteristic
57	C₂²×C₄	#107	1	1	yes	characteristic
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	—	2	1	no	—
60	C₂×C₈	—	2	1	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	4	1	no	—
63	C₂×C₈	—	4	2	no	—
64	C₂×C₈	—	8	4	no	—
65	C₂×C₈	#99	1	1	yes	characteristic
66	C₂×C₈	#99	1	1	yes	characteristic
67	C₄⋊C₄	—	4	1	no	—
68	C₄⋊C₄	—	4	2	no	—
69	C₄²	—	2	2	no	—
70	C₄²	—	2	1	no	—
71	C₄²	—	2	2	no	—
72	C₄²	—	2	1	no	—
73	C₁₆	—	4	2	no	—
74	C₁₆	—	8	1	no	—
75	C₂³	#340	1	1	yes	characteristic
76	Q₈	—	4	4	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	2	1	no	—
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	—	4	1	no	—
88	C₂×C₄	H(3,ℤ/4)	1	1	yes	characteristic
89	C₂×C₄	#342	1	1	yes	characteristic
90	C₂×C₄	#368	1	2	yes	—
91	C₈	—	2	1	no	—
92	C₈	—	4	1	no	—
93	C₈	—	4	2	no	—
94	C₈	—	16	4	no	—
95	C₈	#355	1	1	yes	characteristic
96	C₈	#356	1	1	yes	characteristic
97	C₂²	—	2	1	no	—
98	C₂²	—	2	1	no	—
99	C₂²	#1289	1	1	yes	characteristic
100	C₂²	#1381	1	1	yes	characteristic
101	C₂²	#1385	1	1	yes	characteristic
102	C₄	—	2	2	no	—
103	C₄	—	2	1	no	—
104	C₄	—	4	2	no	—
105	C₄	—	4	1	no	—
106	C₄	—	4	2	no	—
107	C₄	#1392	1	1	yes	characteristic
108	C₄	#1394	1	1	yes	characteristic
109	C₂	—	2	1	no	—
110	C₂	—	2	1	no	—
111	C₂	#7228	1	1	yes	characteristic
112	C₂	#7231	1	1	yes	characteristic
113	C₂	#7238	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13263

C₂

yes

maximal, characteristic

#8623

C₂

yes

maximal

C₂×C₈×Q₈

C₂²

yes

characteristic

C₂×C₄⋊C₁₆

C₄

yes

characteristic

C₂×C₄×C₁₆

C₄

yes

—

—

—

C₄

yes

—

C₂×C₄×Q₈

D₄

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

—

C₈×Q₈

—

C₈×Q₈

—

#432

—

C₂×C₄⋊C₈

C₂×C₄

yes

—

C₂×C₄⋊C₈

Q₈

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

—

C₄⋊C₁₆

—

C₄×C₁₆

—

—

—

—

—

C₂²×Q₈

Q₁₆

yes

characteristic

C₂²×C₈

C₄²

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

D₈

yes

characteristic

C₂×C₄⋊C₄

SD₁₆

yes

—

C₂×C₄²

C₂²⋊C₄

yes

—

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

#95

yes

characteristic

C₂²×C₄

#102

yes

—

C₂²×C₄

#103

yes

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

C₂³

#340

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

yes

C₈

yes

C₂²

—

C₂²

—

C₂²

#1289

yes

characteristic

100

C₂²

#1381

yes

characteristic

101

C₂²

#1385

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

#1392

yes

characteristic

108

C₄

#1394

yes

characteristic

109

C₂

—

110

C₂

—

111

C₂

#7228

yes

characteristic

112

C₂

#7231

yes

characteristic

113

C₂

#7238

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1289
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₂×C₈×Q₈	C₂²
prime radical	C₂³	#340
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₈×Q₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

#1289

commutator

C₂²×C₈

C₄²

Frattini, cosocle

C₂×C₈×Q₈

C₂²

prime radical

C₂³

#340

LCS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₈×Q₈ ◁ G

LECS

G ▷ C₂×C₈×Q₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

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

subgroup	index
#13263	2
#8623	2

subgroup

index

#13263

#8623

Subgroups — #93016

Conjugacy classes of subgroups

Maximal subgroups