Group #93249 — subgroups

Subgroups — #93249

Order 512.

264 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 138 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	#12590	C₂	1	2	yes	maximal
2	#8131	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₆	C₄	1	4	yes	—
4	C₄²×C₈	C₄	1	2	yes	—
5	C₄²×C₈	C₂²	1	1	yes	characteristic
6	#1374	—	2	2	no	—
7	#1374	—	2	2	no	—
8	#1273	—	2	2	no	—
9	C₂²×C₁₆	—	2	4	no	—
10	C₂×C₄×C₈	—	2	2	no	—
11	C₂×C₄×C₈	—	2	2	no	—
12	C₂×C₄×C₈	C₈	1	4	yes	—
13	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₈	D₄	1	2	yes	—
15	C₂×C₄×C₈	D₄	1	1	yes	characteristic
16	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
17	C₄³	C₂×C₄	1	1	yes	characteristic
18	C₄×C₁₆	—	2	4	no	—
19	C₄×C₁₆	C₈	1	8	yes	complemented
20	C₂²×C₈	—	2	4	no	—
21	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
22	C₂²×C₈	C₄⋊C₄	1	2	yes	—
23	C₂×C₄²	C₄²	1	1	yes	characteristic
24	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₄²	C₂×C₈	1	2	yes	—
27	C₂×C₄²	M₁₆	1	2	yes	—
28	C₂×C₁₆	—	2	8	no	—
29	C₂×C₁₆	—	4	4	no	—
30	C₂×C₁₆	—	4	4	no	—
31	C₄×C₈	—	2	4	no	—
32	C₄×C₈	—	4	2	no	—
33	C₄×C₈	—	4	1	no	—
34	C₄×C₈	—	4	1	no	—
35	C₄×C₈	—	4	4	no	—
36	C₄×C₈	—	4	4	no	—
37	C₄×C₈	C₁₆	1	8	yes	complemented
38	C₄×C₈	C₂×C₈	1	4	yes	—
39	C₄×C₈	M₁₆	1	4	yes	—
40	C₂²×C₄	#95	1	1	yes	characteristic
41	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
42	C₂²×C₄	#97	1	1	yes	characteristic
43	C₂²×C₄	C₂²⋊C₈	1	2	yes	—
44	C₂²×C₄	C₄⋊C₈	1	2	yes	—
45	C₂×C₈	—	2	2	no	—
46	C₂×C₈	—	2	2	no	—
47	C₂×C₈	—	2	2	no	—
48	C₂×C₈	—	2	2	no	—
49	C₂×C₈	—	2	8	no	—
50	C₂×C₈	—	4	4	no	—
51	C₂×C₈	—	4	4	no	—
52	C₂×C₈	C₂²⋊C₈	1	4	yes	—
53	C₂×C₈	C₄⋊C₈	1	4	yes	—
54	C₄²	—	2	2	no	—
55	C₄²	—	2	1	no	—
56	C₄²	—	2	1	no	—
57	C₄²	—	2	4	no	—
58	C₄²	C₄×C₈	1	2	yes	—
59	C₄²	#97	1	2	yes	—
60	C₄²	C₂×C₁₆	1	4	yes	—
61	C₄²	M₃₂	1	4	yes	—
62	C₁₆	—	4	8	no	—
63	C₂³	#336	1	1	yes	characteristic
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	C₈²	1	1	yes	characteristic
69	C₂×C₄	#322	1	1	yes	characteristic
70	C₂×C₄	#322	1	2	yes	—
71	C₂×C₄	#336	1	2	yes	—
72	C₂×C₄	H(3,ℤ/4)	1	1	yes	characteristic
73	C₂×C₄	H(3,ℤ/4)	1	2	yes	—
74	C₂×C₄	#338	1	1	yes	characteristic
75	C₂×C₄	C₄×C₁₆	1	2	yes	—
76	C₂×C₄	#346	1	2	yes	—
77	C₂×C₄	#347	1	4	yes	—
78	C₂×C₄	C₂²⋊C₁₆	1	4	yes	—
79	C₂×C₄	C₄⋊C₁₆	1	4	yes	—
80	C₈	—	2	4	no	—
81	C₈	—	2	4	no	—
82	C₈	—	4	8	no	—
83	C₂²	#1273	1	1	yes	characteristic
84	C₂²	#1274	1	1	yes	characteristic
85	C₂²	#1275	1	1	yes	characteristic
86	C₂²	#1374	1	2	yes	—
87	C₂²	#1375	1	2	yes	—
88	C₄	—	2	4	no	—
89	C₄	—	2	4	no	—
90	C₄	#1274	1	2	yes	—
91	C₄	#1275	1	2	yes	—
92	C₄	C₈×C₁₆	1	1	yes	characteristic
93	C₄	#1311	1	1	yes	characteristic
94	C₄	#1312	1	2	yes	—
95	C₄	#1313	1	4	yes	—
96	C₂	#7014	1	1	yes	characteristic
97	C₂	#7136	1	1	yes	characteristic
98	C₂	#7137	1	1	yes	characteristic
99	C₂	#7142	1	2	yes	—
100	C₂	#7145	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#12590

C₂

yes

maximal

#8131

C₂

yes

maximal, characteristic

C₂×C₄×C₁₆

C₄

yes

—

C₄²×C₈

C₄

yes

—

C₄²×C₈

C₂²

yes

—

—

—

C₂²×C₁₆

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₈

yes

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

Q₈

yes

characteristic

C₄³

C₂×C₄

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

C₈

yes

complemented

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂×C₄²

C₄²

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₂×C₈

yes

—

C₂×C₄²

M₁₆

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₁₆

yes

complemented

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₂²×C₄

#95

yes

characteristic

C₂²×C₄

C₄×C₈

yes

characteristic

C₂²×C₄

#97

yes

characteristic

C₂²×C₄

C₂²⋊C₈

yes

—

C₂²×C₄

C₄⋊C₈

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄×C₈

yes

—

C₄²

#97

yes

—

C₄²

C₂×C₁₆

yes

—

C₄²

M₃₂

yes

—

C₁₆

—

C₂³

#336

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₈²

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

#336

yes

—

C₂×C₄

H(3,ℤ/4)

yes

characteristic

C₂×C₄

H(3,ℤ/4)

yes

—

C₂×C₄

#338

yes

characteristic

C₂×C₄

C₄×C₁₆

yes

—

C₂×C₄

#346

yes

—

C₂×C₄

#347

yes

—

C₂×C₄

C₂²⋊C₁₆

yes

—

C₂×C₄

C₄⋊C₁₆

yes

—

C₈

—

C₈

—

C₈

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#1375

yes

—

C₄

—

C₄

—

C₄

#1274

yes

—

C₄

#1275

yes

—

C₄

C₈×C₁₆

yes

C₄

yes

C₄

yes

—

C₄

#1313

yes

—

C₂

yes

C₂

yes

C₂

yes

C₂

yes

—

100

C₂

#7145

yes

—

	subgroup	G/H
center	C₂×C₄²	C₄²
commutator	C₄	C₈×C₁₆
Frattini, cosocle	C₄²×C₈	C₂²
socle, prime radical	C₂³	#336
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄²

C₄²

commutator

C₄

C₈×C₁₆

Frattini, cosocle

C₄²×C₈

C₂²

C₂³

G ▷ C₄ ▷ C₁

C₁ ◁ C₂×C₄² ◁ G

G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
19. C₄×C₁₆	C₈	semidirect
37. C₄×C₈	C₁₆	semidirect

H ≅ G/N

type

19. C₄×C₁₆

C₈

semidirect

37. C₄×C₈

C₁₆

semidirect

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

subgroup	index
#12590	2
#8131	2

subgroup

index

#12590

#8131

Subgroups — #93249

Conjugacy classes of subgroups

Complements

Maximal subgroups