Group #94251 — subgroups

Subgroups — #94251

Order 512.

126 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 62 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	C₂×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	#12024	C₂	1	1	yes	maximal, characteristic
3	#12023	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₃₂	C₄	1	2	yes	—
5	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
6	#1568	—	2	1	no	—
7	#1567	—	2	1	no	—
8	C₄⋊C₃₂	—	2	1	no	—
9	C₄⋊C₃₂	—	2	1	no	—
10	#1372	—	2	1	no	—
11	#1372	—	2	1	no	—
12	#1372	C₄	1	2	yes	—
13	#1371	C₄	1	2	yes	—
14	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₁₆	Q₈	1	1	yes	characteristic
16	#432	—	4	1	no	—
17	#423	—	4	1	no	—
18	C₂×C₄×C₈	D₄	1	1	yes	characteristic
19	C₂×C₃₂	—	2	2	no	—
20	C₂×C₃₂	—	4	2	no	—
21	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₄×C₁₆	D₄	1	1	yes	characteristic
24	C₄×C₁₆	D₄	1	1	yes	characteristic
25	#335	—	2	2	no	—
26	#335	—	2	2	no	—
27	#335	—	4	1	no	—
28	#334	—	4	1	no	—
29	C₂×M₁₆	—	8	1	no	—
30	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₈	Q₁₆	1	1	yes	characteristic
32	C₂×C₄²	D₈	1	1	yes	characteristic
33	C₂×C₁₆	—	2	1	no	—
34	C₂×C₁₆	—	2	1	no	—
35	C₂×C₁₆	—	4	1	no	—
36	C₂×C₁₆	C₄²	1	1	yes	characteristic
37	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
40	C₄⋊C₈	—	4	2	no	—
41	C₄⋊C₈	—	4	2	no	—
42	C₄⋊C₈	—	8	1	no	—
43	C₂²⋊C₈	—	8	1	no	—
44	#97	—	8	1	no	—
45	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
46	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
47	C₄×C₈	SD₁₆	1	1	yes	characteristic
48	C₄×C₈	SD₁₆	1	1	yes	characteristic
49	C₃₂	—	4	2	no	—
50	C₂²×C₄	#107	1	1	yes	characteristic
51	C₂²×C₄	SD₃₂	1	1	yes	characteristic
52	C₂²×C₄	SD₃₂	1	1	yes	characteristic
53	M₁₆	—	16	2	no	—
54	C₂×C₈	—	2	1	no	—
55	C₂×C₈	—	2	1	no	—
56	C₂×C₈	—	4	1	no	—
57	C₂×C₈	—	8	2	no	—
58	C₂×C₈	—	8	2	no	—
59	C₂×C₈	#95	1	1	yes	characteristic
60	C₂×C₈	#103	1	1	yes	characteristic
61	C₂×C₈	#103	1	1	yes	characteristic
62	C₂×C₈	#106	1	1	yes	characteristic
63	C₄²	#102	1	1	yes	characteristic
64	C₄²	#102	1	1	yes	characteristic
65	C₄²	D₁₆	1	1	yes	characteristic
66	C₄²	Q₃₂	1	1	yes	characteristic
67	C₁₆	—	2	1	no	—
68	C₁₆	—	2	1	no	—
69	C₁₆	—	2	1	no	—
70	C₁₆	#100	1	1	yes	characteristic
71	C₁₆	#101	1	1	yes	characteristic
72	C₂³	#367	1	1	yes	characteristic
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	#340	1	1	yes	characteristic
78	C₂×C₄	#357	1	1	yes	characteristic
79	C₂×C₄	#357	1	1	yes	characteristic
80	C₂×C₄	#358	1	1	yes	characteristic
81	C₂×C₄	#358	1	1	yes	characteristic
82	C₂×C₄	#366	1	1	yes	characteristic
83	C₈	—	2	1	no	—
84	C₈	—	2	1	no	—
85	C₈	—	2	1	no	—
86	C₈	—	16	2	no	—
87	C₈	—	16	2	no	—
88	C₈	#338	1	1	yes	characteristic
89	C₈	#343	1	1	yes	characteristic
90	C₂²	—	2	1	no	—
91	C₂²	—	2	1	no	—
92	C₂²	#1380	1	1	yes	characteristic
93	C₂²	#1426	1	2	yes	—
94	C₄	—	2	1	no	—
95	C₄	—	2	1	no	—
96	C₄	—	2	1	no	—
97	C₄	—	2	1	no	—
98	C₄	#1292	1	1	yes	characteristic
99	C₄	#1385	1	1	yes	characteristic
100	C₄	#1417	1	2	yes	—
101	C₂	—	2	2	no	—
102	C₂	#7241	1	1	yes	characteristic
103	C₂	#7473	1	1	yes	characteristic
104	C₂	#7474	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊C₃₂

C₂

yes

maximal, characteristic

#12024

C₂

yes

maximal, characteristic

#12023

C₂

yes

maximal, characteristic

C₂²×C₃₂

C₄

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1568

—

#1567

—

C₄⋊C₃₂

—

C₄⋊C₃₂

—

#1372

—

#1372

—

#1372

C₄

yes

—

#1371

C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

Q₈

yes

characteristic

#432

—

#423

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₃₂

—

C₂×C₃₂

—

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

D₄

yes

—

—

—

—

C₂×M₁₆

—

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

Q₁₆

yes

characteristic

C₂×C₄²

D₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄²

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₄⋊C₄

yes

characteristic

C₂×C₁₆

C₄⋊C₄

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₃₂

—

C₂²×C₄

#107

yes

characteristic

C₂²×C₄

SD₃₂

yes

characteristic

C₂²×C₄

SD₃₂

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄²

yes

C₄²

yes

C₄²

D₁₆

yes

characteristic

C₄²

Q₃₂

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

yes

C₁₆

yes

C₂³

yes

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

—

C₈

—

C₈

—

C₈

yes

C₈

yes

C₂²

—

C₂²

—

C₂²

#1380

yes

characteristic

C₂²

#1426

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

100

C₄

#1417

yes

—

101

C₂

—

102

C₂

#7241

yes

characteristic

103

C₂

#7473

yes

characteristic

104

C₂

#7474

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1380
commutator	C₂×C₁₆	C₄²
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
prime radical	C₂³	#367
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1380

commutator

C₂×C₁₆

C₄²

Frattini, cosocle

C₂×C₄×C₁₆

C₂²

prime radical

C₂³

#367

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
C₂×C₄⋊C₃₂	2
#12024	2
#12023	2

subgroup

index

C₂×C₄⋊C₃₂

#12024

#12023

Subgroups — #94251

Conjugacy classes of subgroups

Maximal subgroups