Group #123372 — subgroups

Subgroups — #123372

Order 512.

147 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 147 conjugacy classes of subgroups — too many to draw. Only its 52 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	#12521	C₂	1	1	yes	maximal, characteristic
2	#12448	C₂	1	2	yes	maximal
3	C₁₆×Q₁₆	C₂	1	1	yes	maximal, characteristic
4	#7100	C₂	1	1	yes	maximal, characteristic
5	#7098	C₂	1	2	yes	maximal
6	#2252	—	2	1	no	—
7	#2252	—	2	1	no	—
8	#2227	—	2	2	no	—
9	Q₈×C₁₆	—	2	1	no	—
10	#1713	C₂²	1	2	yes	—
11	C₈×Q₁₆	C₂²	1	1	yes	characteristic
12	#1370	C₂²	1	1	yes	characteristic
13	#1370	C₂²	1	2	yes	—
14	#1365	—	2	2	no	—
15	Q₈⋊C₁₆	—	2	1	no	—
16	C₈×C₁₆	C₂²	1	1	yes	characteristic
17	#500	—	2	2	no	—
18	#500	—	2	2	no	—
19	#494	—	2	2	no	—
20	C₈×Q₈	—	2	1	no	—
21	C₄×Q₁₆	D₄	1	1	yes	characteristic
22	#366	—	4	2	no	—
23	#366	—	4	1	no	—
24	#366	—	4	1	no	—
25	#366	—	4	2	no	—
26	C₄⋊C₁₆	—	2	1	no	—
27	C₄⋊C₁₆	—	2	2	no	—
28	C₄⋊C₁₆	—	4	1	no	—
29	#358	—	4	2	no	—
30	C₄×C₁₆	—	4	1	no	—
31	C₄×C₁₆	D₄	1	1	yes	characteristic
32	C₄×C₁₆	D₄	1	1	yes	characteristic
33	#335	D₄	1	1	yes	characteristic
34	Q₈⋊C₈	—	2	1	no	—
35	C₈²	C₂³	1	1	yes	characteristic
36	C₂×Q₁₆	—	8	2	no	—
37	C₂×Q₁₆	D₈	1	1	yes	characteristic
38	C₄⋊Q₈	—	4	2	no	—
39	C₄×Q₈	—	2	1	no	—
40	C₂×C₁₆	—	2	1	no	—
41	C₂×C₁₆	—	4	1	no	—
42	C₂×C₁₆	—	4	2	no	—
43	C₂×C₁₆	D₈	1	1	yes	characteristic
44	C₂×C₁₆	D₈	1	1	yes	characteristic
45	#107	—	4	2	no	—
46	#107	—	4	2	no	—
47	#107	—	4	2	no	—
48	#107	—	4	2	no	—
49	#107	D₈	1	1	yes	characteristic
50	C₄⋊C₈	—	2	1	no	—
51	C₄⋊C₈	—	4	1	no	—
52	#103	—	2	1	no	—
53	C₄×C₈	—	4	1	no	—
54	C₄×C₈	C₂×D₄	1	1	yes	characteristic
55	C₄×C₈	C₂×D₄	1	1	yes	characteristic
56	C₄×C₈	#41	1	1	yes	characteristic
57	C₂×Q₈	—	2	1	no	—
58	C₂×Q₈	—	8	2	no	—
59	Q₁₆	—	2	1	no	—
60	Q₁₆	—	16	4	no	—
61	Q₁₆	Q₃₂	1	2	yes	—
62	C₂×C₈	—	2	1	no	—
63	C₂×C₈	—	4	1	no	—
64	C₂×C₈	C₂×D₈	1	1	yes	characteristic
65	C₂×C₈	C₂×D₈	1	1	yes	characteristic
66	C₂×C₈	#135	1	1	yes	characteristic
67	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₄⋊C₄	—	2	1	no	—
69	C₄⋊C₄	—	4	1	no	—
70	C₄⋊C₄	—	8	2	no	—
71	C₄⋊C₄	—	8	2	no	—
72	C₄²	—	4	1	no	—
73	C₄²	#121	1	1	yes	characteristic
74	C₁₆	—	2	2	no	—
75	C₁₆	—	2	1	no	—
76	C₁₆	—	8	2	no	—
77	C₁₆	—	8	1	no	—
78	C₁₆	Q₃₂	1	2	yes	—
79	Q₈	—	2	2	no	—
80	Q₈	—	4	1	no	—
81	Q₈	—	16	4	no	—
82	C₂×C₄	—	4	1	no	—
83	C₂×C₄	—	4	1	no	—
84	C₂×C₄	—	16	2	no	—
85	C₂×C₄	#459	1	1	yes	characteristic
86	C₂×C₄	#466	1	1	yes	characteristic
87	C₂×C₄	#472	1	1	yes	characteristic
88	C₈	—	2	2	no	—
89	C₈	—	2	1	no	—
90	C₈	—	2	1	no	—
91	C₈	—	8	1	no	—
92	C₈	C₂×Q₃₂	1	1	yes	characteristic
93	C₈	C₂×Q₃₂	1	1	yes	characteristic
94	C₈	#508	1	1	yes	characteristic
95	C₈	#510	1	1	yes	characteristic
96	C₂²	#1667	1	1	yes	characteristic
97	C₄	—	2	1	no	—
98	C₄	—	4	2	no	—
99	C₄	—	8	1	no	—
100	C₄	—	32	2	no	—
101	C₄	#2209	1	1	yes	characteristic
102	C₄	#2214	1	1	yes	characteristic
103	C₄	#2216	1	1	yes	characteristic
104	C₄	#2222	1	1	yes	characteristic
105	C₂	#12427	1	1	yes	characteristic
106	C₂	#12431	1	1	yes	characteristic
107	C₂	#12436	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12521

C₂

yes

maximal, characteristic

#12448

C₂

yes

maximal

C₁₆×Q₁₆

C₂

yes

maximal, characteristic

#7100

C₂

yes

maximal, characteristic

#7098

C₂

yes

—

—

—

Q₈×C₁₆

—

#1713

C₂²

yes

—

C₈×Q₁₆

C₂²

yes

characteristic

#1370

C₂²

yes

characteristic

#1370

C₂²

yes

—

#1365

—

Q₈⋊C₁₆

—

C₈×C₁₆

C₂²

yes

—

—

—

C₈×Q₈

—

C₄×Q₁₆

D₄

yes

—

—

—

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#358

—

C₄×C₁₆

—

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

#335

D₄

yes

characteristic

Q₈⋊C₈

—

C₈²

C₂³

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

D₈

yes

characteristic

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₈

yes

characteristic

C₂×C₁₆

D₈

yes

—

—

—

—

D₈

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

Q₃₂

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₈

yes

characteristic

C₂×C₈

C₂×D₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

#121

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

Q₃₂

yes

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

C₈

—

C₈

C₂×Q₃₂

yes

characteristic

C₈

C₂×Q₃₂

yes

C₈

yes

C₈

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

100

C₄

—

101

C₄

#2209

yes

characteristic

102

C₄

#2214

yes

characteristic

103

C₄

#2216

yes

characteristic

104

C₄

#2222

yes

characteristic

105

C₂

#12427

yes

characteristic

106

C₂

#12431

yes

characteristic

107

C₂

#12436

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	#1667
commutator, Frattini, cosocle	C₈²	C₂³
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂²

#1667

commutator, Frattini, cosocle

C₈²

C₂³

LCS

G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G

LECS

G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

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

subgroup	index
#12521	2
#12448	2
C₁₆×Q₁₆	2
#7100	2
#7098	2

subgroup

index

C₁₆×Q₁₆

Subgroups — #123372

Conjugacy classes of subgroups

Maximal subgroups