Group #12409 — subgroups

Subgroups — #12409

Order 256.

135 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 135 conjugacy classes of subgroups — too many to draw. Only its 43 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	#3383	C₂	1	1	yes	maximal, characteristic
2	#2254	C₂	1	1	yes	maximal, characteristic
3	#2227	C₂	1	1	yes	maximal, characteristic
4	#1676	C₂	1	1	yes	maximal, characteristic
5	#1364	C₂	1	1	yes	maximal, characteristic
6	Q₈⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	#1332	C₂	1	1	yes	maximal, characteristic
8	#557	—	2	1	no	—
9	#500	C₂²	1	1	yes	characteristic
10	#500	C₂²	1	1	yes	characteristic
11	#499	—	2	1	no	—
12	#479	—	2	2	no	—
13	#477	—	2	1	no	—
14	#475	—	2	1	no	—
15	C₈×Q₈	C₂²	1	1	yes	characteristic
16	C₄×Q₁₆	—	2	1	no	—
17	C₄×Q₁₆	C₂²	1	1	yes	characteristic
18	#367	—	2	2	no	—
19	#366	—	2	1	no	—
20	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
21	#358	—	2	1	no	—
22	C₄×C₁₆	C₂²	1	1	yes	characteristic
23	#335	C₂²	1	1	yes	characteristic
24	#333	—	2	1	no	—
25	C₂×Q₁₆	D₄	1	1	yes	characteristic
26	C₄⋊Q₈	—	2	1	no	—
27	C₄⋊Q₈	—	2	1	no	—
28	C₄⋊Q₈	—	4	1	no	—
29	#125	—	2	2	no	—
30	#125	—	4	2	no	—
31	C₄×Q₈	—	2	2	no	—
32	C₄×Q₈	—	2	1	no	—
33	C₄×Q₈	—	4	1	no	—
34	C₄×Q₈	D₄	1	1	yes	characteristic
35	C₂×C₁₆	—	2	1	no	—
36	C₂×C₁₆	—	2	1	no	—
37	#107	—	2	1	no	—
38	#107	—	2	1	no	—
39	#107	—	2	1	no	—
40	#107	—	4	1	no	—
41	#107	D₄	1	1	yes	characteristic
42	#107	D₄	1	2	yes	—
43	#106	—	2	2	no	—
44	C₄⋊C₈	—	2	1	no	—
45	C₄⋊C₈	—	2	1	no	—
46	C₄⋊C₈	D₄	1	1	yes	characteristic
47	#103	—	2	2	no	—
48	#103	—	2	1	no	—
49	#103	—	4	1	no	—
50	C₄×C₈	—	2	1	no	—
51	C₄×C₈	C₂³	1	1	yes	characteristic
52	C₂×Q₈	—	2	1	no	—
53	C₂×Q₈	—	4	1	no	—
54	C₂×Q₈	—	4	1	no	—
55	C₂×Q₈	D₈	1	1	yes	characteristic
56	Q₁₆	—	2	1	no	—
57	Q₁₆	Q₁₆	1	2	yes	—
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	—	4	1	no	—
60	C₂×C₈	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈	C₂×D₄	1	1	yes	characteristic
62	C₄⋊C₄	—	2	1	no	—
63	C₄⋊C₄	—	2	1	no	—
64	C₄⋊C₄	—	2	2	no	—
65	C₄⋊C₄	—	4	2	no	—
66	C₄⋊C₄	—	4	2	no	—
67	C₄⋊C₄	—	4	2	no	—
68	C₄⋊C₄	—	4	1	no	—
69	C₄⋊C₄	—	4	1	no	—
70	C₄⋊C₄	—	4	1	no	—
71	C₄⋊C₄	—	4	1	no	—
72	C₄⋊C₄	D₈	1	1	yes	characteristic
73	C₄²	—	2	1	no	—
74	C₄²	—	4	2	no	—
75	C₄²	—	4	1	no	—
76	C₄²	C₂×D₄	1	1	yes	characteristic
77	C₁₆	—	2	2	no	—
78	C₁₆	—	4	1	no	—
79	Q₈	—	2	2	no	—
80	Q₈	—	2	1	no	—
81	Q₈	—	2	1	no	—
82	Q₈	—	4	1	no	—
83	Q₈	—	8	1	no	—
84	Q₈	—	8	1	no	—
85	Q₈	—	16	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	4	1	no	—
89	C₂×C₄	—	4	1	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	8	1	no	—
92	C₂×C₄	#120	1	1	yes	characteristic
93	C₂×C₄	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₈	—	2	1	no	—
96	C₈	—	4	1	no	—
97	C₈	—	8	1	no	—
98	C₈	C₂×Q₁₆	1	1	yes	characteristic
99	C₈	#137	1	1	yes	characteristic
100	C₂²	#447	1	1	yes	characteristic
101	C₄	—	2	1	no	—
102	C₄	—	4	2	no	—
103	C₄	—	4	1	no	—
104	C₄	—	4	1	no	—
105	C₄	—	8	2	no	—
106	C₄	—	8	1	no	—
107	C₄	—	16	1	no	—
108	C₄	#451	1	1	yes	characteristic
109	C₄	#456	1	1	yes	characteristic
110	C₄	#508	1	1	yes	characteristic
111	C₄	#510	1	1	yes	characteristic
112	C₂	#1649	1	1	yes	characteristic
113	C₂	#2189	1	1	yes	characteristic
114	C₂	#2193	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3383

C₂

yes

maximal, characteristic

#2254

C₂

yes

maximal, characteristic

#2227

C₂

yes

maximal, characteristic

#1676

C₂

yes

maximal, characteristic

#1364

C₂

yes

maximal, characteristic

Q₈⋊C₁₆

C₂

yes

maximal, characteristic

#1332

C₂

yes

maximal, characteristic

#557

—

#500

C₂²

yes

characteristic

#500

C₂²

yes

—

—

—

—

C₈×Q₈

C₂²

yes

characteristic

C₄×Q₁₆

—

C₄×Q₁₆

C₂²

yes

characteristic

#367

—

#366

—

C₄⋊C₁₆

C₂²

yes

characteristic

#358

—

C₄×C₁₆

C₂²

yes

characteristic

#335

C₂²

yes

characteristic

#333

—

C₂×Q₁₆

D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

D₄

yes

characteristic

#107

D₄

yes

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₄

yes

—

—

—

C₄×C₈

—

C₄×C₈

C₂³

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₈

yes

characteristic

Q₁₆

—

Q₁₆

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₈

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×D₄

yes

characteristic

C₁₆

—

C₁₆

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#120

yes

characteristic

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

C₂×Q₁₆

yes

characteristic

C₈

#137

yes

characteristic

100

C₂²

#447

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

#451

yes

characteristic

109

C₄

#456

yes

characteristic

110

C₄

#508

yes

characteristic

111

C₄

#510

yes

characteristic

112

C₂

#1649

yes

characteristic

113

C₂

#2189

yes

characteristic

114

C₂

#2193

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂²

#447

commutator, Frattini, cosocle

C₄×C₈

C₂³

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ G

LECS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#3383	2
#2254	2
#2227	2
#1676	2
#1364	2
Q₈⋊C₁₆	2
#1332	2

subgroup

index

Q₈⋊C₁₆

Subgroups — #12409

Conjugacy classes of subgroups

Maximal subgroups