Group #12423 — subgroups

Subgroups — #12423

Order 256.

109 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 109 conjugacy classes of subgroups — too many to draw. Only its 41 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	#2225	C₂	1	1	yes	maximal, characteristic
2	#2209	C₂	1	1	yes	maximal, characteristic
3	#1707	C₂	1	1	yes	maximal, characteristic
4	C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic
5	#1367	C₂	1	1	yes	maximal, characteristic
6	#1364	C₂	1	1	yes	maximal, characteristic
7	#1362	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₃₂	—	2	1	no	—
9	#500	C₂²	1	1	yes	characteristic
10	#498	—	2	1	no	—
11	#494	C₂²	1	1	yes	characteristic
12	#487	—	2	1	no	—
13	C₈×Q₈	—	2	1	no	—
14	C₄×Q₁₆	C₂²	1	1	yes	characteristic
15	#367	—	2	1	no	—
16	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
18	#358	—	2	1	no	—
19	#358	—	2	1	no	—
20	#335	C₂²	1	1	yes	characteristic
21	Q₈⋊C₈	—	2	1	no	—
22	C₈²	C₂²	1	1	yes	characteristic
23	C₂×Q₁₆	—	2	1	no	—
24	C₂×Q₁₆	—	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	—	2	1	no	—
29	Q₃₂	—	4	2	no	—
30	C₂×C₁₆	—	2	1	no	—
31	C₂×C₁₆	—	2	1	no	—
32	#107	—	2	1	no	—
33	#107	—	2	1	no	—
34	#107	D₄	1	1	yes	characteristic
35	#106	—	4	2	no	—
36	C₄⋊C₈	—	2	1	no	—
37	C₄⋊C₈	—	4	1	no	—
38	#103	—	2	1	no	—
39	#103	—	4	1	no	—
40	#103	—	4	1	no	—
41	C₄×C₈	—	4	1	no	—
42	C₄×C₈	D₄	1	1	yes	characteristic
43	C₄×C₈	D₄	1	1	yes	characteristic
44	C₄×C₈	C₂³	1	1	yes	characteristic
45	C₂×Q₈	—	2	1	no	—
46	C₂×Q₈	—	4	1	no	—
47	C₂×Q₈	—	4	1	no	—
48	Q₁₆	—	2	1	no	—
49	Q₁₆	—	4	1	no	—
50	Q₁₆	—	4	1	no	—
51	Q₁₆	—	8	1	no	—
52	Q₁₆	SD₁₆	1	2	yes	—
53	C₂×C₈	—	2	1	no	—
54	C₂×C₈	—	4	1	no	—
55	C₂×C₈	D₈	1	1	yes	characteristic
56	C₂×C₈	D₈	1	1	yes	characteristic
57	C₂×C₈	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈	#41	1	1	yes	characteristic
59	C₄⋊C₄	—	2	1	no	—
60	C₄⋊C₄	—	4	1	no	—
61	C₄⋊C₄	—	4	1	no	—
62	C₄⋊C₄	—	4	1	no	—
63	C₄⋊C₄	—	4	1	no	—
64	C₄⋊C₄	—	4	1	no	—
65	C₄²	—	4	1	no	—
66	C₄²	C₂×D₄	1	1	yes	characteristic
67	C₁₆	—	4	1	no	—
68	C₁₆	—	4	1	no	—
69	Q₈	—	2	2	no	—
70	Q₈	—	4	1	no	—
71	Q₈	—	8	1	no	—
72	Q₈	—	8	1	no	—
73	Q₈	—	8	1	no	—
74	Q₈	—	8	1	no	—
75	C₂×C₄	—	4	1	no	—
76	C₂×C₄	—	4	1	no	—
77	C₂×C₄	—	8	1	no	—
78	C₂×C₄	—	8	1	no	—
79	C₂×C₄	#121	1	1	yes	characteristic
80	C₂×C₄	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₈	—	2	2	no	—
83	C₈	—	2	1	no	—
84	C₈	—	2	1	no	—
85	C₈	—	2	1	no	—
86	C₈	—	8	1	no	—
87	C₈	C₂×SD₁₆	1	1	yes	characteristic
88	C₈	#135	1	1	yes	characteristic
89	C₂²	#459	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	4	2	no	—
92	C₄	—	8	1	no	—
93	C₄	—	16	1	no	—
94	C₄	—	16	1	no	—
95	C₄	#465	1	1	yes	characteristic
96	C₄	#471	1	1	yes	characteristic
97	C₄	#508	1	1	yes	characteristic
98	C₄	#510	1	1	yes	characteristic
99	C₂	#1665	1	1	yes	characteristic
100	C₂	#2211	1	1	yes	characteristic
101	C₂	#2214	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2225

C₂

yes

maximal, characteristic

#2209

C₂

yes

maximal, characteristic

#1707

C₂

yes

maximal, characteristic

C₈×Q₁₆

C₂

yes

maximal, characteristic

#1367

C₂

yes

maximal, characteristic

#1364

C₂

yes

maximal, characteristic

#1362

C₂

yes

maximal, characteristic

C₂×Q₃₂

—

#500

C₂²

yes

characteristic

#498

—

#494

C₂²

yes

characteristic

#487

—

C₈×Q₈

—

C₄×Q₁₆

C₂²

yes

characteristic

#367

—

C₄⋊C₁₆

C₂²

yes

characteristic

C₄⋊C₁₆

C₂²

yes

—

—

C₂²

yes

characteristic

Q₈⋊C₈

—

C₈²

C₂²

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

#107

—

#107

D₄

yes

characteristic

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#103

—

#103

—

C₄×C₈

—

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

SD₁₆

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₈

yes

characteristic

C₂×C₈

D₈

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

#41

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂×D₄

yes

characteristic

C₁₆

—

C₁₆

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#121

yes

characteristic

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

C₂×SD₁₆

yes

C₈

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

C₄

yes

C₂

yes

100

C₂

#2211

yes

characteristic

101

C₂

#2214

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	#459
commutator, Frattini, cosocle	C₄×C₈	C₂³
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂²

#459

commutator, Frattini, cosocle

C₄×C₈

C₂³

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ 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
#2225	2
#2209	2
#1707	2
C₈×Q₁₆	2
#1367	2
#1364	2
#1362	2

subgroup

index

C₈×Q₁₆

Subgroups — #12423

Conjugacy classes of subgroups

Maximal subgroups