Group #83245 — subgroups

Subgroups — #83245

Order 384.

416 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 149 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×C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#870	C₂	1	1	yes	maximal, characteristic
3	#5229	C₂	1	4	yes	maximal
4	#4985	C₂	1	8	yes	maximal
5	#4932	C₂	1	1	yes	maximal, characteristic
6	#2642	—	3	1	no	maximal, Sylow
7	C₂²×C₃×Q₈	C₂²	1	2	yes	—
8	C₃×C₄⋊Q₈	—	2	4	no	—
9	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
10	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂×Dic₃⋊C₄	C₂²	1	4	yes	—
13	C₂×C₄×Dic₃	C₂²	1	2	yes	—
14	#870	—	2	4	no	—
15	C₃⋊#95	C₂²	1	4	yes	—
16	C₃⋊#95	C₂²	1	16	yes	—
17	C₂×C₄⋊Q₈	D₃	1	1	yes	characteristic
18	C₂×#125	—	3	1	no	—
19	#400	—	3	8	no	—
20	#391	—	3	4	no	—
21	#376	—	3	1	no	—
22	C₂×C₃×Q₈	—	2	4	no	—
23	C₂×C₃×Q₈	—	2	8	no	—
24	C₂²×C₃×C₄	C₂³	1	4	yes	—
25	C₂²×C₃×C₄	C₂³	1	2	yes	—
26	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
27	C₂²×Dic₃	C₂³	1	4	yes	—
28	C₂²×Dic₃	C₂³	1	4	yes	—
29	C₃×C₄⋊C₄	—	2	8	no	—
30	C₃×C₄²	D₄	1	4	yes	—
31	C₄⋊Dic₃	—	2	4	no	—
32	Dic₃⋊C₄	—	2	8	no	—
33	C₄×Dic₃	—	2	4	no	—
34	C₂²×Q₈	C₂×D₃	1	2	yes	—
35	C₄⋊Q₈	—	2	4	no	—
36	#125	—	6	4	no	—
37	C₂×C₄⋊C₄	—	3	2	no	—
38	C₂×C₄⋊C₄	—	3	4	no	—
39	C₂×C₄⋊C₄	C₂×D₃	1	4	yes	—
40	C₂×C₄²	—	3	2	no	—
41	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
42	#95	—	3	4	no	—
43	#95	—	3	16	no	—
44	C₂³×C₃	C₂⁴	1	1	yes	characteristic
45	C₃×Q₈	—	4	8	no	—
46	C₂×C₃×C₄	—	2	4	no	—
47	C₂×C₃×C₄	—	2	4	no	—
48	C₂×C₃×C₄	—	2	8	no	—
49	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
50	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
51	C₂×C₃×C₄	#41	1	4	yes	—
52	C₂×Dic₃	—	2	4	no	—
53	C₂×Dic₃	—	2	4	no	—
54	C₂×Dic₃	—	2	8	no	—
55	C₂×Dic₃	—	2	8	no	—
56	C₂×Q₈	—	2	8	no	—
57	C₂×Q₈	—	2	4	no	—
58	C₂²×C₄	—	3	4	no	—
59	C₂²×C₄	—	3	4	no	—
60	C₂²×C₄	C₂²×D₃	1	4	yes	—
61	C₂²×C₄	C₂²×D₃	1	2	yes	—
62	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
63	C₄⋊C₄	—	2	8	no	—
64	C₄⋊C₄	—	6	8	no	—
65	C₄⋊C₄	—	6	4	no	—
66	C₄²	—	6	4	no	—
67	C₄²	#67	1	4	yes	—
68	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
69	C₂²×C₃	C₂×#41	1	2	yes	—
70	C₂²×C₃	2₋¹⁺⁴	1	4	yes	—
71	C₃×C₄	—	2	4	no	—
72	C₃×C₄	—	4	4	no	—
73	C₃×C₄	#124	1	4	yes	—
74	Dic₃	—	4	4	no	—
75	Dic₃	—	4	4	no	—
76	C₂³	C₂³×D₃	1	1	yes	characteristic
77	Q₈	—	4	8	no	—
78	C₂×C₄	—	2	4	no	—
79	C₂×C₄	—	2	8	no	—
80	C₂×C₄	—	2	4	no	—
81	C₂×C₄	—	6	4	no	—
82	C₂×C₄	—	6	8	no	—
83	C₂×C₄	—	6	8	no	—
84	C₂×C₄	—	6	4	no	—
85	C₂×C₄	#237	1	4	yes	—
86	C₂×C₄	C₂×#67	1	4	yes	—
87	C₂×C₄	C₂×#67	1	2	yes	—
88	C₂×C₃	C₂×#124	1	1	yes	characteristic
89	C₂×C₃	#536	1	2	yes	—
90	C₂×C₃	#541	1	4	yes	—
91	C₂²	C₂×#237	1	2	yes	—
92	C₂²	#1007	1	4	yes	—
93	C₂²	C₂²×#67	1	1	yes	characteristic
94	C₄	—	2	4	no	—
95	C₄	—	4	4	no	—
96	C₄	—	12	4	no	—
97	C₄	—	12	4	no	—
98	C₄	#936	1	4	yes	—
99	C₃	#2642	1	1	yes	characteristic, complemented, Sylow
100	C₂	#5731	1	4	yes	—
101	C₂	C₂×#936	1	1	yes	characteristic
102	C₂	#5816	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×#870

C₂

yes

maximal, characteristic

#5229

C₂

yes

maximal

#4985

C₂

yes

maximal

#4932

C₂

yes

maximal, characteristic

#2642

—

maximal, Sylow

C₂²×C₃×Q₈

C₂²

yes

—

C₃×C₄⋊Q₈

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×Dic₃⋊C₄

C₂²

yes

—

C₂×C₄×Dic₃

C₂²

yes

—

#870

—

C₃⋊#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

C₂×C₄⋊Q₈

D₃

yes

—

—

—

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

C₂³

yes

—

C₂²×Dic₃

C₂³

yes

—

C₃×C₄⋊C₄

—

C₃×C₄²

D₄

yes

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂²×Q₈

C₂×D₃

yes

—

C₄⋊Q₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃

yes

characteristic

#95

—

#95

—

C₂³×C₃

C₂⁴

yes

characteristic

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

#67

yes

—

C₂²×C₃

C₂²×D₄

yes

characteristic

C₂²×C₃

C₂×#41

yes

—

C₂²×C₃

2₋¹⁺⁴

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#124

yes

—

Dic₃

—

Dic₃

—

C₂³

C₂³×D₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#237

yes

—

C₂×C₄

C₂×#67

yes

—

C₂×C₄

C₂×#67

yes

—

C₂×C₃

C₂×#124

yes

characteristic

C₂×C₃

#536

yes

—

C₂×C₃

#541

yes

—

C₂²

C₂×#237

yes

—

C₂²

#1007

yes

—

C₂²

C₂²×#67

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

#936

yes

—

C₃

#2642

yes

characteristic, complemented, Sylow

100

C₂

#5731

yes

—

101

C₂

C₂×#936

yes

characteristic

102

C₂

#5816

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator, socle, prime radical, cosocle	C₂³×C₃	C₂⁴
Fitting	C₂×C₃×C₄⋊Q₈	C₂
hypercenter	C₂×C₄⋊Q₈	D₃
hypocenter	C₃	#2642
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃

commutator, socle, prime radical, cosocle

C₂³×C₃

C₂⁴

Fitting

C₂×C₃×C₄⋊Q₈

C₂

hypercenter

C₂×C₄⋊Q₈

D₃

C₃

G ▷ C₂³×C₃ ▷ C₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
99. C₃	#2642	semidirect

H ≅ G/N

type

99. C₃

#2642

semidirect

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

subgroup	index
C₂×C₃×C₄⋊Q₈	2
C₂×#870	2
#5229	2
#4985	2
#4932	2
#2642	3

subgroup

index

C₂×C₃×C₄⋊Q₈

Subgroups — #83245

Conjugacy classes of subgroups

Complements

Maximal subgroups