Group #74354 — subgroups

Subgroups — #74354

Order 384.

348 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 133 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	C₃×#384	C₂	1	1	yes	maximal, characteristic
2	#4986	C₂	1	3	yes	maximal
3	#4982	C₂	1	6	yes	maximal
4	#4979	C₂	1	3	yes	maximal
5	#4937	C₂	1	2	yes	maximal
6	#2843	—	3	1	no	maximal, Sylow
7	C₂²×Dic₆	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄⋊C₄	C₂²	1	3	yes	—
9	C₂×C₃×C₄⋊C₄	C₂²	1	3	yes	—
10	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₃	C₂²	1	3	yes	—
12	C₂×Dic₃⋊C₄	C₂²	1	6	yes	—
13	C₃⋊#95	C₂²	1	6	yes	—
14	C₃⋊#95	C₂²	1	6	yes	—
15	C₃⋊#95	C₂²	1	3	yes	—
16	C₃⋊#95	C₂²	1	3	yes	—
17	#400	—	3	3	no	—
18	#398	—	3	6	no	—
19	#395	—	3	3	no	—
20	#384	D₃	1	1	yes	characteristic
21	#383	—	3	2	no	—
22	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
23	C₂²×C₃×C₄	C₂³	1	3	yes	—
24	C₂²×C₃×C₄	C₂³	1	3	yes	—
25	C₂²×Dic₃	C₂³	1	6	yes	—
26	C₂²×Dic₃	C₂³	1	2	yes	—
27	C₂×Dic₆	—	4	3	no	—
28	C₃×C₄⋊C₄	—	4	3	no	—
29	C₃×C₄⋊C₄	—	4	3	no	—
30	C₃×C₄²	Q₈	1	4	yes	complemented
31	C₄⋊Dic₃	—	4	3	no	—
32	Dic₃⋊C₄	—	4	6	no	—
33	C₂²×Q₈	—	3	1	no	—
34	C₂×C₄⋊C₄	—	3	6	no	—
35	C₂×C₄⋊C₄	—	3	3	no	—
36	C₂×C₄⋊C₄	C₂×D₃	1	3	yes	—
37	C₂×C₄⋊C₄	C₂×D₃	1	3	yes	—
38	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
39	#95	—	3	6	no	—
40	#95	—	3	6	no	—
41	#95	—	3	3	no	—
42	#95	—	3	3	no	—
43	C₂³×C₃	C₂⁴	1	1	yes	characteristic
44	C₂×C₃×C₄	—	2	3	no	—
45	C₂×C₃×C₄	—	2	3	no	—
46	C₂×C₃×C₄	—	2	6	no	—
47	C₂×C₃×C₄	—	2	3	no	—
48	C₂×C₃×C₄	—	2	3	no	—
49	C₂×C₃×C₄	C₂×Q₈	1	6	yes	—
50	C₂×Dic₃	—	2	6	no	—
51	C₂×Dic₃	—	2	6	no	—
52	C₂×Dic₃	—	2	12	no	—
53	Dic₆	—	4	4	no	—
54	C₂×Q₈	—	12	3	no	—
55	C₂²×C₄	—	3	6	no	—
56	C₂²×C₄	—	3	2	no	—
57	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
58	C₂²×C₄	C₂²×D₃	1	3	yes	—
59	C₂²×C₄	C₂²×D₃	1	3	yes	—
60	C₄⋊C₄	—	4	3	no	—
61	C₄⋊C₄	—	4	3	no	—
62	C₄⋊C₄	—	12	6	no	—
63	C₄⋊C₄	—	12	3	no	—
64	C₄²	Dic₆	1	4	yes	complemented
65	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
66	C₂²×C₃	H(5,𝔽₂)	1	3	yes	—
67	C₂²×C₃	2₋¹⁺⁴	1	3	yes	—
68	C₃×C₄	—	2	6	no	—
69	C₃×C₄	—	4	1	no	—
70	C₃×C₄	—	4	3	no	—
71	Dic₃	—	4	2	no	—
72	Dic₃	—	4	6	no	—
73	C₂³	C₂³×D₃	1	1	yes	characteristic
74	Q₈	—	12	4	no	—
75	C₂×C₄	—	2	3	no	—
76	C₂×C₄	—	2	3	no	—
77	C₂×C₄	—	2	6	no	—
78	C₂×C₄	—	2	3	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	6	6	no	—
81	C₂×C₄	—	6	6	no	—
82	C₂×C₄	—	6	12	no	—
83	C₂×C₄	C₂×Dic₆	1	6	yes	—
84	C₂×C₃	#544	1	3	yes	—
85	C₂×C₃	#560	1	1	yes	characteristic
86	C₂×C₃	#563	1	3	yes	—
87	C₂²	C₂²×Dic₆	1	1	yes	characteristic
88	C₂²	#1004	1	3	yes	—
89	C₂²	#1007	1	3	yes	—
90	C₄	—	2	6	no	—
91	C₄	—	4	1	no	—
92	C₄	—	4	3	no	—
93	C₄	—	12	2	no	—
94	C₄	—	12	6	no	—
95	C₃	#2843	1	1	yes	characteristic, complemented, Sylow
96	C₂	#5500	1	3	yes	—
97	C₂	#5720	1	1	yes	characteristic
98	C₂	#5735	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#384

C₂

yes

maximal, characteristic

#4986

C₂

yes

maximal

#4982

C₂

yes

maximal

#4979

C₂

yes

maximal

#4937

C₂

yes

maximal

#2843

—

maximal, Sylow

C₂²×Dic₆

C₂²

yes

characteristic

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×Dic₃⋊C₄

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

—

—

—

D₃

yes

characteristic

#383

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×Dic₃

C₂³

yes

—

C₂²×Dic₃

C₂³

yes

—

C₂×Dic₆

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

Q₈

yes

complemented

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₂²×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₄²

C₂×D₃

yes

—

—

—

—

C₂³×C₃

C₂⁴

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂²×C₄

C₂²×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Dic₆

yes

complemented

C₂²×C₃

C₂²×Q₈

yes

characteristic

C₂²×C₃

H(5,𝔽₂)

yes

—

C₂²×C₃

2₋¹⁺⁴

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

Dic₃

—

Dic₃

—

C₂³

C₂³×D₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×Dic₆

yes

—

C₂×C₃

#544

yes

—

C₂×C₃

#560

yes

characteristic

C₂×C₃

#563

yes

—

C₂²

C₂²×Dic₆

yes

characteristic

C₂²

#1004

yes

—

C₂²

#1007

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₃

#2843

yes

characteristic, complemented, Sylow

C₂

#5500

yes

—

C₂

#5720

yes

characteristic

C₂

#5735

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator, socle, prime radical, cosocle	C₂³×C₃	C₂⁴
Fitting	C₃×#384	C₂
hypercenter	#384	D₃
hypocenter	C₃	#2843
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃

commutator, socle, prime radical, cosocle

C₂³×C₃

C₂⁴

C₂

D₃

C₃

G ▷ C₂³×C₃ ▷ C₃

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
30. C₃×C₄²	Q₈	semidirect
64. C₄²	Dic₆	semidirect
95. C₃	#2843	semidirect

H ≅ G/N

type

30. C₃×C₄²

Q₈

semidirect

64. C₄²

Dic₆

semidirect

95. C₃

#2843

semidirect

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

subgroup	index
C₃×#384	2
#4986	2
#4982	2
#4979	2
#4937	2
#2843	3

subgroup

index

Subgroups — #74354

Conjugacy classes of subgroups

Complements

Maximal subgroups