Subgroups — #88796

Order 432.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6308	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₄⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic
3	C₃²×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#3744	—	3	1	no	maximal
5	#3716	—	3	4	no	maximal
6	C₂×C₃×C₃⋊D₃	—	2	1	no	—
7	C₃³×C₄	C₂²	1	1	yes	characteristic
8	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
9	C₄⋊(C₃⋊D₃)	D₃	1	1	yes	characteristic
10	#634	—	3	1	no	—
11	#634	—	3	4	no	—
12	#634	—	6	4	no	—
13	C₃×D₁₂	—	3	4	no	—
14	C₃²×C₈	—	3	1	no	—
15	C₃×C₃⋊C₈	D₃	1	4	yes	complemented
16	C₂×C₃³	D₄	1	1	yes	characteristic
17	C₃×C₃⋊D₃	—	4	1	no	—
18	#214	—	9	1	no	—
19	#204	—	9	4	no	—
20	C₂×C₃⋊D₃	—	2	1	no	—
21	C₂×C₃×D₃	—	6	4	no	—
22	C₃²×C₄	—	2	4	no	—
23	C₃²×C₄	C₂×D₃	1	4	yes	—
24	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
25	C₃⋊Dic₃	—	6	8	no	—
26	C₃⋊Dic₃	—	6	4	no	—
27	C₃⋊Dic₃	—	6	1	no	—
28	C₃³	SD₁₆	1	1	yes	characteristic, complemented, Sylow
29	C₃×D₄	—	9	1	no	—
30	D₁₂	—	3	4	no	—
31	Dic₆	—	9	4	no	—
32	Dic₆	—	9	1	no	—
33	Dic₆	—	18	4	no	—
34	C₃×C₈	—	3	4	no	—
35	C₃⋊C₈	C₃⋊D₃	1	1	yes	characteristic, complemented
36	C₂×C₃²	—	2	4	no	—
37	C₂×C₃²	D₁₂	1	4	yes	—
38	C₂×C₃²	#67	1	1	yes	characteristic
39	C₃⋊D₃	—	4	1	no	—
40	C₃×D₃	—	12	4	no	—
41	SD₁₆	—	27	1	no	Sylow
42	C₂²×C₃	—	18	1	no	—
43	C₂×D₃	—	6	4	no	—
44	C₃×C₄	—	2	4	no	—
45	C₃×C₄	D₃²	1	4	yes	—
46	C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
47	Dic₃	—	18	8	no	—
48	Dic₃	—	18	4	no	—
49	Dic₃	—	18	1	no	—
50	C₃²	—	2	4	no	—
51	C₃²	#204	1	4	yes	complemented
52	C₃²	#214	1	1	yes	characteristic, complemented
53	Q₈	—	27	1	no	—
54	D₄	—	9	1	no	—
55	C₈	—	3	1	no	—
56	C₂×C₃	—	2	4	no	—
57	C₂×C₃	—	36	1	no	—
58	C₂×C₃	#626	1	4	yes	—
59	C₂×C₃	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
60	D₃	—	12	4	no	—
61	C₂²	—	18	1	no	—
62	C₄	—	54	1	no	—
63	C₄	D₃×C₃⋊D₃	1	1	yes	characteristic
64	C₃	—	2	4	no	—
65	C₃	#3716	1	4	yes	complemented
66	C₃	#3744	1	1	yes	characteristic, complemented
67	C₂	—	36	1	no	—
68	C₂	#6292	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6292
commutator, Fitting, cosocle	C₃³×C₄	C₂²
Frattini, hypercenter	C₄	D₃×C₃⋊D₃
socle	C₂×C₃³	D₄
prime radical	C₃×C₄⋊(C₃⋊D₃)	C₂
hypocenter	C₃³	SD₁₆
LCS	G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #6308	C₂	semidirect
3. C₃²×C₃⋊C₈	C₂	semidirect
15. C₃×C₃⋊C₈	D₃	semidirect
28. C₃³	SD₁₆	semidirect
35. C₃⋊C₈	C₃⋊D₃	semidirect
51. C₃²	#204	semidirect
52. C₃²	#214	semidirect
65. C₃	#3716	semidirect
66. C₃	#3744	semidirect

Maximal subgroups

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

subgroup	index
#6308	2
C₃×C₄⋊(C₃⋊D₃)	2
C₃²×C₃⋊C₈	2
#3744	3
#3716	3