Subgroups — #88793

Order 432.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic
2	C₃²×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic, complemented
4	#3755	—	3	1	no	maximal
5	#3715	—	3	4	no	maximal
6	C₂×C₃×C₃⋊D₃	—	2	1	no	—
7	C₃³×C₄	C₂²	1	1	yes	characteristic
8	C₃²×Dic₃	—	2	1	no	—
9	C₃²×Q₈	—	3	1	no	—
10	C₄⋊(C₃⋊D₃)	D₃	1	1	yes	characteristic
11	C₃×D₁₂	—	3	4	no	—
12	C₃×Dic₆	D₃	1	4	yes	complemented
13	C₃⋊(C₃⋊C₈)	—	3	1	no	—
14	C₃⋊(C₃⋊C₈)	—	3	4	no	—
15	C₃⋊(C₃⋊C₈)	—	6	4	no	—
16	C₂×C₃³	D₄	1	1	yes	characteristic
17	C₃×C₃⋊D₃	—	4	1	no	—
18	#215	—	9	4	no	—
19	#214	—	9	1	no	—
20	C₂×C₃⋊D₃	—	2	1	no	—
21	C₂×C₃×D₃	—	6	4	no	—
22	C₃²×C₄	—	2	4	no	—
23	C₃²×C₄	—	6	1	no	—
24	C₃²×C₄	C₂×D₃	1	4	yes	—
25	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
26	C₃×Dic₃	—	2	4	no	—
27	C₃³	SD₁₆	1	1	yes	characteristic, complemented, Sylow
28	C₃×Q₈	—	3	4	no	—
29	C₃×D₄	—	9	1	no	—
30	D₁₂	—	3	4	no	—
31	Dic₆	C₃⋊D₃	1	1	yes	characteristic, complemented
32	C₃⋊C₈	—	9	4	no	—
33	C₃⋊C₈	—	9	1	no	—
34	C₃⋊C₈	—	18	4	no	—
35	C₂×C₃²	—	2	4	no	—
36	C₂×C₃²	#67	1	4	yes	—
37	C₂×C₃²	#67	1	1	yes	characteristic
38	C₃⋊D₃	—	4	1	no	—
39	C₃×D₃	—	12	4	no	—
40	SD₁₆	—	27	1	no	Sylow
41	C₂²×C₃	—	18	1	no	—
42	C₂×D₃	—	6	4	no	—
43	C₃×C₄	—	2	4	no	—
44	C₃×C₄	—	6	4	no	—
45	C₃×C₄	D₃²	1	4	yes	—
46	C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
47	Dic₃	—	2	1	no	—
48	C₃²	—	2	4	no	—
49	C₃²	#214	1	1	yes	characteristic, complemented
50	C₃²	#215	1	4	yes	complemented
51	Q₈	—	3	1	no	—
52	D₄	—	9	1	no	—
53	C₈	—	27	1	no	—
54	C₂×C₃	—	2	4	no	—
55	C₂×C₃	—	36	1	no	—
56	C₂×C₃	#625	1	4	yes	—
57	C₂×C₃	#638	1	1	yes	characteristic
58	D₃	—	12	4	no	—
59	C₂²	—	18	1	no	—
60	C₄	—	6	1	no	—
61	C₄	D₃×C₃⋊D₃	1	1	yes	characteristic
62	C₃	—	2	4	no	—
63	C₃	#3715	1	4	yes	complemented
64	C₃	#3755	1	1	yes	characteristic, complemented
65	C₂	—	36	1	no	—
66	C₂	#6290	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6290
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
2. C₃²×Dic₆	C₂	semidirect
3. C₃⋊(C₃⋊(C₃⋊C₈))	C₂	semidirect
12. C₃×Dic₆	D₃	semidirect
27. C₃³	SD₁₆	semidirect
31. Dic₆	C₃⋊D₃	semidirect
49. C₃²	#214	semidirect
50. C₃²	#215	semidirect
63. C₃	#3715	semidirect
64. C₃	#3755	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄⋊(C₃⋊D₃)	2
C₃²×Dic₆	2
C₃⋊(C₃⋊(C₃⋊C₈))	2
#3755	3
#3715	3