Subgroups — #88949

Order 432.

270 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1113	C₂	1	1	yes	maximal, characteristic
2	Dic₃×C₃⋊D₃	C₂	1	3	yes	maximal, complemented
3	C₃×#624	C₂	1	3	yes	maximal, complemented
4	C₄×D₃²	—	3	3	no	maximal
5	C₂×C₃×C₃⋊D₃	C₂²	1	3	yes	—
6	#1113	C₄	1	4	yes	complemented
7	C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic, complemented
8	C₃²×Dic₃	C₂²	1	3	yes	complemented
9	C₂×D₃²	—	3	3	no	—
10	C₄×C₃⋊D₃	—	3	3	no	—
11	C₃×C₄×D₃	—	3	6	no	—
12	#624	D₃	1	3	yes	complemented
13	D₃×Dic₃	—	3	6	no	—
14	D₃×Dic₃	—	6	3	no	—
15	C₂×C₃³	C₂³	1	1	yes	characteristic
16	C₃×C₃⋊D₃	C₂×C₄	1	6	yes	complemented
17	C₂×C₄×D₃	—	9	3	no	—
18	C₂×C₃⋊D₃	C₂×D₃	1	3	yes	—
19	C₂×C₃×D₃	—	3	6	no	—
20	C₂×C₃×D₃	—	6	3	no	—
21	D₃²	—	3	12	no	—
22	C₃²×C₄	—	3	3	no	—
23	C₃⋊Dic₃	—	3	3	no	—
24	C₃⋊Dic₃	—	6	3	no	—
25	C₃⋊Dic₃	—	12	1	no	—
26	C₃×Dic₃	—	2	3	no	—
27	C₃×Dic₃	C₂×D₃	1	6	yes	complemented
28	C₃³	C₂²×C₄	1	1	yes	characteristic, complemented, Sylow
29	C₂²×D₃	—	9	3	no	—
30	C₂×C₃×C₄	—	9	3	no	—
31	C₂×Dic₃	—	9	3	no	—
32	C₄×D₃	—	3	6	no	—
33	C₄×D₃	—	9	6	no	—
34	C₄×D₃	—	18	3	no	—
35	C₂×C₃²	—	2	3	no	—
36	C₂×C₃²	—	4	1	no	—
37	C₂×C₃²	C₂²×D₃	1	3	yes	—
38	C₃⋊D₃	C₄×D₃	1	6	yes	complemented
39	C₃×D₃	—	3	12	no	—
40	C₃×D₃	—	6	6	no	—
41	C₂²×C₄	—	27	1	no	Sylow
42	C₂²×C₃	—	9	3	no	—
43	C₂×D₃	—	3	6	no	—
44	C₂×D₃	—	6	3	no	—
45	C₂×D₃	—	9	12	no	—
46	C₃×C₄	—	3	6	no	—
47	C₃×C₄	—	6	3	no	—
48	Dic₃	—	9	3	no	—
49	Dic₃	—	18	3	no	—
50	Dic₃	—	36	1	no	—
51	Dic₃	D₃²	1	3	yes	complemented
52	C₃²	—	2	3	no	—
53	C₃²	—	4	1	no	—
54	C₃²	C₂×C₄×D₃	1	3	yes	complemented
55	C₂³	—	27	1	no	—
56	C₂×C₄	—	9	3	no	—
57	C₂×C₄	—	27	3	no	—
58	C₂×C₃	—	2	3	no	—
59	C₂×C₃	—	4	1	no	—
60	C₂×C₃	—	9	6	no	—
61	C₂×C₃	C₂×D₃²	1	3	yes	—
62	D₃	—	3	12	no	—
63	D₃	—	6	6	no	—
64	C₂²	—	9	3	no	—
65	C₂²	—	27	4	no	—
66	C₄	—	3	3	no	—
67	C₄	—	27	1	no	—
68	C₃	—	2	3	no	—
69	C₃	—	4	1	no	—
70	C₃	C₄×D₃²	1	3	yes	complemented
71	C₂	—	9	6	no	—
72	C₂	D₃³	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃³
commutator, hypocenter	C₃³	C₂²×C₄
Fitting, socle, cosocle	C₂×C₃³	C₂³
prime radical	C₂×#1113	C₂
LCS	G ▷ C₃³
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. Dic₃×C₃⋊D₃	C₂	semidirect
3. C₃×#624	C₂	semidirect
6. #1113	C₄	semidirect
7. C₃⋊(C₃⋊Dic₃)	C₂²	semidirect
8. C₃²×Dic₃	C₂²	semidirect
12. #624	D₃	semidirect
16. C₃×C₃⋊D₃	C₂×C₄	semidirect
27. C₃×Dic₃	C₂×D₃	semidirect
28. C₃³	C₂²×C₄	semidirect
38. C₃⋊D₃	C₄×D₃	semidirect
51. Dic₃	D₃²	semidirect
54. C₃²	C₂×C₄×D₃	semidirect
70. C₃	C₄×D₃²	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1113	2
Dic₃×C₃⋊D₃	2
C₃×#624	2
C₄×D₃²	3