Subgroups — #89042

Order 432.

214 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₃⋊D₃	C₂	1	3	yes	maximal, complemented
2	#6296	C₂	1	1	yes	maximal, characteristic, complemented
3	#6295	C₂	1	3	yes	maximal, complemented
4	#3798	—	3	3	no	maximal
5	C₂×C₃×C₃⋊D₃	C₂²	1	3	yes	—
6	C₃³×C₄	C₂²	1	1	yes	characteristic
7	C₃×C₃⋊Dic₃	C₂²	1	3	yes	—
8	C₄×C₃⋊D₃	D₃	1	3	yes	complemented
9	C₃×C₄×D₃	—	3	6	no	—
10	C₃×C₄×D₃	—	6	3	no	—
11	#627	—	3	3	no	—
12	#626	—	3	6	no	—
13	#625	—	3	3	no	—
14	C₂×C₃³	C₂³	1	1	yes	characteristic
15	C₃×C₃⋊D₃	—	2	3	no	—
16	#235	—	9	3	no	—
17	C₂×C₃⋊D₃	C₂×D₃	1	3	yes	—
18	C₂×C₃×D₃	—	3	6	no	—
19	C₂×C₃×D₃	—	6	3	no	—
20	C₃²×C₄	—	2	3	no	—
21	C₃²×C₄	—	4	1	no	—
22	C₃²×C₄	C₂×D₃	1	3	yes	—
23	C₃⋊Dic₃	C₂×D₃	1	3	yes	—
24	C₃×Dic₃	—	3	6	no	—
25	C₃×Dic₃	—	6	3	no	—
26	C₃³	#41	1	1	yes	characteristic, complemented, Sylow
27	C₂×C₃×C₄	—	9	3	no	—
28	#67	—	9	6	no	—
29	D₁₂	—	9	3	no	—
30	C₄×D₃	—	3	6	no	—
31	C₄×D₃	—	6	3	no	—
32	Dic₆	—	9	3	no	—
33	C₂×C₃²	—	2	3	no	—
34	C₂×C₃²	—	4	1	no	—
35	C₂×C₃²	C₂²×D₃	1	3	yes	—
36	C₃⋊D₃	—	2	3	no	—
37	C₃×D₃	—	6	6	no	—
38	C₃×D₃	—	6	6	no	—
39	#41	—	27	1	no	Sylow
40	C₂²×C₃	—	9	3	no	—
41	C₂×D₃	—	3	6	no	—
42	C₂×D₃	—	6	3	no	—
43	C₃×C₄	—	2	3	no	—
44	C₃×C₄	—	4	1	no	—
45	C₃×C₄	—	9	3	no	—
46	C₃×C₄	D₃²	1	3	yes	—
47	Dic₃	—	3	6	no	—
48	Dic₃	—	6	3	no	—
49	C₃²	—	2	3	no	—
50	C₃²	—	4	1	no	—
51	C₃²	#235	1	3	yes	complemented
52	Q₈	—	27	1	no	—
53	D₄	—	27	3	no	—
54	C₂×C₄	—	9	3	no	—
55	C₂×C₃	—	2	3	no	—
56	C₂×C₃	—	4	1	no	—
57	C₂×C₃	—	18	3	no	—
58	C₂×C₃	C₂×D₃²	1	3	yes	—
59	D₃	—	6	6	no	—
60	D₃	—	6	6	no	—
61	C₂²	—	9	3	no	—
62	C₄	—	9	3	no	—
63	C₄	#1113	1	1	yes	characteristic
64	C₃	—	2	3	no	—
65	C₃	—	4	1	no	—
66	C₃	#3798	1	3	yes	complemented
67	C₂	—	18	3	no	—
68	C₂	C₂×#1113	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄	#1113
commutator, socle, cosocle	C₂×C₃³	C₂³
Frattini	C₂	C₂×#1113
Fitting	C₃³×C₄	C₂²
hypocenter	C₃³	#41
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. C₃×C₄×C₃⋊D₃	C₂	semidirect
2. #6296	C₂	semidirect
3. #6295	C₂	semidirect
8. C₄×C₃⋊D₃	D₃	semidirect
26. C₃³	#41	semidirect
51. C₃²	#235	semidirect
66. C₃	#3798	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
#6296	2
#6295	2
#3798	3