Subgroups — #88807

Order 432.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃²×Dic₃	C₂	1	1	yes	maximal, characteristic
4	#3750	—	3	1	no	maximal
5	Dic₃⋊Dic₃	—	3	4	no	maximal
6	C₂²×C₃³	C₂²	1	1	yes	characteristic
7	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
8	C₃×C₃⋊Dic₃	C₄	1	2	yes	complemented
9	C₃²×Dic₃	—	2	1	no	—
10	C₂×C₃²×C₄	—	3	1	no	—
11	C₂×C₃⋊Dic₃	—	3	1	no	—
12	C₂×C₃⋊Dic₃	—	3	4	no	—
13	C₂×C₃⋊Dic₃	—	6	4	no	—
14	C₂×C₃⋊Dic₃	D₃	1	1	yes	characteristic
15	C₂×C₃×Dic₃	—	3	4	no	—
16	C₂×C₃×Dic₃	D₃	1	4	yes	—
17	C₂×C₃³	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃³	D₄	1	1	yes	characteristic
19	C₂×C₃³	Q₈	1	1	yes	characteristic
20	C₄⋊Dic₃	—	9	1	no	—
21	Dic₃⋊C₄	—	9	4	no	—
22	C₂²×C₃²	—	2	4	no	—
23	C₂²×C₃²	C₂×D₃	1	4	yes	—
24	C₂²×C₃²	C₂×D₃	1	1	yes	characteristic
25	C₃²×C₄	—	6	1	no	—
26	C₃⋊Dic₃	—	6	8	no	—
27	C₃⋊Dic₃	—	6	1	no	—
28	C₃⋊Dic₃	—	6	4	no	—
29	C₃⋊Dic₃	Dic₃	1	2	yes	complemented
30	C₃×Dic₃	—	2	4	no	—
31	C₃×Dic₃	—	3	8	no	—
32	C₃³	C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
33	C₂×C₃×C₄	—	3	4	no	—
34	C₂×C₃×C₄	—	9	1	no	—
35	C₂×Dic₃	—	3	4	no	—
36	C₂×Dic₃	—	9	4	no	—
37	C₂×Dic₃	—	9	1	no	—
38	C₂×Dic₃	—	18	4	no	—
39	C₂×Dic₃	C₃⋊D₃	1	1	yes	characteristic
40	C₂×C₃²	—	2	4	no	—
41	C₂×C₃²	—	2	4	no	—
42	C₂×C₃²	—	2	4	no	—
43	C₂×C₃²	Dic₆	1	4	yes	—
44	C₂×C₃²	Dic₆	1	1	yes	characteristic
45	C₂×C₃²	C₄×D₃	1	4	yes	—
46	C₂×C₃²	D₁₂	1	1	yes	characteristic
47	C₂×C₃²	C₂×Dic₃	1	1	yes	characteristic
48	C₂×C₃²	#67	1	4	yes	—
49	C₄⋊C₄	—	27	1	no	Sylow
50	C₂²×C₃	—	2	4	no	—
51	C₂²×C₃	D₃²	1	4	yes	—
52	C₂²×C₃	C₂×C₃⋊D₃	1	1	yes	characteristic
53	C₃×C₄	—	6	4	no	—
54	C₃×C₄	—	9	2	no	—
55	Dic₃	—	2	1	no	—
56	Dic₃	—	3	8	no	—
57	Dic₃	—	18	8	no	—
58	Dic₃	—	18	4	no	—
59	Dic₃	—	18	1	no	—
60	C₃²	—	2	4	no	—
61	C₃²	Dic₃⋊C₄	1	4	yes	complemented
62	C₃²	C₄⋊Dic₃	1	1	yes	characteristic, complemented
63	C₂×C₄	—	3	1	no	—
64	C₂×C₄	—	9	1	no	—
65	C₂×C₄	—	27	1	no	—
66	C₂×C₃	—	2	4	no	—
67	C₂×C₃	—	2	4	no	—
68	C₂×C₃	—	2	4	no	—
69	C₂×C₃	D₃×Dic₃	1	4	yes	—
70	C₂×C₃	#626	1	4	yes	—
71	C₂×C₃	#627	1	4	yes	—
72	C₂×C₃	#634	1	1	yes	characteristic
73	C₂×C₃	C₄×C₃⋊D₃	1	1	yes	characteristic
74	C₂×C₃	#638	1	1	yes	characteristic
75	C₂²	D₃×C₃⋊D₃	1	1	yes	characteristic
76	C₄	—	6	1	no	—
77	C₄	—	9	2	no	—
78	C₄	—	54	1	no	—
79	C₃	—	2	4	no	—
80	C₃	Dic₃⋊Dic₃	1	4	yes	complemented
81	C₃	#3750	1	1	yes	characteristic, complemented
82	C₂	Dic₃×C₃⋊D₃	1	1	yes	characteristic
83	C₂	#6291	1	1	yes	characteristic
84	C₂	#6293	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
8. C₃×C₃⋊Dic₃	C₄	semidirect
29. C₃⋊Dic₃	Dic₃	semidirect
32. C₃³	C₄⋊C₄	semidirect
61. C₃²	Dic₃⋊C₄	semidirect
62. C₃²	C₄⋊Dic₃	semidirect
80. C₃	Dic₃⋊Dic₃	semidirect
81. C₃	#3750	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃⋊(C₃⋊Dic₃)	2
C₂×C₃×C₃⋊Dic₃	2
C₂×C₃²×Dic₃	2
#3750	3
Dic₃⋊Dic₃	3