Subgroups — #88736

Order 432.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#6233	C₂	1	2	yes	maximal, complemented
3	C₄⋊(C₃⋊D₉)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₃⋊D₉	C₂	1	2	yes	maximal, complemented
5	#6226	C₂	1	1	yes	maximal, characteristic, complemented
6	#3828	—	3	1	no	maximal
7	#3697	—	3	3	no	maximal
8	C₂²×C₃×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₃⋊D₉	C₂²	1	2	yes	—
10	C₃×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₃⋊Dic₉	C₂²	1	2	yes	—
13	C₂×C₃²×C₄	D₃	1	1	yes	characteristic
14	#638	—	3	2	no	—
15	C₄⋊(C₃⋊D₃)	—	3	1	no	—
16	C₄×C₃⋊D₃	—	3	2	no	—
17	#634	—	3	1	no	—
18	C₂×C₄×C₉	D₃	1	3	yes	complemented
19	#611	—	3	6	no	—
20	D₃₆	—	3	3	no	—
21	C₄×D₉	—	3	6	no	—
22	Dic₁₈	—	3	3	no	—
23	C₂×C₃×C₉	—	2	1	no	—
24	C₂×C₃×C₉	C₂³	1	1	yes	characteristic
25	C₃⋊D₉	—	2	2	no	—
26	#235	—	9	3	no	—
27	#235	—	9	1	no	—
28	C₂²×C₃²	C₂×D₃	1	1	yes	characteristic
29	C₂×C₃⋊D₃	—	3	2	no	—
30	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
31	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
32	C₃⋊Dic₃	—	3	2	no	—
33	C₂²×C₉	C₂×D₃	1	3	yes	—
34	C₂×D₉	—	3	6	no	—
35	C₄×C₉	C₂×D₃	1	3	yes	—
36	C₄×C₉	C₂×D₃	1	3	yes	—
37	Dic₉	—	3	6	no	—
38	C₃×C₉	#41	1	1	yes	characteristic, complemented, Sylow
39	C₂×C₃×C₄	D₉	1	3	yes	complemented
40	C₂×C₃×C₄	C₃⋊D₃	1	1	yes	characteristic
41	#67	—	9	6	no	—
42	#67	—	9	2	no	—
43	D₁₂	—	9	3	no	—
44	D₁₂	—	9	1	no	—
45	C₄×D₃	—	9	6	no	—
46	C₄×D₃	—	9	2	no	—
47	Dic₆	—	9	3	no	—
48	Dic₆	—	9	1	no	—
49	C₂×C₃²	—	2	1	no	—
50	C₂×C₃²	C₂²×D₃	1	1	yes	characteristic
51	C₃⋊D₃	—	6	2	no	—
52	C₂×C₉	—	2	3	no	—
53	C₂×C₉	C₂²×D₃	1	3	yes	—
54	D₉	—	6	6	no	—
55	#41	—	27	1	no	Sylow
56	C₂²×C₃	C₂×D₉	1	3	yes	—
57	C₂²×C₃	C₂×C₃⋊D₃	1	1	yes	characteristic
58	C₂×D₃	—	9	6	no	—
59	C₂×D₃	—	9	2	no	—
60	C₃×C₄	C₂×D₉	1	3	yes	—
61	C₃×C₄	C₂×D₉	1	3	yes	—
62	C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
63	C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
64	Dic₃	—	9	6	no	—
65	Dic₃	—	9	2	no	—
66	C₃²	#235	1	1	yes	characteristic
67	C₉	#235	1	3	yes	complemented
68	Q₈	—	27	1	no	—
69	D₄	—	27	2	no	—
70	D₄	—	27	1	no	—
71	C₂×C₄	—	27	2	no	—
72	C₂×C₄	C₃⋊D₉	1	1	yes	characteristic, complemented
73	C₂×C₃	—	2	3	no	—
74	C₂×C₃	—	2	1	no	—
75	C₂×C₃	C₂²×D₉	1	3	yes	—
76	C₂×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
77	D₃	—	18	6	no	—
78	D₃	—	18	2	no	—
79	C₂²	—	27	2	no	—
80	C₂²	C₂×C₃⋊D₉	1	1	yes	characteristic
81	C₄	—	27	2	no	—
82	C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
83	C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
84	C₃	#3697	1	3	yes	complemented
85	C₃	#3828	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	54	2	no	—
88	C₂	C₂²×C₃⋊D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₃⋊D₉
commutator, cosocle	C₂×C₃×C₉	C₂³
Frattini	C₂×C₃	C₂²×C₃⋊D₃
Fitting	C₂×C₃×C₄×C₉	C₂
socle	C₂×C₃²	C₂²×D₃
hypercenter	C₂×C₄	C₃⋊D₉
hypocenter	C₃×C₉	#41
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₄×C₉	C₂	semidirect
2. #6233	C₂	semidirect
3. C₄⋊(C₃⋊D₉)	C₂	semidirect
4. C₄×C₃⋊D₉	C₂	semidirect
5. #6226	C₂	semidirect
18. C₂×C₄×C₉	D₃	semidirect
38. C₃×C₉	#41	semidirect
39. C₂×C₃×C₄	D₉	semidirect
67. C₉	#235	semidirect
72. C₂×C₄	C₃⋊D₉	semidirect
84. C₃	#3697	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₉	2
#6233	2
C₄⋊(C₃⋊D₉)	2
C₄×C₃⋊D₉	2
#6226	2
#3828	3
#3697	3