Subgroups — #88536

Order 432.

170 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃⋊D₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃⋊Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#3752	—	3	1	no	maximal
5	(C₂×D₉)⋊C₄	—	3	3	no	maximal
6	C₂²×C₃×C₉	C₂²	1	1	yes	characteristic
7	C₂×C₃⋊D₉	—	2	1	no	—
8	C₂×C₃⋊D₉	—	2	1	no	—
9	C₂×C₃⋊D₉	C₄	1	2	yes	complemented
10	C₃×C₄×C₉	—	2	1	no	—
11	C₃⋊Dic₉	—	2	1	no	—
12	C₂²×C₃⋊D₃	—	3	1	no	—
13	C₂×C₃²×C₄	D₃	1	1	yes	characteristic
14	C₂×C₃⋊Dic₃	—	3	1	no	—
15	C₂²×D₉	—	3	3	no	—
16	C₂×C₄×C₉	D₃	1	3	yes	complemented
17	C₂×Dic₉	—	3	3	no	—
18	C₂×C₃×C₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₃×C₉	D₄	1	1	yes	characteristic
20	C₂×C₃×C₉	D₄	1	1	yes	characteristic
21	C₃⋊D₉	—	2	2	no	—
22	(C₂×D₃)⋊C₄	—	9	3	no	—
23	(C₂×D₃)⋊C₄	—	9	1	no	—
24	C₂²×C₃²	C₂×D₃	1	1	yes	characteristic
25	C₂×C₃⋊D₃	—	3	2	no	—
26	C₂×C₃⋊D₃	—	6	1	no	—
27	C₂×C₃⋊D₃	—	6	1	no	—
28	C₃²×C₄	—	2	1	no	—
29	C₃⋊Dic₃	—	6	1	no	—
30	C₂²×C₉	C₂×D₃	1	3	yes	—
31	C₂×D₉	—	3	6	no	—
32	C₂×D₉	—	6	3	no	—
33	C₂×D₉	—	6	3	no	—
34	C₄×C₉	—	2	3	no	—
35	Dic₉	—	6	3	no	—
36	C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
37	C₂²×D₃	—	9	3	no	—
38	C₂²×D₃	—	9	1	no	—
39	C₂×C₃×C₄	D₉	1	3	yes	complemented
40	C₂×C₃×C₄	C₃⋊D₃	1	1	yes	characteristic
41	C₂×Dic₃	—	9	3	no	—
42	C₂×Dic₃	—	9	1	no	—
43	C₂×C₃²	C₄×D₃	1	1	yes	characteristic
44	C₂×C₃²	D₁₂	1	1	yes	characteristic
45	C₂×C₃²	#67	1	1	yes	characteristic
46	C₃⋊D₃	—	6	2	no	—
47	C₂×C₉	C₄×D₃	1	3	yes	—
48	C₂×C₉	D₁₂	1	3	yes	—
49	C₂×C₉	#67	1	3	yes	—
50	D₉	—	6	6	no	—
51	C₂²⋊C₄	—	27	1	no	Sylow
52	C₂²×C₃	C₂×D₉	1	3	yes	—
53	C₂²×C₃	C₂×C₃⋊D₃	1	1	yes	characteristic
54	C₂×D₃	—	9	6	no	—
55	C₂×D₃	—	9	2	no	—
56	C₂×D₃	—	18	3	no	—
57	C₂×D₃	—	18	3	no	—
58	C₂×D₃	—	18	1	no	—
59	C₂×D₃	—	18	1	no	—
60	C₃×C₄	—	2	3	no	—
61	C₃×C₄	—	2	1	no	—
62	Dic₃	—	18	3	no	—
63	Dic₃	—	18	1	no	—
64	C₃²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
65	C₉	(C₂×D₃)⋊C₄	1	3	yes	complemented
66	C₂³	—	27	1	no	—
67	C₂×C₄	—	27	1	no	—
68	C₂×C₄	C₃⋊D₉	1	1	yes	characteristic, complemented
69	C₂×C₃	C₄×D₉	1	3	yes	—
70	C₂×C₃	D₃₆	1	3	yes	—
71	C₂×C₃	#611	1	3	yes	—
72	C₂×C₃	C₄×C₃⋊D₃	1	1	yes	characteristic
73	C₂×C₃	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
74	C₂×C₃	#638	1	1	yes	characteristic
75	D₃	—	18	6	no	—
76	D₃	—	18	2	no	—
77	C₂²	—	27	2	no	—
78	C₂²	—	54	1	no	—
79	C₂²	—	54	1	no	—
80	C₂²	C₂×C₃⋊D₉	1	1	yes	characteristic
81	C₄	—	2	1	no	—
82	C₄	—	54	1	no	—
83	C₃	(C₂×D₉)⋊C₄	1	3	yes	complemented
84	C₃	#3752	1	1	yes	characteristic
85	C₂	—	54	2	no	—
86	C₂	C₄×C₃⋊D₉	1	1	yes	characteristic
87	C₂	C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
88	C₂	#6233	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄×C₉	C₂	semidirect
3. C₂×C₃⋊Dic₉	C₂	semidirect
9. C₂×C₃⋊D₉	C₄	semidirect
16. C₂×C₄×C₉	D₃	semidirect
36. C₃×C₉	C₂²⋊C₄	semidirect
39. C₂×C₃×C₄	D₉	semidirect
65. C₉	(C₂×D₃)⋊C₄	semidirect
68. C₂×C₄	C₃⋊D₉	semidirect
83. C₃	(C₂×D₉)⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃⋊D₉	2
C₂×C₃×C₄×C₉	2
C₂×C₃⋊Dic₉	2
#3752	3
(C₂×D₉)⋊C₄	3