Subgroups — C₃×C₂²⋊(C₄²⋊C₉) #13152253

Order 1728.

594 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₄²	C₃	1	1	yes	maximal, characteristic
2	C₂²⋊(C₄²⋊C₉)	C₃	1	3	yes	maximal, direct product factor
3	C₃×C₂²⋊(C₂²⋊C₉)	—	4	1	no	maximal
4	C₃×C₄²⋊C₉	—	4	4	no	maximal
5	C₂³×C₃²×C₄	—	3	1	no	—
6	C₂×C₃²×C₄²	—	3	4	no	—
7	C₂²×C₃×C₄²	C₉	1	3	yes	complemented
8	C₂²×C₃×C₄²	C₃²	1	1	yes	characteristic
9	C₂⁴×C₃²	A₄	1	1	yes	characteristic
10	C₂²×C₃²×C₄	—	3	8	no	—
11	C₂²×C₃²×C₄	—	3	6	no	—
12	C₂²⋊(C₂²⋊C₉)	—	4	3	no	—
13	C₃²×C₄²	—	3	4	no	—
14	C₃²×C₄²	A₄	1	4	yes	—
15	C₄²⋊C₉	—	4	12	no	—
16	C₃×C₂²⋊C₉	—	16	4	no	—
17	C₃×C₂²⋊C₉	—	16	1	no	—
18	C₂³×C₃×C₄	—	3	3	no	—
19	C₂³×C₃×C₄	—	3	1	no	—
20	C₂×C₃×C₄²	—	3	12	no	—
21	C₂×C₃×C₄²	—	3	4	no	—
22	C₂³×C₃²	—	3	4	no	—
23	C₂³×C₃²	—	3	1	no	—
24	C₂×C₃²×C₄	—	3	24	no	—
25	C₂×C₃²×C₄	—	3	4	no	—
26	C₂²×C₄²	C₃×C₉	1	1	yes	characteristic, complemented, Sylow
27	C₂⁴×C₃	C₂²⋊C₉	1	3	yes	—
28	C₂⁴×C₃	C₃×A₄	1	1	yes	characteristic
29	C₂²×C₃×C₄	—	3	6	no	—
30	C₂²×C₃×C₄	—	3	18	no	—
31	C₂²×C₃×C₄	—	3	24	no	—
32	C₂²×C₃×C₄	—	3	8	no	—
33	C₃×C₄²	—	3	4	no	—
34	C₃×C₄²	—	3	12	no	—
35	C₃×C₄²	C₂²⋊C₉	1	12	yes	complemented
36	C₃×C₄²	C₃×A₄	1	4	yes	—
37	C₂²×C₃²	—	3	6	no	—
38	C₂²×C₃²	—	3	4	no	—
39	C₂²×C₃²	C₄²⋊C₃	1	4	yes	—
40	C₂²×C₃²	C₂²⋊A₄	1	1	yes	characteristic
41	C₃²×C₄	—	3	8	no	—
42	C₂²⋊C₉	—	16	3	no	—
43	C₂²⋊C₉	—	16	12	no	—
44	C₂³×C₄	—	3	1	no	—
45	C₂×C₄²	—	3	4	no	—
46	C₃×C₉	—	64	1	no	Sylow
47	C₂³×C₃	—	3	1	no	—
48	C₂³×C₃	—	3	12	no	—
49	C₂³×C₃	—	3	4	no	—
50	C₂³×C₃	—	3	3	no	—
51	C₂×C₃×C₄	—	3	72	no	—
52	C₂×C₃×C₄	—	3	24	no	—
53	C₂×C₃×C₄	—	3	4	no	—
54	C₂×C₃×C₄	—	3	12	no	—
55	C₂×C₃²	—	3	1	no	—
56	C₂×C₃²	—	3	4	no	—
57	C₂⁴	C₃×C₂²⋊C₉	1	1	yes	characteristic
58	C₂²×C₄	—	3	6	no	—
59	C₂²×C₄	—	3	8	no	—
60	C₄²	—	3	4	no	—
61	C₄²	C₃×C₂²⋊C₉	1	4	yes	complemented
62	C₂²×C₃	—	3	18	no	—
63	C₂²×C₃	—	3	6	no	—
64	C₂²×C₃	—	3	12	no	—
65	C₂²×C₃	—	3	4	no	—
66	C₂²×C₃	C₄²⋊C₉	1	12	yes	complemented
67	C₂²×C₃	C₃×C₄²⋊C₃	1	4	yes	—
68	C₂²×C₃	C₂²⋊(C₂²⋊C₉)	1	3	yes	—
69	C₂²×C₃	C₃×C₂²⋊A₄	1	1	yes	characteristic
70	C₃×C₄	—	3	24	no	—
71	C₃×C₄	—	3	8	no	—
72	C₃²	C₂²⋊(C₄²⋊C₃)	1	1	yes	characteristic
73	C₉	—	64	3	no	—
74	C₂³	—	3	1	no	—
75	C₂³	—	3	4	no	—
76	C₂×C₄	—	3	24	no	—
77	C₂×C₄	—	3	4	no	—
78	C₂×C₃	—	3	3	no	—
79	C₂×C₃	—	3	12	no	—
80	C₂×C₃	—	3	1	no	—
81	C₂×C₃	—	3	4	no	—
82	C₂²	—	3	6	no	—
83	C₂²	—	3	4	no	—
84	C₂²	C₃×C₄²⋊C₉	1	4	yes	complemented
85	C₂²	C₃×C₂²⋊(C₂²⋊C₉)	1	1	yes	characteristic
86	C₄	—	3	8	no	—
87	C₃	C₂²⋊(C₄²⋊C₉)	1	3	yes	direct product factor
88	C₃	C₃×C₂²⋊(C₄²⋊C₃)	1	1	yes	characteristic
89	C₂	—	3	1	no	—
90	C₂	—	3	4	no	—

	subgroup	G/H
center, hypercenter	C₃²	C₂²⋊(C₄²⋊C₃)
commutator, hypocenter	C₂²×C₄²	C₃×C₉
Frattini	C₂²×C₃	C₃×C₂²⋊A₄
Fitting	C₂²×C₃²×C₄²	C₃
socle, prime radical	C₂⁴×C₃²	A₄
cosocle	C₂²×C₃×C₄²	C₃²
LCS	G ▷ C₂²×C₄²
UCS	C₁ ◁ C₃²

Complements

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

N	H ≅ G/N	type
2. C₂²⋊(C₄²⋊C₉)	C₃	direct (N × H)
7. C₂²×C₃×C₄²	C₉	semidirect
26. C₂²×C₄²	C₃×C₉	semidirect
35. C₃×C₄²	C₂²⋊C₉	semidirect
61. C₄²	C₃×C₂²⋊C₉	semidirect
66. C₂²×C₃	C₄²⋊C₉	semidirect
84. C₂²	C₃×C₄²⋊C₉	semidirect
87. C₃	C₂²⋊(C₄²⋊C₉)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃²×C₄²	3
C₂²⋊(C₄²⋊C₉)	3
C₃×C₂²⋊(C₂²⋊C₉)	4
C₃×C₄²⋊C₉	4

Subgroups — C3×C22⋊(C42⋊C9) #13152253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₂²⋊(C₄²⋊C₉) #13152253