Subgroups — #88547

Order 432.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	#6226	C₂	1	1	yes	maximal, characteristic
3	C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#3756	—	3	1	no	maximal
5	#3674	—	3	3	no	maximal
6	C₃×C₄×C₉	—	2	1	no	—
7	C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₃⋊Dic₉	—	2	1	no	—
9	C₃²×Q₈	D₃	1	1	yes	characteristic
10	#634	—	3	1	no	—
11	C₃⋊(C₃⋊C₈)	—	3	1	no	—
12	Q₈×C₉	D₃	1	3	yes	—
13	Dic₁₈	—	3	3	no	—
14	C₉⋊C₈	—	3	3	no	—
15	C₂×C₃×C₉	D₄	1	1	yes	characteristic
16	#216	—	9	3	no	—
17	#216	—	9	1	no	—
18	C₃²×C₄	—	2	1	no	—
19	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
20	C₃⋊Dic₃	—	6	1	no	—
21	C₄×C₉	—	2	3	no	—
22	C₄×C₉	C₂×D₃	1	3	yes	—
23	Dic₉	—	6	3	no	—
24	C₃×C₉	Q₁₆	1	1	yes	characteristic, complemented, Sylow
25	C₃×Q₈	D₉	1	3	yes	—
26	C₃×Q₈	C₃⋊D₃	1	1	yes	characteristic
27	Dic₆	—	9	3	no	—
28	Dic₆	—	9	1	no	—
29	C₃⋊C₈	—	9	3	no	—
30	C₃⋊C₈	—	9	1	no	—
31	C₂×C₃²	#67	1	1	yes	characteristic
32	C₂×C₉	#67	1	3	yes	—
33	Q₁₆	—	27	1	no	Sylow
34	C₃×C₄	—	2	3	no	—
35	C₃×C₄	—	2	1	no	—
36	C₃×C₄	C₂×D₉	1	3	yes	—
37	C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
38	Dic₃	—	18	3	no	—
39	Dic₃	—	18	1	no	—
40	C₃²	#216	1	1	yes	characteristic
41	C₉	#216	1	3	yes	complemented
42	Q₈	—	27	1	no	—
43	Q₈	C₃⋊D₉	1	1	yes	characteristic
44	C₈	—	27	1	no	—
45	C₂×C₃	#611	1	3	yes	—
46	C₂×C₃	#638	1	1	yes	characteristic
47	C₄	—	2	1	no	—
48	C₄	—	54	1	no	—
49	C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
50	C₃	#3674	1	3	yes	complemented
51	C₃	#3756	1	1	yes	characteristic
52	C₂	#6233	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6233
commutator, cosocle	C₃×C₄×C₉	C₂²
Frattini	C₃×C₄	C₂×C₃⋊D₃
Fitting	C₃×Q₈×C₉	C₂
socle, prime radical	C₂×C₃²	#67
hypercenter	Q₈	C₃⋊D₉
hypocenter	C₃×C₉	Q₁₆
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₈

Complements

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

N	H ≅ G/N	type
24. C₃×C₉	Q₁₆	semidirect
41. C₉	#216	semidirect
50. C₃	#3674	semidirect

Maximal subgroups

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

subgroup	index
C₃×Q₈×C₉	2
#6226	2
C₃⋊(C₉⋊C₈)	2
#3756	3
#3674	3