Subgroups — #88946

Order 432.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⋊#157	C₂	1	2	yes	maximal
2	C₂×C₃×#157	C₂	1	1	yes	maximal, characteristic
3	#3777	—	3	1	no	maximal
4	C₂×C₃×C₃⋊D₃	C₂²	1	1	yes	characteristic
5	C₃⋊#157	—	2	2	no	—
6	C₃×#157	C₄	1	2	yes	complemented
7	C₂×#157	—	3	2	no	—
8	C₂×#157	D₃	1	1	yes	characteristic
9	C₂×C₃³	Q₈	1	1	yes	characteristic
10	C₃×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
11	C₃×C₃⋊D₃	D₄	1	1	yes	characteristic
12	C₄⋊Dic₃	—	9	1	no	maximal
13	C₂×C₃⋊D₃	C₂×D₃	1	1	yes	characteristic
14	C₂×C₃×D₃	—	12	1	no	—
15	#157	—	6	2	no	—
16	#157	Dic₃	1	2	yes	complemented
17	C₃³	C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
18	C₂×C₃×C₄	—	9	1	no	—
19	C₂×Dic₃	—	9	2	no	—
20	C₂×C₃²	—	4	1	no	—
21	C₂×C₃²	—	8	1	no	—
22	C₂×C₃²	Dic₆	1	1	yes	characteristic
23	C₃⋊D₃	D₁₂	1	1	yes	characteristic
24	C₃⋊D₃	C₂×Dic₃	1	1	yes	characteristic
25	C₃×D₃	—	12	1	no	—
26	C₃×D₃	—	12	1	no	—
27	C₄⋊C₄	—	27	1	no	Sylow
28	C₂²×C₃	—	9	1	no	—
29	C₂×D₃	—	12	1	no	—
30	C₃×C₄	—	9	2	no	—
31	Dic₃	—	18	2	no	—
32	C₃²	—	4	1	no	—
33	C₃²	—	8	1	no	—
34	C₃²	C₄⋊Dic₃	1	1	yes	characteristic, complemented
35	C₂×C₄	—	9	1	no	—
36	C₂×C₄	—	27	2	no	—
37	C₂×C₃	—	4	1	no	—
38	C₂×C₃	—	8	1	no	—
39	C₂×C₃	—	9	1	no	—
40	C₂×C₃	—	9	1	no	—
41	C₂×C₃	PSU(3,𝔽₂)	1	1	yes	characteristic
42	D₃	—	12	1	no	—
43	D₃	—	12	1	no	—
44	C₂²	—	9	1	no	—
45	C₄	—	9	2	no	—
46	C₄	—	54	2	no	—
47	C₃	—	4	1	no	—
48	C₃	—	8	1	no	—
49	C₃	#3777	1	1	yes	characteristic, complemented
50	C₂	—	9	1	no	—
51	C₂	—	9	1	no	—
52	C₂	C₃⋊PSU(3,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₃⋊PSU(3,𝔽₂)
commutator	C₃×C₃⋊D₃	C₂×C₄
Fitting, socle	C₂×C₃³	Q₈
prime radical, cosocle	C₂×C₃×C₃⋊D₃	C₂²
hypocenter	C₃³	C₄⋊C₄
LCS	G ▷ C₃×C₃⋊D₃ ▷ C₃³
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
6. C₃×#157	C₄	semidirect
16. #157	Dic₃	semidirect
17. C₃³	C₄⋊C₄	semidirect
34. C₃²	C₄⋊Dic₃	semidirect
49. C₃	#3777	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃⋊#157	2
C₂×C₃×#157	2
#3777	3
C₄⋊Dic₃	9