Subgroups — #88988

Order 432.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₃⋊#622	C₂	1	2	yes	maximal, complemented
3	#3787	—	3	1	no	maximal
4	C₂×C₃×C₃⋊D₃	C₄	1	1	yes	characteristic
5	C₃³×C₄	C₄	1	1	yes	characteristic
6	C₃×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
7	C₄×C₃⋊D₃	D₃	1	1	yes	characteristic
8	C₃×C₄×D₃	—	6	2	no	—
9	#622	—	3	2	no	—
10	C₂×C₃³	C₂×C₄	1	1	yes	characteristic
11	C₃×C₃⋊D₃	C₈	1	2	yes	complemented
12	C₂×C₃⋊C₈	—	9	1	no	maximal
13	C₂×C₃⋊D₃	Dic₃	1	1	yes	characteristic
14	C₂×C₃×D₃	—	6	2	no	—
15	C₃²×C₄	—	2	2	no	—
16	C₃²×C₄	—	4	2	no	—
17	C₃²×C₄	Dic₃	1	1	yes	characteristic
18	C₃⋊Dic₃	C₂×D₃	1	1	yes	characteristic
19	C₃×Dic₃	—	6	2	no	—
20	C₃³	C₂×C₈	1	1	yes	characteristic, complemented, Sylow
21	C₂×C₃×C₄	—	9	1	no	—
22	C₄×D₃	—	6	2	no	—
23	C₃⋊C₈	—	9	2	no	—
24	C₂×C₃²	—	2	2	no	—
25	C₂×C₃²	—	4	2	no	—
26	C₂×C₃²	C₂×Dic₃	1	1	yes	characteristic
27	C₃⋊D₃	C₃⋊C₈	1	2	yes	complemented
28	C₃×D₃	—	6	4	no	—
29	C₂×C₈	—	27	1	no	Sylow
30	C₂²×C₃	—	9	1	no	—
31	C₂×D₃	—	6	2	no	—
32	C₃×C₄	—	2	2	no	—
33	C₃×C₄	—	4	2	no	—
34	C₃×C₄	—	9	1	no	—
35	C₃×C₄	#157	1	1	yes	characteristic
36	Dic₃	—	6	2	no	—
37	C₃²	—	2	2	no	—
38	C₃²	—	4	2	no	—
39	C₃²	C₂×C₃⋊C₈	1	1	yes	characteristic, complemented
40	C₂×C₄	—	9	1	no	—
41	C₈	—	27	2	no	—
42	C₂×C₃	—	2	2	no	—
43	C₂×C₃	—	4	2	no	—
44	C₂×C₃	—	9	2	no	—
45	C₂×C₃	C₂×#157	1	1	yes	characteristic
46	D₃	—	6	4	no	—
47	C₂²	—	9	1	no	—
48	C₄	—	9	1	no	—
49	C₄	C₃⋊#157	1	1	yes	characteristic
50	C₃	—	2	2	no	—
51	C₃	—	4	2	no	—
52	C₃	#3787	1	1	yes	characteristic, complemented
53	C₂	—	9	2	no	—
54	C₂	C₂×C₃⋊#157	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄	C₃⋊#157
commutator, hypocenter	C₃³	C₂×C₈
Frattini	C₂	C₂×C₃⋊#157
Fitting	C₃³×C₄	C₄
socle	C₂×C₃³	C₂×C₄
prime radical	C₂×C₃×C₃⋊D₃	C₄
cosocle	C₃×C₃⋊Dic₃	C₂²
LCS	G ▷ C₃³
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₃⋊#622	C₂	semidirect
11. C₃×C₃⋊D₃	C₈	semidirect
20. C₃³	C₂×C₈	semidirect
27. C₃⋊D₃	C₃⋊C₈	semidirect
39. C₃²	C₂×C₃⋊C₈	semidirect
52. C₃	#3787	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₃⋊D₃	2
C₃⋊#622	2
#3787	3
C₂×C₃⋊C₈	9