Subgroups — #88859

Order 432.

476 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃³×C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic, complemented
4	#3752	—	3	13	no	maximal
5	C₂²×C₃³	C₂²	1	1	yes	characteristic
6	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
7	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
8	C₂×C₃⋊(C₃⋊D₃)	C₄	1	2	yes	complemented
9	C₃³×C₄	—	2	1	no	—
10	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
11	C₂²×C₃⋊D₃	—	3	13	no	—
12	C₂×C₃²×C₄	D₃	1	13	yes	complemented
13	C₂×C₃⋊Dic₃	—	3	13	no	—
14	C₂×C₃³	C₂×C₄	1	1	yes	characteristic
15	C₂×C₃³	D₄	1	1	yes	characteristic
16	C₂×C₃³	D₄	1	1	yes	characteristic
17	C₃⋊(C₃⋊D₃)	—	2	2	no	—
18	(C₂×D₃)⋊C₄	—	9	13	no	—
19	C₂²×C₃²	C₂×D₃	1	13	yes	—
20	C₂×C₃⋊D₃	—	3	26	no	—
21	C₂×C₃⋊D₃	—	6	13	no	—
22	C₂×C₃⋊D₃	—	6	13	no	—
23	C₃²×C₄	—	2	13	no	—
24	C₃⋊Dic₃	—	6	13	no	—
25	C₃³	C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
26	C₂²×D₃	—	9	13	no	—
27	C₂×C₃×C₄	C₃⋊D₃	1	13	yes	complemented
28	C₂×Dic₃	—	9	13	no	—
29	C₂×C₃²	C₄×D₃	1	13	yes	—
30	C₂×C₃²	D₁₂	1	13	yes	—
31	C₂×C₃²	#67	1	13	yes	—
32	C₃⋊D₃	—	6	26	no	—
33	C₂²⋊C₄	—	27	1	no	Sylow
34	C₂²×C₃	C₂×C₃⋊D₃	1	13	yes	—
35	C₂×D₃	—	9	26	no	—
36	C₂×D₃	—	18	13	no	—
37	C₂×D₃	—	18	13	no	—
38	C₃×C₄	—	2	13	no	—
39	Dic₃	—	18	13	no	—
40	C₃²	(C₂×D₃)⋊C₄	1	13	yes	complemented
41	C₂³	—	27	1	no	—
42	C₂×C₄	—	27	1	no	—
43	C₂×C₄	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
44	C₂×C₃	C₄×C₃⋊D₃	1	13	yes	—
45	C₂×C₃	C₄⋊(C₃⋊D₃)	1	13	yes	—
46	C₂×C₃	#638	1	13	yes	—
47	D₃	—	18	26	no	—
48	C₂²	—	27	2	no	—
49	C₂²	—	54	1	no	—
50	C₂²	—	54	1	no	—
51	C₂²	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
52	C₄	—	2	1	no	—
53	C₄	—	54	1	no	—
54	C₃	#3752	1	13	yes	complemented
55	C₂	—	54	2	no	—
56	C₂	C₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
57	C₂	C₄⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic
58	C₂	#6312	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₃³×C₄	C₂	semidirect
3. C₂×C₃⋊(C₃⋊Dic₃)	C₂	semidirect
8. C₂×C₃⋊(C₃⋊D₃)	C₄	semidirect
12. C₂×C₃²×C₄	D₃	semidirect
25. C₃³	C₂²⋊C₄	semidirect
27. C₂×C₃×C₄	C₃⋊D₃	semidirect
40. C₃²	(C₂×D₃)⋊C₄	semidirect
43. C₂×C₄	C₃⋊(C₃⋊D₃)	semidirect
54. C₃	#3752	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃⋊(C₃⋊D₃)	2
C₂×C₃³×C₄	2
C₂×C₃⋊(C₃⋊Dic₃)	2
#3752	3