Subgroups — #88799

Order 432.

156 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6308	C₂	1	1	yes	maximal, characteristic
2	C₃²×Dic₆	C₂	1	1	yes	maximal, characteristic
3	C₃×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#3756	—	3	1	no	maximal
5	#3719	—	3	4	no	maximal
6	C₃³×C₄	C₂²	1	1	yes	characteristic
7	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
8	C₃²×Dic₃	—	2	1	no	—
9	C₃²×Q₈	—	3	1	no	—
10	#634	—	3	1	no	—
11	#634	—	3	4	no	—
12	#634	—	6	4	no	—
13	C₃×Dic₆	D₃	1	4	yes	—
14	C₃⋊(C₃⋊C₈)	D₃	1	1	yes	characteristic
15	C₃×C₃⋊C₈	—	3	4	no	—
16	C₂×C₃³	D₄	1	1	yes	characteristic
17	#216	—	9	4	no	—
18	Dic₁₂	—	9	1	no	—
19	C₃²×C₄	—	2	4	no	—
20	C₃²×C₄	—	6	1	no	—
21	C₃²×C₄	C₂×D₃	1	4	yes	—
22	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
23	C₃⋊Dic₃	—	6	8	no	—
24	C₃⋊Dic₃	—	6	1	no	—
25	C₃⋊Dic₃	—	6	4	no	—
26	C₃×Dic₃	—	2	4	no	—
27	C₃³	Q₁₆	1	1	yes	characteristic, complemented, Sylow
28	C₃×Q₈	—	3	4	no	—
29	Dic₆	—	9	4	no	—
30	Dic₆	—	9	1	no	—
31	Dic₆	—	18	4	no	—
32	Dic₆	C₃⋊D₃	1	1	yes	characteristic
33	C₃×C₈	—	9	1	no	—
34	C₃⋊C₈	—	3	4	no	—
35	C₂×C₃²	—	2	4	no	—
36	C₂×C₃²	D₁₂	1	1	yes	characteristic
37	C₂×C₃²	#67	1	4	yes	—
38	Q₁₆	—	27	1	no	Sylow
39	C₃×C₄	—	2	4	no	—
40	C₃×C₄	—	6	4	no	—
41	C₃×C₄	D₃²	1	4	yes	—
42	C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
43	Dic₃	—	2	1	no	—
44	Dic₃	—	18	8	no	—
45	Dic₃	—	18	4	no	—
46	Dic₃	—	18	1	no	—
47	C₃²	—	2	4	no	—
48	C₃²	Dic₁₂	1	1	yes	characteristic, complemented
49	C₃²	#216	1	4	yes	complemented
50	Q₈	—	3	1	no	—
51	Q₈	—	27	1	no	—
52	C₈	—	9	1	no	—
53	C₂×C₃	—	2	4	no	—
54	C₂×C₃	#626	1	4	yes	—
55	C₂×C₃	#638	1	1	yes	characteristic
56	C₄	—	6	1	no	—
57	C₄	—	54	1	no	—
58	C₄	D₃×C₃⋊D₃	1	1	yes	characteristic
59	C₃	—	2	4	no	—
60	C₃	#3719	1	4	yes	complemented
61	C₃	#3756	1	1	yes	characteristic, complemented
62	C₂	#6291	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6291
commutator, Fitting, cosocle	C₃³×C₄	C₂²
Frattini, hypercenter	C₄	D₃×C₃⋊D₃
socle, prime radical	C₂×C₃³	D₄
hypocenter	C₃³	Q₁₆
LCS	G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
27. C₃³	Q₁₆	semidirect
48. C₃²	Dic₁₂	semidirect
49. C₃²	#216	semidirect
60. C₃	#3719	semidirect
61. C₃	#3756	semidirect

Maximal subgroups

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

subgroup	index
#6308	2
C₃²×Dic₆	2
C₃×C₃⋊(C₃⋊C₈)	2
#3756	3
#3719	3