Subgroups — #88795

Order 432.

184 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₃⋊(C₃⋊D₃))	C₂	1	1	yes	maximal, characteristic
2	C₃²×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#3755	—	3	1	no	maximal
5	#3717	—	3	4	no	maximal
6	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
7	C₃³×C₄	C₂²	1	1	yes	characteristic
8	C₃²×Dic₃	—	2	1	no	—
9	C₃²×Q₈	—	3	1	no	—
10	C₄⋊(C₃⋊D₃)	—	3	1	no	—
11	C₄⋊(C₃⋊D₃)	—	3	4	no	—
12	C₄⋊(C₃⋊D₃)	—	6	4	no	—
13	C₃×Dic₆	D₃	1	4	yes	complemented
14	C₃⋊(C₃⋊C₈)	D₃	1	1	yes	characteristic, complemented
15	C₃×C₃⋊C₈	—	3	4	no	—
16	C₂×C₃³	D₄	1	1	yes	characteristic
17	C₃⋊(C₃⋊D₃)	—	4	1	no	—
18	#215	—	9	4	no	—
19	#204	—	9	1	no	—
20	C₂×C₃⋊D₃	—	6	8	no	—
21	C₂×C₃⋊D₃	—	6	1	no	—
22	C₂×C₃⋊D₃	—	6	4	no	—
23	C₃²×C₄	—	2	4	no	—
24	C₃²×C₄	—	6	1	no	—
25	C₃²×C₄	C₂×D₃	1	4	yes	—
26	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
27	C₃×Dic₃	—	2	4	no	—
28	C₃³	SD₁₆	1	1	yes	characteristic, complemented, Sylow
29	C₃×Q₈	—	3	4	no	—
30	D₁₂	—	9	4	no	—
31	D₁₂	—	9	1	no	—
32	D₁₂	—	18	4	no	—
33	Dic₆	C₃⋊D₃	1	1	yes	characteristic, complemented
34	C₃×C₈	—	9	1	no	—
35	C₃⋊C₈	—	3	4	no	—
36	C₂×C₃²	—	2	4	no	—
37	C₂×C₃²	D₁₂	1	1	yes	characteristic
38	C₂×C₃²	#67	1	4	yes	—
39	C₃⋊D₃	—	12	8	no	—
40	C₃⋊D₃	—	12	1	no	—
41	C₃⋊D₃	—	12	4	no	—
42	SD₁₆	—	27	1	no	Sylow
43	C₂×D₃	—	18	8	no	—
44	C₂×D₃	—	18	4	no	—
45	C₂×D₃	—	18	1	no	—
46	C₃×C₄	—	2	4	no	—
47	C₃×C₄	—	6	4	no	—
48	C₃×C₄	D₃²	1	4	yes	—
49	C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
50	Dic₃	—	2	1	no	—
51	C₃²	—	2	4	no	—
52	C₃²	#204	1	1	yes	characteristic, complemented
53	C₃²	#215	1	4	yes	complemented
54	Q₈	—	3	1	no	—
55	D₄	—	27	1	no	—
56	C₈	—	9	1	no	—
57	C₂×C₃	—	2	4	no	—
58	C₂×C₃	#626	1	4	yes	—
59	C₂×C₃	#638	1	1	yes	characteristic
60	D₃	—	36	8	no	—
61	D₃	—	36	4	no	—
62	D₃	—	36	1	no	—
63	C₂²	—	54	1	no	—
64	C₄	—	6	1	no	—
65	C₄	D₃×C₃⋊D₃	1	1	yes	characteristic
66	C₃	—	2	4	no	—
67	C₃	#3717	1	4	yes	complemented
68	C₃	#3755	1	1	yes	characteristic, complemented
69	C₂	—	108	1	no	—
70	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	C₂×C₃³	D₄
prime radical	C₄⋊(C₃⋊(C₃⋊D₃))	C₂
hypocenter	C₃³	SD₁₆
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
2. C₃²×Dic₆	C₂	semidirect
3. C₃×C₃⋊(C₃⋊C₈)	C₂	semidirect
13. C₃×Dic₆	D₃	semidirect
14. C₃⋊(C₃⋊C₈)	D₃	semidirect
28. C₃³	SD₁₆	semidirect
33. Dic₆	C₃⋊D₃	semidirect
52. C₃²	#204	semidirect
53. C₃²	#215	semidirect
67. C₃	#3717	semidirect
68. C₃	#3755	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₃⋊(C₃⋊D₃))	2
C₃²×Dic₆	2
C₃×C₃⋊(C₃⋊C₈)	2
#3755	3
#3717	3