Subgroups — (((C₃⋊D₃)⋊C₃)⋊C₂)⋊C₄ #88591

Order 432.

99 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×((C₃⋊D₃)⋊C₃)⋊C₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#1088	C₂	1	1	yes	maximal, characteristic, complemented
3	#6199	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#272	C₂²	1	1	yes	characteristic
5	C₂×(C₃⋊D₃)⋊C₃	—	2	1	no	—
6	((C₃⋊D₃)⋊C₃)⋊C₂	—	2	1	no	—
7	((C₃⋊D₃)⋊C₃)⋊C₂	C₄	1	2	yes	complemented
8	#1088	—	2	1	no	—
9	(C₃⋊Dic₃)⋊C₃	—	2	1	no	—
10	C₂×D₃²	—	6	1	no	—
11	D₃×Dic₃	—	6	1	no	—
12	C₂×H(3,𝔽₃)	D₄	1	1	yes	characteristic
13	#272	C₂×C₄	1	1	yes	characteristic
14	#272	D₄	1	1	yes	characteristic
15	(C₃⋊D₃)⋊C₃	—	2	2	no	—
16	(C₂×D₃)⋊C₄	—	9	1	no	maximal
17	C₂×C₃⋊D₃	—	2	1	no	—
18	C₂×C₃×D₃	—	6	1	no	—
19	C₂×C₃×D₃	—	6	1	no	—
20	C₂×C₃×D₃	—	6	1	no	—
21	D₃²	—	6	2	no	—
22	D₃²	—	6	2	no	—
23	C₃⋊Dic₃	—	2	1	no	—
24	C₃×Dic₃	—	6	1	no	—
25	H(3,𝔽₃)	C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
26	C₂²×D₃	—	9	1	no	—
27	C₂²×D₃	—	18	1	no	—
28	C₂×C₃×C₄	—	9	1	no	—
29	C₂×Dic₃	—	9	1	no	—
30	C₄×D₃	—	18	1	no	—
31	C₂×C₃²	—	2	1	no	—
32	C₂×C₃²	—	2	1	no	—
33	C₃⋊D₃	—	2	2	no	—
34	C₃×D₃	—	6	1	no	—
35	C₃×D₃	—	6	1	no	—
36	C₃×D₃	—	6	1	no	—
37	C₃×D₃	—	6	1	no	—
38	C₃×D₃	—	6	2	no	—
39	C₂²⋊C₄	—	27	1	no	Sylow
40	C₂²×C₃	—	9	1	no	—
41	C₂²×C₃	—	18	1	no	—
42	C₂×D₃	—	6	1	no	—
43	C₂×D₃	—	6	1	no	—
44	C₂×D₃	—	6	1	no	—
45	C₂×D₃	—	9	2	no	—
46	C₂×D₃	—	18	2	no	—
47	C₂×D₃	—	18	1	no	—
48	C₂×D₃	—	18	2	no	—
49	C₂×D₃	—	36	1	no	—
50	C₃×C₄	—	18	1	no	—
51	C₃×C₄	—	18	1	no	—
52	Dic₃	—	6	1	no	—
53	Dic₃	—	18	1	no	—
54	C₃²	—	2	1	no	—
55	C₃²	—	2	1	no	—
56	C₂³	—	27	1	no	—
57	C₂×C₄	—	9	1	no	—
58	C₂×C₄	—	27	1	no	—
59	C₂×C₃	—	6	1	no	—
60	C₂×C₃	—	9	1	no	—
61	C₂×C₃	—	9	1	no	—
62	C₂×C₃	—	12	1	no	—
63	C₂×C₃	—	18	2	no	—
64	C₂×C₃	GO⁺(4,𝔽₂)	1	1	yes	characteristic
65	D₃	—	6	1	no	—
66	D₃	—	6	1	no	—
67	D₃	—	6	1	no	—
68	D₃	—	6	1	no	—
69	D₃	—	6	2	no	—
70	D₃	—	36	2	no	—
71	C₂²	—	9	1	no	—
72	C₂²	—	18	1	no	—
73	C₂²	—	27	2	no	—
74	C₂²	—	54	1	no	—
75	C₄	—	18	1	no	—
76	C₄	—	18	1	no	—
77	C₃	—	6	1	no	—
78	C₃	—	12	1	no	—
79	C₃	D₃²⋊C₄	1	1	yes	characteristic
80	C₂	—	9	1	no	—
81	C₂	—	9	1	no	—
82	C₂	—	18	2	no	—
83	C₂	#1088⋊C₂	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂	#1088⋊C₂
commutator	#272	C₂×C₄
Frattini, socle	C₂×C₃	GO⁺(4,𝔽₂)
Fitting	C₂×H(3,𝔽₃)	D₄
prime radical	C₂×((C₃⋊D₃)⋊C₃)⋊C₂	C₂
cosocle	C₂×#272	C₂²
hypocenter	H(3,𝔽₃)	C₂²⋊C₄
LCS	G ▷ #272 ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. C₂×#1088	C₂	semidirect
3. #6199	C₂	semidirect
7. ((C₃⋊D₃)⋊C₃)⋊C₂	C₄	semidirect
25. H(3,𝔽₃)	C₂²⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×((C₃⋊D₃)⋊C₃)⋊C₂	2
C₂×#1088	2
#6199	2
(C₂×D₃)⋊C₄	9

Subgroups — (((C3⋊D3)⋊C3)⋊C2)⋊C4 #88591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — (((C₃⋊D₃)⋊C₃)⋊C₂)⋊C₄ #88591