Subgroups — C₂²×C₃×C₃⋊C₈ #64043

Order 288.

179 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₃⋊C₈	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃×C₈	—	3	1	no	maximal
4	C₂²×C₃⋊C₈	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₂³×C₃²	C₄	1	1	yes	characteristic
6	C₂×C₃²×C₄	C₄	1	3	yes	—
7	C₂×C₃²×C₄	C₂²	1	3	yes	—
8	C₃×C₃⋊C₈	C₂²	1	4	yes	direct product factor
9	C₂²×C₃×C₄	—	2	1	no	—
10	C₂²×C₃×C₄	D₃	1	1	yes	characteristic
11	C₂²×C₃×C₄	C₂×C₃	1	1	yes	characteristic
12	C₂×C₃×C₈	—	3	6	no	—
13	C₂×C₃⋊C₈	C₂×C₃	1	6	yes	direct product factor
14	C₂²×C₃²	C₈	1	4	yes	complemented
15	C₂²×C₃²	C₂×C₄	1	3	yes	—
16	C₃²×C₄	C₂×C₄	1	3	yes	—
17	C₃²×C₄	C₂³	1	1	yes	characteristic
18	C₂²×C₈	—	3	1	no	Sylow
19	C₂³×C₃	—	2	1	no	—
20	C₂³×C₃	Dic₃	1	1	yes	characteristic
21	C₂³×C₃	C₃×C₄	1	1	yes	characteristic
22	C₂×C₃×C₄	—	2	3	no	—
23	C₂×C₃×C₄	—	2	3	no	—
24	C₂×C₃×C₄	Dic₃	1	3	yes	—
25	C₂×C₃×C₄	C₃×C₄	1	3	yes	—
26	C₂×C₃×C₄	C₂×D₃	1	3	yes	—
27	C₂×C₃×C₄	C₂²×C₃	1	3	yes	—
28	C₃×C₈	—	3	4	no	—
29	C₃⋊C₈	C₂²×C₃	1	4	yes	direct product factor
30	C₂×C₃²	C₂×C₈	1	6	yes	complemented
31	C₂×C₃²	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄	C₃×D₃	1	1	yes	characteristic
33	C₂×C₈	—	3	6	no	—
34	C₂²×C₃	—	2	4	no	—
35	C₂²×C₃	—	2	3	no	—
36	C₂²×C₃	C₃⋊C₈	1	4	yes	direct product factor
37	C₂²×C₃	C₃×C₈	1	4	yes	complemented
38	C₂²×C₃	C₂×Dic₃	1	3	yes	—
39	C₂²×C₃	C₂×C₃×C₄	1	3	yes	—
40	C₃×C₄	—	2	3	no	—
41	C₃×C₄	—	2	1	no	—
42	C₃×C₄	C₂×Dic₃	1	3	yes	—
43	C₃×C₄	C₂×C₃×C₄	1	3	yes	—
44	C₃×C₄	C₂²×D₃	1	1	yes	characteristic
45	C₃×C₄	C₂³×C₃	1	1	yes	characteristic
46	C₃²	C₂²×C₈	1	1	yes	characteristic, complemented, Sylow
47	C₂³	C₃×Dic₃	1	1	yes	characteristic
48	C₂×C₄	C₃×Dic₃	1	3	yes	—
49	C₂×C₄	C₂×C₃×D₃	1	3	yes	—
50	C₈	—	3	4	no	—
51	C₂×C₃	—	2	6	no	—
52	C₂×C₃	—	2	1	no	—
53	C₂×C₃	C₂×C₃⋊C₈	1	6	yes	direct product factor
54	C₂×C₃	C₂×C₃×C₈	1	6	yes	complemented
55	C₂×C₃	C₂²×Dic₃	1	1	yes	characteristic
56	C₂×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
57	C₂²	C₃×C₃⋊C₈	1	4	yes	direct product factor
58	C₂²	C₂×C₃×Dic₃	1	3	yes	—
59	C₄	C₂×C₃×Dic₃	1	3	yes	—
60	C₄	C₂²×C₃×D₃	1	1	yes	characteristic
61	C₃	—	2	1	no	—
62	C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic, direct product factor
63	C₃	C₂²×C₃×C₈	1	1	yes	characteristic, complemented
64	C₂	C₂×C₃×C₃⋊C₈	1	6	yes	direct product factor
65	C₂	C₂²×C₃×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₃×C₄	D₃
commutator, hypocenter	C₃	C₂²×C₃×C₈
Frattini	C₄	C₂²×C₃×D₃
Fitting	C₂²×C₃²×C₄	C₂
socle, prime radical	C₂³×C₃²	C₄
cosocle	C₃×C₄	C₂³×C₃
LCS	G ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₃⋊C₈	C₂	direct (N × H)
4. C₂²×C₃⋊C₈	C₃	direct (N × H)
8. C₃×C₃⋊C₈	C₂²	direct (N × H)
13. C₂×C₃⋊C₈	C₂×C₃	direct (N × H)
14. C₂²×C₃²	C₈	semidirect
29. C₃⋊C₈	C₂²×C₃	direct (N × H)
30. C₂×C₃²	C₂×C₈	semidirect
36. C₂²×C₃	C₃⋊C₈	direct (N × H)
37. C₂²×C₃	C₃×C₈	semidirect
46. C₃²	C₂²×C₈	semidirect
53. C₂×C₃	C₂×C₃⋊C₈	direct (N × H)
54. C₂×C₃	C₂×C₃×C₈	semidirect
57. C₂²	C₃×C₃⋊C₈	direct (N × H)
62. C₃	C₂²×C₃⋊C₈	direct (N × H)
63. C₃	C₂²×C₃×C₈	semidirect
64. C₂	C₂×C₃×C₃⋊C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃²×C₄	2
C₂×C₃×C₃⋊C₈	2
C₂²×C₃×C₈	3
C₂²×C₃⋊C₈	3

Subgroups — C22×C3×C3⋊C8 #64043

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₃⋊C₈ #64043