Subgroups — C₂×C₃×C₃⋊(C₃⋊C₈) #88838

Order 432.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, hypercenter	C₂×C₃×C₄	C₃⋊D₃
commutator, hypocenter	C₃²	C₂×C₃×C₈
Frattini	C₄	C₂×C₃×C₃⋊D₃
Fitting	C₂×C₃³×C₄	C₂
socle, prime radical	C₂²×C₃³	C₄
cosocle	C₃²×C₄	C₂²×C₃
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂×C₃×C₄

Complements

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

N	H ≅ G/N	type
2. C₃×C₃⋊(C₃⋊C₈)	C₂	direct (N × H)
3. C₂×C₃⋊(C₃⋊C₈)	C₃	direct (N × H)
11. C₃⋊(C₃⋊C₈)	C₂×C₃	direct (N × H)
13. C₂×C₃³	C₈	semidirect
26. C₃³	C₂×C₈	semidirect
34. C₂×C₃²	C₃⋊C₈	semidirect
35. C₂×C₃²	C₃×C₈	semidirect
49. C₃²	C₂×C₃⋊C₈	semidirect
50. C₃²	C₂×C₃×C₈	semidirect
55. C₂×C₃	C₃×C₃⋊C₈	semidirect
56. C₂×C₃	C₃⋊(C₃⋊C₈)	direct (N × H)
63. C₃	C₂×C₃×C₃⋊C₈	semidirect
64. C₃	C₂×C₃⋊(C₃⋊C₈)	direct (N × H)
65. C₂	C₃×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
C₂×C₃³×C₄	2
C₃×C₃⋊(C₃⋊C₈)	2
C₂×C₃⋊(C₃⋊C₈)	3
C₂×C₃×C₃⋊C₈	3

Subgroups — C2×C3×C3⋊(C3⋊C8) #88838

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₃⋊(C₃⋊C₈) #88838