Subgroups — C₂×C₃²×C₃⋊C₈ #88822

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	C₃²×C₃⋊C₈	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃²×C₈	—	3	1	no	maximal
4	C₂×C₃×C₃⋊C₈	C₃	1	4	yes	maximal, direct product factor
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	1	yes	characteristic
10	C₂×C₃²×C₄	C₂×C₃	1	4	yes	—
11	C₃²×C₈	—	3	2	no	—
12	C₃×C₃⋊C₈	C₂×C₃	1	8	yes	direct product factor
13	C₂×C₃³	C₈	1	2	yes	complemented
14	C₂×C₃³	C₂×C₄	1	1	yes	characteristic
15	C₂×C₃×C₈	—	3	4	no	—
16	C₂×C₃⋊C₈	C₃²	1	1	yes	characteristic, direct product factor
17	C₂²×C₃²	—	2	4	no	—
18	C₂²×C₃²	Dic₃	1	1	yes	characteristic
19	C₂²×C₃²	C₃×C₄	1	4	yes	—
20	C₃²×C₄	—	2	4	no	—
21	C₃²×C₄	—	2	4	no	—
22	C₃²×C₄	Dic₃	1	1	yes	characteristic
23	C₃²×C₄	C₃×C₄	1	4	yes	—
24	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
25	C₃²×C₄	C₂²×C₃	1	4	yes	—
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	C₂×C₃²	1	1	yes	characteristic
30	C₃×C₈	—	3	8	no	—
31	C₃⋊C₈	C₂×C₃²	1	2	yes	direct product factor
32	C₂×C₃²	—	2	8	no	—
33	C₂×C₃²	—	2	4	no	—
34	C₂×C₃²	C₃⋊C₈	1	2	yes	direct product factor
35	C₂×C₃²	C₃×C₈	1	8	yes	complemented
36	C₂×C₃²	C₂×Dic₃	1	1	yes	characteristic
37	C₂×C₃²	C₂×C₃×C₄	1	4	yes	—
38	C₂×C₈	—	3	1	no	Sylow
39	C₂²×C₃	—	2	4	no	—
40	C₂²×C₃	C₃×Dic₃	1	4	yes	—
41	C₂²×C₃	C₃²×C₄	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	C₃²×C₄	1	1	yes	characteristic
46	C₃×C₄	C₂×C₃×D₃	1	4	yes	—
47	C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
48	C₃²	—	2	4	no	—
49	C₃²	C₂×C₃⋊C₈	1	1	yes	characteristic, direct product factor
50	C₃²	C₂×C₃×C₈	1	4	yes	complemented
51	C₂×C₄	C₃²×D₃	1	1	yes	characteristic
52	C₈	—	3	2	no	—
53	C₂×C₃	—	2	8	no	—
54	C₂×C₃	—	2	4	no	—
55	C₂×C₃	C₃×C₃⋊C₈	1	8	yes	direct product factor
56	C₂×C₃	C₃²×C₈	1	2	yes	complemented
57	C₂×C₃	C₂×C₃×Dic₃	1	4	yes	—
58	C₂×C₃	C₂×C₃²×C₄	1	1	yes	characteristic
59	C₂²	C₃²×Dic₃	1	1	yes	characteristic
60	C₄	C₃²×Dic₃	1	1	yes	characteristic
61	C₄	C₂×C₃²×D₃	1	1	yes	characteristic
62	C₃	—	2	4	no	—
63	C₃	C₂×C₃×C₃⋊C₈	1	4	yes	direct product factor
64	C₃	C₂×C₃²×C₈	1	1	yes	characteristic, complemented
65	C₂	C₃²×C₃⋊C₈	1	2	yes	direct product factor
66	C₂	C₂×C₃²×Dic₃	1	1	yes	characteristic

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

Complements

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

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

Subgroups — C2×C32×C3⋊C8 #88822

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×C₃⋊C₈ #88822