Subgroups — C₃²×#213 #88828

Order 432.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×D₈	—	3	1	no	maximal
5	C₃×#213	C₃	1	4	yes	maximal, direct product factor
6	C₂²×C₃³	—	2	1	no	—
7	C₂×C₃²×D₃	—	2	1	no	—
8	C₃³×C₄	C₂²	1	1	yes	characteristic
9	C₃²×D₄	—	2	4	no	—
10	C₃²×D₄	—	3	1	no	—
11	C₃²×D₄	D₃	1	1	yes	characteristic, complemented
12	C₃²×D₄	C₂×C₃	1	4	yes	complemented
13	C₃×D₁₂	C₂×C₃	1	4	yes	complemented
14	C₃²×C₈	—	3	1	no	—
15	C₃×C₃⋊C₈	C₂×C₃	1	4	yes	complemented
16	C₂×C₃³	—	4	1	no	—
17	C₂×C₃³	D₄	1	1	yes	characteristic
18	C₃²×D₃	—	4	1	no	—
19	C₃×D₈	—	3	4	no	—
20	#213	C₃²	1	1	yes	characteristic, direct product factor
21	C₂²×C₃²	—	2	8	no	—
22	C₂²×C₃²	—	2	1	no	—
23	C₂²×C₃²	—	2	4	no	—
24	C₂²×C₃²	—	6	1	no	—
25	C₂×C₃×D₃	—	2	4	no	—
26	C₃²×C₄	—	2	4	no	—
27	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
28	C₃²×C₄	C₂²×C₃	1	4	yes	—
29	C₃³	D₈	1	1	yes	characteristic, complemented, Sylow
30	C₃×D₄	—	2	4	no	—
31	C₃×D₄	—	3	4	no	—
32	C₃×D₄	C₃×D₃	1	4	yes	complemented
33	C₃×D₄	C₂×C₃²	1	1	yes	characteristic, complemented
34	D₁₂	C₂×C₃²	1	1	yes	characteristic, complemented
35	C₃×C₈	—	3	4	no	—
36	C₃⋊C₈	C₂×C₃²	1	1	yes	characteristic, complemented
37	C₂×C₃²	—	2	4	no	—
38	C₂×C₃²	—	4	8	no	—
39	C₂×C₃²	—	4	1	no	—
40	C₂×C₃²	—	4	4	no	—
41	C₂×C₃²	—	12	1	no	—
42	C₂×C₃²	#67	1	1	yes	characteristic
43	C₂×C₃²	C₃×D₄	1	4	yes	—
44	C₃×D₃	—	4	4	no	—
45	D₈	—	3	1	no	Sylow
46	C₂²×C₃	—	2	8	no	—
47	C₂²×C₃	—	2	4	no	—
48	C₂²×C₃	—	2	1	no	—
49	C₂²×C₃	—	6	4	no	—
50	C₂×D₃	—	2	1	no	—
51	C₃×C₄	—	2	4	no	—
52	C₃×C₄	C₂×C₃×D₃	1	4	yes	—
53	C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
54	C₃²	—	2	4	no	—
55	C₃²	#213	1	1	yes	characteristic, direct product factor
56	C₃²	C₃×D₈	1	4	yes	complemented
57	D₄	—	3	1	no	—
58	D₄	C₃²×D₃	1	1	yes	characteristic, complemented
59	C₈	—	3	1	no	—
60	C₂×C₃	—	2	4	no	—
61	C₂×C₃	—	4	8	no	—
62	C₂×C₃	—	4	4	no	—
63	C₂×C₃	—	4	1	no	—
64	C₂×C₃	—	12	4	no	—
65	C₂×C₃	C₃×#67	1	4	yes	—
66	C₂×C₃	C₃²×D₄	1	1	yes	characteristic
67	D₃	—	4	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	—	6	1	no	—
70	C₄	C₂×C₃²×D₃	1	1	yes	characteristic
71	C₃	—	2	4	no	—
72	C₃	C₃×#213	1	4	yes	direct product factor
73	C₃	C₃²×D₈	1	1	yes	characteristic, complemented
74	C₂	—	4	1	no	—
75	C₂	—	12	1	no	—
76	C₂	C₃²×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²	#67
commutator, cosocle	C₃×C₄	C₂²×C₃²
Frattini	C₄	C₂×C₃²×D₃
Fitting	C₃³×D₄	C₂
socle	C₂×C₃³	D₄
hypercenter	C₃²×D₄	D₃
hypocenter	C₃	C₃²×D₈
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃² ◁ C₃²×C₄ ◁ C₃²×D₄

Complements

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

N	H ≅ G/N	type
1. C₃³×D₄	C₂	semidirect
2. C₃²×D₁₂	C₂	semidirect
3. C₃²×C₃⋊C₈	C₂	semidirect
5. C₃×#213	C₃	direct (N × H)
11. C₃²×D₄	D₃	semidirect
12. C₃²×D₄	C₂×C₃	semidirect
13. C₃×D₁₂	C₂×C₃	semidirect
15. C₃×C₃⋊C₈	C₂×C₃	semidirect
20. #213	C₃²	direct (N × H)
29. C₃³	D₈	semidirect
32. C₃×D₄	C₃×D₃	semidirect
33. C₃×D₄	C₂×C₃²	semidirect
34. D₁₂	C₂×C₃²	semidirect
36. C₃⋊C₈	C₂×C₃²	semidirect
55. C₃²	#213	direct (N × H)
56. C₃²	C₃×D₈	semidirect
58. D₄	C₃²×D₃	semidirect
72. C₃	C₃×#213	direct (N × H)
73. C₃	C₃²×D₈	semidirect

Maximal subgroups

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

subgroup	index
C₃³×D₄	2
C₃²×D₁₂	2
C₃²×C₃⋊C₈	2
C₃²×D₈	3
C₃×#213	3

Subgroups — C32×#213 #88828

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#213 #88828