Subgroups — C₃²×#216 #88831

Order 432.

156 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃²×Dic₆	C₂	1	1	yes	maximal, characteristic
3	C₃²×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₃²×Q₁₆	—	3	1	no	maximal
5	C₃×#216	C₃	1	4	yes	maximal, direct product factor
6	C₃³×C₄	—	2	1	no	—
7	C₃³×C₄	C₂²	1	1	yes	characteristic
8	C₃²×Dic₃	—	2	1	no	—
9	C₃²×Q₈	—	2	4	no	—
10	C₃²×Q₈	—	3	1	no	—
11	C₃²×Q₈	D₃	1	1	yes	characteristic
12	C₃²×Q₈	C₂×C₃	1	4	yes	—
13	C₃×Dic₆	C₂×C₃	1	4	yes	—
14	C₃²×C₈	—	3	1	no	—
15	C₃×C₃⋊C₈	C₂×C₃	1	4	yes	—
16	C₂×C₃³	D₄	1	1	yes	characteristic
17	C₃×Q₁₆	—	3	4	no	—
18	#216	C₃²	1	1	yes	characteristic, direct product factor
19	C₃²×C₄	—	2	8	no	—
20	C₃²×C₄	—	2	1	no	—
21	C₃²×C₄	—	2	4	no	—
22	C₃²×C₄	—	2	4	no	—
23	C₃²×C₄	—	6	1	no	—
24	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
25	C₃²×C₄	C₂²×C₃	1	4	yes	—
26	C₃×Dic₃	—	2	4	no	—
27	C₃³	Q₁₆	1	1	yes	characteristic, complemented, Sylow
28	C₃×Q₈	—	2	4	no	—
29	C₃×Q₈	—	3	4	no	—
30	C₃×Q₈	C₃×D₃	1	4	yes	—
31	C₃×Q₈	C₂×C₃²	1	1	yes	characteristic
32	Dic₆	C₂×C₃²	1	1	yes	characteristic
33	C₃×C₈	—	3	4	no	—
34	C₃⋊C₈	C₂×C₃²	1	1	yes	characteristic
35	C₂×C₃²	—	2	4	no	—
36	C₂×C₃²	#67	1	1	yes	characteristic
37	C₂×C₃²	C₃×D₄	1	4	yes	—
38	Q₁₆	—	3	1	no	Sylow
39	C₃×C₄	—	2	8	no	—
40	C₃×C₄	—	2	4	no	—
41	C₃×C₄	—	2	1	no	—
42	C₃×C₄	—	2	4	no	—
43	C₃×C₄	—	6	4	no	—
44	C₃×C₄	C₂×C₃×D₃	1	4	yes	—
45	C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
46	Dic₃	—	2	1	no	—
47	C₃²	—	2	4	no	—
48	C₃²	#216	1	1	yes	characteristic, direct product factor
49	C₃²	C₃×Q₁₆	1	4	yes	complemented
50	Q₈	—	3	1	no	—
51	Q₈	C₃²×D₃	1	1	yes	characteristic
52	C₈	—	3	1	no	—
53	C₂×C₃	—	2	4	no	—
54	C₂×C₃	C₃×#67	1	4	yes	—
55	C₂×C₃	C₃²×D₄	1	1	yes	characteristic
56	C₄	—	2	1	no	—
57	C₄	—	6	1	no	—
58	C₄	C₂×C₃²×D₃	1	1	yes	characteristic
59	C₃	—	2	4	no	—
60	C₃	C₃×#216	1	4	yes	direct product factor
61	C₃	C₃²×Q₁₆	1	1	yes	characteristic, complemented
62	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×Q₈	C₂
socle, prime radical	C₂×C₃³	D₄
hypercenter	C₃²×Q₈	D₃
hypocenter	C₃	C₃²×Q₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃² ◁ C₃²×C₄ ◁ C₃²×Q₈

Complements

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

N	H ≅ G/N	type
5. C₃×#216	C₃	direct (N × H)
18. #216	C₃²	direct (N × H)
27. C₃³	Q₁₆	semidirect
48. C₃²	#216	direct (N × H)
49. C₃²	C₃×Q₁₆	semidirect
60. C₃	C₃×#216	direct (N × H)
61. C₃	C₃²×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₃³×Q₈	2
C₃²×Dic₆	2
C₃²×C₃⋊C₈	2
C₃²×Q₁₆	3
C₃×#216	3

Subgroups — C32×#216 #88831

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#216 #88831