Subgroups — #89019

Order 432.

292 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic
2	C₃×#634	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#6293	C₂	1	2	yes	maximal, complemented
5	#6289	C₂	1	2	yes	maximal
6	Q₈×C₃⋊D₃	—	3	1	no	maximal
7	#3797	—	3	4	no	maximal
8	C₂×C₃⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
9	C₃³×C₄	C₂²	1	1	yes	characteristic
10	C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
11	C₃×C₃⋊Dic₃	C₂²	1	2	yes	—
12	C₃²×Dic₃	C₂²	1	2	yes	—
13	C₃²×Q₈	—	3	1	no	—
14	C₄×C₃⋊D₃	—	3	1	no	—
15	C₄×C₃⋊D₃	—	3	2	no	—
16	C₄×C₃⋊D₃	—	3	4	no	—
17	C₄×C₃⋊D₃	—	6	4	no	—
18	#634	—	3	2	no	—
19	#634	D₃	1	1	yes	characteristic, complemented
20	C₃×Dic₆	—	3	4	no	—
21	C₃×Dic₆	D₃	1	4	yes	complemented
22	#627	—	3	8	no	—
23	#624	—	3	8	no	—
24	C₂×C₃³	C₂³	1	1	yes	characteristic
25	C₃⋊(C₃⋊D₃)	Q₈	1	2	yes	complemented
26	D₃×Q₈	—	9	4	no	—
27	D₃×Q₈	—	9	1	no	—
28	C₂×C₃⋊D₃	—	3	1	no	—
29	C₂×C₃⋊D₃	—	3	4	no	—
30	C₂×C₃⋊D₃	—	6	4	no	—
31	C₃²×C₄	—	2	4	no	—
32	C₃²×C₄	—	3	2	no	—
33	C₃²×C₄	C₂×D₃	1	4	yes	—
34	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
35	C₃⋊Dic₃	—	3	1	no	—
36	C₃⋊Dic₃	—	3	4	no	—
37	C₃⋊Dic₃	—	6	4	no	—
38	C₃⋊Dic₃	C₂×D₃	1	2	yes	—
39	C₃×Dic₃	—	3	8	no	—
40	C₃×Dic₃	C₂×D₃	1	8	yes	—
41	C₃³	C₂×Q₈	1	1	yes	characteristic, complemented, Sylow
42	C₃×Q₈	—	3	4	no	—
43	C₃×Q₈	—	9	1	no	—
44	C₄×D₃	—	9	8	no	—
45	C₄×D₃	—	9	4	no	—
46	C₄×D₃	—	9	1	no	—
47	C₄×D₃	—	9	2	no	—
48	C₄×D₃	—	18	4	no	—
49	Dic₆	—	3	4	no	—
50	Dic₆	—	9	8	no	—
51	Dic₆	—	9	2	no	—
52	Dic₆	C₃⋊D₃	1	1	yes	characteristic, complemented
53	C₂×C₃²	—	2	4	no	—
54	C₂×C₃²	C₂²×D₃	1	4	yes	—
55	C₂×C₃²	C₂²×D₃	1	1	yes	characteristic
56	C₃⋊D₃	—	3	2	no	—
57	C₃⋊D₃	—	3	8	no	—
58	C₃⋊D₃	—	6	8	no	—
59	C₂×Q₈	—	27	1	no	Sylow
60	C₂×D₃	—	9	4	no	—
61	C₂×D₃	—	9	1	no	—
62	C₂×D₃	—	18	4	no	—
63	C₃×C₄	—	2	4	no	—
64	C₃×C₄	—	3	8	no	—
65	C₃×C₄	—	9	2	no	—
66	C₃×C₄	D₃²	1	4	yes	—
67	C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
68	Dic₃	—	3	8	no	—
69	Dic₃	—	9	4	no	—
70	Dic₃	—	9	1	no	—
71	Dic₃	—	18	4	no	—
72	Dic₃	C₂×C₃⋊D₃	1	2	yes	—
73	C₃²	—	2	4	no	—
74	C₃²	D₃×Q₈	1	4	yes	complemented
75	C₃²	D₃×Q₈	1	1	yes	characteristic, complemented
76	Q₈	—	3	1	no	—
77	Q₈	—	9	1	no	—
78	Q₈	—	27	2	no	—
79	C₂×C₄	—	27	2	no	—
80	C₂×C₄	—	27	1	no	—
81	C₂×C₃	—	2	4	no	—
82	C₂×C₃	C₂×D₃²	1	4	yes	—
83	C₂×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
84	D₃	—	9	8	no	—
85	D₃	—	9	2	no	—
86	D₃	—	18	8	no	—
87	C₂²	—	27	1	no	—
88	C₄	—	3	2	no	—
89	C₄	—	9	2	no	—
90	C₄	—	27	1	no	—
91	C₄	D₃×C₃⋊D₃	1	1	yes	characteristic
92	C₃	—	2	4	no	—
93	C₃	#3797	1	4	yes	complemented
94	C₃	Q₈×C₃⋊D₃	1	1	yes	characteristic, complemented
95	C₂	—	27	2	no	—
96	C₂	C₂×D₃×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₃×C₃⋊D₃
commutator, socle, cosocle	C₂×C₃³	C₂³
Fitting	C₃³×C₄	C₂²
prime radical	C₂×C₃⋊(C₃⋊D₃)	C₂²
hypercenter	C₄	D₃×C₃⋊D₃
hypocenter	C₃³	C₂×Q₈
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₃×#634	C₂	semidirect
3. C₃²×Dic₆	C₂	semidirect
4. #6293	C₂	semidirect
19. #634	D₃	semidirect
21. C₃×Dic₆	D₃	semidirect
25. C₃⋊(C₃⋊D₃)	Q₈	semidirect
41. C₃³	C₂×Q₈	semidirect
52. Dic₆	C₃⋊D₃	semidirect
74. C₃²	D₃×Q₈	semidirect
75. C₃²	D₃×Q₈	semidirect
93. C₃	#3797	semidirect
94. C₃	Q₈×C₃⋊D₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₃⋊(C₃⋊D₃)	2
C₃×#634	2
C₃²×Dic₆	2
#6293	2
#6289	2
Q₈×C₃⋊D₃	3
#3797	3