Subgroups — #63573

Order 288.

85 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#817	—	3	1	no	maximal
5	#808	—	3	1	no	maximal
6	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
7	C₂×C₃⋊Dic₃	—	2	1	no	—
8	C₂×C₃×Dic₃	—	2	1	no	—
9	C₃×C₃⋊C₈	C₄	1	2	yes	complemented
10	C₂×C₃×C₈	—	3	1	no	—
11	C₃×C₄⋊C₄	—	3	1	no	—
12	C₄⋊Dic₃	—	3	1	no	—
13	C₄⋊Dic₃	—	3	1	no	—
14	C₄⋊Dic₃	—	6	1	no	—
15	C₄⋊Dic₃	D₃	1	1	yes	characteristic
16	C₂×C₃⋊C₈	D₃	1	1	yes	characteristic
17	C₂²×C₃²	D₄	1	1	yes	characteristic
18	C₃²×C₄	C₂×C₄	1	1	yes	characteristic
19	C₃²×C₄	Q₈	1	1	yes	characteristic
20	C₃⋊Dic₃	—	4	1	no	—
21	C₃×Dic₃	—	4	1	no	—
22	#106	—	9	1	no	Sylow
23	C₂×C₃×C₄	—	2	1	no	—
24	C₂×C₃×C₄	—	6	1	no	—
25	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
26	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
27	C₂×Dic₃	—	2	1	no	—
28	C₂×Dic₃	—	6	2	no	—
29	C₂×Dic₃	—	6	1	no	—
30	C₂×Dic₃	—	6	1	no	—
31	C₃×C₈	—	3	2	no	—
32	C₃⋊C₈	Dic₃	1	2	yes	complemented
33	C₂×C₃²	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₃²	SD₁₆	1	1	yes	characteristic
35	C₂×C₃²	SD₁₆	1	1	yes	characteristic
36	C₂×C₈	—	3	1	no	—
37	C₄⋊C₄	—	3	1	no	—
38	C₄⋊C₄	—	9	1	no	—
39	C₂²×C₃	—	2	1	no	—
40	C₂²×C₃	D₁₂	1	1	yes	characteristic
41	C₂²×C₃	#67	1	1	yes	characteristic
42	C₃×C₄	—	2	1	no	—
43	C₃×C₄	—	2	1	no	—
44	C₃×C₄	—	12	1	no	—
45	C₃×C₄	Dic₆	1	1	yes	characteristic
46	C₃×C₄	Dic₆	1	1	yes	characteristic
47	C₃×C₄	C₄×D₃	1	1	yes	characteristic
48	C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
49	Dic₃	—	4	1	no	—
50	Dic₃	—	12	2	no	—
51	Dic₃	—	12	1	no	—
52	Dic₃	—	12	1	no	—
53	C₃²	#106	1	1	yes	characteristic, complemented, Sylow
54	C₂×C₄	—	6	1	no	—
55	C₂×C₄	—	18	1	no	—
56	C₂×C₄	D₃²	1	1	yes	characteristic
57	C₈	—	3	2	no	—
58	C₂×C₃	—	2	1	no	—
59	C₂×C₃	—	2	1	no	—
60	C₂×C₃	—	2	1	no	—
61	C₂×C₃	#204	1	1	yes	characteristic
62	C₂×C₃	#204	1	1	yes	characteristic
63	C₂×C₃	Dic₃⋊C₄	1	1	yes	characteristic
64	C₂×C₃	C₄⋊Dic₃	1	1	yes	characteristic
65	C₂×C₃	#214	1	1	yes	characteristic
66	C₂×C₃	#215	1	1	yes	characteristic
67	C₂²	#626	1	1	yes	characteristic
68	C₄	—	12	1	no	—
69	C₄	—	36	1	no	—
70	C₄	D₃×Dic₃	1	1	yes	characteristic
71	C₄	#627	1	1	yes	characteristic
72	C₃	—	2	1	no	—
73	C₃	#808	1	1	yes	characteristic, complemented
74	C₃	#817	1	1	yes	characteristic, complemented
75	C₂	#3716	1	1	yes	characteristic
76	C₂	#3717	1	1	yes	characteristic
77	C₂	Dic₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#626
commutator	C₃²×C₄	C₂×C₄
Frattini, hypercenter	C₂×C₄	D₃²
Fitting, cosocle	C₂×C₃²×C₄	C₂²
socle, prime radical	C₂²×C₃²	D₄
hypocenter	C₃²	#106
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
9. C₃×C₃⋊C₈	C₄	semidirect
32. C₃⋊C₈	Dic₃	semidirect
53. C₃²	#106	semidirect
73. C₃	#808	semidirect
74. C₃	#817	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₃⋊Dic₃)	2
C₃×C₄⋊Dic₃	2
C₂×C₃×C₃⋊C₈	2
#817	3
#808	3