Subgroups — #10592634

Order 576.

558 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊Dic₃)	C₂	1	6	yes	maximal
3	#4935	—	3	4	no	maximal
4	C₂²×C₃²×C₄	C₂²	1	3	yes	—
5	C₂²×C₃⋊Dic₃	C₂²	1	4	yes	—
6	C₃²×C₄²	C₄	1	4	yes	complemented
7	C₄⋊(C₃⋊Dic₃)	—	2	12	no	—
8	C₂×C₃×C₄²	D₃	1	4	yes	—
9	C₂×C₄⋊Dic₃	—	3	24	no	—
10	C₂³×C₃²	C₂³	1	1	yes	characteristic
11	C₂×C₃²×C₄	C₂×C₄	1	6	yes	—
12	C₂×C₃²×C₄	D₄	1	6	yes	—
13	C₂×C₃²×C₄	Q₈	1	6	yes	—
14	C₂×C₃⋊Dic₃	—	2	12	no	—
15	#384	—	9	1	no	Sylow
16	C₂²×C₃×C₄	C₂×D₃	1	12	yes	—
17	C₂²×Dic₃	—	3	16	no	—
18	C₃×C₄²	Dic₃	1	16	yes	complemented
19	C₄⋊Dic₃	—	6	48	no	—
20	C₂²×C₃²	C₂²×C₄	1	1	yes	characteristic
21	C₂²×C₃²	C₂×D₄	1	3	yes	—
22	C₂²×C₃²	C₂×Q₈	1	3	yes	—
23	C₃²×C₄	C₄⋊C₄	1	12	yes	complemented
24	C₃⋊Dic₃	—	4	4	no	—
25	C₂×C₄⋊C₄	—	9	6	no	—
26	C₂×C₄²	C₃⋊D₃	1	1	yes	characteristic
27	C₂³×C₃	C₂²×D₃	1	4	yes	—
28	C₂×C₃×C₄	Dic₆	1	24	yes	—
29	C₂×C₃×C₄	D₁₂	1	24	yes	—
30	C₂×C₃×C₄	C₂×Dic₃	1	24	yes	—
31	C₂×Dic₃	—	6	48	no	—
32	C₂×C₃²	C₂×C₄⋊C₄	1	3	yes	—
33	C₂×C₃²	#127	1	1	yes	characteristic
34	C₂×C₃²	C₄⋊Q₈	1	3	yes	—
35	C₂²×C₄	—	9	4	no	—
36	C₂²×C₄	C₂×C₃⋊D₃	1	3	yes	—
37	C₄⋊C₄	—	18	12	no	—
38	C₄²	C₃⋊Dic₃	1	4	yes	complemented
39	C₂²×C₃	C₂×Dic₆	1	12	yes	—
40	C₂²×C₃	C₂×D₁₂	1	12	yes	—
41	C₂²×C₃	C₂²×Dic₃	1	4	yes	—
42	C₃×C₄	C₄⋊Dic₃	1	48	yes	complemented
43	Dic₃	—	12	16	no	—
44	C₃²	#384	1	1	yes	characteristic, complemented, Sylow
45	C₂³	C₂²×C₃⋊D₃	1	1	yes	characteristic
46	C₂×C₄	—	18	12	no	—
47	C₂×C₄	#634	1	6	yes	—
48	C₂×C₄	C₄⋊(C₃⋊D₃)	1	6	yes	—
49	C₂×C₄	C₂×C₃⋊Dic₃	1	6	yes	—
50	C₂×C₃	#869	1	12	yes	—
51	C₂×C₃	#874	1	4	yes	—
52	C₂×C₃	C₂×C₄⋊Dic₃	1	12	yes	—
53	C₂²	C₂×#634	1	3	yes	—
54	C₂²	C₂×C₄⋊(C₃⋊D₃)	1	3	yes	—
55	C₂²	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
56	C₄	—	36	4	no	—
57	C₄	C₄⋊(C₃⋊Dic₃)	1	12	yes	complemented
58	C₃	#4935	1	4	yes	complemented
59	C₂	#64078	1	3	yes	—
60	C₂	#64083	1	1	yes	characteristic
61	C₂	C₂×C₄⋊(C₃⋊Dic₃)	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×C₃⋊D₃
commutator	C₂²×C₃²	C₂²×C₄
Fitting	C₂×C₃²×C₄²	C₂
socle, prime radical, cosocle	C₂³×C₃²	C₂³
hypercenter	C₂×C₄²	C₃⋊D₃
hypocenter	C₃²	#384
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
6. C₃²×C₄²	C₄	semidirect
18. C₃×C₄²	Dic₃	semidirect
23. C₃²×C₄	C₄⋊C₄	semidirect
38. C₄²	C₃⋊Dic₃	semidirect
42. C₃×C₄	C₄⋊Dic₃	semidirect
44. C₃²	#384	semidirect
57. C₄	C₄⋊(C₃⋊Dic₃)	semidirect
58. C₃	#4935	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄²	2
C₂×C₄⋊(C₃⋊Dic₃)	2
#4935	3