Subgroups — #10592554

Order 576.

294 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#64097	C₂	1	2	yes	maximal
2	C₃²×#106	C₂	1	1	yes	maximal, characteristic
3	#63644	C₂	1	1	yes	maximal, characteristic
4	C₈×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
5	#63635	C₂	1	2	yes	maximal
6	#4855	—	3	4	no	maximal
7	C₂×#634	—	2	2	no	—
8	C₂×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
10	C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
11	#3750	—	2	2	no	—
12	C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
14	#888	—	3	8	no	—
15	C₃×#106	D₃	1	4	yes	—
16	#817	—	3	4	no	—
17	C₈×Dic₃	—	3	4	no	—
18	#808	—	3	8	no	—
19	C₂×C₃²×C₄	—	2	2	no	—
20	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
21	C₂×C₃⋊Dic₃	—	2	2	no	—
22	C₂×C₃⋊Dic₃	D₄	1	2	yes	—
23	#634	—	4	4	no	—
24	C₃²×C₈	Q₈	1	2	yes	complemented
25	C₃⋊(C₃⋊C₈)	Q₈	1	2	yes	complemented
26	#498	—	9	1	no	Sylow
27	C₂×Dic₆	—	6	8	no	—
28	C₂×C₃×C₈	C₂×D₃	1	4	yes	—
29	C₃×C₄⋊C₄	C₂×D₃	1	8	yes	—
30	C₄⋊Dic₃	—	3	8	no	—
31	Dic₃⋊C₄	—	6	8	no	—
32	C₄×Dic₃	—	3	4	no	—
33	C₂×C₃⋊C₈	—	3	4	no	—
34	C₂²×C₃²	C₂×D₄	1	1	yes	characteristic
35	C₃²×C₄	—	4	2	no	—
36	C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
37	C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
38	C₃⋊Dic₃	—	4	2	no	—
39	C₃⋊Dic₃	SD₁₆	1	4	yes	—
40	C₄⋊Q₈	—	9	2	no	—
41	#106	—	9	1	no	—
42	#106	—	9	2	no	—
43	#106	C₃⋊D₃	1	1	yes	characteristic
44	C₄×C₈	—	9	1	no	—
45	C₂×C₃×C₄	—	2	8	no	—
46	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
47	C₂×Dic₃	—	3	8	no	—
48	C₂×Dic₃	—	6	8	no	—
49	Dic₆	—	12	16	no	—
50	C₃×C₈	Dic₆	1	8	yes	complemented
51	C₃⋊C₈	—	3	8	no	—
52	C₂×C₃²	C₄⋊Q₈	1	1	yes	characteristic
53	C₂×C₃²	C₂×SD₁₆	1	2	yes	—
54	C₂×Q₈	—	18	2	no	—
55	C₂×C₈	—	9	1	no	—
56	C₂×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
57	C₄⋊C₄	—	9	2	no	—
58	C₄⋊C₄	—	18	2	no	—
59	C₄⋊C₄	C₂×C₃⋊D₃	1	2	yes	—
60	C₄²	—	9	1	no	—
61	C₂²×C₃	D₃×D₄	1	4	yes	—
62	C₃×C₄	—	4	8	no	—
63	C₃×C₄	C₂×Dic₆	1	4	yes	—
64	C₃×C₄	D₃×Q₈	1	4	yes	—
65	Dic₃	—	3	16	no	—
66	Dic₃	—	12	8	no	—
67	C₃²	#498	1	1	yes	characteristic, complemented, Sylow
68	Q₈	—	36	4	no	—
69	C₂×C₄	—	2	2	no	—
70	C₂×C₄	—	9	2	no	—
71	C₂×C₄	—	18	2	no	—
72	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
73	C₈	—	9	2	no	—
74	C₈	#634	1	2	yes	complemented
75	C₂×C₃	#888	1	4	yes	—
76	C₂×C₃	D₃×SD₁₆	1	8	yes	—
77	C₂²	D₄×C₃⋊D₃	1	1	yes	characteristic
78	C₄	—	4	2	no	—
79	C₄	—	9	4	no	—
80	C₄	—	36	2	no	—
81	C₄	C₂×#634	1	1	yes	characteristic
82	C₄	Q₈×C₃⋊D₃	1	1	yes	characteristic
83	C₃	#4855	1	4	yes	complemented
84	C₂	#64097	1	1	yes	characteristic
85	C₂	SD₁₆×C₃⋊D₃	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×C₃⋊D₃
commutator, cosocle	C₂×C₃²×C₄	C₂³
Frattini	C₂×C₄	C₂²×C₃⋊D₃
Fitting	C₃²×#106	C₂
socle, prime radical	C₂²×C₃²	C₂×D₄
hypercenter	#106	C₃⋊D₃
hypocenter	C₃²	#498
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
24. C₃²×C₈	Q₈	semidirect
25. C₃⋊(C₃⋊C₈)	Q₈	semidirect
50. C₃×C₈	Dic₆	semidirect
67. C₃²	#498	semidirect
74. C₈	#634	semidirect
83. C₃	#4855	semidirect

Maximal subgroups

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

subgroup	index
#64097	2
C₃²×#106	2
#63644	2
C₈×C₃⋊Dic₃	2
#63635	2
#4855	3