Subgroups — #10592764

Order 576.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#125	C₂	1	1	yes	maximal, characteristic
2	#64079	C₂	1	1	yes	maximal, characteristic
3	#63635	C₂	1	2	yes	maximal
4	#63634	C₂	1	2	yes	maximal
5	C₄×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#5065	—	3	4	no	maximal
7	C₃²×C₄⋊C₄	—	2	2	no	—
8	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
9	C₃²×C₄²	C₂²	1	1	yes	characteristic
10	C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
11	#3750	—	2	2	no	—
12	C₂×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
14	C₃×#125	D₃	1	4	yes	—
15	#870	—	3	4	no	—
16	#808	—	3	8	no	—
17	#807	—	3	8	no	—
18	C₄×C₃⋊C₈	—	3	4	no	—
19	C₂×C₃²×C₄	—	2	2	no	—
20	C₂×C₃²×C₄	D₄	1	2	yes	—
21	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
22	C₂×C₃⋊Dic₃	—	2	2	no	—
23	C₃⋊(C₃⋊C₈)	Q₈	1	2	yes	—
24	C₃⋊(C₃⋊C₈)	Q₈	1	2	yes	—
25	#499	—	9	1	no	Sylow
26	C₃×C₄⋊C₄	—	2	8	no	—
27	C₃×C₄⋊C₄	C₂×D₃	1	8	yes	—
28	C₃×C₄²	C₂×D₃	1	4	yes	—
29	C₄⋊Dic₃	—	3	8	no	—
30	Dic₃⋊C₄	—	6	8	no	—
31	C₂×C₃⋊C₈	—	3	4	no	—
32	C₂×C₃⋊C₈	—	3	4	no	—
33	C₂²×C₃²	C₂×D₄	1	1	yes	characteristic
34	C₃²×C₄	—	2	2	no	—
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	#125	—	9	1	no	—
40	#125	C₃⋊D₃	1	1	yes	characteristic
41	#107	—	9	2	no	—
42	#106	—	9	2	no	—
43	C₄×C₈	—	9	1	no	—
44	C₂×C₃×C₄	—	2	8	no	—
45	C₂×C₃×C₄	#67	1	8	yes	—
46	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
47	C₂×Dic₃	—	6	8	no	—
48	C₃⋊C₈	—	3	8	no	—
49	C₃⋊C₈	—	3	8	no	—
50	C₂×C₃²	C₄⋊Q₈	1	1	yes	characteristic
51	C₂×C₃²	#135	1	2	yes	—
52	C₂×C₈	—	9	1	no	—
53	C₂×C₈	—	9	1	no	—
54	C₄⋊C₄	—	2	2	no	—
55	C₄⋊C₄	—	9	2	no	—
56	C₄⋊C₄	—	18	2	no	—
57	C₄⋊C₄	C₂×C₃⋊D₃	1	2	yes	—
58	C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
59	C₂²×C₃	C₂×#67	1	4	yes	—
60	C₃×C₄	—	2	8	no	—
61	C₃×C₄	—	4	8	no	—
62	C₃×C₄	D₃×Q₈	1	4	yes	—
63	C₃×C₄	D₃×Q₈	1	4	yes	—
64	Dic₃	—	12	8	no	—
65	C₃²	#499	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	18	2	no	—
68	C₂×C₄	#638	1	2	yes	—
69	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
70	C₈	—	9	2	no	—
71	C₈	—	9	2	no	—
72	C₂×C₃	#944	1	4	yes	—
73	C₂×C₃	#950	1	8	yes	—
74	C₂²	C₂×#638	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	4	2	no	—
77	C₄	—	36	2	no	—
78	C₄	Q₈×C₃⋊D₃	1	1	yes	characteristic
79	C₄	Q₈×C₃⋊D₃	1	1	yes	characteristic
80	C₃	#5065	1	4	yes	complemented
81	C₂	#64153	1	1	yes	characteristic
82	C₂	#64159	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#638
commutator, cosocle	C₂×C₃²×C₄	C₂³
Frattini	C₂×C₄	C₂²×C₃⋊D₃
Fitting	C₃²×#125	C₂
socle, prime radical	C₂²×C₃²	C₂×D₄
hypercenter	#125	C₃⋊D₃
hypocenter	C₃²	#499
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
65. C₃²	#499	semidirect
80. C₃	#5065	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#125	2
#64079	2
#63635	2
#63634	2
C₄×C₃⋊(C₃⋊C₈)	2
#5065	3