Subgroups — #10592768

Order 576.

270 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	#64078	C₂	1	1	yes	maximal, characteristic
3	#63637	C₂	1	4	yes	maximal
4	#63630	C₂	1	1	yes	maximal, characteristic
5	#5069	—	3	4	no	maximal
6	C₂×#634	C₂²	1	2	yes	—
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)	—	2	2	no	—
11	C₂×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
12	C₃×#125	D₃	1	4	yes	—
13	#869	—	3	4	no	—
14	#810	—	3	16	no	—
15	#803	—	3	4	no	—
16	C₂×C₃²×C₄	—	2	2	no	—
17	C₂×C₃²×C₄	D₄	1	2	yes	—
18	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
19	C₂×C₃⋊Dic₃	—	2	2	no	—
20	#634	—	2	4	no	—
21	C₃⋊(C₃⋊C₈)	—	2	2	no	—
22	#491	—	9	1	no	Sylow
23	C₂×Dic₆	—	3	8	no	—
24	C₃×C₄⋊C₄	—	2	8	no	—
25	C₃×C₄⋊C₄	C₂×D₃	1	8	yes	—
26	C₃×C₄²	C₂×D₃	1	4	yes	—
27	C₄⋊Dic₃	—	6	8	no	—
28	C₂×C₃⋊C₈	—	3	8	no	—
29	C₂²×C₃²	C₂×D₄	1	1	yes	characteristic
30	C₃²×C₄	—	2	2	no	—
31	C₃²×C₄	—	4	2	no	—
32	C₃²×C₄	#41	1	2	yes	—
33	C₃⋊Dic₃	—	4	2	no	—
34	C₄⋊Q₈	—	9	1	no	—
35	#125	C₃⋊D₃	1	1	yes	characteristic
36	#103	—	9	4	no	—
37	#97	—	9	1	no	—
38	C₂×C₃×C₄	—	2	8	no	—
39	C₂×C₃×C₄	#67	1	8	yes	—
40	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
41	C₂×Dic₃	—	6	8	no	—
42	Dic₆	—	6	16	no	—
43	C₃⋊C₈	—	6	8	no	—
44	C₂×C₃²	#124	1	1	yes	characteristic
45	C₂×C₃²	#137	1	2	yes	—
46	C₂×Q₈	—	9	2	no	—
47	C₂×C₈	—	9	2	no	—
48	C₄⋊C₄	—	2	2	no	—
49	C₄⋊C₄	—	18	2	no	—
50	C₄⋊C₄	C₂×C₃⋊D₃	1	2	yes	—
51	C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
52	C₂²×C₃	C₂×#67	1	4	yes	—
53	C₃×C₄	—	2	8	no	—
54	C₃×C₄	—	4	8	no	—
55	C₃×C₄	#239	1	8	yes	—
56	Dic₃	—	12	8	no	—
57	C₃²	#491	1	1	yes	characteristic, complemented, Sylow
58	Q₈	—	18	4	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	18	2	no	—
61	C₂×C₄	#638	1	2	yes	—
62	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
63	C₈	—	18	2	no	—
64	C₂×C₃	#947	1	4	yes	—
65	C₂×C₃	#951	1	8	yes	—
66	C₂²	C₂×#638	1	1	yes	characteristic
67	C₄	—	2	2	no	—
68	C₄	—	4	2	no	—
69	C₄	—	36	2	no	—
70	C₄	#3832	1	2	yes	—
71	C₃	#5069	1	4	yes	complemented
72	C₂	#64156	1	1	yes	characteristic
73	C₂	#64160	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	#491
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
57. C₃²	#491	semidirect
71. C₃	#5069	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
#64078	2
#63637	2
#63630	2
#5069	3