Subgroups — #10596157

Order 576.

516 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	#64099	C₂	1	4	yes	maximal
3	#64098	C₂	1	4	yes	maximal
4	#64098	C₂	1	4	yes	maximal
5	#64079	C₂	1	2	yes	maximal
6	C₃⋊#564	—	3	4	no	maximal
7	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
8	C₃²×C₄⋊C₄	C₂²	1	4	yes	—
9	C₃²×C₄²	C₂²	1	1	yes	characteristic
10	C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
11	C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
12	C₄⋊(C₃⋊Dic₃)	C₂²	1	4	yes	—
13	#3750	C₂²	1	4	yes	—
14	#3750	C₂²	1	4	yes	—
15	#3750	C₂²	1	4	yes	—
16	#3750	C₂²	1	4	yes	—
17	C₄×C₃⋊Dic₃	C₂²	1	4	yes	—
18	C₃×#125	D₃	1	4	yes	—
19	#890	—	3	16	no	—
20	#889	—	3	16	no	—
21	#889	—	3	16	no	—
22	#870	—	3	8	no	—
23	C₂×C₃²×C₄	C₂³	1	4	yes	—
24	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
25	C₂×C₃²×C₄	C₂³	1	2	yes	—
26	C₂×C₃⋊Dic₃	C₂³	1	4	yes	—
27	C₂×C₃⋊Dic₃	C₂³	1	4	yes	—
28	#564	—	9	1	no	Sylow
29	C₃×C₄⋊C₄	C₂×D₃	1	8	yes	—
30	C₃×C₄⋊C₄	C₂×D₃	1	16	yes	—
31	C₃×C₄²	C₂×D₃	1	4	yes	—
32	C₄⋊Dic₃	—	3	8	no	—
33	C₄⋊Dic₃	—	3	8	no	—
34	C₄⋊Dic₃	—	3	16	no	—
35	Dic₃⋊C₄	—	3	16	no	—
36	Dic₃⋊C₄	—	3	16	no	—
37	Dic₃⋊C₄	—	3	16	no	—
38	Dic₃⋊C₄	—	3	16	no	—
39	C₄×Dic₃	—	3	16	no	—
40	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
41	C₃²×C₄	—	2	1	no	—
42	C₃²×C₄	—	2	4	no	—
43	C₃²×C₄	—	2	2	no	—
44	C₃⋊Dic₃	—	2	4	no	—
45	C₃⋊Dic₃	—	2	4	no	—
46	#125	—	9	4	no	—
47	#125	—	9	4	no	—
48	#125	—	9	4	no	—
49	#125	—	9	2	no	—
50	#125	C₃⋊D₃	1	1	yes	characteristic
51	C₂×C₃×C₄	C₂²×D₃	1	16	yes	—
52	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
53	C₂×C₃×C₄	C₂²×D₃	1	8	yes	—
54	C₂×Dic₃	—	3	16	no	—
55	C₂×Dic₃	—	3	16	no	—
56	C₂×C₃²	2₋¹⁺⁴	1	1	yes	characteristic
57	C₂×C₃²	2₋¹⁺⁴	1	2	yes	—
58	C₄⋊C₄	—	9	4	no	—
59	C₄⋊C₄	—	9	4	no	—
60	C₄⋊C₄	—	9	4	no	—
61	C₄⋊C₄	—	9	4	no	—
62	C₄⋊C₄	—	9	4	no	—
63	C₄⋊C₄	—	9	2	no	—
64	C₄⋊C₄	—	9	2	no	—
65	C₄⋊C₄	C₂×C₃⋊D₃	1	2	yes	—
66	C₄⋊C₄	C₂×C₃⋊D₃	1	4	yes	—
67	C₄²	—	9	4	no	—
68	C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
69	C₂²×C₃	C₂³×D₃	1	4	yes	—
70	C₃×C₄	—	2	4	no	—
71	C₃×C₄	—	2	16	no	—
72	C₃×C₄	—	2	8	no	—
73	Dic₃	—	6	16	no	—
74	Dic₃	—	6	16	no	—
75	C₃²	#564	1	1	yes	characteristic, complemented, Sylow
76	C₂×C₄	—	9	4	no	—
77	C₂×C₄	—	9	4	no	—
78	C₂×C₄	C₂²×C₃⋊D₃	1	4	yes	—
79	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
80	C₂×C₄	C₂²×C₃⋊D₃	1	2	yes	—
81	C₂×C₃	#1007	1	4	yes	—
82	C₂×C₃	#1010	1	8	yes	—
83	C₂²	C₂³×C₃⋊D₃	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	2	4	no	—
86	C₄	—	2	2	no	—
87	C₄	—	18	4	no	—
88	C₄	—	18	4	no	—
89	C₃	C₃⋊#564	1	4	yes	complemented
90	C₂	#64364	1	1	yes	characteristic
91	C₂	#64367	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₃⋊D₃
commutator, socle, prime radical, cosocle	C₂²×C₃²	C₂⁴
Fitting	C₃²×#125	C₂
hypercenter	#125	C₃⋊D₃
hypocenter	C₃²	#564
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
75. C₃²	#564	semidirect
89. C₃	C₃⋊#564	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
#64099	2
#64098	2
#64079	2
C₃⋊#564	3