Subgroups — #10592766

Order 576.

330 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, complemented
2	#64083	C₂	1	1	yes	maximal, characteristic
3	#63636	C₂	1	4	yes	maximal, complemented
4	#63630	C₂	1	1	yes	maximal, characteristic, complemented
5	#5067	—	3	4	no	maximal
6	C₂×C₄⋊(C₃⋊D₃)	—	2	2	no	—
7	C₂×C₄⋊(C₃⋊D₃)	C₂²	1	2	yes	—
8	C₃²×C₄⋊C₄	—	2	2	no	—
9	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
10	C₃²×C₄²	C₂²	1	1	yes	characteristic
11	C₂×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
12	C₃×#125	D₃	1	4	yes	complemented
13	#874	—	3	4	no	—
14	#809	—	3	16	no	—
15	#803	—	3	4	no	—
16	C₂²×C₃⋊D₃	—	2	2	no	—
17	C₂×C₃²×C₄	—	2	2	no	—
18	C₂×C₃²×C₄	D₄	1	2	yes	—
19	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
20	C₄⋊(C₃⋊D₃)	—	2	4	no	—
21	C₄⋊(C₃⋊D₃)	—	4	4	no	—
22	C₃⋊(C₃⋊C₈)	—	2	2	no	—
23	#490	—	9	1	no	Sylow
24	C₂×D₁₂	—	3	8	no	—
25	C₂×D₁₂	—	6	8	no	—
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	C₂×C₃⋊C₈	—	3	8	no	—
30	C₂²×C₃²	C₂×D₄	1	1	yes	characteristic
31	C₂×C₃⋊D₃	—	4	2	no	—
32	C₂×C₃⋊D₃	—	4	4	no	—
33	C₃²×C₄	—	2	2	no	—
34	C₃²×C₄	—	4	2	no	—
35	C₃²×C₄	#41	1	2	yes	—
36	#127	—	9	1	no	—
37	#125	C₃⋊D₃	1	1	yes	characteristic, complemented
38	#102	—	9	4	no	—
39	#97	—	9	1	no	—
40	C₂²×D₃	—	6	8	no	—
41	C₂×C₃×C₄	—	2	8	no	—
42	C₂×C₃×C₄	#67	1	8	yes	—
43	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
44	D₁₂	—	6	16	no	—
45	D₁₂	—	12	16	no	—
46	C₃⋊C₈	—	6	8	no	—
47	C₂×C₃²	#124	1	1	yes	characteristic
48	C₂×C₃²	AGL(1,ℤ/8)	1	2	yes	—
49	C₃⋊D₃	—	8	2	no	—
50	C₂×D₄	—	9	2	no	—
51	C₂×D₄	—	18	2	no	—
52	C₂×C₈	—	9	2	no	—
53	C₄⋊C₄	—	2	2	no	—
54	C₄⋊C₄	C₂×C₃⋊D₃	1	2	yes	—
55	C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
56	C₂²×C₃	C₂×#67	1	4	yes	—
57	C₂×D₃	—	12	8	no	—
58	C₂×D₃	—	12	16	no	—
59	C₃×C₄	—	2	8	no	—
60	C₃×C₄	—	4	8	no	—
61	C₃×C₄	#239	1	8	yes	—
62	C₃²	#490	1	1	yes	characteristic, complemented, Sylow
63	C₂³	—	18	2	no	—
64	D₄	—	18	4	no	—
65	D₄	—	36	4	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	#638	1	2	yes	—
68	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
69	C₈	—	18	2	no	—
70	C₂×C₃	#947	1	4	yes	—
71	C₂×C₃	#949	1	8	yes	—
72	D₃	—	24	8	no	—
73	C₂²	—	36	2	no	—
74	C₂²	—	36	4	no	—
75	C₂²	C₂×#638	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	4	2	no	—
78	C₄	#3832	1	2	yes	—
79	C₃	#5067	1	4	yes	complemented
80	C₂	—	72	2	no	—
81	C₂	#64156	1	1	yes	characteristic
82	C₂	#64158	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	C₂²×C₃²	C₂×D₄
prime radical	#64083	C₂
hypercenter	#125	C₃⋊D₃
hypocenter	C₃²	#490
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
1. C₃²×#125	C₂	semidirect
3. #63636	C₂	semidirect
4. #63630	C₂	semidirect
12. C₃×#125	D₃	semidirect
37. #125	C₃⋊D₃	semidirect
62. C₃²	#490	semidirect
79. C₃	#5067	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
#64083	2
#63636	2
#63630	2
#5067	3