Subgroups — #10592770

Order 576.

330 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#127	C₂	1	1	yes	maximal, characteristic
2	#64079	C₂	1	1	yes	maximal, characteristic, complemented
3	#63659	C₂	1	4	yes	maximal, complemented
4	#63630	C₂	1	1	yes	maximal, characteristic, complemented
5	#5071	—	3	4	no	maximal
6	C₂×C₃²×D₄	—	2	2	no	—
7	C₂×C₃²×D₄	C₂²	1	2	yes	—
8	C₃²×C₄²	C₂²	1	1	yes	characteristic
9	C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
10	#3750	—	2	2	no	—
11	C₂×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
12	C₃×#127	D₃	1	4	yes	—
13	#870	—	3	4	no	—
14	#832	—	3	16	no	—
15	#803	—	3	4	no	—
16	C₂³×C₃²	—	2	2	no	—
17	C₃²×D₄	—	2	4	no	—
18	C₃²×D₄	—	4	4	no	—
19	C₂×C₃²×C₄	D₄	1	2	yes	—
20	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
21	C₂×C₃⋊Dic₃	—	2	2	no	—
22	C₃⋊(C₃⋊C₈)	—	2	2	no	—
23	#490	—	9	1	no	Sylow
24	C₂×C₃×D₄	—	2	8	no	—
25	C₂×C₃×D₄	C₂×D₃	1	8	yes	—
26	C₃×C₄²	C₂×D₃	1	4	yes	—
27	C₄⋊Dic₃	—	3	8	no	—
28	Dic₃⋊C₄	—	6	8	no	—
29	C₂×C₃⋊C₈	—	3	8	no	—
30	C₂²×C₃²	—	4	2	no	—
31	C₂²×C₃²	—	4	4	no	—
32	C₂²×C₃²	C₂×D₄	1	1	yes	characteristic
33	C₃²×C₄	—	2	2	no	—
34	C₃²×C₄	#41	1	2	yes	—
35	C₃⋊Dic₃	—	4	2	no	—
36	#127	C₃⋊D₃	1	1	yes	characteristic
37	#125	—	9	1	no	—
38	#102	—	9	4	no	—
39	#97	—	9	1	no	—
40	C₂³×C₃	—	2	8	no	—
41	C₃×D₄	—	2	16	no	—
42	C₃×D₄	—	4	16	no	—
43	C₂×C₃×C₄	#67	1	8	yes	—
44	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
45	C₂×Dic₃	—	6	8	no	—
46	C₃⋊C₈	—	6	8	no	—
47	C₂×C₃²	—	8	2	no	—
48	C₂×C₃²	#124	1	1	yes	characteristic
49	C₂×C₃²	AGL(1,ℤ/8)	1	2	yes	—
50	C₂×D₄	—	2	2	no	—
51	C₂×D₄	C₂×C₃⋊D₃	1	2	yes	—
52	C₂×C₈	—	9	2	no	—
53	C₄⋊C₄	—	9	2	no	—
54	C₄⋊C₄	—	18	2	no	—
55	C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
56	C₂²×C₃	—	4	16	no	—
57	C₂²×C₃	—	4	8	no	—
58	C₂²×C₃	C₂×#67	1	4	yes	—
59	C₃×C₄	—	2	8	no	—
60	C₃×C₄	#237	1	8	yes	—
61	Dic₃	—	12	8	no	—
62	C₃²	#490	1	1	yes	characteristic, complemented, Sylow
63	C₂³	—	2	2	no	—
64	D₄	—	2	4	no	—
65	D₄	—	4	4	no	—
66	C₂×C₄	—	18	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	—	8	8	no	—
71	C₂×C₃	#932	1	8	yes	—
72	C₂×C₃	#936	1	4	yes	—
73	C₂²	—	4	4	no	—
74	C₂²	—	4	2	no	—
75	C₂²	C₂×#638	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	36	2	no	—
78	C₄	#3830	1	2	yes	—
79	C₃	#5071	1	4	yes	complemented
80	C₂	—	8	2	no	—
81	C₂	#64141	1	2	yes	—
82	C₂	#64145	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#638
commutator, cosocle	C₂×C₃²×C₄	C₂³
Frattini	C₂×C₄	C₂²×C₃⋊D₃
Fitting, prime radical	C₃²×#127	C₂
socle	C₂²×C₃²	C₂×D₄
hypercenter	#127	C₃⋊D₃
hypocenter	C₃²	#490
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
2. #64079	C₂	semidirect
3. #63659	C₂	semidirect
4. #63630	C₂	semidirect
62. C₃²	#490	semidirect
79. C₃	#5071	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#127	2
#64079	2
#63659	2
#63630	2
#5071	3