Subgroups — #10588305

Order 576.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#120⋊C₉	C₂	1	1	yes	maximal, characteristic
2	#4536	—	3	1	no	maximal
3	C₂²⋊(C₂²⋊C₉)	C₄	1	1	yes	characteristic, complemented
4	(C₂²⋊C₉)⋊C₄	—	4	1	no	maximal
5	C₃×#120	D₃	1	1	yes	characteristic
6	#834	—	3	1	no	—
7	#833	—	3	1	no	—
8	C₂×C₂²⋊C₉	—	4	1	no	—
9	AΣL(1,𝔽₁₆)	—	9	1	no	Sylow
10	C₂⁴×C₃	Dic₃	1	1	yes	characteristic
11	C₂×C₃×D₄	—	3	1	no	—
12	C₃×C₂²⋊C₄	—	3	1	no	—
13	#217	—	6	1	no	—
14	#208	—	6	1	no	—
15	C₂²⋊C₉	—	4	1	no	—
16	C₂²⋊C₉	—	16	1	no	—
17	Dic₉	—	16	1	no	—
18	#120	D₉	1	1	yes	characteristic
19	#100	—	9	1	no	—
20	#99	—	9	1	no	—
21	C₂³×C₃	—	3	1	no	—
22	C₂³×C₃	—	12	1	no	—
23	C₂³×C₃	S₄	1	1	yes	characteristic
24	C₃×D₄	—	12	1	no	—
25	C₂×C₃×C₄	—	3	1	no	—
26	C₂×Dic₃	—	12	1	no	—
27	C₃⋊C₈	—	12	1	no	—
28	C₂×C₉	—	16	1	no	—
29	C₂⁴	Dic₉	1	1	yes	characteristic, complemented
30	C₂×D₄	—	3	1	no	—
31	M₁₆	—	18	1	no	—
32	C₂²⋊C₄	—	3	1	no	—
33	C₂²⋊C₄	—	18	1	no	—
34	C₂²×C₃	—	4	1	no	—
35	C₂²×C₃	—	6	1	no	—
36	C₂²×C₃	—	6	1	no	—
37	C₂²×C₃	—	12	1	no	—
38	C₂²×C₃	—	12	1	no	—
39	C₂²×C₃	SL(2,ℤ/4)	1	1	yes	characteristic
40	C₃×C₄	—	6	1	no	—
41	Dic₃	—	24	1	no	—
42	C₉	—	16	1	no	Sylow
43	C₂³	—	3	1	no	—
44	C₂³	—	12	1	no	—
45	C₂³	(C₂²⋊C₉)⋊C₂	1	1	yes	characteristic
46	D₄	—	12	1	no	—
47	C₂×C₄	—	3	1	no	—
48	C₂×C₄	—	36	1	no	—
49	C₈	—	36	1	no	—
50	C₂×C₃	—	3	1	no	—
51	C₂×C₃	—	4	1	no	—
52	C₂×C₃	—	12	1	no	—
53	C₂²	—	4	1	no	—
54	C₂²	—	6	1	no	—
55	C₂²	—	6	1	no	—
56	C₂²	—	12	1	no	—
57	C₂²	—	12	1	no	—
58	C₂²	(C₂²⋊C₉)⋊C₄	1	1	yes	characteristic
59	C₄	—	6	1	no	—
60	C₄	—	72	1	no	—
61	C₃	#4625	1	1	yes	characteristic
62	C₂	—	3	1	no	—
63	C₂	—	4	1	no	—
64	C₂	—	12	1	no	—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, hypocenter	C₂²⋊(C₂²⋊C₉)	C₄
Frattini	C₂³×C₃	S₄
Fitting, prime radical	C₃×#120	D₃
socle	C₂²×C₃	SL(2,ℤ/4)
cosocle	#120⋊C₉	C₂
LCS	G ▷ C₂²⋊(C₂²⋊C₉)
UCS	C₁

Complements

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

N	H ≅ G/N	type
3. C₂²⋊(C₂²⋊C₉)	C₄	semidirect
29. C₂⁴	Dic₉	semidirect

Maximal subgroups

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

subgroup	index
#120⋊C₉	2
#4536	3
(C₂²⋊C₉)⋊C₄	4