Subgroups — #10589377

Order 576.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₈×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	#63639	C₂	1	1	yes	maximal, characteristic
4	#4511	—	3	4	no	maximal
5	C₂×C₃²×C₈	C₂²	1	1	yes	characteristic
6	C₄×C₃⋊Dic₃	C₄	1	1	yes	characteristic
7	C₂×C₃⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
8	C₃²×C₁₆	—	2	1	no	—
9	C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
10	C₃×M₃₂	D₃	1	4	yes	—
11	C₈×Dic₃	—	3	4	no	—
12	#812	—	3	4	no	—
13	C₂×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
14	C₂×C₃⋊Dic₃	—	2	1	no	—
15	C₃²×C₈	D₄	1	1	yes	characteristic
16	C₃²×C₈	Q₈	1	1	yes	characteristic
17	C₃⋊(C₃⋊C₈)	C₈	1	2	yes	—
18	#364	—	9	1	no	Sylow
19	C₂×C₃×C₈	C₂×D₃	1	4	yes	—
20	C₄×Dic₃	—	3	4	no	—
21	C₂×C₃⋊C₈	—	3	4	no	—
22	C₃×C₁₆	—	2	4	no	—
23	C₃⋊C₁₆	—	6	4	no	—
24	C₂²×C₃²	M₁₆	1	1	yes	characteristic
25	C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
26	C₃²×C₄	C₂×C₈	1	1	yes	characteristic
27	C₃⋊Dic₃	—	2	2	no	—
28	M₃₂	—	9	1	no	—
29	M₃₂	C₃⋊D₃	1	1	yes	characteristic
30	C₄×C₈	—	9	1	no	—
31	C₂×C₃×C₄	C₄×D₃	1	4	yes	—
32	C₂×Dic₃	—	6	4	no	—
33	C₃×C₈	Dic₆	1	4	yes	—
34	C₃×C₈	#67	1	4	yes	—
35	C₃⋊C₈	—	3	8	no	—
36	C₂×C₃²	—	2	1	no	—
37	C₂×C₃²	C₄⋊C₈	1	1	yes	characteristic
38	C₂×C₈	—	9	1	no	—
39	C₂×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
40	C₄²	—	9	1	no	—
41	C₁₆	—	2	1	no	—
42	C₁₆	—	18	1	no	—
43	C₂²×C₃	ΓL(1,𝔽₂₅)	1	4	yes	—
44	C₃×C₄	D₃×C₈	1	4	yes	—
45	C₃×C₄	Dic₃⋊C₄	1	4	yes	—
46	Dic₃	—	6	8	no	—
47	C₃²	#364	1	1	yes	characteristic, complemented, Sylow
48	C₂×C₄	—	18	1	no	—
49	C₂×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
50	C₈	—	9	2	no	—
51	C₈	#634	1	1	yes	characteristic
52	C₈	#638	1	1	yes	characteristic
53	C₂×C₃	—	2	4	no	—
54	C₂×C₃	Dic₃⋊C₈	1	4	yes	—
55	C₂²	#3743	1	1	yes	characteristic
56	C₄	—	18	2	no	—
57	C₄	C₈×C₃⋊D₃	1	1	yes	characteristic
58	C₄	#3750	1	1	yes	characteristic
59	C₃	#4511	1	4	yes	complemented
60	C₂	—	2	1	no	—
61	C₂	#63641	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#638
commutator	C₃²×C₄	C₂×C₈
Frattini	C₂×C₈	C₂×C₃⋊D₃
Fitting	C₃²×M₃₂	C₂
socle	C₂×C₃²	C₄⋊C₈
prime radical	C₂²×C₃²	M₁₆
cosocle	C₂×C₃²×C₈	C₂²
hypercenter	M₃₂	C₃⋊D₃
hypocenter	C₃²	#364
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
47. C₃²	#364	semidirect
59. C₃	#4511	semidirect

Maximal subgroups

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

subgroup	index
C₃²×M₃₂	2
C₈×C₃⋊Dic₃	2
#63639	2
#4511	3