Subgroups — #10590082

Order 576.

61 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#157	C₂	1	1	yes	maximal, characteristic
2	#63764	C₂	1	2	yes	maximal
3	C₄×#157	C₄	1	1	yes	characteristic
4	#3787	C₄	1	1	yes	characteristic
5	C₈×C₃⋊D₃	C₂²	1	1	yes	characteristic
6	#3708	—	2	2	no	—
7	C₂×#157	C₈	1	1	yes	characteristic
8	C₂×#157	C₈	1	1	yes	characteristic
9	C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
10	#622	—	2	1	no	—
11	C₃²×C₈	Q₈	1	1	yes	characteristic
12	C₃⋊(C₃⋊C₈)	D₄	1	1	yes	characteristic
13	C₄⋊C₁₆	—	9	1	no	maximal, Sylow
14	D₃×C₈	—	12	1	no	—
15	C₂×C₃⋊D₃	C₂×C₈	1	1	yes	characteristic
16	#157	—	2	1	no	—
17	#157	C₁₆	1	2	yes	complemented
18	C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
19	C₃⋊Dic₃	M₁₆	1	1	yes	characteristic
20	C₂×C₁₆	—	9	2	no	—
21	C₄×C₈	—	9	1	no	—
22	C₄×D₃	—	12	1	no	—
23	C₃×C₈	—	4	1	no	—
24	C₃⋊C₈	—	12	1	no	—
25	C₂×C₃²	C₄⋊C₈	1	1	yes	characteristic
26	C₃⋊D₃	C₂×C₁₆	1	1	yes	characteristic
27	C₃⋊D₃	M₃₂	1	1	yes	characteristic
28	C₂×C₈	—	9	1	no	—
29	C₂×C₈	—	9	1	no	—
30	C₄²	—	9	1	no	—
31	C₁₆	—	18	2	no	—
32	C₂×D₃	—	12	1	no	—
33	C₃×C₄	—	4	1	no	—
34	Dic₃	—	12	1	no	—
35	C₃²	C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
36	C₂×C₄	—	9	1	no	—
37	C₂×C₄	—	9	1	no	—
38	C₂×C₄	—	9	1	no	—
39	C₈	—	9	1	no	—
40	C₈	—	18	1	no	—
41	C₈	PSU(3,𝔽₂)	1	1	yes	characteristic
42	C₂×C₃	—	4	1	no	—
43	D₃	—	12	1	no	—
44	D₃	—	12	1	no	—
45	C₂²	—	9	1	no	—
46	C₄	—	9	2	no	—
47	C₄	—	9	1	no	—
48	C₄	—	18	1	no	—
49	C₄	#3777	1	1	yes	characteristic
50	C₃	—	4	1	no	—
51	C₂	—	9	1	no	—
52	C₂	—	9	1	no	—
53	C₂	#63744	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₈	PSU(3,𝔽₂)
commutator	C₃⋊D₃	C₂×C₁₆
Fitting	C₃²×C₈	Q₈
socle	C₂×C₃²	C₄⋊C₈
prime radical	C₂×C₃⋊D₃	C₂×C₈
cosocle	C₈×C₃⋊D₃	C₂²
hypocenter	C₃²	C₄⋊C₁₆
LCS	G ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₈

Complements

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

N	H ≅ G/N	type
17. #157	C₁₆	semidirect
35. C₃²	C₄⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₈×#157	2
#63764	2
C₄⋊C₁₆	9