Subgroups — #10592839

Order 576.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#131	C₂	1	1	yes	maximal, characteristic
2	#63639	C₂	1	3	yes	maximal, complemented
3	C₂×C₃⋊(C₃⋊C₁₆)	C₂	1	3	yes	maximal, complemented
4	#5140	—	3	4	no	maximal
5	C₃²×#41	C₄	1	1	yes	characteristic
6	C₃²×M₁₆	C₄	1	3	yes	—
7	C₂×C₃²×C₈	C₂²	1	3	yes	—
8	C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
9	C₃⋊(C₃⋊C₁₆)	C₂²	1	3	yes	—
10	C₃×#131	D₃	1	4	yes	—
11	#812	—	3	12	no	—
12	C₂×C₃⋊C₁₆	—	3	12	no	—
13	C₃²×Q₈	C₈	1	1	yes	characteristic
14	C₃²×D₄	C₈	1	3	yes	—
15	C₂×C₃²×C₄	C₂×C₄	1	3	yes	—
16	C₃²×C₈	C₂×C₄	1	3	yes	—
17	C₃²×C₈	C₂³	1	1	yes	characteristic
18	#504	—	9	1	no	Sylow
19	C₃×#41	Dic₃	1	4	yes	—
20	C₃×M₁₆	Dic₃	1	12	yes	—
21	C₂×C₃×C₈	C₂×D₃	1	12	yes	—
22	C₃⋊C₁₆	—	3	4	no	—
23	C₃⋊C₁₆	—	3	12	no	—
24	C₂²×C₃²	C₂×C₈	1	3	yes	—
25	C₃²×C₄	C₂×C₈	1	3	yes	—
26	C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
27	#131	C₃⋊D₃	1	1	yes	characteristic
28	M₃₂	—	9	3	no	—
29	C₂×C₁₆	—	9	3	no	—
30	C₃×Q₈	C₃⋊C₈	1	4	yes	—
31	C₃×D₄	C₃⋊C₈	1	12	yes	—
32	C₂×C₃×C₄	C₂×Dic₃	1	12	yes	—
33	C₃×C₈	C₂×Dic₃	1	12	yes	—
34	C₃×C₈	C₂²×D₃	1	4	yes	—
35	C₂×C₃²	—	2	3	no	—
36	C₂×C₃²	C₂²×C₈	1	1	yes	characteristic
37	#41	C₃⋊Dic₃	1	1	yes	characteristic
38	M₁₆	C₃⋊Dic₃	1	3	yes	—
39	C₂×C₈	C₂×C₃⋊D₃	1	3	yes	—
40	C₁₆	—	9	1	no	—
41	C₁₆	—	9	3	no	—
42	C₂²×C₃	C₂×C₃⋊C₈	1	12	yes	—
43	C₃×C₄	C₂×C₃⋊C₈	1	12	yes	—
44	C₃×C₄	C₂²×Dic₃	1	4	yes	—
45	C₃²	#504	1	1	yes	characteristic, complemented, Sylow
46	Q₈	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
47	D₄	C₃⋊(C₃⋊C₈)	1	3	yes	—
48	C₂×C₄	C₂×C₃⋊Dic₃	1	3	yes	—
49	C₈	C₂×C₃⋊Dic₃	1	3	yes	—
50	C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
51	C₂×C₃	—	2	12	no	—
52	C₂×C₃	C₂²×C₃⋊C₈	1	4	yes	—
53	C₂²	C₂×C₃⋊(C₃⋊C₈)	1	3	yes	—
54	C₄	C₂×C₃⋊(C₃⋊C₈)	1	3	yes	—
55	C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
56	C₃	#5140	1	4	yes	complemented
57	C₂	—	2	3	no	—
58	C₂	C₂²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂²×C₃⋊D₃
commutator, socle	C₂×C₃²	C₂²×C₈
Fitting	C₃²×#131	C₂
prime radical	C₃²×#41	C₄
cosocle	C₃²×C₈	C₂³
hypercenter	#131	C₃⋊D₃
hypocenter	C₃²	#504
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₈ ◁ #131

Complements

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

N	H ≅ G/N	type
2. #63639	C₂	semidirect
3. C₂×C₃⋊(C₃⋊C₁₆)	C₂	semidirect
45. C₃²	#504	semidirect
56. C₃	#5140	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#131	2
#63639	2
C₂×C₃⋊(C₃⋊C₁₆)	2
#5140	3