Subgroups — #10592398

Order 576.

366 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#3746	C₂	1	4	yes	maximal
2	#64078	C₂	1	2	yes	maximal
3	C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
4	#4699	—	3	4	no	maximal
5	C₂×#634	C₂²	1	4	yes	—
6	C₂×C₃²×C₈	C₂²	1	2	yes	—
7	C₃²×C₄²	C₂²	1	1	yes	characteristic
8	C₄⋊(C₃⋊Dic₃)	—	2	4	no	—
9	#3746	—	2	8	no	—
10	C₂×Dic₁₂	—	3	16	no	—
11	#869	—	3	8	no	—
12	C₃×C₄×C₈	D₃	1	4	yes	—
13	C₂×C₃²×C₄	D₄	1	2	yes	—
14	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
15	C₂×C₃⋊Dic₃	—	2	4	no	—
16	#634	—	2	8	no	—
17	C₃²×C₈	D₄	1	4	yes	—
18	#494	—	9	1	no	Sylow
19	C₂×Dic₆	—	3	16	no	—
20	C₂×C₃×C₈	C₂×D₃	1	8	yes	—
21	C₃×C₄²	C₂×D₃	1	4	yes	—
22	C₄⋊Dic₃	—	6	16	no	—
23	Dic₁₂	—	6	32	no	—
24	C₂²×C₃²	C₂×D₄	1	1	yes	characteristic
25	C₃²×C₄	Q₁₆	1	4	yes	complemented
26	C₃²×C₄	C₂×D₄	1	2	yes	—
27	C₃⋊Dic₃	—	4	4	no	—
28	C₂×Q₁₆	—	9	4	no	—
29	C₄⋊Q₈	—	9	2	no	—
30	C₄×C₈	C₃⋊D₃	1	1	yes	characteristic
31	C₂×C₃×C₄	D₁₂	1	8	yes	—
32	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
33	C₂×Dic₃	—	6	16	no	—
34	Dic₆	—	6	32	no	—
35	C₃×C₈	D₁₂	1	16	yes	—
36	C₂×C₃²	#127	1	1	yes	characteristic
37	C₂×C₃²	C₂×Q₁₆	1	2	yes	—
38	C₂×Q₈	—	9	4	no	—
39	Q₁₆	—	18	8	no	—
40	C₂×C₈	C₂×C₃⋊D₃	1	2	yes	—
41	C₄⋊C₄	—	18	4	no	—
42	C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
43	C₂²×C₃	C₂×D₁₂	1	4	yes	—
44	C₃×C₄	Dic₁₂	1	16	yes	complemented
45	C₃×C₄	C₂×D₁₂	1	8	yes	—
46	Dic₃	—	12	16	no	—
47	C₃²	#494	1	1	yes	characteristic, complemented, Sylow
48	Q₈	—	18	8	no	—
49	C₂×C₄	—	18	4	no	—
50	C₂×C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
51	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
52	C₈	C₄⋊(C₃⋊D₃)	1	4	yes	—
53	C₂×C₃	#874	1	4	yes	—
54	C₂×C₃	C₂×Dic₁₂	1	8	yes	—
55	C₂²	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
56	C₄	—	36	4	no	—
57	C₄	#3746	1	4	yes	complemented
58	C₄	C₂×C₄⋊(C₃⋊D₃)	1	2	yes	—
59	C₃	#4699	1	4	yes	complemented
60	C₂	#64083	1	1	yes	characteristic
61	C₂	C₂×#3746	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₄⋊(C₃⋊D₃)
commutator, cosocle	C₂×C₃²×C₄	C₂³
Frattini	C₂×C₄	C₂²×C₃⋊D₃
Fitting	C₃²×C₄×C₈	C₂
socle, prime radical	C₂²×C₃²	C₂×D₄
hypercenter	C₄×C₈	C₃⋊D₃
hypocenter	C₃²	#494
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
25. C₃²×C₄	Q₁₆	semidirect
44. C₃×C₄	Dic₁₂	semidirect
47. C₃²	#494	semidirect
57. C₄	#3746	semidirect
59. C₃	#4699	semidirect

Maximal subgroups

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

subgroup	index
C₂×#3746	2
#64078	2
C₃²×C₄×C₈	2
#4699	3