Subgroups — #10589382

Order 576.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#3745	C₂	1	1	yes	maximal, characteristic
2	C₃²×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#63646	C₂	1	1	yes	maximal, characteristic, complemented
4	#4516	—	3	4	no	maximal
5	C₂×C₄⋊(C₃⋊D₃)	—	2	1	no	—
6	C₂×C₃²×C₈	C₂²	1	1	yes	characteristic
7	#3748	—	2	1	no	—
8	#3745	—	2	1	no	—
9	#3745	C₄	1	2	yes	—
10	C₃²×C₁₆	—	2	1	no	—
11	C₂×D₂₄	—	3	4	no	—
12	C₃×M₃₂	D₃	1	4	yes	complemented
13	#819	—	3	4	no	—
14	C₂²×C₃⋊D₃	—	4	1	no	—
15	C₂×C₃²×C₄	D₄	1	1	yes	characteristic
16	C₄⋊(C₃⋊D₃)	—	2	2	no	—
17	C₄⋊(C₃⋊D₃)	—	4	1	no	—
18	C₃²×C₈	C₂×C₄	1	1	yes	characteristic
19	C₃²×C₈	D₄	1	1	yes	characteristic
20	C₃⋊(C₃⋊C₈)	—	4	1	no	—
21	#361	—	9	1	no	Sylow
22	C₂×D₁₂	—	6	4	no	—
23	C₂×C₃×C₈	C₂×D₃	1	4	yes	—
24	#208	—	6	4	no	—
25	D₂₄	—	3	8	no	—
26	D₂₄	—	6	4	no	—
27	C₃×C₁₆	—	2	4	no	—
28	C₂²×C₃²	SD₁₆	1	1	yes	characteristic
29	C₂×C₃⋊D₃	—	4	2	no	—
30	C₂×C₃⋊D₃	—	8	2	no	—
31	C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
32	C₃²×C₄	D₈	1	1	yes	characteristic
33	C₂×D₈	—	9	1	no	—
34	M₃₂	C₃⋊D₃	1	1	yes	characteristic, complemented
35	#108	—	9	1	no	—
36	C₂²×D₃	—	12	4	no	—
37	C₂×C₃×C₄	D₁₂	1	4	yes	—
38	D₁₂	—	6	8	no	—
39	D₁₂	—	12	4	no	—
40	C₃×C₈	C₄×D₃	1	4	yes	—
41	C₃×C₈	#67	1	4	yes	—
42	C₃⋊C₈	—	12	4	no	—
43	C₂×C₃²	—	2	1	no	—
44	C₂×C₃²	#102	1	1	yes	characteristic
45	C₃⋊D₃	—	8	2	no	—
46	C₂×D₄	—	18	1	no	—
47	D₈	—	9	2	no	—
48	D₈	—	18	1	no	—
49	M₁₆	—	18	1	no	—
50	C₂×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
51	C₁₆	—	2	1	no	—
52	C₂²×C₃	#204	1	4	yes	—
53	C₂×D₃	—	12	8	no	—
54	C₂×D₃	—	24	8	no	—
55	C₃×C₄	D₂₄	1	4	yes	—
56	C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
57	C₃²	#361	1	1	yes	characteristic, complemented, Sylow
58	C₂³	—	36	1	no	—
59	D₄	—	18	2	no	—
60	D₄	—	36	1	no	—
61	C₂×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
62	C₈	—	36	1	no	—
63	C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
64	C₈	#638	1	1	yes	characteristic
65	C₂×C₃	—	2	4	no	—
66	C₂×C₃	#821	1	4	yes	—
67	D₃	—	24	8	no	—
68	C₂²	—	36	2	no	—
69	C₂²	—	72	2	no	—
70	C₂²	#3744	1	1	yes	characteristic
71	C₄	#3745	1	1	yes	characteristic
72	C₄	#3752	1	1	yes	characteristic
73	C₃	#4516	1	4	yes	complemented
74	C₂	—	2	1	no	—
75	C₂	—	72	2	no	—
76	C₂	#63648	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#63648
commutator	C₃²×C₈	C₂×C₄
Frattini	C₂×C₈	C₂×C₃⋊D₃
Fitting	C₃²×M₃₂	C₂
socle	C₂×C₃²	#102
prime radical	C₂×#3745	C₂
cosocle	C₂×C₃²×C₈	C₂²
hypercenter	M₃₂	C₃⋊D₃
hypocenter	C₃²	#361
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. C₃²×M₃₂	C₂	semidirect
3. #63646	C₂	semidirect
12. C₃×M₃₂	D₃	semidirect
34. M₃₂	C₃⋊D₃	semidirect
57. C₃²	#361	semidirect
73. C₃	#4516	semidirect

Maximal subgroups

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

subgroup	index
C₂×#3745	2
C₃²×M₃₂	2
#63646	2
#4516	3