Subgroups — #11773071

Order 1152.

156 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×M₆₄	C₂	1	1	yes	maximal, characteristic
2	#10589372	C₂	1	1	yes	maximal, characteristic
3	#10589371	C₂	1	1	yes	maximal, characteristic
4	#67601	—	3	4	no	maximal
5	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
6	#63646	—	2	1	no	—
7	#63645	—	2	1	no	—
8	C₃²×C₃₂	C₄	1	2	yes	complemented
9	C₃×M₆₄	D₃	1	4	yes	—
10	#4506	—	3	4	no	—
11	#4505	—	3	4	no	—
12	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
13	C₄⋊(C₃⋊Dic₃)	—	4	1	no	—
14	#3748	—	4	1	no	—
15	C₃²×C₁₆	C₂×C₄	1	1	yes	characteristic
16	C₃²×C₁₆	Q₈	1	1	yes	characteristic
17	#1426	—	9	1	no	Sylow
18	C₂×C₃×C₁₆	C₂×D₃	1	4	yes	—
19	#819	—	6	4	no	—
20	#818	—	6	4	no	—
21	C₃×C₃₂	Dic₃	1	8	yes	complemented
22	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
23	C₂×C₃⋊Dic₃	—	8	1	no	—
24	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
25	C₃²×C₈	Q₁₆	1	1	yes	characteristic
26	C₃⋊(C₃⋊C₈)	—	8	1	no	—
27	M₆₄	C₃⋊D₃	1	1	yes	characteristic
28	#368	—	9	1	no	—
29	#367	—	9	1	no	—
30	C₂×C₃×C₈	D₁₂	1	4	yes	—
31	C₄⋊Dic₃	—	12	4	no	—
32	#208	—	12	4	no	—
33	C₃×C₁₆	Dic₆	1	4	yes	—
34	C₃×C₁₆	C₂×Dic₃	1	4	yes	—
35	C₂²×C₃²	SD₃₂	1	1	yes	characteristic
36	C₃²×C₄	#107	1	1	yes	characteristic
37	C₃²×C₄	SD₃₂	1	1	yes	characteristic
38	C₃⋊Dic₃	—	16	1	no	—
39	C₂×C₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
40	#108	—	18	1	no	—
41	#107	—	18	1	no	—
42	C₃₂	C₃⋊Dic₃	1	2	yes	complemented
43	C₂×C₃×C₄	D₂₄	1	4	yes	—
44	C₂×Dic₃	—	24	4	no	—
45	C₃×C₈	Dic₁₂	1	4	yes	—
46	C₃×C₈	C₄⋊Dic₃	1	4	yes	—
47	C₃⋊C₈	—	24	4	no	—
48	C₂×C₃²	—	2	1	no	—
49	C₂×C₃²	#367	1	1	yes	characteristic
50	M₁₆	—	36	1	no	—
51	C₂×C₈	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
52	C₄⋊C₄	—	36	1	no	—
53	C₁₆	#634	1	1	yes	characteristic
54	C₁₆	C₂×C₃⋊Dic₃	1	1	yes	characteristic
55	C₂²×C₃	#800	1	4	yes	—
56	C₃×C₄	#800	1	4	yes	—
57	C₃×C₄	#818	1	4	yes	—
58	Dic₃	—	48	4	no	—
59	C₃²	#1426	1	1	yes	characteristic, complemented, Sylow
60	C₂×C₄	—	72	1	no	—
61	C₂×C₄	#3745	1	1	yes	characteristic
62	C₈	—	72	1	no	—
63	C₈	#3746	1	1	yes	characteristic
64	C₈	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
65	C₂×C₃	—	2	4	no	—
66	C₂×C₃	#4505	1	4	yes	—
67	C₂²	#63627	1	1	yes	characteristic
68	C₄	—	144	1	no	—
69	C₄	#63627	1	1	yes	characteristic
70	C₄	#63645	1	1	yes	characteristic
71	C₃	#67601	1	4	yes	complemented
72	C₂	—	2	1	no	—
73	C₂	#10589371	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10589371
commutator	C₃²×C₁₆	C₂×C₄
Frattini	C₂×C₁₆	C₂×C₃⋊D₃
Fitting	C₃²×M₆₄	C₂
socle	C₂×C₃²	#367
prime radical	C₂²×C₃²	SD₃₂
cosocle	C₂×C₃²×C₁₆	C₂²
hypercenter	M₆₄	C₃⋊D₃
hypocenter	C₃²	#1426
LCS	G ▷ C₃²×C₁₆ ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
8. C₃²×C₃₂	C₄	semidirect
21. C₃×C₃₂	Dic₃	semidirect
42. C₃₂	C₃⋊Dic₃	semidirect
59. C₃²	#1426	semidirect
71. C₃	#67601	semidirect

Maximal subgroups

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

subgroup	index
C₃²×M₆₄	2
#10589372	2
#10589371	2
#67601	3