Subgroups — #10709931

Order 768.

262 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#69046	C₂	1	4	yes	maximal
2	#68065	C₂	1	2	yes	maximal
3	C₃×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
4	#12523	—	3	1	no	maximal, Sylow
5	#4699	—	2	2	no	—
6	#4682	—	2	2	no	—
7	#4682	C₂²	1	4	yes	—
8	C₃×C₄×C₁₆	C₂²	1	2	yes	—
9	C₃×C₈²	C₂²	1	1	yes	characteristic
10	#4505	—	2	8	no	—
11	#2254	—	3	4	no	—
12	#1713	—	3	2	no	—
13	C₈×C₁₆	D₃	1	1	yes	characteristic
14	C₂×Dic₁₂	—	4	4	no	—
15	#869	—	2	4	no	—
16	C₂×C₃×C₁₆	D₄	1	4	yes	—
17	C₃×C₄×C₈	D₄	1	2	yes	—
18	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
19	#818	—	2	8	no	—
20	#818	—	4	4	no	—
21	#500	—	3	4	no	—
22	#500	—	6	2	no	—
23	#494	—	6	2	no	—
24	#367	—	6	8	no	—
25	C₄×C₁₆	C₂×D₃	1	2	yes	—
26	C₈²	C₂×D₃	1	1	yes	characteristic
27	C₂×Dic₆	—	4	4	no	—
28	C₂×C₃×C₈	D₈	1	4	yes	—
29	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
30	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
31	C₄⋊Dic₃	—	4	4	no	—
32	C₄⋊Dic₃	—	4	4	no	—
33	Dic₁₂	—	8	8	no	—
34	C₃×C₁₆	Q₁₆	1	8	yes	complemented
35	C₂×Q₁₆	—	12	4	no	—
36	C₄⋊Q₈	—	6	4	no	—
37	C₂×C₁₆	D₁₂	1	4	yes	—
38	#107	—	6	8	no	—
39	#107	—	12	4	no	—
40	C₄×C₈	D₁₂	1	2	yes	—
41	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
42	C₂×C₃×C₄	#127	1	1	yes	characteristic
43	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
44	C₂×Dic₃	—	8	4	no	—
45	Dic₆	—	8	8	no	—
46	C₃×C₈	SD₃₂	1	8	yes	—
47	C₃×C₈	C₂×Q₁₆	1	4	yes	—
48	C₂×Q₈	—	12	4	no	—
49	Q₁₆	—	24	8	no	—
50	C₂×C₈	D₂₄	1	4	yes	—
51	C₂×C₈	C₂×D₁₂	1	2	yes	—
52	C₄⋊C₄	—	12	4	no	—
53	C₄⋊C₄	—	12	4	no	—
54	C₄²	C₂×D₁₂	1	1	yes	characteristic
55	C₁₆	Dic₁₂	1	8	yes	complemented
56	C₂²×C₃	#493	1	1	yes	characteristic
57	C₃×C₄	#494	1	2	yes	—
58	C₃×C₄	C₂×SD₃₂	1	4	yes	—
59	Dic₃	—	16	4	no	—
60	Q₈	—	24	8	no	—
61	C₂×C₄	—	24	4	no	—
62	C₂×C₄	#874	1	1	yes	characteristic
63	C₂×C₄	C₂×D₂₄	1	2	yes	—
64	C₈	#800	1	8	yes	—
65	C₈	C₂×Dic₁₂	1	4	yes	—
66	C₂×C₃	#1713	1	1	yes	characteristic
67	C₂×C₃	#2248	1	2	yes	—
68	C₂²	#4695	1	1	yes	characteristic
69	C₄	—	48	4	no	—
70	C₄	#4699	1	2	yes	—
71	C₄	C₂×#800	1	4	yes	—
72	C₃	#12523	1	1	yes	characteristic, complemented, Sylow
73	C₂	#68065	1	1	yes	characteristic
74	C₂	#69061	1	2	yes	—

	subgroup	G/H
center	C₂²	#4695
commutator, cosocle	C₃×C₄×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₃
Fitting	C₃×C₈×C₁₆	C₂
socle, prime radical	C₂²×C₃	#493
hypercenter	C₈×C₁₆	D₃
hypocenter	C₃	#12523
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
34. C₃×C₁₆	Q₁₆	semidirect
55. C₁₆	Dic₁₂	semidirect
72. C₃	#12523	semidirect

Maximal subgroups

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

subgroup	index
#69046	2
#68065	2
C₃×C₈×C₁₆	2
#12523	3