Subgroups — #10709928

Order 768.

238 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#69046	C₂	1	2	yes	maximal
2	#69044	C₂	1	2	yes	maximal
3	#68064	C₂	1	2	yes	maximal
4	C₃×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
5	#12522	—	3	1	no	maximal, Sylow
6	#4682	C₂²	1	4	yes	—
7	#4681	—	2	4	no	—
8	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
9	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
10	C₃×C₈²	C₂²	1	1	yes	characteristic
11	#4505	—	2	4	no	—
12	#4504	—	2	4	no	—
13	#2254	—	3	2	no	—
14	#2252	—	3	2	no	—
15	#1710	—	3	2	no	—
16	C₈×C₁₆	D₃	1	1	yes	characteristic
17	#869	—	2	4	no	—
18	C₂×C₃×C₁₆	D₄	1	2	yes	—
19	C₂×C₃×C₁₆	D₄	1	2	yes	—
20	C₃×C₄×C₈	D₄	1	2	yes	—
21	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
22	#818	—	2	4	no	—
23	#818	—	2	4	no	—
24	#816	—	4	8	no	—
25	#500	—	3	4	no	—
26	#487	—	6	4	no	—
27	#367	—	6	4	no	—
28	#366	—	6	4	no	—
29	C₄×C₁₆	C₂×D₃	1	1	yes	characteristic
30	C₄×C₁₆	C₂×D₃	1	1	yes	characteristic
31	C₈²	C₂×D₃	1	1	yes	characteristic
32	C₂×Dic₆	—	4	4	no	—
33	C₂×C₃×C₈	D₈	1	4	yes	—
34	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
35	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
36	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
37	C₄⋊Dic₃	—	4	4	no	—
38	C₄⋊Dic₃	—	4	4	no	—
39	C₃×C₁₆	Q₁₆	1	4	yes	—
40	C₃×C₁₆	Q₁₆	1	4	yes	—
41	C₄⋊Q₈	—	6	4	no	—
42	C₂×C₁₆	D₁₂	1	2	yes	—
43	C₂×C₁₆	D₁₂	1	2	yes	—
44	#107	—	6	4	no	—
45	#107	—	6	4	no	—
46	#103	—	12	8	no	—
47	C₄×C₈	D₁₂	1	2	yes	—
48	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
49	C₂×C₃×C₄	#127	1	1	yes	characteristic
50	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
51	C₂×Dic₃	—	8	4	no	—
52	Dic₆	—	8	8	no	—
53	C₃×C₈	—	2	4	no	—
54	C₃×C₈	C₂×Q₁₆	1	2	yes	—
55	C₃×C₈	C₂×Q₁₆	1	2	yes	—
56	C₂×Q₈	—	12	4	no	—
57	C₂×C₈	D₂₄	1	4	yes	—
58	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
59	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
60	C₄⋊C₄	—	12	4	no	—
61	C₄⋊C₄	—	12	4	no	—
62	C₄²	C₂×D₁₂	1	1	yes	characteristic
63	C₁₆	Dic₁₂	1	4	yes	—
64	C₁₆	Dic₁₂	1	4	yes	—
65	C₂²×C₃	#493	1	1	yes	characteristic
66	C₃×C₄	#494	1	1	yes	characteristic
67	C₃×C₄	#494	1	1	yes	characteristic
68	C₃×C₄	#508	1	4	yes	—
69	Dic₃	—	16	4	no	—
70	Q₈	—	24	8	no	—
71	C₂×C₄	—	24	4	no	—
72	C₂×C₄	#874	1	1	yes	characteristic
73	C₂×C₄	C₂×D₂₄	1	2	yes	—
74	C₈	—	2	4	no	—
75	C₈	C₂×Dic₁₂	1	2	yes	—
76	C₈	C₂×Dic₁₂	1	2	yes	—
77	C₂×C₃	#1713	1	1	yes	characteristic
78	C₂×C₃	#2249	1	2	yes	—
79	C₂²	#4695	1	1	yes	characteristic
80	C₄	—	48	4	no	—
81	C₄	#4699	1	1	yes	characteristic
82	C₄	#4699	1	1	yes	characteristic
83	C₄	#4904	1	4	yes	—
84	C₃	#12522	1	1	yes	characteristic, complemented, Sylow
85	C₂	#68065	1	1	yes	characteristic
86	C₂	#69062	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	#12522
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
84. C₃	#12522	semidirect

Maximal subgroups

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

subgroup	index
#69046	2
#69044	2
#68064	2
C₃×C₈×C₁₆	2
#12522	3