Subgroups — #10709895

Order 768.

246 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#69065	C₂	1	4	yes	maximal
2	#68065	C₂	1	1	yes	maximal, characteristic
3	#68064	C₂	1	1	yes	maximal, characteristic
4	C₃×#1311	C₂	1	1	yes	maximal, characteristic
5	#12514	—	3	1	no	maximal, Sylow
6	#4699	C₂²	1	2	yes	—
7	#4682	—	2	2	no	—
8	#4682	C₂²	1	2	yes	—
9	#4681	—	2	2	no	—
10	C₃×C₄×C₁₆	C₂²	1	2	yes	—
11	C₃×C₈²	C₂²	1	1	yes	characteristic
12	#4503	—	2	8	no	—
13	#2241	—	3	4	no	—
14	#1713	—	3	1	no	—
15	#1710	—	3	1	no	—
16	#1311	D₃	1	1	yes	characteristic
17	C₂×Dic₁₂	—	2	4	no	—
18	#869	—	2	2	no	—
19	#869	—	2	2	no	—
20	C₂×C₃×C₁₆	D₄	1	4	yes	—
21	C₃×C₄×C₈	D₄	1	2	yes	—
22	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
23	#818	—	2	4	no	—
24	#818	—	4	4	no	—
25	#816	—	4	4	no	—
26	#500	—	3	2	no	—
27	#500	—	6	2	no	—
28	#494	—	3	2	no	—
29	#487	—	6	2	no	—
30	#358	—	6	8	no	—
31	C₄×C₁₆	C₂×D₃	1	2	yes	—
32	C₈²	C₂×D₃	1	1	yes	characteristic
33	C₂×Dic₆	—	4	2	no	—
34	C₂×Dic₆	—	4	2	no	—
35	C₂×C₃×C₈	D₈	1	4	yes	—
36	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
37	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
38	C₄⋊Dic₃	—	4	2	no	—
39	C₄⋊Dic₃	—	4	2	no	—
40	C₄⋊Dic₃	—	4	4	no	—
41	Dic₁₂	—	4	8	no	—
42	C₃×C₁₆	—	2	4	no	—
43	C₂×Q₁₆	—	6	4	no	—
44	C₄⋊Q₈	—	6	2	no	—
45	C₄⋊Q₈	—	6	2	no	—
46	C₂×C₁₆	D₁₂	1	4	yes	—
47	#107	—	6	4	no	—
48	#107	—	12	4	no	—
49	#103	—	12	4	no	—
50	C₄×C₈	D₁₂	1	2	yes	—
51	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
52	C₂×C₃×C₄	#127	1	1	yes	characteristic
53	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
54	C₂×Dic₃	—	8	2	no	—
55	C₂×Dic₃	—	8	2	no	—
56	Dic₆	—	8	4	no	—
57	Dic₆	—	8	4	no	—
58	C₃×C₈	—	2	4	no	—
59	C₃×C₈	#135	1	4	yes	—
60	C₂×Q₈	—	12	2	no	—
61	C₂×Q₈	—	12	2	no	—
62	Q₁₆	—	12	8	no	—
63	C₂×C₈	D₂₄	1	4	yes	—
64	C₂×C₈	C₂×D₁₂	1	2	yes	—
65	C₄⋊C₄	—	12	2	no	—
66	C₄⋊C₄	—	12	2	no	—
67	C₄⋊C₄	—	12	4	no	—
68	C₄²	C₂×D₁₂	1	1	yes	characteristic
69	C₁₆	—	2	4	no	—
70	C₂²×C₃	#493	1	1	yes	characteristic
71	C₃×C₄	#495	1	2	yes	—
72	C₃×C₄	#510	1	4	yes	—
73	Dic₃	—	16	2	no	—
74	Dic₃	—	16	2	no	—
75	Q₈	—	24	4	no	—
76	Q₈	—	24	4	no	—
77	C₂×C₄	—	24	2	no	—
78	C₂×C₄	—	24	2	no	—
79	C₂×C₄	#874	1	1	yes	characteristic
80	C₂×C₄	C₂×D₂₄	1	2	yes	—
81	C₈	—	2	4	no	—
82	C₈	#904	1	4	yes	—
83	C₂×C₃	#1709	1	1	yes	characteristic
84	C₂×C₃	#2251	1	2	yes	—
85	C₂²	#4695	1	1	yes	characteristic
86	C₄	—	48	2	no	—
87	C₄	—	48	2	no	—
88	C₄	#4696	1	2	yes	—
89	C₄	#4909	1	4	yes	—
90	C₃	#12514	1	1	yes	characteristic, complemented, Sylow
91	C₂	#68060	1	1	yes	characteristic
92	C₂	#69084	1	2	yes	—

	subgroup	G/H
center	C₂²	#4695
commutator, cosocle	C₃×C₄×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₃
Fitting	C₃×#1311	C₂
socle, prime radical	C₂²×C₃	#493
hypercenter	#1311	D₃
hypocenter	C₃	#12514
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

Complements

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

N	H ≅ G/N	type
90. C₃	#12514	semidirect

Maximal subgroups

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

subgroup	index
#69065	2
#68065	2
#68064	2
C₃×#1311	2
#12514	3