Subgroups — #10683071

Order 768.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2262	C₂	1	1	yes	maximal, characteristic
2	#67593	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃⋊C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#7539	—	3	1	no	maximal, Sylow
5	C₃×#508	—	2	1	no	—
6	C₃×Q₆₄	C₄	1	1	yes	characteristic
7	C₃×SD₆₄	—	2	1	no	—
8	C₃×D₃₂	C₄	1	1	yes	characteristic
9	C₂×C₃×C₃₂	C₂²	1	1	yes	characteristic
10	#4506	—	2	1	no	—
11	C₃⋊C₆₄	—	2	1	no	—
12	#2262	D₃	1	1	yes	characteristic
13	C₂×C₆₄	—	3	1	no	—
14	#1425	—	3	1	no	—
15	C₃×#135	—	4	1	no	—
16	C₃×Q₃₂	—	2	1	no	—
17	C₃×SD₃₂	—	4	1	no	—
18	C₃×D₁₆	—	2	1	no	—
19	C₂×C₃×C₁₆	D₄	1	1	yes	characteristic
20	#819	—	4	1	no	—
21	C₃×C₃₂	C₂×C₄	1	1	yes	characteristic
22	C₃×C₃₂	D₄	1	1	yes	characteristic
23	#508	—	2	1	no	—
24	Q₆₄	Dic₃	1	1	yes	characteristic
25	SD₆₄	—	2	1	no	—
26	D₃₂	Dic₃	1	1	yes	characteristic
27	C₂×C₃₂	C₂×D₃	1	1	yes	characteristic
28	#368	—	6	1	no	—
29	C₆₄	—	6	1	no	—
30	C₃×#41	—	8	1	no	—
31	C₃×Q₁₆	—	4	1	no	—
32	C₃×SD₁₆	—	8	1	no	—
33	C₃×D₈	—	4	1	no	—
34	C₂×C₃×C₈	D₈	1	1	yes	characteristic
35	#208	—	8	1	no	—
36	C₃×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
37	C₃×C₁₆	SD₁₆	1	1	yes	characteristic
38	#135	—	4	1	no	—
39	Q₃₂	—	2	1	no	—
40	SD₃₂	—	4	1	no	—
41	D₁₆	—	2	1	no	—
42	C₂×C₁₆	#67	1	1	yes	characteristic
43	#108	—	12	1	no	—
44	C₃₂	C₂×Dic₃	1	1	yes	characteristic
45	C₃₂	#67	1	1	yes	characteristic
46	C₃×Q₈	—	8	1	no	—
47	C₃×D₄	—	8	1	no	—
48	C₃×D₄	—	16	1	no	—
49	C₂×C₃×C₄	—	16	1	no	—
50	C₂×C₃×C₄	D₁₆	1	1	yes	characteristic
51	C₃×C₈	#102	1	1	yes	characteristic
52	C₃×C₈	SD₃₂	1	1	yes	characteristic
53	C₃⋊C₈	—	16	1	no	—
54	#41	—	8	1	no	—
55	Q₁₆	—	4	1	no	—
56	SD₁₆	—	8	1	no	—
57	D₈	—	4	1	no	—
58	M₁₆	—	24	1	no	—
59	C₂×C₈	#213	1	1	yes	characteristic
60	C₁₆	#214	1	1	yes	characteristic
61	C₁₆	#217	1	1	yes	characteristic
62	C₂²×C₃	—	16	1	no	—
63	C₂²×C₃	SD₆₄	1	1	yes	characteristic
64	C₃×C₄	—	16	1	no	—
65	C₃×C₄	#357	1	1	yes	characteristic
66	C₃×C₄	D₃₂	1	1	yes	characteristic
67	Q₈	—	8	1	no	—
68	D₄	—	8	1	no	—
69	D₄	—	16	1	no	—
70	C₂×C₄	—	16	1	no	—
71	C₂×C₄	#826	1	1	yes	characteristic
72	C₈	—	48	1	no	—
73	C₈	#827	1	1	yes	characteristic
74	C₈	#832	1	1	yes	characteristic
75	C₂×C₃	—	2	1	no	—
76	C₂×C₃	—	32	1	no	—
77	C₂×C₃	#1415	1	1	yes	characteristic
78	C₂²	—	16	1	no	—
79	C₂²	#4520	1	1	yes	characteristic
80	C₄	—	16	1	no	—
81	C₄	#4519	1	1	yes	characteristic
82	C₄	#4562	1	1	yes	characteristic
83	C₃	#7539	1	1	yes	characteristic, complemented, Sylow
84	C₂	—	2	1	no	—
85	C₂	—	32	1	no	—
86	C₂	#67824	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#4519
commutator	C₃×C₃₂	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₃
Fitting, prime radical	C₃×#2262	C₂
socle	C₂×C₃	#1415
cosocle	C₂×C₃×C₃₂	C₂²
hypercenter	#2262	D₃
hypocenter	C₃	#7539
LCS	G ▷ C₃×C₃₂ ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ #2262

Complements

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

N	H ≅ G/N	type
2. #67593	C₂	semidirect
3. C₂×C₃⋊C₆₄	C₂	semidirect
83. C₃	#7539	semidirect

Maximal subgroups

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

subgroup	index
C₃×#2262	2
#67593	2
C₂×C₃⋊C₆₄	2
#7539	3