Subgroups — #67475

Order 384.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4682	C₂	1	1	yes	maximal, characteristic
2	C₃×#335	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	#1364	—	3	1	no	maximal, Sylow
5	#869	—	2	1	no	—
6	C₃×C₄⋊C₈	—	2	1	no	—
7	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
8	#818	—	2	1	no	—
9	#818	C₄	1	2	yes	—
10	C₂×C₃⋊C₁₆	—	2	1	no	—
11	#500	—	3	1	no	—
12	C₄⋊C₁₆	—	3	1	no	—
13	#335	D₃	1	1	yes	characteristic
14	C₂×Dic₆	—	4	1	no	—
15	C₂×C₃×C₈	—	4	1	no	—
16	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂×C₃×C₈	D₄	1	1	yes	characteristic
18	C₃×C₄²	D₄	1	1	yes	characteristic
19	C₄⋊Dic₃	—	2	2	no	—
20	C₄⋊Dic₃	—	4	1	no	—
21	C₃⋊C₁₆	—	4	1	no	—
22	C₄⋊Q₈	—	6	1	no	—
23	C₂×C₁₆	—	6	1	no	—
24	#107	—	3	2	no	—
25	#107	—	6	1	no	—
26	C₄⋊C₈	—	2	1	no	—
27	C₄×C₈	C₂×D₃	1	1	yes	characteristic
28	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₃×C₄	D₈	1	1	yes	characteristic
30	C₂×C₃×C₄	SD₁₆	1	1	yes	characteristic
31	C₂×Dic₃	—	4	2	no	—
32	Dic₆	—	8	1	no	—
33	Dic₆	—	8	1	no	—
34	C₃×C₈	—	2	1	no	—
35	C₃×C₈	—	8	1	no	—
36	C₃×C₈	SD₁₆	1	1	yes	characteristic
37	C₃×C₈	Q₁₆	1	1	yes	characteristic
38	C₂×Q₈	—	12	1	no	—
39	C₂×C₈	—	4	1	no	—
40	C₂×C₈	C₄×D₃	1	1	yes	characteristic
41	C₂×C₈	D₁₂	1	1	yes	characteristic
42	C₄⋊C₄	—	6	2	no	—
43	C₄⋊C₄	—	12	1	no	—
44	C₄²	#67	1	1	yes	characteristic
45	C₁₆	—	12	1	no	—
46	C₂²×C₃	#102	1	1	yes	characteristic
47	C₃×C₄	—	2	1	no	—
48	C₃×C₄	#101	1	1	yes	characteristic
49	C₃×C₄	#103	1	1	yes	characteristic
50	C₃×C₄	D₁₆	1	1	yes	characteristic
51	C₃×C₄	SD₃₂	1	1	yes	characteristic
52	Dic₃	—	8	2	no	—
53	Q₈	—	24	1	no	—
54	Q₈	—	24	1	no	—
55	C₂×C₄	—	12	2	no	—
56	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
57	C₂×C₄	#213	1	1	yes	characteristic
58	C₂×C₄	#215	1	1	yes	characteristic
59	C₈	—	2	1	no	—
60	C₈	—	8	1	no	—
61	C₈	#204	1	1	yes	characteristic
62	C₈	Dic₁₂	1	1	yes	characteristic
63	C₂×C₃	#332	1	1	yes	characteristic
64	C₂×C₃	#357	1	1	yes	characteristic
65	C₂×C₃	#362	1	1	yes	characteristic
66	C₂²	#809	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	24	2	no	—
69	C₄	#816	1	1	yes	characteristic
70	C₄	#824	1	1	yes	characteristic
71	C₄	#826	1	1	yes	characteristic
72	C₄	#828	1	1	yes	characteristic
73	C₃	#1364	1	1	yes	characteristic, complemented, Sylow
74	C₂	#4482	1	1	yes	characteristic
75	C₂	#4491	1	1	yes	characteristic
76	C₂	#4496	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#809
commutator	C₂×C₃×C₈	C₂×C₄
Frattini	C₄×C₈	C₂×D₃
Fitting	C₃×#335	C₂
socle, prime radical	C₂²×C₃	#102
cosocle	C₃×C₄×C₈	C₂²
hypercenter	#335	D₃
hypocenter	C₃	#1364
LCS	G ▷ C₂×C₃×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

Complements

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

N	H ≅ G/N	type
73. C₃	#1364	semidirect

Maximal subgroups

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

subgroup	index
#4682	2
C₃×#335	2
C₄⋊(C₃⋊C₁₆)	2
#1364	3