Subgroups — #67801

Order 384.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#500	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#4458	C₂	1	1	yes	maximal, characteristic
4	#1364	—	3	1	no	maximal, Sylow
5	C₃×C₄⋊Q₈	—	2	1	no	—
6	C₃×#107	—	2	1	no	—
7	C₃×#107	C₄	1	2	yes	—
8	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃⋊C₁₆	—	2	1	no	—
10	C₄⋊(C₃⋊C₈)	—	2	1	no	—
11	#500	D₃	1	1	yes	characteristic
12	C₄⋊C₁₆	—	3	1	no	—
13	#335	—	3	1	no	—
14	C₂×C₃×Q₈	—	4	1	no	—
15	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃×C₈	D₄	1	1	yes	characteristic
17	C₃×C₄⋊C₄	—	2	2	no	—
18	C₃×C₄⋊C₄	—	4	1	no	—
19	C₃×C₄²	D₄	1	1	yes	characteristic
20	C₂×C₃⋊C₈	—	4	1	no	—
21	C₃⋊C₁₆	—	4	1	no	—
22	C₄⋊Q₈	—	2	1	no	—
23	C₂×C₁₆	—	6	1	no	—
24	#107	—	2	1	no	—
25	#107	Dic₃	1	2	yes	—
26	C₄⋊C₈	—	6	1	no	—
27	C₄×C₈	C₂×D₃	1	1	yes	characteristic
28	C₃×Q₈	—	8	1	no	—
29	C₃×Q₈	—	8	1	no	—
30	C₂×C₃×C₄	—	4	2	no	—
31	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₃×C₄	D₈	1	1	yes	characteristic
33	C₂×C₃×C₄	SD₁₆	1	1	yes	characteristic
34	C₃×C₈	—	2	1	no	—
35	C₃×C₈	SD₁₆	1	1	yes	characteristic
36	C₃×C₈	Q₁₆	1	1	yes	characteristic
37	C₃⋊C₈	—	8	1	no	—
38	C₂×Q₈	—	4	1	no	—
39	C₂×C₈	—	12	1	no	—
40	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
41	C₂×C₈	#67	1	1	yes	characteristic
42	C₄⋊C₄	—	2	2	no	—
43	C₄⋊C₄	—	4	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	—	8	2	no	—
49	C₃×C₄	#101	1	1	yes	characteristic
50	C₃×C₄	#103	1	1	yes	characteristic
51	C₃×C₄	D₁₆	1	1	yes	characteristic
52	C₃×C₄	SD₃₂	1	1	yes	characteristic
53	Q₈	—	8	1	no	—
54	Q₈	—	8	1	no	—
55	C₂×C₄	—	4	2	no	—
56	C₂×C₄	#213	1	1	yes	characteristic
57	C₂×C₄	#214	1	1	yes	characteristic
58	C₂×C₄	#217	1	1	yes	characteristic
59	C₈	—	2	1	no	—
60	C₈	—	24	1	no	—
61	C₈	#215	1	1	yes	characteristic
62	C₈	#216	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	#832	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	8	2	no	—
69	C₄	#826	1	1	yes	characteristic
70	C₄	#827	1	1	yes	characteristic
71	C₄	#835	1	1	yes	characteristic
72	C₄	#836	1	1	yes	characteristic
73	C₃	#1364	1	1	yes	characteristic, complemented, Sylow
74	C₂	#4547	1	1	yes	characteristic
75	C₂	#4562	1	1	yes	characteristic
76	C₂	#4565	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#832
commutator	C₂×C₃×C₈	C₂×C₄
Frattini	C₄×C₈	C₂×D₃
Fitting	C₃×#500	C₂
socle, prime radical	C₂²×C₃	#102
cosocle	C₃×C₄×C₈	C₂²
hypercenter	#500	D₃
hypocenter	C₃	#1364
LCS	G ▷ C₂×C₃×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

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
C₃×#500	2
C₄⋊(C₃⋊C₁₆)	2
#4458	2
#1364	3