Subgroups — #67792

Order 384.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#493	C₂	1	1	yes	maximal, characteristic
2	#4545	C₂	1	2	yes	maximal, complemented
3	#1408	—	3	1	no	maximal, Sylow
4	C₂×C₃×D₈	—	2	2	no	—
5	C₃×#127	C₄	1	1	yes	characteristic, complemented
6	C₃×#127	C₂²	1	1	yes	characteristic
7	C₃×C₄×C₈	C₄	1	1	yes	characteristic, complemented
8	#834	—	2	2	no	—
9	#493	D₃	1	1	yes	characteristic
10	#353	—	3	2	no	—
11	C₂×C₃×D₄	—	2	1	no	—
12	C₂×C₃×D₄	—	4	1	no	—
13	C₂×C₃×D₄	—	4	1	no	—
14	C₂×C₃×D₄	D₄	1	2	yes	—
15	C₃×D₈	—	4	4	no	—
16	C₂×C₃×C₈	—	2	1	no	—
17	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
18	#217	—	4	2	no	—
19	C₂×D₈	—	2	2	no	—
20	#127	Dic₃	1	1	yes	characteristic, complemented
21	#127	C₂×D₃	1	1	yes	characteristic
22	#99	—	6	2	no	—
23	C₄×C₈	Dic₃	1	1	yes	characteristic, complemented
24	C₂³×C₃	—	2	2	no	—
25	C₂³×C₃	—	4	1	no	—
26	C₃×D₄	—	4	2	no	—
27	C₃×D₄	—	4	2	no	—
28	C₃×D₄	—	8	2	no	—
29	C₃×D₄	—	8	2	no	—
30	C₂×C₃×C₄	—	2	1	no	—
31	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×Dic₃	—	8	2	no	—
33	C₃×C₈	—	2	2	no	—
34	C₂×D₄	—	2	1	no	—
35	C₂×D₄	—	4	1	no	—
36	C₂×D₄	—	4	1	no	—
37	C₂×D₄	#67	1	2	yes	—
38	D₈	—	4	4	no	—
39	C₂×C₈	—	2	1	no	—
40	C₂²⋊C₄	—	12	2	no	—
41	C₄²	C₂×Dic₃	1	1	yes	characteristic
42	C₂²×C₃	—	4	2	no	—
43	C₂²×C₃	—	8	2	no	—
44	C₂²×C₃	—	8	2	no	—
45	C₂²×C₃	—	8	1	no	—
46	C₂²×C₃	#99	1	1	yes	characteristic
47	C₃×C₄	—	2	2	no	—
48	C₃×C₄	—	2	1	no	—
49	Dic₃	—	16	2	no	—
50	C₂³	—	2	2	no	—
51	C₂³	—	4	1	no	—
52	D₄	—	4	2	no	—
53	D₄	—	4	2	no	—
54	D₄	—	8	2	no	—
55	D₄	—	8	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	24	2	no	—
58	C₂×C₄	#217	1	1	yes	characteristic
59	C₈	—	2	2	no	—
60	C₂×C₃	—	2	1	no	—
61	C₂×C₃	—	8	2	no	—
62	C₂×C₃	—	16	1	no	—
63	C₂×C₃	#353	1	1	yes	characteristic
64	C₂²	—	4	2	no	—
65	C₂²	—	8	2	no	—
66	C₂²	—	8	2	no	—
67	C₂²	—	8	1	no	—
68	C₂²	#834	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	—	2	1	no	—
71	C₄	—	48	2	no	—
72	C₃	#1408	1	1	yes	characteristic, complemented, Sylow
73	C₂	—	2	1	no	—
74	C₂	—	8	2	no	—
75	C₂	—	16	1	no	—
76	C₂	#4545	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4545
commutator	C₃×C₄²	C₂×C₄
Frattini	#127	C₂×D₃
Fitting, prime radical	C₃×#493	C₂
socle	C₂×C₃	#353
cosocle	C₃×#127	C₂²
hypercenter	#493	D₃
hypocenter	C₃	#1408
LCS	G ▷ C₃×C₄² ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #127 ◁ #493

Complements

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

N	H ≅ G/N	type
2. #4545	C₂	semidirect
5. C₃×#127	C₄	semidirect
7. C₃×C₄×C₈	C₄	semidirect
20. #127	Dic₃	semidirect
23. C₄×C₈	Dic₃	semidirect
72. C₃	#1408	semidirect

Maximal subgroups

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

subgroup	index
C₃×#493	2
#4545	2
#1408	3