Subgroups — #67784

Order 384.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#490	C₂	1	1	yes	maximal, characteristic
2	#4545	C₂	1	2	yes	maximal
3	#1412	—	3	1	no	maximal, Sylow
4	C₃×#127	C₂²	1	1	yes	characteristic
5	C₃×#125	C₄	1	1	yes	characteristic, complemented
6	C₃×#102	—	2	2	no	—
7	C₃×#97	C₄	1	1	yes	characteristic, complemented
8	#834	—	2	2	no	—
9	#490	D₃	1	1	yes	characteristic
10	#353	—	3	2	no	—
11	C₂×C₃×D₄	—	4	1	no	—
12	C₂×C₃×D₄	D₄	1	2	yes	—
13	C₂×C₃×C₈	—	2	1	no	—
14	C₃×C₄⋊C₄	—	2	1	no	—
15	C₃×C₄⋊C₄	—	4	1	no	—
16	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
17	#217	—	4	2	no	—
18	#127	C₂×D₃	1	1	yes	characteristic
19	#125	Dic₃	1	1	yes	characteristic, complemented
20	#102	—	2	2	no	—
21	#99	—	6	2	no	—
22	#97	Dic₃	1	1	yes	characteristic, complemented
23	C₂³×C₃	—	2	2	no	—
24	C₃×D₄	—	4	2	no	—
25	C₃×D₄	—	8	2	no	—
26	C₂×C₃×C₄	—	2	1	no	—
27	C₂×C₃×C₄	—	4	1	no	—
28	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×Dic₃	—	8	2	no	—
30	C₃×C₈	—	4	1	no	—
31	C₂×D₄	—	4	1	no	—
32	C₂×D₄	#67	1	2	yes	—
33	C₂×C₈	—	2	1	no	—
34	C₄⋊C₄	—	2	1	no	—
35	C₄⋊C₄	—	4	1	no	—
36	C₂²⋊C₄	—	12	2	no	—
37	C₄²	C₂×Dic₃	1	1	yes	characteristic
38	C₂²×C₃	—	4	2	no	—
39	C₂²×C₃	—	8	2	no	—
40	C₂²×C₃	#99	1	1	yes	characteristic
41	C₃×C₄	—	2	1	no	—
42	C₃×C₄	—	4	1	no	—
43	C₃×C₄	—	8	1	no	—
44	Dic₃	—	16	2	no	—
45	C₂³	—	2	2	no	—
46	D₄	—	4	2	no	—
47	D₄	—	8	2	no	—
48	C₂×C₄	—	2	1	no	—
49	C₂×C₄	—	4	1	no	—
50	C₂×C₄	—	24	2	no	—
51	C₂×C₄	#217	1	1	yes	characteristic
52	C₈	—	4	1	no	—
53	C₂×C₃	—	2	1	no	—
54	C₂×C₃	—	8	2	no	—
55	C₂×C₃	#354	1	1	yes	characteristic
56	C₂²	—	4	2	no	—
57	C₂²	—	8	2	no	—
58	C₂²	#834	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	—	4	1	no	—
61	C₄	—	8	1	no	—
62	C₄	—	48	2	no	—
63	C₃	#1412	1	1	yes	characteristic, complemented, Sylow
64	C₂	—	2	1	no	—
65	C₂	—	8	2	no	—
66	C₂	#4541	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4541
commutator	C₃×C₄²	C₂×C₄
Frattini	#127	C₂×D₃
Fitting	C₃×#490	C₂
socle	C₂×C₃	#354
prime radical, cosocle	C₃×#127	C₂²
hypercenter	#490	D₃
hypocenter	C₃	#1412
LCS	G ▷ C₃×C₄² ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #127 ◁ #490

Complements

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

N	H ≅ G/N	type
5. C₃×#125	C₄	semidirect
7. C₃×#97	C₄	semidirect
19. #125	Dic₃	semidirect
22. #97	Dic₃	semidirect
63. C₃	#1412	semidirect

Maximal subgroups

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

subgroup	index
C₃×#490	2
#4545	2
#1412	3