Subgroups — #67783

Order 384.

88 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	#4466	C₂	1	2	yes	maximal, complemented
3	#1355	—	3	1	no	maximal, Sylow
4	C₃×#127	C₄	1	1	yes	characteristic
5	C₃×#125	C₄	1	1	yes	characteristic
6	C₃×#102	—	2	2	no	—
7	C₃×#97	C₂²	1	1	yes	characteristic
8	#812	—	2	2	no	—
9	#490	D₃	1	1	yes	characteristic
10	#347	—	3	2	no	—
11	C₂×C₃×D₄	—	2	1	no	—
12	C₂×C₃×D₄	—	4	1	no	—
13	C₂×C₃×C₈	D₄	1	2	yes	—
14	C₃×C₄⋊C₄	—	2	1	no	—
15	C₃×C₄⋊C₄	—	4	1	no	—
16	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
17	C₃⋊C₁₆	—	2	4	no	—
18	#127	Dic₃	1	1	yes	characteristic
19	#125	Dic₃	1	1	yes	characteristic
20	M₃₂	—	6	2	no	—
21	#102	—	2	2	no	—
22	#97	C₂×D₃	1	1	yes	characteristic
23	C₂³×C₃	—	4	1	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	C₃×C₈	—	2	2	no	—
30	C₂×D₄	—	2	1	no	—
31	C₂×D₄	—	4	1	no	—
32	C₂×C₈	#67	1	2	yes	—
33	C₄⋊C₄	—	2	1	no	—
34	C₄⋊C₄	—	4	1	no	—
35	C₄²	C₂×Dic₃	1	1	yes	characteristic
36	C₁₆	—	6	4	no	—
37	C₂²×C₃	—	8	2	no	—
38	C₂²×C₃	—	8	1	no	—
39	C₂²×C₃	#100	1	1	yes	characteristic
40	C₃×C₄	—	4	1	no	—
41	C₃×C₄	—	8	1	no	—
42	C₃×C₄	#104	1	2	yes	—
43	C₂³	—	4	1	no	—
44	D₄	—	4	2	no	—
45	D₄	—	8	2	no	—
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	—	4	1	no	—
48	C₂×C₄	#217	1	1	yes	characteristic
49	C₈	—	2	2	no	—
50	C₂×C₃	—	2	1	no	—
51	C₂×C₃	—	16	1	no	—
52	C₂×C₃	#329	1	1	yes	characteristic
53	C₂²	—	8	2	no	—
54	C₂²	—	8	1	no	—
55	C₂²	#833	1	1	yes	characteristic
56	C₄	—	4	1	no	—
57	C₄	—	8	1	no	—
58	C₄	#837	1	2	yes	—
59	C₃	#1355	1	1	yes	characteristic, complemented, Sylow
60	C₂	—	2	1	no	—
61	C₂	—	16	1	no	—
62	C₂	#4540	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4540
commutator	C₃×C₄²	C₂×C₄
Frattini	#97	C₂×D₃
Fitting	C₃×#490	C₂
socle	C₂×C₃	#329
prime radical	C₃×#127	C₄
cosocle	C₃×#97	C₂²
hypercenter	#490	D₃
hypocenter	C₃	#1355
LCS	G ▷ C₃×C₄² ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #490

Complements

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

N	H ≅ G/N	type
2. #4466	C₂	semidirect
59. C₃	#1355	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
#4466	2
#1355	3