Subgroups — #67785

Order 384.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#491	C₂	1	1	yes	maximal, characteristic
2	#4466	C₂	1	2	yes	maximal
3	#1356	—	3	1	no	maximal, Sylow
4	C₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
5	C₃×#125	C₄	1	1	yes	characteristic
6	C₃×#103	—	2	2	no	—
7	C₃×#97	C₂²	1	1	yes	characteristic
8	#812	—	2	2	no	—
9	#491	D₃	1	1	yes	characteristic
10	#347	—	3	2	no	—
11	C₂×C₃×Q₈	—	2	1	no	—
12	C₂×C₃×C₈	D₄	1	2	yes	—
13	C₃×C₄⋊C₄	—	2	1	no	—
14	C₃×C₄⋊C₄	—	4	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	C₄⋊Q₈	Dic₃	1	1	yes	characteristic
19	#125	Dic₃	1	1	yes	characteristic
20	M₃₂	—	6	2	no	—
21	#103	—	2	2	no	—
22	#97	C₂×D₃	1	1	yes	characteristic
23	C₃×Q₈	—	4	2	no	—
24	C₂×C₃×C₄	—	2	1	no	—
25	C₂×C₃×C₄	—	4	1	no	—
26	C₂×C₃×C₄	—	4	1	no	—
27	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
28	C₃×C₈	—	2	2	no	—
29	C₂×Q₈	—	2	1	no	—
30	C₂×C₈	#67	1	2	yes	—
31	C₄⋊C₄	—	2	1	no	—
32	C₄⋊C₄	—	4	1	no	—
33	C₄⋊C₄	—	4	1	no	—
34	C₄²	C₂×Dic₃	1	1	yes	characteristic
35	C₁₆	—	6	4	no	—
36	C₂²×C₃	#100	1	1	yes	characteristic
37	C₃×C₄	—	4	1	no	—
38	C₃×C₄	—	8	1	no	—
39	C₃×C₄	—	8	1	no	—
40	C₃×C₄	#104	1	2	yes	—
41	Q₈	—	4	2	no	—
42	C₂×C₄	—	2	1	no	—
43	C₂×C₄	—	4	1	no	—
44	C₂×C₄	—	4	1	no	—
45	C₂×C₄	#217	1	1	yes	characteristic
46	C₈	—	2	2	no	—
47	C₂×C₃	—	2	1	no	—
48	C₂×C₃	#329	1	1	yes	characteristic
49	C₂²	#833	1	1	yes	characteristic
50	C₄	—	4	1	no	—
51	C₄	—	8	1	no	—
52	C₄	—	8	1	no	—
53	C₄	#837	1	2	yes	—
54	C₃	#1356	1	1	yes	characteristic, complemented, Sylow
55	C₂	—	2	1	no	—
56	C₂	#4540	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4540
commutator	C₃×C₄²	C₂×C₄
Frattini	#97	C₂×D₃
Fitting	C₃×#491	C₂
socle	C₂×C₃	#329
prime radical	C₂²×C₃	#100
cosocle	C₃×#97	C₂²
hypercenter	#491	D₃
hypocenter	C₃	#1356
LCS	G ▷ C₃×C₄² ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #491

Complements

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

N	H ≅ G/N	type
54. C₃	#1356	semidirect

Maximal subgroups

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

subgroup	index
C₃×#491	2
#4466	2
#1356	3