Subgroups — #69439

Order 384.

256 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#376	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊C₈)	C₂	1	6	yes	maximal
3	#1843	—	3	1	no	maximal, Sylow
4	C₂×C₃×C₄²	C₂²	1	3	yes	—
5	C₂²×C₃⋊C₈	C₂²	1	4	yes	—
6	C₃×#95	C₄	1	4	yes	—
7	C₄⋊(C₃⋊C₈)	—	2	12	no	—
8	C₂×C₄⋊C₈	—	3	6	no	—
9	#376	D₃	1	1	yes	characteristic
10	C₂²×C₃×C₄	C₂×C₄	1	6	yes	—
11	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
12	C₃×C₄²	D₄	1	6	yes	—
13	C₃×C₄²	Q₈	1	6	yes	—
14	C₂×C₃⋊C₈	—	2	12	no	—
15	C₂²×C₈	—	3	4	no	—
16	C₂×C₄²	C₂×D₃	1	3	yes	—
17	C₄⋊C₈	—	6	12	no	—
18	#95	Dic₃	1	4	yes	—
19	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
20	C₂×C₃×C₄	—	2	12	no	—
21	C₂×C₃×C₄	C₄⋊C₄	1	12	yes	—
22	C₂×C₃×C₄	C₂×D₄	1	3	yes	—
23	C₂×C₃×C₄	C₂×Q₈	1	3	yes	—
24	C₃⋊C₈	—	4	4	no	—
25	C₂²×C₄	C₂×Dic₃	1	6	yes	—
26	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
27	C₂×C₈	—	6	12	no	—
28	C₄²	Dic₆	1	6	yes	—
29	C₄²	D₁₂	1	6	yes	—
30	C₂²×C₃	C₂×C₄⋊C₄	1	3	yes	—
31	C₂²×C₃	#131	1	4	yes	—
32	C₃×C₄	—	2	12	no	—
33	C₃×C₄	#127	1	1	yes	characteristic
34	C₃×C₄	C₄⋊Q₈	1	3	yes	—
35	C₂³	C₂²×Dic₃	1	1	yes	characteristic
36	C₂×C₄	—	2	12	no	—
37	C₂×C₄	C₄⋊Dic₃	1	12	yes	—
38	C₂×C₄	C₂×Dic₆	1	3	yes	—
39	C₂×C₄	C₂×D₁₂	1	3	yes	—
40	C₈	—	12	4	no	—
41	C₂×C₃	#384	1	1	yes	characteristic
42	C₂×C₃	#424	1	6	yes	—
43	C₂²	C₂×C₄⋊Dic₃	1	3	yes	—
44	C₂²	#948	1	4	yes	—
45	C₄	—	2	12	no	—
46	C₄	#869	1	3	yes	—
47	C₄	#874	1	1	yes	characteristic
48	C₃	#1843	1	1	yes	characteristic, complemented, Sylow
49	C₂	#4935	1	1	yes	characteristic
50	C₂	#4999	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃
commutator, socle, prime radical	C₂³×C₃	C₂²×C₄
Fitting	C₃×#376	C₂
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#376	D₃
hypocenter	C₃	#1843
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
48. C₃	#1843	semidirect

Maximal subgroups

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

subgroup	index
C₃×#376	2
C₂×C₄⋊(C₃⋊C₈)	2
#1843	3