Subgroups — #87296

Order 384.

1424 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#585	C₂	1	1	yes	maximal, characteristic
2	#5819	C₂	1	15	yes	maximal, complemented
3	C₂×#948	C₂	1	15	yes	maximal, complemented
4	#3573	—	3	1	no	maximal, Sylow
5	C₃×2₋¹⁺⁴	C₄	1	6	yes	—
6	C₃×H(5,𝔽₂)	C₄	1	10	yes	—
7	C₂×C₃×#41	C₂²	1	15	yes	—
8	#948	C₂²	1	20	yes	complemented
9	#948	C₂²	1	60	yes	complemented
10	C₂×#208	C₂²	1	45	yes	complemented
11	C₂²×C₃⋊C₈	C₂²	1	15	yes	complemented
12	#585	D₃	1	1	yes	characteristic
13	#568	—	3	15	no	—
14	C₂×#131	—	3	15	no	—
15	C₃×#41	C₂×C₄	1	60	yes	—
16	C₃×#41	C₂³	1	20	yes	—
17	C₂×C₃×Q₈	C₂×C₄	1	15	yes	—
18	C₂×C₃×D₄	C₂×C₄	1	45	yes	—
19	C₂²×C₃×C₄	C₂³	1	15	yes	—
20	#208	C₂³	1	60	yes	—
21	C₂×C₃⋊C₈	C₂³	1	15	yes	—
22	C₂×C₃⋊C₈	C₂³	1	45	yes	—
23	2₋¹⁺⁴	Dic₃	1	6	yes	—
24	H(5,𝔽₂)	Dic₃	1	10	yes	—
25	C₂×#41	C₂×D₃	1	15	yes	—
26	#131	—	3	20	no	—
27	#131	—	3	60	no	—
28	C₂×M₁₆	—	3	45	no	—
29	C₂²×C₈	—	3	15	no	—
30	C₂³×C₃	C₂²×C₄	1	15	yes	—
31	C₃×Q₈	C₂²×C₄	1	20	yes	—
32	C₃×D₄	C₂²×C₄	1	60	yes	—
33	C₂×C₃×C₄	C₂²×C₄	1	45	yes	—
34	C₂×C₃×C₄	C₂⁴	1	15	yes	—
35	C₃⋊C₈	C₂⁴	1	1	yes	characteristic
36	C₃⋊C₈	C₂⁴	1	15	yes	—
37	#41	C₂×Dic₃	1	60	yes	—
38	#41	C₂²×D₃	1	20	yes	—
39	C₂×Q₈	C₂×Dic₃	1	15	yes	—
40	C₂×D₄	C₂×Dic₃	1	45	yes	—
41	C₂²×C₄	C₂²×D₃	1	15	yes	—
42	M₁₆	—	3	60	no	—
43	C₂×C₈	—	3	15	no	—
44	C₂×C₈	—	3	45	no	—
45	C₂²×C₃	—	4	15	no	—
46	C₂²×C₃	C₂³×C₄	1	15	yes	—
47	C₃×C₄	C₂³×C₄	1	15	yes	—
48	C₃×C₄	C₂⁵	1	1	yes	characteristic
49	C₂³	C₂²×Dic₃	1	15	yes	—
50	Q₈	C₂²×Dic₃	1	20	yes	—
51	D₄	C₂²×Dic₃	1	60	yes	—
52	C₂×C₄	C₂²×Dic₃	1	45	yes	—
53	C₂×C₄	C₂³×D₃	1	15	yes	—
54	C₈	—	3	1	no	—
55	C₈	—	3	15	no	—
56	C₂×C₃	—	2	15	no	—
57	C₂×C₃	C₂⁴×C₄	1	1	yes	characteristic
58	C₂²	—	4	15	no	—
59	C₂²	C₂³×Dic₃	1	15	yes	—
60	C₄	C₂³×Dic₃	1	15	yes	—
61	C₄	C₂⁴×D₃	1	1	yes	characteristic
62	C₃	#3573	1	1	yes	characteristic, complemented, Sylow
63	C₂	—	2	15	no	—
64	C₂	C₂⁴×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂⁴×D₃
commutator, socle	C₂×C₃	C₂⁴×C₄
Fitting, prime radical	C₃×#585	C₂
cosocle	C₃×C₄	C₂⁵
hypercenter	#585	D₃
hypocenter	C₃	#3573
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ #585

Complements

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

N	H ≅ G/N	type
2. #5819	C₂	semidirect
3. C₂×#948	C₂	semidirect
8. #948	C₂²	semidirect
9. #948	C₂²	semidirect
10. C₂×#208	C₂²	semidirect
11. C₂²×C₃⋊C₈	C₂²	semidirect
62. C₃	#3573	semidirect

Maximal subgroups

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

subgroup	index
C₃×#585	2
#5819	2
C₂×#948	2
#3573	3