Subgroups — #67438

Order 384.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₈²	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃⋊C₁₆	C₂	1	2	yes	maximal
3	#1312	—	3	1	no	maximal, Sylow
4	C₃×C₄×C₈	C₄	1	1	yes	characteristic
5	C₃×C₄×C₈	C₄	1	1	yes	characteristic
6	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₃⋊C₁₆	C₄	1	4	yes	—
8	C₄×C₁₆	—	3	2	no	—
9	C₈²	D₃	1	1	yes	characteristic
10	C₂×C₃×C₈	C₈	1	2	yes	—
11	C₂×C₃×C₈	C₈	1	2	yes	—
12	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
13	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
14	C₃⋊C₁₆	—	2	4	no	—
15	C₂×C₁₆	—	3	4	no	—
16	C₄×C₈	Dic₃	1	1	yes	characteristic
17	C₄×C₈	Dic₃	1	1	yes	characteristic
18	C₄×C₈	C₂×D₃	1	1	yes	characteristic
19	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
20	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
21	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
22	C₃×C₈	—	2	2	no	—
23	C₃×C₈	C₁₆	1	4	yes	complemented
24	C₃×C₈	M₁₆	1	4	yes	—
25	C₂×C₈	C₃⋊C₈	1	2	yes	—
26	C₂×C₈	C₃⋊C₈	1	2	yes	—
27	C₂×C₈	C₄×D₃	1	2	yes	—
28	C₄²	C₂×Dic₃	1	1	yes	characteristic
29	C₁₆	—	6	4	no	—
30	C₂²×C₃	C₄×C₈	1	1	yes	characteristic
31	C₃×C₄	#97	1	2	yes	—
32	C₃×C₄	C₂×C₁₆	1	2	yes	—
33	C₃×C₄	M₃₂	1	2	yes	—
34	C₂×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
35	C₂×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
36	C₂×C₄	C₄×Dic₃	1	1	yes	characteristic
37	C₈	—	2	2	no	—
38	C₈	C₃⋊C₁₆	1	4	yes	complemented
39	C₈	ΓL(1,𝔽₂₅)	1	4	yes	—
40	C₂×C₃	#322	1	1	yes	characteristic
41	C₂×C₃	C₄×C₁₆	1	1	yes	characteristic
42	C₂×C₃	#346	1	1	yes	characteristic
43	C₂²	C₄×C₃⋊C₈	1	1	yes	characteristic
44	C₄	C₂×C₃⋊C₁₆	1	2	yes	—
45	C₄	#812	1	2	yes	—
46	C₄	#815	1	2	yes	—
47	C₃	#1312	1	1	yes	characteristic, complemented, Sylow
48	C₂	#4455	1	1	yes	characteristic
49	C₂	C₄×C₃⋊C₁₆	1	1	yes	characteristic
50	C₂	#4461	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₈	C₂×D₃
commutator	C₂×C₃	C₄×C₁₆
Fitting	C₃×C₈²	C₂
socle, prime radical	C₂²×C₃	C₄×C₈
cosocle	C₃×C₄×C₈	C₂²
hypercenter	C₈²	D₃
hypocenter	C₃	#1312
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
23. C₃×C₈	C₁₆	semidirect
38. C₈	C₃⋊C₁₆	semidirect
47. C₃	#1312	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₈²	2
C₄×C₃⋊C₁₆	2
#1312	3