Subgroups — #67440

Order 384.

76 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	#1370	—	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	—	2	2	no	—
8	C₄⋊C₁₆	—	3	2	no	—
9	C₈²	D₃	1	1	yes	characteristic
10	C₂×C₃×C₈	—	2	1	no	—
11	C₂×C₃×C₈	C₈	1	1	yes	characteristic
12	C₂×C₃×C₈	C₈	1	1	yes	characteristic
13	C₂×C₃×C₈	D₄	1	1	yes	characteristic
14	C₂×C₃×C₈	Q₈	1	1	yes	characteristic
15	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
16	C₃⋊C₁₆	—	4	2	no	—
17	C₂×C₁₆	—	6	2	no	—
18	C₄×C₈	Dic₃	1	1	yes	characteristic
19	C₄×C₈	Dic₃	1	1	yes	characteristic
20	C₄×C₈	C₂×D₃	1	1	yes	characteristic
21	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
22	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
23	C₂×C₃×C₄	M₁₆	1	1	yes	characteristic
24	C₃×C₈	—	2	2	no	—
25	C₃×C₈	—	2	1	no	—
26	C₃×C₈	—	2	1	no	—
27	C₃×C₈	C₁₆	1	2	yes	complemented
28	C₃×C₈	D₈	1	1	yes	characteristic
29	C₃×C₈	Q₁₆	1	1	yes	characteristic
30	C₂×C₈	—	2	1	no	—
31	C₂×C₈	C₃⋊C₈	1	1	yes	characteristic
32	C₂×C₈	C₃⋊C₈	1	1	yes	characteristic
33	C₂×C₈	Dic₆	1	1	yes	characteristic
34	C₂×C₈	D₁₂	1	1	yes	characteristic
35	C₄²	C₂×Dic₃	1	1	yes	characteristic
36	C₁₆	—	12	2	no	—
37	C₂²×C₃	C₄⋊C₈	1	1	yes	characteristic
38	C₃×C₄	—	2	1	no	—
39	C₃×C₄	#107	1	1	yes	characteristic
40	C₃×C₄	#108	1	1	yes	characteristic
41	C₃×C₄	C₂×C₁₆	1	1	yes	characteristic
42	C₃×C₄	M₃₂	1	1	yes	characteristic
43	C₂×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
44	C₂×C₄	#208	1	1	yes	characteristic
45	C₂×C₄	C₄⋊Dic₃	1	1	yes	characteristic
46	C₈	—	2	2	no	—
47	C₈	—	2	1	no	—
48	C₈	—	2	1	no	—
49	C₈	C₃⋊C₁₆	1	2	yes	complemented
50	C₈	D₂₄	1	1	yes	characteristic
51	C₈	Dic₁₂	1	1	yes	characteristic
52	C₂×C₃	#335	1	1	yes	characteristic
53	C₂×C₃	C₄⋊C₁₆	1	1	yes	characteristic
54	C₂×C₃	#364	1	1	yes	characteristic
55	C₂²	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
56	C₄	—	2	1	no	—
57	C₄	C₂×C₃⋊C₁₆	1	1	yes	characteristic
58	C₄	#812	1	1	yes	characteristic
59	C₄	#818	1	1	yes	characteristic
60	C₄	#819	1	1	yes	characteristic
61	C₃	#1370	1	1	yes	characteristic, complemented, Sylow
62	C₂	#4458	1	1	yes	characteristic
63	C₂	C₄⋊(C₃⋊C₁₆)	1	1	yes	characteristic
64	C₂	#4463	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₁₂
commutator	C₃×C₄	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₃
Fitting	C₃×C₈²	C₂
socle, prime radical	C₂²×C₃	C₄⋊C₈
cosocle	C₃×C₄×C₈	C₂²
hypercenter	C₈²	D₃
hypocenter	C₃	#1370
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
27. C₃×C₈	C₁₆	semidirect
49. C₈	C₃⋊C₁₆	semidirect
61. C₃	#1370	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
#1370	3