Subgroups — #67555

Order 384.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#1370	—	3	1	no	maximal, Sylow
5	C₂×C₃×C₁₆	—	2	1	no	—
6	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₃⋊C₁₆	—	2	1	no	—
8	C₄×C₃⋊C₈	C₄	1	1	yes	characteristic
9	C₄×C₃⋊C₈	C₄	1	1	yes	characteristic
10	C₄⋊C₁₆	—	3	1	no	—
11	C₄⋊C₁₆	D₃	1	1	yes	characteristic
12	C₈²	—	3	1	no	—
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	C₂×C₃⋊C₈	—	2	1	no	—
17	C₂×C₃⋊C₈	C₈	1	1	yes	characteristic
18	C₂×C₃⋊C₈	C₈	1	1	yes	characteristic
19	C₃×C₁₆	—	4	1	no	—
20	C₃⋊C₁₆	—	4	1	no	—
21	C₂×C₁₆	—	2	1	no	—
22	C₂×C₁₆	—	6	1	no	—
23	C₄×C₈	—	3	1	no	—
24	C₄×C₈	—	3	1	no	—
25	C₄×C₈	C₂×D₃	1	1	yes	characteristic
26	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
28	C₂×C₃×C₄	M₁₆	1	1	yes	characteristic
29	C₃×C₈	—	2	1	no	—
30	C₃×C₈	D₈	1	1	yes	characteristic
31	C₃×C₈	Q₁₆	1	1	yes	characteristic
32	C₃⋊C₈	—	2	2	no	—
33	C₃⋊C₈	—	2	1	no	—
34	C₃⋊C₈	C₁₆	1	2	yes	complemented
35	C₂×C₈	—	3	1	no	—
36	C₂×C₈	—	3	1	no	—
37	C₂×C₈	—	6	1	no	—
38	C₂×C₈	Dic₆	1	1	yes	characteristic
39	C₂×C₈	#67	1	1	yes	characteristic
40	C₄²	C₄×D₃	1	1	yes	characteristic
41	C₁₆	—	4	1	no	—
42	C₁₆	—	12	1	no	—
43	C₂²×C₃	C₄⋊C₈	1	1	yes	characteristic
44	C₃×C₄	—	2	1	no	—
45	C₃×C₄	#107	1	1	yes	characteristic
46	C₃×C₄	#108	1	1	yes	characteristic
47	C₃×C₄	C₂×C₁₆	1	1	yes	characteristic
48	C₃×C₄	M₃₂	1	1	yes	characteristic
49	C₂×C₄	D₃×C₈	1	1	yes	characteristic
50	C₂×C₄	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
51	C₂×C₄	Dic₃⋊C₄	1	1	yes	characteristic
52	C₈	—	2	1	no	—
53	C₈	—	3	2	no	—
54	C₈	—	6	2	no	—
55	C₈	—	6	1	no	—
56	C₈	#213	1	1	yes	characteristic
57	C₈	#216	1	1	yes	characteristic
58	C₂×C₃	#335	1	1	yes	characteristic
59	C₂×C₃	C₄⋊C₁₆	1	1	yes	characteristic
60	C₂×C₃	#364	1	1	yes	characteristic
61	C₂²	Dic₃⋊C₈	1	1	yes	characteristic
62	C₄	—	2	1	no	—
63	C₄	D₃×C₁₆	1	1	yes	characteristic
64	C₄	#798	1	1	yes	characteristic
65	C₄	#807	1	1	yes	characteristic
66	C₄	#822	1	1	yes	characteristic
67	C₃	#1370	1	1	yes	characteristic, complemented, Sylow
68	C₂	#4479	1	1	yes	characteristic
69	C₂	Dic₃⋊C₁₆	1	1	yes	characteristic
70	C₂	#4511	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#67
commutator	C₃×C₄	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₃
Fitting	C₃×C₄⋊C₁₆	C₂
socle, prime radical	C₂²×C₃	C₄⋊C₈
cosocle	C₃×C₄×C₈	C₂²
hypercenter	C₄⋊C₁₆	D₃
hypocenter	C₃	#1370
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
34. C₃⋊C₈	C₁₆	semidirect
67. 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×C₄⋊C₁₆	2
C₄⋊(C₃⋊C₁₆)	2
C₈×C₃⋊C₈	2
#1370	3