Subgroups — #67499

Order 384.

92 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	#1312	—	3	1	no	maximal, Sylow
5	C₂×C₃×C₁₆	C₄	1	2	yes	—
6	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₃⋊C₁₆	C₄	1	2	yes	—
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	C₂×C₄	1	1	yes	characteristic
14	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
15	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃⋊C₈	C₈	1	2	yes	—
17	C₂×C₃⋊C₈	C₈	1	2	yes	—
18	C₃×C₁₆	—	2	2	no	—
19	C₃⋊C₁₆	—	2	2	no	—
20	C₂×C₁₆	—	3	2	no	—
21	C₂×C₁₆	Dic₃	1	2	yes	—
22	C₄×C₈	—	3	1	no	—
23	C₄×C₈	—	3	1	no	—
24	C₄×C₈	C₂×D₃	1	1	yes	characteristic
25	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
26	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
27	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
28	C₃×C₈	M₁₆	1	2	yes	—
29	C₃×C₈	M₁₆	1	2	yes	—
30	C₃⋊C₈	—	2	2	no	—
31	C₃⋊C₈	C₁₆	1	4	yes	complemented
32	C₂×C₈	—	3	2	no	—
33	C₂×C₈	—	3	2	no	—
34	C₂×C₈	C₄×D₃	1	1	yes	characteristic
35	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
36	C₄²	C₄×D₃	1	1	yes	characteristic
37	C₁₆	—	2	2	no	—
38	C₁₆	—	6	2	no	—
39	C₂²×C₃	C₄×C₈	1	1	yes	characteristic
40	C₃×C₄	#97	1	1	yes	characteristic
41	C₃×C₄	#97	1	1	yes	characteristic
42	C₃×C₄	C₂×C₁₆	1	2	yes	—
43	C₃×C₄	M₃₂	1	2	yes	—
44	C₂×C₄	D₃×C₈	1	1	yes	characteristic
45	C₂×C₄	D₃×C₈	1	1	yes	characteristic
46	C₂×C₄	C₄×Dic₃	1	1	yes	characteristic
47	C₈	—	3	4	no	—
48	C₈	—	6	2	no	—
49	C₈	ΓL(1,𝔽₂₅)	1	2	yes	—
50	C₈	#208	1	2	yes	—
51	C₂×C₃	#322	1	1	yes	characteristic
52	C₂×C₃	C₄×C₁₆	1	1	yes	characteristic
53	C₂×C₃	#346	1	1	yes	characteristic
54	C₂²	C₈×Dic₃	1	1	yes	characteristic
55	C₄	D₃×C₁₆	1	2	yes	—
56	C₄	#798	1	2	yes	—
57	C₄	#803	1	1	yes	characteristic
58	C₄	#815	1	1	yes	characteristic
59	C₃	#1312	1	1	yes	characteristic, complemented, Sylow
60	C₂	#4454	1	1	yes	characteristic
61	C₂	Dic₃×C₁₆	1	1	yes	characteristic
62	C₂	#4502	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
31. C₃⋊C₈	C₁₆	semidirect
59. 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×C₄×C₁₆	2
C₄×C₃⋊C₁₆	2
C₈×C₃⋊C₈	2
#1312	3