Subgroups — #63546

Order 288.

67 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3745	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×D₂₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#826	—	3	1	no	maximal
5	D₄₈	—	3	1	no	maximal
6	C₄⋊(C₃⋊D₃)	—	2	1	no	—
7	C₃×D₁₂	—	2	1	no	—
8	C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₃×D₈	—	3	1	no	—
10	D₂₄	—	3	1	no	—
11	D₂₄	—	3	1	no	—
12	D₂₄	—	6	1	no	—
13	D₂₄	D₃	1	1	yes	characteristic, complemented
14	C₃×C₁₆	—	3	1	no	—
15	C₃⋊C₁₆	D₃	1	1	yes	characteristic, complemented
16	C₂×C₃⋊D₃	—	4	1	no	—
17	C₂×C₃×D₃	—	4	1	no	—
18	C₃²×C₄	D₄	1	1	yes	characteristic
19	D₁₆	—	9	1	no	Sylow
20	C₃×D₄	—	6	1	no	—
21	D₁₂	—	2	1	no	—
22	D₁₂	—	6	2	no	—
23	D₁₂	—	6	1	no	—
24	D₁₂	—	6	1	no	—
25	C₃×C₈	—	2	1	no	—
26	C₃×C₈	C₂×D₃	1	1	yes	characteristic
27	C₃×C₈	C₂×D₃	1	1	yes	characteristic
28	C₂×C₃²	D₈	1	1	yes	characteristic
29	C₃⋊D₃	—	8	1	no	—
30	C₃×D₃	—	8	1	no	—
31	D₈	—	3	1	no	—
32	D₈	—	9	1	no	—
33	C₁₆	—	3	1	no	—
34	C₂²×C₃	—	12	1	no	—
35	C₂×D₃	—	4	1	no	—
36	C₂×D₃	—	12	2	no	—
37	C₂×D₃	—	12	1	no	—
38	C₂×D₃	—	12	1	no	—
39	C₃×C₄	—	2	1	no	—
40	C₃×C₄	D₁₂	1	1	yes	characteristic
41	C₃×C₄	#67	1	1	yes	characteristic
42	C₃²	D₁₆	1	1	yes	characteristic, complemented, Sylow
43	D₄	—	6	1	no	—
44	D₄	—	18	1	no	—
45	C₈	D₃²	1	1	yes	characteristic
46	C₂×C₃	—	2	1	no	—
47	C₂×C₃	—	24	1	no	—
48	C₂×C₃	D₂₄	1	1	yes	characteristic
49	C₂×C₃	#213	1	1	yes	characteristic
50	D₃	—	8	1	no	—
51	D₃	—	24	2	no	—
52	D₃	—	24	1	no	—
53	D₃	—	24	1	no	—
54	C₂²	—	12	1	no	—
55	C₂²	—	36	1	no	—
56	C₄	#626	1	1	yes	characteristic
57	C₃	—	2	1	no	—
58	C₃	D₄₈	1	1	yes	characteristic, complemented
59	C₃	#826	1	1	yes	characteristic, complemented
60	C₂	—	24	1	no	—
61	C₂	—	72	1	no	—
62	C₂	#3714	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#3714
commutator, Fitting, cosocle	C₃²×C₈	C₂²
Frattini, hypercenter	C₈	D₃²
socle	C₂×C₃²	D₈
hypocenter	C₃²	D₁₆
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
1. #3745	C₂	semidirect
2. C₃×D₂₄	C₂	semidirect
3. C₃×C₃⋊C₁₆	C₂	semidirect
13. D₂₄	D₃	semidirect
15. C₃⋊C₁₆	D₃	semidirect
42. C₃²	D₁₆	semidirect
58. C₃	D₄₈	semidirect
59. C₃	#826	semidirect

Maximal subgroups

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

subgroup	index
#3745	2
C₃×D₂₄	2
C₃×C₃⋊C₁₆	2
#826	3
D₄₈	3