Subgroups — #63648

Order 288.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic, complemented
4	#821	—	3	4	no	maximal
5	C₂²×C₃⋊D₃	—	2	1	no	—
6	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
7	C₂×C₃⋊Dic₃	—	2	1	no	—
8	C₄⋊(C₃⋊D₃)	—	2	1	no	—
9	C₄⋊(C₃⋊D₃)	C₄	1	2	yes	complemented
10	C₃²×C₈	—	2	1	no	—
11	C₂×D₁₂	—	3	4	no	—
12	C₂×C₃×C₈	D₃	1	4	yes	complemented
13	C₄⋊Dic₃	—	3	4	no	—
14	C₂²×C₃²	D₄	1	1	yes	characteristic
15	C₂×C₃⋊D₃	—	2	2	no	—
16	C₂×C₃⋊D₃	—	4	1	no	—
17	C₂×C₃⋊D₃	—	4	1	no	—
18	C₃²×C₄	C₂×C₄	1	1	yes	characteristic
19	C₃²×C₄	D₄	1	1	yes	characteristic
20	C₃⋊Dic₃	—	4	1	no	—
21	#102	—	9	1	no	Sylow
22	C₂²×D₃	—	6	4	no	—
23	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
24	C₂×Dic₃	—	6	4	no	—
25	D₁₂	—	3	8	no	—
26	D₁₂	—	6	4	no	—
27	C₃×C₈	—	2	4	no	—
28	C₂×C₃²	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₃²	D₈	1	1	yes	characteristic
30	C₂×C₃²	SD₁₆	1	1	yes	characteristic
31	C₃⋊D₃	—	4	2	no	—
32	C₂×D₄	—	9	1	no	—
33	C₂×C₈	C₃⋊D₃	1	1	yes	characteristic, complemented
34	C₄⋊C₄	—	9	1	no	—
35	C₂²×C₃	D₁₂	1	4	yes	—
36	C₂×D₃	—	6	8	no	—
37	C₂×D₃	—	12	4	no	—
38	C₂×D₃	—	12	4	no	—
39	C₃×C₄	C₄×D₃	1	4	yes	—
40	C₃×C₄	#67	1	4	yes	—
41	Dic₃	—	12	4	no	—
42	C₃²	#102	1	1	yes	characteristic, complemented, Sylow
43	C₂³	—	18	1	no	—
44	D₄	—	9	2	no	—
45	D₄	—	18	1	no	—
46	C₂×C₄	—	18	1	no	—
47	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
48	C₈	—	2	1	no	—
49	C₂×C₃	#204	1	4	yes	—
50	C₂×C₃	D₂₄	1	4	yes	—
51	C₂×C₃	(C₂×D₃)⋊C₄	1	4	yes	—
52	D₃	—	12	8	no	—
53	C₂²	—	18	2	no	—
54	C₂²	—	36	1	no	—
55	C₂²	—	36	1	no	—
56	C₂²	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
57	C₄	—	36	1	no	—
58	C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
59	C₄	#638	1	1	yes	characteristic
60	C₃	#821	1	4	yes	complemented
61	C₂	—	36	2	no	—
62	C₂	#3744	1	1	yes	characteristic
63	C₂	#3745	1	1	yes	characteristic
64	C₂	#3752	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₄⋊(C₃⋊D₃)
commutator	C₃²×C₄	C₂×C₄
Frattini	C₂×C₄	C₂×C₃⋊D₃
Fitting	C₂×C₃²×C₈	C₂
socle	C₂²×C₃²	D₄
prime radical	C₂×C₄⋊(C₃⋊D₃)	C₂
cosocle	C₂×C₃²×C₄	C₂²
hypercenter	C₂×C₈	C₃⋊D₃
hypocenter	C₃²	#102
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃²×C₈	C₂	semidirect
3. C₄⋊(C₃⋊Dic₃)	C₂	semidirect
9. C₄⋊(C₃⋊D₃)	C₄	semidirect
12. C₂×C₃×C₈	D₃	semidirect
33. C₂×C₈	C₃⋊D₃	semidirect
42. C₃²	#102	semidirect
60. C₃	#821	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₃⋊D₃)	2
C₂×C₃²×C₈	2
C₄⋊(C₃⋊Dic₃)	2
#821	3