Subgroups — #63647

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	C₂×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₃)⋊C₈	—	3	4	no	maximal
5	C₂²×C₃⋊D₃	C₄	1	1	yes	characteristic
6	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
7	C₂×C₃⋊Dic₃	C₄	1	1	yes	characteristic
8	C₄×C₃⋊D₃	—	2	1	no	—
9	C₄×C₃⋊D₃	—	2	1	no	—
10	C₃²×C₈	—	2	1	no	—
11	C₃⋊(C₃⋊C₈)	—	2	1	no	—
12	C₂×C₄×D₃	—	3	4	no	—
13	C₂×C₃×C₈	D₃	1	4	yes	complemented
14	C₂×C₃⋊C₈	—	3	4	no	—
15	C₂²×C₃²	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃⋊D₃	—	2	1	no	—
17	C₂×C₃⋊D₃	—	2	1	no	—
18	C₂×C₃⋊D₃	C₈	1	2	yes	complemented
19	C₃²×C₄	D₄	1	1	yes	characteristic
20	C₃²×C₄	D₄	1	1	yes	characteristic
21	C₃⋊Dic₃	—	2	1	no	—
22	C₂²⋊C₈	—	9	1	no	Sylow
23	C₂²×D₃	—	3	4	no	—
24	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
25	C₂×Dic₃	—	3	4	no	—
26	C₄×D₃	—	6	4	no	—
27	C₄×D₃	—	6	4	no	—
28	C₃×C₈	—	2	4	no	—
29	C₃⋊C₈	—	6	4	no	—
30	C₂×C₃²	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₃²	C₂×C₈	1	1	yes	characteristic
32	C₂×C₃²	M₁₆	1	1	yes	characteristic
33	C₃⋊D₃	—	2	2	no	—
34	C₂²×C₄	—	9	1	no	—
35	C₂×C₈	—	9	1	no	—
36	C₂×C₈	C₃⋊D₃	1	1	yes	characteristic, complemented
37	C₂²×C₃	C₄×D₃	1	4	yes	—
38	C₂×D₃	—	3	8	no	—
39	C₂×D₃	—	6	4	no	—
40	C₂×D₃	—	6	4	no	—
41	C₃×C₄	D₁₂	1	4	yes	—
42	C₃×C₄	#67	1	4	yes	—
43	Dic₃	—	6	4	no	—
44	C₃²	C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
45	C₂³	—	9	1	no	—
46	C₂×C₄	—	9	1	no	—
47	C₂×C₄	—	18	1	no	—
48	C₂×C₄	—	18	1	no	—
49	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
50	C₈	—	2	1	no	—
51	C₈	—	18	1	no	—
52	C₂×C₃	D₃×C₈	1	4	yes	—
53	C₂×C₃	ΓL(1,𝔽₂₅)	1	4	yes	—
54	C₂×C₃	(C₂×D₃)⋊C₄	1	4	yes	—
55	D₃	—	6	8	no	—
56	C₂²	—	9	2	no	—
57	C₂²	—	18	1	no	—
58	C₂²	—	18	1	no	—
59	C₂²	C₄×C₃⋊D₃	1	1	yes	characteristic
60	C₄	—	18	1	no	—
61	C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
62	C₄	#638	1	1	yes	characteristic
63	C₃	(C₂×D₃)⋊C₈	1	4	yes	complemented
64	C₂	—	18	2	no	—
65	C₂	C₈×C₃⋊D₃	1	1	yes	characteristic
66	C₂	#3743	1	1	yes	characteristic
67	C₂	#3752	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×C₃⋊D₃
commutator	C₂×C₃²	C₂×C₈
Fitting	C₂×C₃²×C₈	C₂
socle	C₂²×C₃²	C₂×C₄
prime radical	C₂²×C₃⋊D₃	C₄
cosocle	C₂×C₃²×C₄	C₂²
hypercenter	C₂×C₈	C₃⋊D₃
hypocenter	C₃²	C₂²⋊C₈
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ 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. C₂×C₃⋊(C₃⋊C₈)	C₂	semidirect
13. C₂×C₃×C₈	D₃	semidirect
18. C₂×C₃⋊D₃	C₈	semidirect
36. C₂×C₈	C₃⋊D₃	semidirect
44. C₃²	C₂²⋊C₈	semidirect
63. C₃	(C₂×D₃)⋊C₈	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
C₂×C₃⋊(C₃⋊C₈)	2
(C₂×D₃)⋊C₈	3