Subgroups — #64162

Order 288.

390 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃²	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#638	C₂	1	3	yes	maximal, complemented
3	#3757	C₂	1	3	yes	maximal, complemented
4	#953	—	3	4	no	maximal
5	C₂³×C₃²	—	2	6	no	—
6	C₂³×C₃²	C₂²	1	3	yes	—
7	C₂²×C₃⋊D₃	C₂²	1	1	yes	characteristic, complemented
8	#638	—	2	6	no	—
9	C₂×C₃⋊Dic₃	C₂²	1	3	yes	complemented
10	C₂⁴×C₃	D₃	1	4	yes	complemented
11	C₂×#67	—	3	12	no	—
12	#217	—	3	12	no	—
13	C₂²×C₃²	—	2	8	no	—
14	C₂²×C₃²	—	2	6	no	—
15	C₂²×C₃²	D₄	1	6	yes	complemented
16	C₂²×C₃²	C₂³	1	1	yes	characteristic
17	C₂×C₃⋊D₃	—	2	3	no	—
18	C₃⋊Dic₃	—	2	3	no	—
19	#120	—	9	1	no	Sylow
20	C₂³×C₃	—	2	24	no	—
21	C₂³×C₃	C₂×D₃	1	12	yes	—
22	C₂²×D₃	—	3	4	no	—
23	#67	—	6	24	no	—
24	C₂×Dic₃	—	3	12	no	—
25	C₂×C₃²	—	2	6	no	—
26	C₂×C₃²	C₂×D₄	1	3	yes	—
27	C₃⋊D₃	—	4	1	no	—
28	C₂⁴	C₃⋊D₃	1	1	yes	characteristic, complemented
29	C₂×D₄	—	9	3	no	—
30	C₂²⋊C₄	—	9	3	no	—
31	C₂²×C₃	—	2	32	no	—
32	C₂²×C₃	—	2	24	no	—
33	C₂²×C₃	#67	1	24	yes	complemented
34	C₂²×C₃	C₂²×D₃	1	4	yes	—
35	C₂×D₃	—	6	12	no	—
36	Dic₃	—	6	12	no	—
37	C₃²	#120	1	1	yes	characteristic, complemented, Sylow
38	C₂³	—	2	6	no	—
39	C₂³	—	9	1	no	—
40	C₂³	C₂×C₃⋊D₃	1	3	yes	—
41	D₄	—	18	6	no	—
42	C₂×C₄	—	9	3	no	—
43	C₂×C₃	—	2	24	no	—
44	C₂×C₃	C₂×#67	1	12	yes	—
45	D₃	—	12	4	no	—
46	C₂²	—	2	8	no	—
47	C₂²	—	2	6	no	—
48	C₂²	—	18	3	no	—
49	C₂²	#638	1	6	yes	complemented
50	C₂²	C₂²×C₃⋊D₃	1	1	yes	characteristic
51	C₄	—	18	3	no	—
52	C₃	#953	1	4	yes	complemented
53	C₂	—	2	6	no	—
54	C₂	—	36	1	no	—
55	C₂	C₂×#638	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×C₃⋊D₃
commutator, socle, cosocle	C₂²×C₃²	C₂³
Fitting	C₂⁴×C₃²	C₂
hypercenter	C₂⁴	C₃⋊D₃
hypocenter	C₃²	#120
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₃²	C₂	semidirect
2. C₂×#638	C₂	semidirect
3. #3757	C₂	semidirect
7. C₂²×C₃⋊D₃	C₂²	semidirect
9. C₂×C₃⋊Dic₃	C₂²	semidirect
10. C₂⁴×C₃	D₃	semidirect
15. C₂²×C₃²	D₄	semidirect
28. C₂⁴	C₃⋊D₃	semidirect
33. C₂²×C₃	#67	semidirect
37. C₃²	#120	semidirect
49. C₂²	#638	semidirect
52. C₃	#953	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×C₃²	2
C₂×#638	2
#3757	2
#953	3