Subgroups — #64110

Order 288.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3828	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#3748	C₂	1	1	yes	maximal, characteristic, complemented
4	#3743	C₂	1	2	yes	maximal, complemented
5	C₈×C₃⋊D₃	C₂	1	2	yes	maximal, complemented
6	#901	—	3	4	no	maximal
7	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
8	#638	C₄	1	2	yes	—
9	C₄⋊(C₃⋊D₃)	C₄	1	1	yes	characteristic
10	C₄×C₃⋊D₃	C₂²	1	2	yes	—
11	#634	C₄	1	1	yes	characteristic
12	C₃²×C₈	C₂²	1	1	yes	characteristic
13	C₃²×C₈	C₂²	1	1	yes	characteristic
14	C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
15	#235	—	3	4	no	—
16	C₂×C₃×C₈	D₃	1	4	yes	complemented
17	#208	—	3	4	no	—
18	ΓL(1,𝔽₂₅)	—	3	8	no	—
19	D₃×C₈	—	3	8	no	—
20	C₂²×C₃²	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃⋊D₃	C₂×C₄	1	2	yes	—
22	C₃²×C₄	C₂×C₄	1	1	yes	characteristic
23	C₃²×C₄	C₂³	1	1	yes	characteristic
24	C₃⋊Dic₃	C₂×C₄	1	2	yes	—
25	#131	—	9	1	no	Sylow
26	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
27	#67	—	3	8	no	—
28	D₁₂	—	3	4	no	—
29	C₄×D₃	—	3	8	no	—
30	Dic₆	—	3	4	no	—
31	C₃×C₈	C₂×D₃	1	4	yes	—
32	C₃×C₈	C₂×D₃	1	4	yes	—
33	C₃⋊C₈	—	3	8	no	—
34	C₂×C₃²	—	2	1	no	—
35	C₂×C₃²	C₂²×C₄	1	1	yes	characteristic
36	C₃⋊D₃	—	2	2	no	—
37	#41	—	9	1	no	—
38	M₁₆	—	9	1	no	—
39	M₁₆	—	9	2	no	—
40	C₂×C₈	—	9	2	no	—
41	C₂×C₈	C₃⋊D₃	1	1	yes	characteristic, complemented
42	C₂²×C₃	C₄×D₃	1	4	yes	—
43	C₂×D₃	—	3	8	no	—
44	C₃×C₄	C₄×D₃	1	4	yes	—
45	C₃×C₄	C₂²×D₃	1	4	yes	—
46	Dic₃	—	3	8	no	—
47	C₃²	#131	1	1	yes	characteristic, complemented, Sylow
48	Q₈	—	9	1	no	—
49	D₄	—	9	2	no	—
50	D₄	—	9	1	no	—
51	C₂×C₄	—	9	2	no	—
52	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
53	C₈	—	9	2	no	—
54	C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
55	C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
56	C₂×C₃	—	2	4	no	—
57	C₂×C₃	C₂×C₄×D₃	1	4	yes	—
58	D₃	—	6	8	no	—
59	C₂²	—	9	2	no	—
60	C₂²	C₄×C₃⋊D₃	1	1	yes	characteristic
61	C₄	—	9	2	no	—
62	C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
63	C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
64	C₃	#901	1	4	yes	complemented
65	C₂	—	2	1	no	—
66	C₂	—	18	2	no	—
67	C₂	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×C₃⋊D₃
commutator, socle	C₂×C₃²	C₂²×C₄
Frattini	C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×C₈	C₂
prime radical	#3828	C₂
cosocle	C₃²×C₄	C₂³
hypercenter	C₂×C₈	C₃⋊D₃
hypocenter	C₃²	#131
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₈ ◁ 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. #3748	C₂	semidirect
4. #3743	C₂	semidirect
5. C₈×C₃⋊D₃	C₂	semidirect
16. C₂×C₃×C₈	D₃	semidirect
41. C₂×C₈	C₃⋊D₃	semidirect
47. C₃²	#131	semidirect
64. C₃	#901	semidirect

Maximal subgroups

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

subgroup	index
#3828	2
C₂×C₃²×C₈	2
#3748	2
#3743	2
C₈×C₃⋊D₃	2
#901	3