Subgroups — #64366

Order 288.

498 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	#3832	C₂	1	2	yes	maximal, complemented
3	D₄×C₃⋊D₃	C₂	1	6	yes	maximal, complemented
4	#3828	C₂	1	3	yes	maximal, complemented
5	C₂×C₄⋊(C₃⋊D₃)	C₂	1	3	yes	maximal, complemented
6	#1009	—	3	4	no	maximal
7	C₂²×C₃⋊D₃	C₂²	1	6	yes	complemented
8	C₃²×Q₈	C₂²	1	1	yes	characteristic, complemented
9	C₃²×D₄	C₂²	1	3	yes	complemented
10	C₂×C₃²×C₄	C₂²	1	3	yes	complemented
11	#638	C₂²	1	6	yes	complemented
12	C₄⋊(C₃⋊D₃)	C₂²	1	3	yes	complemented
13	C₄⋊(C₃⋊D₃)	C₂²	1	6	yes	complemented
14	C₄×C₃⋊D₃	C₂²	1	6	yes	complemented
15	#634	C₂²	1	1	yes	characteristic, complemented
16	C₃×#41	D₃	1	4	yes	complemented
17	#239	—	3	8	no	—
18	D₃×D₄	—	3	24	no	—
19	#235	—	3	12	no	—
20	C₂×D₁₂	—	3	12	no	—
21	C₂²×C₃²	C₂³	1	3	yes	—
22	C₂×C₃⋊D₃	—	4	6	no	—
23	C₂×C₃⋊D₃	C₂³	1	6	yes	—
24	C₃²×C₄	C₂³	1	1	yes	characteristic
25	C₃²×C₄	C₂³	1	3	yes	—
26	C₃⋊Dic₃	C₂³	1	2	yes	—
27	H(5,𝔽₂)	—	9	1	no	Sylow
28	C₂²×D₃	—	3	24	no	—
29	C₃×Q₈	C₂×D₃	1	4	yes	complemented
30	C₃×D₄	C₂×D₃	1	12	yes	complemented
31	C₂×C₃×C₄	C₂×D₃	1	12	yes	complemented
32	#67	—	3	24	no	—
33	D₁₂	—	3	12	no	—
34	D₁₂	—	3	24	no	—
35	C₄×D₃	—	3	24	no	—
36	Dic₆	—	3	4	no	—
37	C₂×C₃²	—	2	3	no	—
38	C₂×C₃²	C₂⁴	1	1	yes	characteristic
39	C₃⋊D₃	—	2	6	no	—
40	#41	—	9	2	no	—
41	#41	—	9	3	no	—
42	#41	C₃⋊D₃	1	1	yes	characteristic, complemented
43	C₂×D₄	—	9	6	no	—
44	C₂×D₄	—	9	3	no	—
45	C₂²×C₃	C₂²×D₃	1	12	yes	—
46	C₂×D₃	—	3	24	no	—
47	C₂×D₃	—	12	24	no	—
48	C₃×C₄	C₂²×D₃	1	4	yes	—
49	C₃×C₄	C₂²×D₃	1	12	yes	—
50	Dic₃	—	3	8	no	—
51	C₃²	H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
52	C₂³	—	9	6	no	—
53	Q₈	—	9	1	no	—
54	Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
55	D₄	—	9	6	no	—
56	D₄	—	9	6	no	—
57	D₄	—	9	3	no	—
58	D₄	C₂×C₃⋊D₃	1	3	yes	complemented
59	C₂×C₄	—	9	6	no	—
60	C₂×C₄	C₂×C₃⋊D₃	1	3	yes	complemented
61	C₂×C₃	—	2	12	no	—
62	C₂×C₃	C₂³×D₃	1	4	yes	—
63	D₃	—	6	24	no	—
64	C₂²	—	9	6	no	—
65	C₂²	—	36	6	no	—
66	C₂²	C₂²×C₃⋊D₃	1	3	yes	—
67	C₄	—	9	2	no	—
68	C₄	C₂²×C₃⋊D₃	1	3	yes	—
69	C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
70	C₃	#1009	1	4	yes	complemented
71	C₂	—	2	3	no	—
72	C₂	—	18	6	no	—
73	C₂	C₂³×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×C₃⋊D₃
commutator, socle, cosocle	C₂×C₃²	C₂⁴
Fitting	C₃²×#41	C₂
hypercenter	#41	C₃⋊D₃
hypocenter	C₃²	H(5,𝔽₂)
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ #41

Complements

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

N	H ≅ G/N	type
1. C₃²×#41	C₂	semidirect
2. #3832	C₂	semidirect
3. D₄×C₃⋊D₃	C₂	semidirect
4. #3828	C₂	semidirect
5. C₂×C₄⋊(C₃⋊D₃)	C₂	semidirect
7. C₂²×C₃⋊D₃	C₂²	semidirect
8. C₃²×Q₈	C₂²	semidirect
9. C₃²×D₄	C₂²	semidirect
10. C₂×C₃²×C₄	C₂²	semidirect
11. #638	C₂²	semidirect
12. C₄⋊(C₃⋊D₃)	C₂²	semidirect
13. C₄⋊(C₃⋊D₃)	C₂²	semidirect
14. C₄×C₃⋊D₃	C₂²	semidirect
15. #634	C₂²	semidirect
16. C₃×#41	D₃	semidirect
29. C₃×Q₈	C₂×D₃	semidirect
30. C₃×D₄	C₂×D₃	semidirect
31. C₂×C₃×C₄	C₂×D₃	semidirect
42. #41	C₃⋊D₃	semidirect
51. C₃²	H(5,𝔽₂)	semidirect
54. Q₈	C₂×C₃⋊D₃	semidirect
58. D₄	C₂×C₃⋊D₃	semidirect
60. C₂×C₄	C₂×C₃⋊D₃	semidirect
70. C₃	#1009	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#41	2
#3832	2
D₄×C₃⋊D₃	2
#3828	2
C₂×C₄⋊(C₃⋊D₃)	2
#1009	3