Subgroups — #64320

Order 288.

352 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3832	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#239	C₂	1	2	yes	maximal, complemented
3	#3802	C₂	1	3	yes	maximal, complemented
4	D₃×D₁₂	C₂	1	6	yes	maximal, complemented
5	#3798	C₂	1	3	yes	maximal, complemented
6	#1009	—	3	2	no	maximal
7	C₂×D₃²	C₂²	1	6	yes	complemented
8	C₃²×Q₈	C₂²	1	1	yes	characteristic, complemented
9	C₄⋊(C₃⋊D₃)	C₂²	1	3	yes	complemented
10	C₄×C₃⋊D₃	C₂²	1	3	yes	complemented
11	C₃×D₁₂	C₂²	1	6	yes	complemented
12	C₃×C₄×D₃	C₂²	1	6	yes	complemented
13	#627	C₂²	1	1	yes	characteristic, complemented
14	#626	C₂²	1	6	yes	complemented
15	#625	C₂²	1	3	yes	complemented
16	C₃×#41	—	3	2	no	—
17	#239	—	3	2	no	—
18	#239	—	6	1	no	—
19	#239	D₃	1	2	yes	complemented
20	D₃×D₄	—	3	6	no	—
21	D₃×D₄	—	3	6	no	—
22	#235	—	3	6	no	—
23	C₂×D₁₂	—	3	6	no	—
24	C₂×C₃⋊D₃	C₂³	1	3	yes	—
25	C₂×C₃×D₃	C₂³	1	6	yes	—
26	D₃²	—	4	6	no	—
27	C₃²×C₄	C₂³	1	3	yes	—
28	C₃⋊Dic₃	C₂³	1	1	yes	characteristic
29	C₃×Dic₃	C₂³	1	2	yes	—
30	H(5,𝔽₂)	—	9	1	no	Sylow
31	C₂²×D₃	—	3	12	no	—
32	C₃×Q₈	—	2	1	no	—
33	C₃×Q₈	C₂×D₃	1	2	yes	complemented
34	C₃×D₄	—	3	6	no	—
35	C₂×C₃×C₄	—	3	6	no	—
36	#67	—	3	6	no	—
37	#67	—	3	6	no	—
38	D₁₂	—	3	6	no	—
39	D₁₂	—	3	6	no	—
40	D₁₂	—	6	3	no	—
41	D₁₂	C₂×D₃	1	6	yes	complemented
42	C₄×D₃	—	3	6	no	—
43	C₄×D₃	—	6	3	no	—
44	C₄×D₃	C₂×D₃	1	6	yes	complemented
45	Dic₆	—	3	2	no	—
46	C₂×C₃²	C₂⁴	1	1	yes	characteristic
47	C₃⋊D₃	—	2	3	no	—
48	C₃×D₃	—	2	6	no	—
49	#41	—	3	2	no	—
50	#41	—	9	3	no	—
51	#41	—	9	1	no	—
52	C₂×D₄	—	9	6	no	—
53	C₂×D₄	—	9	3	no	—
54	C₂²×C₃	—	3	6	no	—
55	C₂×D₃	—	3	6	no	—
56	C₂×D₃	—	6	3	no	—
57	C₂×D₃	—	12	12	no	—
58	C₂×D₃	C₂²×D₃	1	6	yes	—
59	C₃×C₄	—	2	3	no	—
60	C₃×C₄	—	3	2	no	—
61	C₃×C₄	C₂²×D₃	1	6	yes	—
62	Dic₃	—	3	2	no	—
63	Dic₃	—	6	1	no	—
64	Dic₃	C₂²×D₃	1	2	yes	—
65	C₃²	H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
66	C₂³	—	9	6	no	—
67	Q₈	—	9	1	no	—
68	Q₈	D₃²	1	1	yes	characteristic, complemented
69	D₄	—	3	6	no	—
70	D₄	—	9	6	no	—
71	D₄	—	9	3	no	—
72	D₄	—	9	3	no	—
73	C₂×C₄	—	3	6	no	—
74	C₂×C₄	—	9	3	no	—
75	C₂×C₃	—	2	1	no	—
76	C₂×C₃	—	6	6	no	—
77	C₂×C₃	C₂³×D₃	1	2	yes	—
78	D₃	—	2	6	no	—
79	D₃	—	6	6	no	—
80	D₃	—	12	3	no	—
81	C₂²	—	3	6	no	—
82	C₂²	—	9	3	no	—
83	C₂²	—	36	6	no	—
84	C₄	—	3	2	no	—
85	C₄	—	9	1	no	—
86	C₄	C₂×D₃²	1	3	yes	—
87	C₃	—	2	1	no	—
88	C₃	#1009	1	2	yes	complemented
89	C₂	—	6	6	no	—
90	C₂	—	18	3	no	—
91	C₂	C₂²×D₃²	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×D₃²
commutator, socle, cosocle	C₂×C₃²	C₂⁴
Fitting	C₃²×Q₈	C₂²
hypercenter	Q₈	D₃²
hypocenter	C₃²	H(5,𝔽₂)
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ Q₈

Complements

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

N	H ≅ G/N	type
1. #3832	C₂	semidirect
2. C₃×#239	C₂	semidirect
3. #3802	C₂	semidirect
4. D₃×D₁₂	C₂	semidirect
5. #3798	C₂	semidirect
7. C₂×D₃²	C₂²	semidirect
8. C₃²×Q₈	C₂²	semidirect
9. C₄⋊(C₃⋊D₃)	C₂²	semidirect
10. C₄×C₃⋊D₃	C₂²	semidirect
11. C₃×D₁₂	C₂²	semidirect
12. C₃×C₄×D₃	C₂²	semidirect
13. #627	C₂²	semidirect
14. #626	C₂²	semidirect
15. #625	C₂²	semidirect
19. #239	D₃	semidirect
33. C₃×Q₈	C₂×D₃	semidirect
41. D₁₂	C₂×D₃	semidirect
44. C₄×D₃	C₂×D₃	semidirect
65. C₃²	H(5,𝔽₂)	semidirect
68. Q₈	D₃²	semidirect
88. C₃	#1009	semidirect

Maximal subgroups

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

subgroup	index
#3832	2
C₃×#239	2
#3802	2
D₃×D₁₂	2
#3798	2
#1009	3