Subgroups — #64361

Order 288.

498 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#638	C₂	1	4	yes	maximal, complemented
3	#3830	C₂	1	4	yes	maximal, complemented
4	D₄×C₃⋊D₃	C₂	1	4	yes	maximal, complemented
5	#3828	C₂	1	2	yes	maximal, complemented
6	#1004	—	3	4	no	maximal
7	C₂³×C₃²	C₂²	1	2	yes	complemented
8	C₂²×C₃⋊D₃	C₂²	1	4	yes	complemented
9	C₃²×D₄	C₂²	1	4	yes	complemented
10	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic, complemented
11	#638	C₂²	1	4	yes	complemented
12	#638	C₂²	1	8	yes	complemented
13	C₂×C₃⋊Dic₃	C₂²	1	4	yes	complemented
14	C₄⋊(C₃⋊D₃)	C₂²	1	2	yes	complemented
15	C₄×C₃⋊D₃	C₂²	1	4	yes	complemented
16	#634	C₂²	1	2	yes	complemented
17	C₂×C₃×D₄	D₃	1	4	yes	complemented
18	C₂×#67	—	3	16	no	—
19	#237	—	3	16	no	—
20	D₃×D₄	—	3	16	no	—
21	#235	—	3	8	no	—
22	C₂²×C₃²	—	4	2	no	—
23	C₂²×C₃²	C₂³	1	4	yes	—
24	C₂²×C₃²	C₂³	1	1	yes	characteristic
25	C₂×C₃⋊D₃	—	4	4	no	—
26	C₂×C₃⋊D₃	C₂³	1	4	yes	—
27	C₃²×C₄	C₂³	1	2	yes	—
28	C₃⋊Dic₃	C₂³	1	4	yes	—
29	H(5,𝔽₂)	—	9	1	no	Sylow
30	C₂³×C₃	C₂×D₃	1	8	yes	complemented
31	C₂²×D₃	—	3	16	no	—
32	C₃×D₄	C₂×D₃	1	16	yes	complemented
33	C₂×C₃×C₄	C₂×D₃	1	4	yes	complemented
34	#67	—	3	16	no	—
35	#67	—	3	32	no	—
36	C₂×Dic₃	—	3	16	no	—
37	D₁₂	—	3	8	no	—
38	C₄×D₃	—	3	16	no	—
39	Dic₆	—	3	8	no	—
40	C₂×C₃²	—	2	4	no	—
41	C₂×C₃²	—	2	1	no	—
42	C₂×C₃²	C₂⁴	1	1	yes	characteristic
43	C₃⋊D₃	—	2	4	no	—
44	#41	—	9	4	no	—
45	#41	—	9	2	no	—
46	C₂×D₄	—	9	4	no	—
47	C₂×D₄	—	9	4	no	—
48	C₂×D₄	C₃⋊D₃	1	1	yes	characteristic, complemented
49	C₂²×C₃	—	4	8	no	—
50	C₂²×C₃	C₂²×D₃	1	16	yes	—
51	C₂²×C₃	C₂²×D₃	1	4	yes	—
52	C₂×D₃	—	3	16	no	—
53	C₂×D₃	—	12	16	no	—
54	C₃×C₄	C₂²×D₃	1	8	yes	—
55	Dic₃	—	3	16	no	—
56	C₃²	H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
57	C₂³	—	9	4	no	—
58	C₂³	C₂×C₃⋊D₃	1	2	yes	complemented
59	Q₈	—	9	2	no	—
60	D₄	—	9	8	no	—
61	D₄	—	9	4	no	—
62	D₄	—	9	2	no	—
63	D₄	C₂×C₃⋊D₃	1	4	yes	complemented
64	C₂×C₄	—	9	4	no	—
65	C₂×C₄	—	9	4	no	—
66	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
67	C₂×C₃	—	2	16	no	—
68	C₂×C₃	—	2	4	no	—
69	C₂×C₃	C₂³×D₃	1	4	yes	—
70	D₃	—	6	16	no	—
71	C₂²	—	4	2	no	—
72	C₂²	—	9	4	no	—
73	C₂²	—	36	4	no	—
74	C₂²	C₂²×C₃⋊D₃	1	4	yes	—
75	C₂²	C₂²×C₃⋊D₃	1	1	yes	characteristic
76	C₄	—	9	4	no	—
77	C₄	C₂²×C₃⋊D₃	1	2	yes	—
78	C₃	#1004	1	4	yes	complemented
79	C₂	—	2	1	no	—
80	C₂	—	2	4	no	—
81	C₂	—	18	4	no	—
82	C₂	C₂³×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×C₃⋊D₃
commutator, socle, cosocle	C₂×C₃²	C₂⁴
Fitting	C₂×C₃²×D₄	C₂
hypercenter	C₂×D₄	C₃⋊D₃
hypocenter	C₃²	H(5,𝔽₂)
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×D₄	C₂	semidirect
2. C₂×#638	C₂	semidirect
3. #3830	C₂	semidirect
4. D₄×C₃⋊D₃	C₂	semidirect
5. #3828	C₂	semidirect
7. C₂³×C₃²	C₂²	semidirect
8. C₂²×C₃⋊D₃	C₂²	semidirect
9. C₃²×D₄	C₂²	semidirect
10. C₂×C₃²×C₄	C₂²	semidirect
11. #638	C₂²	semidirect
12. #638	C₂²	semidirect
13. C₂×C₃⋊Dic₃	C₂²	semidirect
14. C₄⋊(C₃⋊D₃)	C₂²	semidirect
15. C₄×C₃⋊D₃	C₂²	semidirect
16. #634	C₂²	semidirect
17. C₂×C₃×D₄	D₃	semidirect
30. C₂³×C₃	C₂×D₃	semidirect
32. C₃×D₄	C₂×D₃	semidirect
33. C₂×C₃×C₄	C₂×D₃	semidirect
48. C₂×D₄	C₃⋊D₃	semidirect
56. C₃²	H(5,𝔽₂)	semidirect
58. C₂³	C₂×C₃⋊D₃	semidirect
63. D₄	C₂×C₃⋊D₃	semidirect
66. C₂×C₄	C₂×C₃⋊D₃	semidirect
78. C₃	#1004	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×D₄	2
C₂×#638	2
#3830	2
D₄×C₃⋊D₃	2
#3828	2
#1004	3