Subgroups — C₂³×#627 #10596705

Order 576.

1818 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃×Dic₃	C₂	1	2	yes	maximal
3	C₂²×#627	C₂	1	28	yes	maximal, direct product factor
4	C₂³×Dic₆	—	3	2	no	maximal
5	C₂⁴×C₃²	C₂²	1	1	yes	characteristic
6	C₂²×C₃⋊Dic₃	C₂²	1	14	yes	—
7	C₂²×C₃×Dic₃	C₂²	1	28	yes	—
8	C₂×#627	C₂²	1	112	yes	direct product factor
9	C₂³×C₃×C₄	—	3	2	no	—
10	C₂³×Dic₃	—	3	2	no	—
11	C₂³×Dic₃	—	6	1	no	—
12	C₂³×Dic₃	D₃	1	2	yes	—
13	C₂²×Dic₆	—	3	56	no	—
14	C₂³×C₃²	Q₈	1	8	yes	complemented
15	C₂³×C₃²	C₂³	1	7	yes	—
16	C₂×C₃⋊Dic₃	C₂³	1	28	yes	—
17	C₂×C₃×Dic₃	C₂³	1	56	yes	—
18	#627	C₂³	1	64	yes	direct product factor
19	C₂³×Q₈	—	9	1	no	Sylow
20	C₂⁴×C₃	—	2	1	no	—
21	C₂⁴×C₃	C₂×D₃	1	2	yes	—
22	C₂²×C₃×C₄	—	3	28	no	—
23	C₂²×Dic₃	—	3	28	no	—
24	C₂²×Dic₃	—	6	14	no	—
25	C₂²×Dic₃	C₂×D₃	1	28	yes	—
26	C₂×Dic₆	—	3	224	no	—
27	C₂²×C₃²	C₂×Q₈	1	28	yes	complemented
28	C₂²×C₃²	C₂⁴	1	7	yes	—
29	C₃⋊Dic₃	C₂⁴	1	8	yes	—
30	C₃×Dic₃	C₂⁴	1	16	yes	—
31	C₂²×Q₈	—	9	28	no	—
32	C₂³×C₄	—	3	2	no	—
33	C₂³×C₄	—	9	1	no	—
34	C₂³×C₃	—	2	8	no	—
35	C₂³×C₃	—	2	7	no	—
36	C₂³×C₃	Dic₆	1	16	yes	complemented
37	C₂³×C₃	C₂²×D₃	1	14	yes	—
38	C₂×C₃×C₄	—	3	56	no	—
39	C₂×Dic₃	—	3	56	no	—
40	C₂×Dic₃	—	6	28	no	—
41	C₂×Dic₃	C₂²×D₃	1	56	yes	—
42	Dic₆	—	3	128	no	—
43	C₂×C₃²	C₂²×Q₈	1	14	yes	complemented
44	C₂×C₃²	C₂⁵	1	1	yes	characteristic
45	C₂⁴	D₃²	1	1	yes	characteristic
46	C₂×Q₈	—	9	112	no	—
47	C₂²×C₄	—	3	28	no	—
48	C₂²×C₄	—	9	14	no	—
49	C₂²×C₃	—	2	7	no	—
50	C₂²×C₃	—	2	28	no	—
51	C₂²×C₃	C₂×Dic₆	1	56	yes	complemented
52	C₂²×C₃	C₂³×D₃	1	14	yes	—
53	C₃×C₄	—	3	16	no	—
54	Dic₃	—	3	16	no	—
55	Dic₃	—	6	8	no	—
56	Dic₃	C₂³×D₃	1	16	yes	—
57	C₃²	C₂³×Q₈	1	1	yes	characteristic, complemented, Sylow
58	C₂³	#627	1	8	yes	direct product factor
59	C₂³	C₂×D₃²	1	7	yes	—
60	Q₈	—	9	64	no	—
61	C₂×C₄	—	3	56	no	—
62	C₂×C₄	—	9	28	no	—
63	C₂×C₃	—	2	14	no	—
64	C₂×C₃	—	2	1	no	—
65	C₂×C₃	C₂²×Dic₆	1	28	yes	complemented
66	C₂×C₃	C₂⁴×D₃	1	2	yes	—
67	C₂²	C₂×#627	1	28	yes	direct product factor
68	C₂²	C₂²×D₃²	1	7	yes	—
69	C₄	—	3	16	no	—
70	C₄	—	9	8	no	—
71	C₃	—	2	1	no	—
72	C₃	C₂³×Dic₆	1	2	yes	complemented
73	C₂	C₂²×#627	1	14	yes	direct product factor
74	C₂	C₂³×D₃²	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁴	D₃²
commutator, cosocle	C₂×C₃²	C₂⁵
Frattini	C₂	C₂³×D₃²
Fitting, socle, prime radical	C₂⁴×C₃²	C₂²
hypocenter	C₃²	C₂³×Q₈
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
3. C₂²×#627	C₂	direct (N × H)
8. C₂×#627	C₂²	direct (N × H)
14. C₂³×C₃²	Q₈	semidirect
18. #627	C₂³	direct (N × H)
27. C₂²×C₃²	C₂×Q₈	semidirect
36. C₂³×C₃	Dic₆	semidirect
43. C₂×C₃²	C₂²×Q₈	semidirect
51. C₂²×C₃	C₂×Dic₆	semidirect
57. C₃²	C₂³×Q₈	semidirect
58. C₂³	#627	direct (N × H)
65. C₂×C₃	C₂²×Dic₆	semidirect
67. C₂²	C₂×#627	direct (N × H)
72. C₃	C₂³×Dic₆	semidirect
73. C₂	C₂²×#627	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃⋊Dic₃	2
C₂³×C₃×Dic₃	2
C₂²×#627	2
C₂³×Dic₆	3

Subgroups — C23×#627 #10596705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#627 #10596705