Subgroups — C₄²×C₃⋊(C₃⋊C₈) #11783008

Order 1152.

1224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄³	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₃⋊(C₃⋊C₈)	C₂	1	6	yes	maximal
3	C₄²×C₃⋊C₈	—	3	4	no	maximal
4	C₂×C₃²×C₄²	C₄	1	4	yes	—
5	C₂×C₃²×C₄²	C₂²	1	3	yes	—
6	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	4	yes	—
7	C₄×C₃⋊(C₃⋊C₈)	C₄	1	24	yes	direct product factor
8	C₃×C₄³	D₃	1	4	yes	—
9	C₂×C₄×C₃⋊C₈	—	3	24	no	—
10	C₂²×C₃²×C₄	C₂×C₄	1	6	yes	—
11	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
12	C₃²×C₄²	C₈	1	16	yes	complemented
13	C₃²×C₄²	C₂×C₄	1	12	yes	—
14	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	24	yes	—
15	C₄²×C₈	—	9	1	no	Sylow
16	C₂×C₃×C₄²	Dic₃	1	16	yes	—
17	C₂×C₃×C₄²	C₂×D₃	1	12	yes	—
18	C₂²×C₃⋊C₈	—	3	16	no	—
19	C₄×C₃⋊C₈	—	3	96	no	—
20	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
21	C₂×C₃²×C₄	C₄²	1	12	yes	—
22	C₂×C₃²×C₄	C₂×C₈	1	24	yes	—
23	C₂×C₃²×C₄	C₂²×C₄	1	6	yes	—
24	C₃⋊(C₃⋊C₈)	C₄²	1	16	yes	direct product factor
25	C₂×C₄×C₈	—	9	6	no	—
26	C₄³	C₃⋊D₃	1	1	yes	characteristic
27	C₂²×C₃×C₄	C₂×Dic₃	1	24	yes	—
28	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
29	C₃×C₄²	C₃⋊C₈	1	64	yes	complemented
30	C₃×C₄²	C₄×D₃	1	48	yes	—
31	C₂×C₃⋊C₈	—	3	96	no	—
32	C₂²×C₃²	C₂×C₄²	1	3	yes	—
33	C₂²×C₃²	C₂²×C₈	1	4	yes	—
34	C₃²×C₄	C₄×C₈	1	24	yes	complemented
35	C₃²×C₄	C₂×C₄²	1	4	yes	—
36	C₂²×C₈	—	9	4	no	—
37	C₂×C₄²	C₃⋊Dic₃	1	4	yes	—
38	C₂×C₄²	C₂×C₃⋊D₃	1	3	yes	—
39	C₄×C₈	—	9	24	no	—
40	C₂³×C₃	C₂²×Dic₃	1	4	yes	—
41	C₂×C₃×C₄	C₂×C₃⋊C₈	1	96	yes	—
42	C₂×C₃×C₄	C₄×Dic₃	1	48	yes	—
43	C₂×C₃×C₄	C₂×C₄×D₃	1	24	yes	—
44	C₃⋊C₈	—	3	64	no	—
45	C₂×C₃²	C₄³	1	1	yes	characteristic
46	C₂×C₃²	C₂×C₄×C₈	1	6	yes	—
47	C₂²×C₄	C₂×C₃⋊Dic₃	1	6	yes	—
48	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
49	C₂×C₈	—	9	24	no	—
50	C₄²	C₃⋊(C₃⋊C₈)	1	16	yes	direct product factor
51	C₄²	C₄×C₃⋊D₃	1	12	yes	—
52	C₂²×C₃	C₂²×C₃⋊C₈	1	16	yes	—
53	C₂²×C₃	C₂×C₄×Dic₃	1	12	yes	—
54	C₃×C₄	C₄×C₃⋊C₈	1	96	yes	complemented
55	C₃×C₄	C₄²×D₃	1	16	yes	—
56	C₃²	C₄²×C₈	1	1	yes	characteristic, complemented, Sylow
57	C₂³	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
58	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	24	yes	—
59	C₂×C₄	C₄×C₃⋊Dic₃	1	12	yes	—
60	C₂×C₄	C₂×C₄×C₃⋊D₃	1	6	yes	—
61	C₈	—	9	16	no	—
62	C₂×C₃	C₂×C₄×C₃⋊C₈	1	24	yes	—
63	C₂×C₃	C₄²×Dic₃	1	4	yes	—
64	C₂²	C₂²×C₃⋊(C₃⋊C₈)	1	4	yes	—
65	C₂²	C₂×C₄×C₃⋊Dic₃	1	3	yes	—
66	C₄	C₄×C₃⋊(C₃⋊C₈)	1	24	yes	direct product factor
67	C₄	C₄²×C₃⋊D₃	1	4	yes	—
68	C₃	C₄²×C₃⋊C₈	1	4	yes	complemented
69	C₂	C₂×C₄×C₃⋊(C₃⋊C₈)	1	6	yes	—
70	C₂	C₄²×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄³	C₃⋊D₃
commutator, hypocenter	C₃²	C₄²×C₈
Frattini	C₂²×C₄	C₂²×C₃⋊D₃
Fitting	C₃²×C₄³	C₂
socle, prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂²×C₃²×C₄	C₂³
LCS	G ▷ C₃²
UCS	C₁ ◁ C₄³

Complements

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

N	H ≅ G/N	type
7. C₄×C₃⋊(C₃⋊C₈)	C₄	direct (N × H)
12. C₃²×C₄²	C₈	semidirect
24. C₃⋊(C₃⋊C₈)	C₄²	direct (N × H)
29. C₃×C₄²	C₃⋊C₈	semidirect
34. C₃²×C₄	C₄×C₈	semidirect
50. C₄²	C₃⋊(C₃⋊C₈)	direct (N × H)
54. C₃×C₄	C₄×C₃⋊C₈	semidirect
56. C₃²	C₄²×C₈	semidirect
66. C₄	C₄×C₃⋊(C₃⋊C₈)	direct (N × H)
68. C₃	C₄²×C₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄³	2
C₂×C₄×C₃⋊(C₃⋊C₈)	2
C₄²×C₃⋊C₈	3

Subgroups — C42×C3⋊(C3⋊C8) #11783008

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₃⋊(C₃⋊C₈) #11783008