Subgroups — C₂⁴×C₃⋊(C₃⋊C₈) #11918422

Order 1152.

5928 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃²×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃⋊(C₃⋊C₈)	C₂	1	30	yes	maximal, direct product factor
3	C₂⁴×C₃⋊C₈	—	3	4	no	maximal
4	C₂⁵×C₃²	C₄	1	1	yes	characteristic
5	C₂³×C₃²×C₄	C₄	1	15	yes	—
6	C₂³×C₃²×C₄	C₂²	1	15	yes	—
7	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	140	yes	direct product factor
8	C₂⁴×C₃×C₄	D₃	1	4	yes	—
9	C₂³×C₃⋊C₈	—	3	120	no	—
10	C₂⁴×C₃²	C₈	1	16	yes	complemented
11	C₂⁴×C₃²	C₂×C₄	1	15	yes	—
12	C₂²×C₃²×C₄	C₂×C₄	1	105	yes	—
13	C₂²×C₃²×C₄	C₂³	1	35	yes	—
14	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	120	yes	direct product factor
15	C₂⁴×C₈	—	9	1	no	Sylow
16	C₂⁵×C₃	Dic₃	1	4	yes	—
17	C₂³×C₃×C₄	Dic₃	1	60	yes	—
18	C₂³×C₃×C₄	C₂×D₃	1	60	yes	—
19	C₂²×C₃⋊C₈	—	3	560	no	—
20	C₂³×C₃²	C₂×C₈	1	120	yes	complemented
21	C₂³×C₃²	C₂²×C₄	1	35	yes	—
22	C₂×C₃²×C₄	C₂²×C₄	1	105	yes	—
23	C₂×C₃²×C₄	C₂⁴	1	15	yes	—
24	C₃⋊(C₃⋊C₈)	C₂⁴	1	16	yes	direct product factor
25	C₂⁴×C₄	C₃⋊D₃	1	1	yes	characteristic
26	C₂³×C₈	—	9	30	no	—
27	C₂⁴×C₃	C₃⋊C₈	1	64	yes	complemented
28	C₂⁴×C₃	C₂×Dic₃	1	60	yes	—
29	C₂²×C₃×C₄	C₂×Dic₃	1	420	yes	—
30	C₂²×C₃×C₄	C₂²×D₃	1	140	yes	—
31	C₂×C₃⋊C₈	—	3	480	no	—
32	C₂²×C₃²	C₂²×C₈	1	140	yes	complemented
33	C₂²×C₃²	C₂³×C₄	1	15	yes	—
34	C₃²×C₄	C₂³×C₄	1	15	yes	—
35	C₃²×C₄	C₂⁵	1	1	yes	characteristic
36	C₂⁵	C₃⋊Dic₃	1	1	yes	characteristic
37	C₂³×C₄	C₃⋊Dic₃	1	15	yes	—
38	C₂³×C₄	C₂×C₃⋊D₃	1	15	yes	—
39	C₂²×C₈	—	9	140	no	—
40	C₂³×C₃	C₂×C₃⋊C₈	1	480	yes	complemented
41	C₂³×C₃	C₂²×Dic₃	1	140	yes	—
42	C₂×C₃×C₄	C₂²×Dic₃	1	420	yes	—
43	C₂×C₃×C₄	C₂³×D₃	1	60	yes	—
44	C₃⋊C₈	—	3	64	no	—
45	C₂×C₃²	C₂³×C₈	1	30	yes	complemented
46	C₂×C₃²	C₂⁴×C₄	1	1	yes	characteristic
47	C₂⁴	C₃⋊(C₃⋊C₈)	1	16	yes	direct product factor
48	C₂⁴	C₂×C₃⋊Dic₃	1	15	yes	—
49	C₂²×C₄	C₂×C₃⋊Dic₃	1	105	yes	—
50	C₂²×C₄	C₂²×C₃⋊D₃	1	35	yes	—
51	C₂×C₈	—	9	120	no	—
52	C₂²×C₃	C₂²×C₃⋊C₈	1	560	yes	complemented
53	C₂²×C₃	C₂³×Dic₃	1	60	yes	—
54	C₃×C₄	C₂³×Dic₃	1	60	yes	—
55	C₃×C₄	C₂⁴×D₃	1	4	yes	—
56	C₃²	C₂⁴×C₈	1	1	yes	characteristic, complemented, Sylow
57	C₂³	C₂×C₃⋊(C₃⋊C₈)	1	120	yes	direct product factor
58	C₂³	C₂²×C₃⋊Dic₃	1	35	yes	—
59	C₂×C₄	C₂²×C₃⋊Dic₃	1	105	yes	—
60	C₂×C₄	C₂³×C₃⋊D₃	1	15	yes	—
61	C₈	—	9	16	no	—
62	C₂×C₃	C₂³×C₃⋊C₈	1	120	yes	complemented
63	C₂×C₃	C₂⁴×Dic₃	1	4	yes	—
64	C₂²	C₂²×C₃⋊(C₃⋊C₈)	1	140	yes	direct product factor
65	C₂²	C₂³×C₃⋊Dic₃	1	15	yes	—
66	C₄	C₂³×C₃⋊Dic₃	1	15	yes	—
67	C₄	C₂⁴×C₃⋊D₃	1	1	yes	characteristic
68	C₃	C₂⁴×C₃⋊C₈	1	4	yes	complemented
69	C₂	C₂³×C₃⋊(C₃⋊C₈)	1	30	yes	direct product factor
70	C₂	C₂⁴×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁴×C₄	C₃⋊D₃
commutator, hypocenter	C₃²	C₂⁴×C₈
Frattini	C₄	C₂⁴×C₃⋊D₃
Fitting	C₂⁴×C₃²×C₄	C₂
socle, prime radical	C₂⁵×C₃²	C₄
cosocle	C₃²×C₄	C₂⁵
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
2. C₂³×C₃⋊(C₃⋊C₈)	C₂	direct (N × H)
7. C₂²×C₃⋊(C₃⋊C₈)	C₂²	direct (N × H)
10. C₂⁴×C₃²	C₈	semidirect
14. C₂×C₃⋊(C₃⋊C₈)	C₂³	direct (N × H)
20. C₂³×C₃²	C₂×C₈	semidirect
24. C₃⋊(C₃⋊C₈)	C₂⁴	direct (N × H)
27. C₂⁴×C₃	C₃⋊C₈	semidirect
32. C₂²×C₃²	C₂²×C₈	semidirect
40. C₂³×C₃	C₂×C₃⋊C₈	semidirect
45. C₂×C₃²	C₂³×C₈	semidirect
47. C₂⁴	C₃⋊(C₃⋊C₈)	direct (N × H)
52. C₂²×C₃	C₂²×C₃⋊C₈	semidirect
56. C₃²	C₂⁴×C₈	semidirect
57. C₂³	C₂×C₃⋊(C₃⋊C₈)	direct (N × H)
62. C₂×C₃	C₂³×C₃⋊C₈	semidirect
64. C₂²	C₂²×C₃⋊(C₃⋊C₈)	direct (N × H)
68. C₃	C₂⁴×C₃⋊C₈	semidirect
69. C₂	C₂³×C₃⋊(C₃⋊C₈)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₃²×C₄	2
C₂³×C₃⋊(C₃⋊C₈)	2
C₂⁴×C₃⋊C₈	3

Subgroups — C24×C3⋊(C3⋊C8) #11918422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₃⋊(C₃⋊C₈) #11918422