Subgroups — C₄⋊C₄×3₋¹⁺² #88561

Order 432.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×3₋¹⁺²	C₂	1	2	yes	maximal
2	C₂×C₄×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
3	C₃²×C₄⋊C₄	C₃	1	1	yes	maximal, characteristic
4	C₉×C₄⋊C₄	C₃	1	3	yes	maximal, complemented
5	C₂²×3₋¹⁺²	C₂²	1	1	yes	characteristic
6	C₄×3₋¹⁺²	—	2	2	no	—
7	C₄×3₋¹⁺²	C₄	1	2	yes	complemented
8	C₂×C₃²×C₄	C₂×C₃	1	2	yes	—
9	C₂×C₃²×C₄	C₂×C₃	1	1	yes	characteristic
10	C₂×C₄×C₉	C₂×C₃	1	6	yes	—
11	C₂×C₄×C₉	C₂×C₃	1	3	yes	—
12	C₂×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
13	C₂×3₋¹⁺²	D₄	1	1	yes	characteristic
14	C₂×3₋¹⁺²	Q₈	1	1	yes	characteristic
15	C₃×C₄⋊C₄	—	3	1	no	—
16	C₃×C₄⋊C₄	C₃²	1	1	yes	characteristic
17	C₂²×C₃²	C₂²×C₃	1	1	yes	characteristic
18	C₃²×C₄	—	2	2	no	—
19	C₃²×C₄	C₃×C₄	1	2	yes	—
20	C₂²×C₉	C₂²×C₃	1	3	yes	—
21	C₄×C₉	—	2	6	no	—
22	C₄×C₉	C₃×C₄	1	6	yes	complemented
23	3₋¹⁺²	C₄⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
24	C₂×C₃×C₄	—	3	1	no	—
25	C₂×C₃×C₄	—	3	2	no	—
26	C₂×C₃×C₄	C₂×C₃²	1	2	yes	—
27	C₂×C₃×C₄	C₂×C₃²	1	1	yes	characteristic
28	C₂×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
29	C₂×C₃²	C₃×D₄	1	1	yes	characteristic
30	C₂×C₃²	C₃×Q₈	1	1	yes	characteristic
31	C₂×C₉	C₂×C₃×C₄	1	3	yes	—
32	C₂×C₉	C₃×D₄	1	3	yes	—
33	C₂×C₉	C₃×Q₈	1	3	yes	—
34	C₄⋊C₄	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
35	C₂²×C₃	—	3	1	no	—
36	C₂²×C₃	C₂²×C₃²	1	1	yes	characteristic
37	C₃×C₄	—	2	2	no	—
38	C₃×C₄	—	3	2	no	—
39	C₃×C₄	—	6	2	no	—
40	C₃×C₄	C₃²×C₄	1	2	yes	—
41	C₃²	C₃×C₄⋊C₄	1	1	yes	characteristic
42	C₉	C₃×C₄⋊C₄	1	3	yes	complemented
43	C₂×C₄	C₂×3₋¹⁺²	1	2	yes	—
44	C₂×C₄	C₂×3₋¹⁺²	1	1	yes	characteristic
45	C₂×C₃	—	3	1	no	—
46	C₂×C₃	—	3	1	no	—
47	C₂×C₃	—	3	1	no	—
48	C₂×C₃	C₂×C₃²×C₄	1	1	yes	characteristic
49	C₂×C₃	C₃²×D₄	1	1	yes	characteristic
50	C₂×C₃	C₃²×Q₈	1	1	yes	characteristic
51	C₂²	C₂²×3₋¹⁺²	1	1	yes	characteristic
52	C₄	—	2	2	no	—
53	C₄	C₄×3₋¹⁺²	1	2	yes	complemented
54	C₃	—	3	1	no	—
55	C₃	C₃²×C₄⋊C₄	1	1	yes	characteristic
56	C₂	C₂×C₄×3₋¹⁺²	1	1	yes	characteristic
57	C₂	D₄×3₋¹⁺²	1	1	yes	characteristic
58	C₂	Q₈×3₋¹⁺²	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle, cosocle	C₂²×C₃	C₂²×C₃²
commutator	C₂×C₃	C₂×C₃²×C₄
prime radical	C₂²×C₃²	C₂²×C₃
LCS	G ▷ C₂×C₃ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₉×C₄⋊C₄	C₃	semidirect
7. C₄×3₋¹⁺²	C₄	semidirect
22. C₄×C₉	C₃×C₄	semidirect
23. 3₋¹⁺²	C₄⋊C₄	direct (N × H)
34. C₄⋊C₄	3₋¹⁺²	direct (N × H)
42. C₉	C₃×C₄⋊C₄	semidirect
53. C₄	C₄×3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×3₋¹⁺²	2
C₃²×C₄⋊C₄	3
C₉×C₄⋊C₄	3

Subgroups — C4⋊C4×3−1+2 #88561

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊C₄×3₋¹⁺² #88561