Subgroups — C₂×C₃²×C₄⋊C₄ #64165

Order 288.

276 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₄	C₂	1	2	yes	maximal
2	C₂²×C₃²×C₄	C₂	1	1	yes	maximal, characteristic
3	C₃²×C₄⋊C₄	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×C₄⋊C₄	C₃	1	4	yes	maximal, direct product factor
5	C₂³×C₃²	C₂²	1	1	yes	characteristic
6	C₂×C₃²×C₄	—	2	4	no	—
7	C₂×C₃²×C₄	C₄	1	4	yes	complemented
8	C₂×C₃²×C₄	C₂²	1	4	yes	—
9	C₂×C₃²×C₄	C₂²	1	2	yes	—
10	C₂²×C₃×C₄	C₂×C₃	1	8	yes	—
11	C₂²×C₃×C₄	C₂×C₃	1	4	yes	—
12	C₃×C₄⋊C₄	C₂×C₃	1	16	yes	direct product factor
13	C₂²×C₃²	C₂×C₄	1	2	yes	—
14	C₂²×C₃²	D₄	1	2	yes	—
15	C₂²×C₃²	Q₈	1	2	yes	—
16	C₂²×C₃²	C₂³	1	1	yes	characteristic
17	C₃²×C₄	—	2	4	no	—
18	C₃²×C₄	C₂×C₄	1	4	yes	complemented
19	C₂×C₄⋊C₄	C₃²	1	1	yes	characteristic, direct product factor, Sylow
20	C₂³×C₃	C₂²×C₃	1	4	yes	—
21	C₂×C₃×C₄	—	2	16	no	—
22	C₂×C₃×C₄	C₃×C₄	1	16	yes	complemented
23	C₂×C₃×C₄	C₂²×C₃	1	16	yes	—
24	C₂×C₃×C₄	C₂²×C₃	1	8	yes	—
25	C₂×C₃²	C₄⋊C₄	1	4	yes	direct product factor
26	C₂×C₃²	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
28	C₂×C₃²	C₂×Q₈	1	1	yes	characteristic
29	C₂²×C₄	C₂×C₃²	1	2	yes	—
30	C₂²×C₄	C₂×C₃²	1	1	yes	characteristic
31	C₄⋊C₄	C₂×C₃²	1	4	yes	direct product factor
32	C₂²×C₃	C₂×C₃×C₄	1	8	yes	—
33	C₂²×C₃	C₃×D₄	1	8	yes	—
34	C₂²×C₃	C₃×Q₈	1	8	yes	—
35	C₂²×C₃	C₂³×C₃	1	4	yes	—
36	C₃×C₄	—	2	16	no	—
37	C₃×C₄	C₂×C₃×C₄	1	16	yes	complemented
38	C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
39	C₂³	C₂²×C₃²	1	1	yes	characteristic
40	C₂×C₄	—	2	4	no	—
41	C₂×C₄	C₃²×C₄	1	4	yes	complemented
42	C₂×C₄	C₂²×C₃²	1	4	yes	—
43	C₂×C₄	C₂²×C₃²	1	2	yes	—
44	C₂×C₃	C₃×C₄⋊C₄	1	16	yes	direct product factor
45	C₂×C₃	C₂²×C₃×C₄	1	4	yes	—
46	C₂×C₃	C₂×C₃×D₄	1	4	yes	—
47	C₂×C₃	C₂×C₃×Q₈	1	4	yes	—
48	C₂²	C₂×C₃²×C₄	1	2	yes	—
49	C₂²	C₃²×D₄	1	2	yes	—
50	C₂²	C₃²×Q₈	1	2	yes	—
51	C₂²	C₂³×C₃²	1	1	yes	characteristic
52	C₄	—	2	4	no	—
53	C₄	C₂×C₃²×C₄	1	4	yes	complemented
54	C₃	C₂×C₃×C₄⋊C₄	1	4	yes	direct product factor
55	C₂	C₃²×C₄⋊C₄	1	4	yes	direct product factor
56	C₂	C₂²×C₃²×C₄	1	1	yes	characteristic
57	C₂	C₂×C₃²×D₄	1	1	yes	characteristic
58	C₂	C₂×C₃²×Q₈	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃²	C₂²
commutator	C₂	C₂²×C₃²×C₄
Frattini, cosocle	C₂²	C₂³×C₃²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃² ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃²×C₄⋊C₄	C₂	direct (N × H)
4. C₂×C₃×C₄⋊C₄	C₃	direct (N × H)
7. C₂×C₃²×C₄	C₄	semidirect
12. C₃×C₄⋊C₄	C₂×C₃	direct (N × H)
18. C₃²×C₄	C₂×C₄	semidirect
19. C₂×C₄⋊C₄	C₃²	direct (N × H)
22. C₂×C₃×C₄	C₃×C₄	semidirect
25. C₂×C₃²	C₄⋊C₄	direct (N × H)
31. C₄⋊C₄	C₂×C₃²	direct (N × H)
37. C₃×C₄	C₂×C₃×C₄	semidirect
38. C₃²	C₂×C₄⋊C₄	direct (N × H)
41. C₂×C₄	C₃²×C₄	semidirect
44. C₂×C₃	C₃×C₄⋊C₄	direct (N × H)
53. C₄	C₂×C₃²×C₄	semidirect
54. C₃	C₂×C₃×C₄⋊C₄	direct (N × H)
55. C₂	C₃²×C₄⋊C₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃²×C₄	2
C₃²×C₄⋊C₄	2
C₂×C₃×C₄⋊C₄	3

Subgroups — C2×C32×C4⋊C4 #64165

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×C₄⋊C₄ #64165