Subgroups — C₂²×C₃×C₄⋊C₄ #5843

Order 192.

418 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄	C₂	1	2	yes	maximal
2	C₂³×C₃×C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄⋊C₄	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₄⋊C₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂⁴×C₃	C₂²	1	1	yes	characteristic
6	C₂²×C₃×C₄	—	2	8	no	—
7	C₂²×C₃×C₄	C₄	1	8	yes	complemented
8	C₂²×C₃×C₄	C₂²	1	12	yes	—
9	C₂²×C₃×C₄	C₂²	1	6	yes	—
10	C₃×C₄⋊C₄	C₂²	1	16	yes	direct product factor
11	C₂³×C₄	C₂×C₃	1	2	yes	—
12	C₂³×C₄	C₂×C₃	1	1	yes	characteristic
13	C₂×C₄⋊C₄	C₂×C₃	1	12	yes	direct product factor
14	C₂³×C₃	C₂×C₄	1	4	yes	—
15	C₂³×C₃	D₄	1	4	yes	—
16	C₂³×C₃	Q₈	1	4	yes	—
17	C₂³×C₃	C₂³	1	3	yes	—
18	C₂×C₃×C₄	—	2	24	no	—
19	C₂×C₃×C₄	C₂×C₄	1	24	yes	complemented
20	C₂×C₃×C₄	C₂³	1	8	yes	—
21	C₂×C₃×C₄	C₂³	1	4	yes	—
22	C₂⁴	C₂²×C₃	1	1	yes	characteristic
23	C₂²×C₄	—	2	8	no	—
24	C₂²×C₄	C₃×C₄	1	8	yes	complemented
25	C₂²×C₄	C₂²×C₃	1	12	yes	—
26	C₂²×C₄	C₂²×C₃	1	6	yes	—
27	C₄⋊C₄	C₂²×C₃	1	16	yes	direct product factor
28	C₂²×C₃	C₄⋊C₄	1	16	yes	direct product factor
29	C₂²×C₃	C₂²×C₄	1	6	yes	—
30	C₂²×C₃	C₂×D₄	1	6	yes	—
31	C₂²×C₃	C₂×Q₈	1	6	yes	—
32	C₂²×C₃	C₂⁴	1	1	yes	characteristic
33	C₃×C₄	—	2	8	no	—
34	C₃×C₄	C₂²×C₄	1	8	yes	complemented
35	C₂³	C₂×C₃×C₄	1	4	yes	—
36	C₂³	C₃×D₄	1	4	yes	—
37	C₂³	C₃×Q₈	1	4	yes	—
38	C₂³	C₂³×C₃	1	3	yes	—
39	C₂×C₄	—	2	24	no	—
40	C₂×C₄	C₂×C₃×C₄	1	24	yes	complemented
41	C₂×C₄	C₂³×C₃	1	8	yes	—
42	C₂×C₄	C₂³×C₃	1	4	yes	—
43	C₂×C₃	C₂×C₄⋊C₄	1	12	yes	direct product factor
44	C₂×C₃	C₂³×C₄	1	1	yes	characteristic
45	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
46	C₂×C₃	C₂²×Q₈	1	1	yes	characteristic
47	C₂²	C₃×C₄⋊C₄	1	16	yes	direct product factor
48	C₂²	C₂²×C₃×C₄	1	6	yes	—
49	C₂²	C₂×C₃×D₄	1	6	yes	—
50	C₂²	C₂×C₃×Q₈	1	6	yes	—
51	C₂²	C₂⁴×C₃	1	1	yes	characteristic
52	C₄	—	2	8	no	—
53	C₄	C₂²×C₃×C₄	1	8	yes	complemented
54	C₃	C₂²×C₄⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
55	C₂	C₂×C₃×C₄⋊C₄	1	12	yes	direct product factor
56	C₂	C₂³×C₃×C₄	1	1	yes	characteristic
57	C₂	C₂²×C₃×D₄	1	1	yes	characteristic
58	C₂	C₂²×C₃×Q₈	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₃	C₂²
commutator	C₂	C₂³×C₃×C₄
Frattini, cosocle	C₂²	C₂⁴×C₃
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×C₄⋊C₄	C₂	direct (N × H)
4. C₂²×C₄⋊C₄	C₃	direct (N × H)
7. C₂²×C₃×C₄	C₄	semidirect
10. C₃×C₄⋊C₄	C₂²	direct (N × H)
13. C₂×C₄⋊C₄	C₂×C₃	direct (N × H)
19. C₂×C₃×C₄	C₂×C₄	semidirect
24. C₂²×C₄	C₃×C₄	semidirect
27. C₄⋊C₄	C₂²×C₃	direct (N × H)
28. C₂²×C₃	C₄⋊C₄	direct (N × H)
34. C₃×C₄	C₂²×C₄	semidirect
40. C₂×C₄	C₂×C₃×C₄	semidirect
43. C₂×C₃	C₂×C₄⋊C₄	direct (N × H)
47. C₂²	C₃×C₄⋊C₄	direct (N × H)
53. C₄	C₂²×C₃×C₄	semidirect
54. C₃	C₂²×C₄⋊C₄	direct (N × H)
55. C₂	C₂×C₃×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
C₂³×C₃×C₄	2
C₂×C₃×C₄⋊C₄	2
C₂²×C₄⋊C₄	3

Subgroups — C22×C3×C4⋊C4 #5843

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₄⋊C₄ #5843