Subgroups — C₂²×C₃×2₋¹⁺⁴ #87582

Order 384.

5872 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×2₋¹⁺⁴	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₃×#41	C₂	1	10	yes	maximal, complemented
3	C₂³×C₃×Q₈	C₂	1	5	yes	maximal, complemented
4	C₂²×2₋¹⁺⁴	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₃×2₋¹⁺⁴	C₂²	1	256	yes	direct product factor
6	C₂×C₃×#41	C₂²	1	240	yes	complemented
7	C₂²×C₃×Q₈	C₂²	1	10	yes	—
8	C₂²×C₃×Q₈	C₂²	1	120	yes	complemented
9	C₂²×C₃×D₄	C₂²	1	10	yes	—
10	C₂³×C₃×C₄	C₂²	1	15	yes	—
11	C₂×2₋¹⁺⁴	C₂×C₃	1	48	yes	direct product factor
12	C₂²×#41	C₂×C₃	1	10	yes	complemented
13	C₂³×Q₈	C₂×C₃	1	5	yes	complemented
14	C₂⁴×C₃	C₂³	1	5	yes	—
15	C₃×#41	C₂³	1	640	yes	complemented
16	C₂×C₃×Q₈	C₂³	1	120	yes	—
17	C₂×C₃×Q₈	C₂³	1	320	yes	complemented
18	C₂×C₃×D₄	C₂³	1	120	yes	—
19	C₂²×C₃×C₄	C₂³	1	180	yes	—
20	C₂²×C₃×C₄	C₂³	1	10	yes	—
21	2₋¹⁺⁴	C₂²×C₃	1	256	yes	direct product factor
22	C₂×#41	C₂²×C₃	1	240	yes	complemented
23	C₂²×Q₈	C₂²×C₃	1	10	yes	—
24	C₂²×Q₈	C₂²×C₃	1	120	yes	complemented
25	C₂²×D₄	C₂²×C₃	1	10	yes	—
26	C₂³×C₄	C₂²×C₃	1	15	yes	—
27	C₂³×C₃	—	2	20	no	—
28	C₂³×C₃	C₂⁴	1	30	yes	—
29	C₂³×C₃	C₂⁴	1	1	yes	characteristic
30	C₃×Q₈	C₂⁴	1	160	yes	—
31	C₃×D₄	C₂⁴	1	160	yes	—
32	C₂×C₃×C₄	C₂⁴	1	60	yes	—
33	C₂×C₃×C₄	C₂⁴	1	240	yes	—
34	C₂⁴	C₂³×C₃	1	5	yes	—
35	#41	C₂³×C₃	1	640	yes	complemented
36	C₂×Q₈	C₂³×C₃	1	120	yes	—
37	C₂×Q₈	C₂³×C₃	1	320	yes	complemented
38	C₂×D₄	C₂³×C₃	1	120	yes	—
39	C₂²×C₄	C₂³×C₃	1	10	yes	—
40	C₂²×C₄	C₂³×C₃	1	180	yes	—
41	C₂²×C₃	—	2	60	no	—
42	C₂²×C₃	2₋¹⁺⁴	1	4	yes	direct product factor
43	C₂²×C₃	C₂⁵	1	20	yes	—
44	C₂²×C₃	C₂⁵	1	3	yes	—
45	C₃×C₄	C₂⁵	1	40	yes	—
46	C₂³	—	2	20	no	—
47	C₂³	C₂⁴×C₃	1	30	yes	—
48	C₂³	C₂⁴×C₃	1	1	yes	characteristic
49	Q₈	C₂⁴×C₃	1	160	yes	—
50	D₄	C₂⁴×C₃	1	160	yes	—
51	C₂×C₄	C₂⁴×C₃	1	60	yes	—
52	C₂×C₄	C₂⁴×C₃	1	240	yes	—
53	C₂×C₃	—	2	20	no	—
54	C₂×C₃	C₂×2₋¹⁺⁴	1	6	yes	direct product factor
55	C₂×C₃	C₂⁶	1	1	yes	characteristic
56	C₂²	—	2	60	no	—
57	C₂²	C₃×2₋¹⁺⁴	1	4	yes	direct product factor
58	C₂²	C₂⁵×C₃	1	20	yes	—
59	C₂²	C₂⁵×C₃	1	3	yes	—
60	C₄	C₂⁵×C₃	1	40	yes	—
61	C₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic, direct product factor, Sylow
62	C₂	—	2	20	no	—
63	C₂	C₂×C₃×2₋¹⁺⁴	1	6	yes	direct product factor
64	C₂	C₂⁶×C₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁶×C₃
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×2₋¹⁺⁴	C₂	direct (N × H)
2. C₂²×C₃×#41	C₂	semidirect
3. C₂³×C₃×Q₈	C₂	semidirect
4. C₂²×2₋¹⁺⁴	C₃	direct (N × H)
5. C₃×2₋¹⁺⁴	C₂²	direct (N × H)
6. C₂×C₃×#41	C₂²	semidirect
8. C₂²×C₃×Q₈	C₂²	semidirect
11. C₂×2₋¹⁺⁴	C₂×C₃	direct (N × H)
12. C₂²×#41	C₂×C₃	semidirect
13. C₂³×Q₈	C₂×C₃	semidirect
15. C₃×#41	C₂³	semidirect
17. C₂×C₃×Q₈	C₂³	semidirect
21. 2₋¹⁺⁴	C₂²×C₃	direct (N × H)
22. C₂×#41	C₂²×C₃	semidirect
24. C₂²×Q₈	C₂²×C₃	semidirect
35. #41	C₂³×C₃	semidirect
37. C₂×Q₈	C₂³×C₃	semidirect
42. C₂²×C₃	2₋¹⁺⁴	direct (N × H)
54. C₂×C₃	C₂×2₋¹⁺⁴	direct (N × H)
57. C₂²	C₃×2₋¹⁺⁴	direct (N × H)
61. C₃	C₂²×2₋¹⁺⁴	direct (N × H)
63. C₂	C₂×C₃×2₋¹⁺⁴	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×2₋¹⁺⁴	2
C₂²×C₃×#41	2
C₂³×C₃×Q₈	2
C₂²×2₋¹⁺⁴	3

Subgroups — C22×C3×2−1+4 #87582

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×2₋¹⁺⁴ #87582