Subgroups — C₂²×Dic₂₄ #82004

Order 384.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₁₂	C₂	1	2	yes	maximal
2	C₂²×C₃×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×Dic₂₄	C₂	1	12	yes	maximal, direct product factor
4	C₂²×Q₃₂	—	3	1	no	maximal, Sylow
5	C₂²×Dic₆	—	2	2	no	—
6	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
7	C₂×Dic₁₂	—	2	6	no	—
8	C₂×Dic₁₂	C₂²	1	12	yes	—
9	C₂×C₃×C₁₆	C₂²	1	6	yes	—
10	Dic₂₄	C₂²	1	16	yes	direct product factor
11	C₂²×Q₁₆	—	3	2	no	—
12	C₂×Q₃₂	—	3	12	no	—
13	C₂²×C₁₆	D₃	1	1	yes	characteristic
14	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
15	C₂²×Dic₃	—	4	2	no	—
16	C₂×Dic₆	—	2	12	no	—
17	C₂×Dic₆	—	4	6	no	—
18	C₂×C₃×C₈	D₄	1	3	yes	—
19	C₂×C₃×C₈	C₂³	1	3	yes	—
20	Dic₁₂	—	2	12	no	—
21	Dic₁₂	C₂³	1	8	yes	—
22	C₃×C₁₆	C₂³	1	4	yes	—
23	C₂²×Q₈	—	6	2	no	—
24	C₂×Q₁₆	—	3	12	no	—
25	C₂×Q₁₆	—	6	6	no	—
26	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
27	Q₃₂	—	3	16	no	—
28	C₂×C₁₆	C₂×D₃	1	6	yes	—
29	C₂³×C₃	D₈	1	1	yes	characteristic
30	C₂×C₃×C₄	D₈	1	3	yes	—
31	C₂×C₃×C₄	C₂×D₄	1	3	yes	—
32	C₂×Dic₃	—	4	12	no	—
33	Dic₆	—	2	8	no	—
34	Dic₆	—	4	12	no	—
35	C₃×C₈	C₂×D₄	1	3	yes	—
36	C₃×C₈	C₂⁴	1	1	yes	characteristic
37	C₂×Q₈	—	6	12	no	—
38	C₂×Q₈	—	12	6	no	—
39	C₂²×C₄	—	12	2	no	—
40	C₂²×C₄	D₁₂	1	1	yes	characteristic
41	Q₁₆	—	3	8	no	—
42	Q₁₆	—	6	12	no	—
43	C₂×C₈	D₁₂	1	3	yes	—
44	C₂×C₈	C₂²×D₃	1	3	yes	—
45	C₁₆	C₂²×D₃	1	4	yes	—
46	C₂²×C₃	Q₃₂	1	4	yes	complemented
47	C₂²×C₃	C₂×D₈	1	3	yes	—
48	C₃×C₄	C₂×D₈	1	3	yes	—
49	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
50	Dic₃	—	4	8	no	—
51	C₂³	D₂₄	1	1	yes	characteristic
52	Q₈	—	6	8	no	—
53	Q₈	—	12	12	no	—
54	C₂×C₄	—	12	12	no	—
55	C₂×C₄	D₂₄	1	3	yes	—
56	C₂×C₄	C₂×D₁₂	1	3	yes	—
57	C₈	C₂×D₁₂	1	3	yes	—
58	C₈	C₂³×D₃	1	1	yes	characteristic
59	C₂×C₃	C₂×Q₃₂	1	6	yes	complemented
60	C₂×C₃	C₂²×D₈	1	1	yes	characteristic
61	C₂²	Dic₂₄	1	4	yes	direct product factor
62	C₂²	C₂×D₂₄	1	3	yes	—
63	C₄	—	12	8	no	—
64	C₄	C₂×D₂₄	1	3	yes	—
65	C₄	C₂²×D₁₂	1	1	yes	characteristic
66	C₃	C₂²×Q₃₂	1	1	yes	characteristic, complemented, Sylow
67	C₂	C₂×Dic₂₄	1	6	yes	direct product factor
68	C₂	C₂²×D₂₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₂₄
commutator, cosocle	C₃×C₈	C₂⁴
Frattini	C₈	C₂³×D₃
Fitting	C₂²×C₃×C₁₆	C₂
socle, prime radical	C₂³×C₃	D₈
hypercenter	C₂²×C₁₆	D₃
hypocenter	C₃	C₂²×Q₃₂
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×Dic₂₄	C₂	direct (N × H)
10. Dic₂₄	C₂²	direct (N × H)
46. C₂²×C₃	Q₃₂	semidirect
59. C₂×C₃	C₂×Q₃₂	semidirect
61. C₂²	Dic₂₄	direct (N × H)
66. C₃	C₂²×Q₃₂	semidirect
67. C₂	C₂×Dic₂₄	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×Dic₁₂	2
C₂²×C₃×C₁₆	2
C₂×Dic₂₄	2
C₂²×Q₃₂	3

Subgroups — C22×Dic24 #82004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Dic₂₄ #82004