Subgroups — C₂³×D₃×Q₈ #87560

Order 384.

6264 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₃×Q₈	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄×D₃	C₂	1	3	yes	maximal
4	C₂³×Dic₆	C₂	1	3	yes	maximal, complemented
5	C₂⁴×Q₈	—	3	1	no	maximal, Sylow
6	C₂⁴×D₃	C₂²	1	1	yes	characteristic
7	C₂²×C₃×Q₈	C₂²	1	28	yes	complemented
8	C₂³×C₃×C₄	C₂²	1	3	yes	—
9	C₂³×Dic₃	C₂²	1	3	yes	—
10	C₂×D₃×Q₈	C₂²	1	448	yes	direct product factor
11	C₂²×C₄×D₃	C₂²	1	84	yes	—
12	C₂²×Dic₆	C₂²	1	84	yes	complemented
13	C₂³×Q₈	—	3	56	no	—
14	C₂³×Q₈	—	3	3	no	—
15	C₂³×Q₈	D₃	1	1	yes	characteristic, direct product factor
16	C₂⁴×C₄	—	3	3	no	—
17	C₂⁴×C₃	C₂³	1	1	yes	characteristic
18	C₂³×D₃	Q₈	1	16	yes	direct product factor
19	C₂³×D₃	C₂³	1	14	yes	—
20	C₂×C₃×Q₈	C₂³	1	112	yes	complemented
21	C₂²×C₃×C₄	C₂³	1	42	yes	—
22	C₂²×Dic₃	C₂³	1	42	yes	—
23	D₃×Q₈	C₂³	1	512	yes	direct product factor
24	C₂×C₄×D₃	C₂³	1	336	yes	—
25	C₂×Dic₆	C₂³	1	336	yes	complemented
26	C₂⁵	—	3	1	no	—
27	C₂²×Q₈	—	3	448	no	—
28	C₂²×Q₈	—	3	84	no	—
29	C₂²×Q₈	C₂×D₃	1	28	yes	direct product factor
30	C₂³×C₄	—	3	3	no	—
31	C₂³×C₄	—	3	84	no	—
32	C₂³×C₄	C₂×D₃	1	3	yes	—
33	C₂³×C₃	C₂×Q₈	1	8	yes	complemented
34	C₂³×C₃	C₂⁴	1	7	yes	—
35	C₂²×D₃	C₂×Q₈	1	112	yes	direct product factor
36	C₂²×D₃	C₂⁴	1	28	yes	—
37	C₃×Q₈	C₂⁴	1	64	yes	complemented
38	C₂×C₃×C₄	C₂⁴	1	84	yes	—
39	C₂×Dic₃	C₂⁴	1	84	yes	—
40	C₄×D₃	C₂⁴	1	192	yes	—
41	Dic₆	C₂⁴	1	192	yes	complemented
42	C₂⁴	—	3	16	no	—
43	C₂⁴	—	3	14	no	—
44	C₂⁴	C₂²×D₃	1	1	yes	characteristic
45	C₂×Q₈	—	3	512	no	—
46	C₂×Q₈	—	3	336	no	—
47	C₂×Q₈	C₂²×D₃	1	112	yes	direct product factor
48	C₂²×C₄	—	3	42	no	—
49	C₂²×C₄	—	3	336	no	—
50	C₂²×C₄	C₂²×D₃	1	42	yes	—
51	C₂²×C₃	C₂²×Q₈	1	28	yes	complemented
52	C₂²×C₃	C₂⁵	1	7	yes	—
53	C₂×D₃	C₂²×Q₈	1	112	yes	direct product factor
54	C₂×D₃	C₂⁵	1	8	yes	—
55	C₃×C₄	C₂⁵	1	24	yes	—
56	Dic₃	C₂⁵	1	24	yes	—
57	C₂³	—	3	112	no	—
58	C₂³	—	3	28	no	—
59	C₂³	D₃×Q₈	1	8	yes	direct product factor
60	C₂³	C₂³×D₃	1	7	yes	—
61	Q₈	—	3	192	no	—
62	Q₈	C₂³×D₃	1	64	yes	direct product factor
63	C₂×C₄	—	3	84	no	—
64	C₂×C₄	—	3	192	no	—
65	C₂×C₄	C₂³×D₃	1	84	yes	—
66	C₂×C₃	C₂³×Q₈	1	14	yes	complemented
67	C₂×C₃	C₂⁶	1	1	yes	characteristic
68	D₃	C₂³×Q₈	1	16	yes	direct product factor
69	C₂²	—	3	112	no	—
70	C₂²	—	3	8	no	—
71	C₂²	C₂×D₃×Q₈	1	28	yes	direct product factor
72	C₂²	C₂⁴×D₃	1	7	yes	—
73	C₄	—	3	24	no	—
74	C₄	C₂⁴×D₃	1	24	yes	—
75	C₃	C₂⁴×Q₈	1	1	yes	characteristic, complemented, Sylow
76	C₂	—	3	16	no	—
77	C₂	C₂²×D₃×Q₈	1	14	yes	direct product factor
78	C₂	C₂⁵×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₃
commutator, cosocle	C₂×C₃	C₂⁶
Frattini	C₂	C₂⁵×D₃
Fitting	C₂³×C₃×Q₈	C₂
socle	C₂⁴×C₃	C₂³
prime radical	C₂⁴×D₃	C₂²
hypercenter	C₂³×Q₈	D₃
hypocenter	C₃	C₂⁴×Q₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×Q₈	C₂	semidirect
2. C₂²×D₃×Q₈	C₂	direct (N × H)
4. C₂³×Dic₆	C₂	semidirect
7. C₂²×C₃×Q₈	C₂²	semidirect
10. C₂×D₃×Q₈	C₂²	direct (N × H)
12. C₂²×Dic₆	C₂²	semidirect
15. C₂³×Q₈	D₃	direct (N × H)
18. C₂³×D₃	Q₈	direct (N × H)
20. C₂×C₃×Q₈	C₂³	semidirect
23. D₃×Q₈	C₂³	direct (N × H)
25. C₂×Dic₆	C₂³	semidirect
29. C₂²×Q₈	C₂×D₃	direct (N × H)
33. C₂³×C₃	C₂×Q₈	semidirect
35. C₂²×D₃	C₂×Q₈	direct (N × H)
37. C₃×Q₈	C₂⁴	semidirect
41. Dic₆	C₂⁴	semidirect
47. C₂×Q₈	C₂²×D₃	direct (N × H)
51. C₂²×C₃	C₂²×Q₈	semidirect
53. C₂×D₃	C₂²×Q₈	direct (N × H)
59. C₂³	D₃×Q₈	direct (N × H)
62. Q₈	C₂³×D₃	direct (N × H)
66. C₂×C₃	C₂³×Q₈	semidirect
68. D₃	C₂³×Q₈	direct (N × H)
71. C₂²	C₂×D₃×Q₈	direct (N × H)
75. C₃	C₂⁴×Q₈	semidirect
77. C₂	C₂²×D₃×Q₈	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×Q₈	2
C₂²×D₃×Q₈	2
C₂³×C₄×D₃	2
C₂³×Dic₆	2
C₂⁴×Q₈	3

Subgroups — C23×D3×Q8 #87560

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₃×Q₈ #87560