Subgroups — D₈×SL(2,𝔽₃) #73267

Order 384.

149 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×SL(2,𝔽₃)	C₂	1	2	yes	maximal, complemented
2	C₈×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×D₈	C₃	1	1	yes	maximal, characteristic, complemented, Sylow
4	C₂²×SL(2,𝔽₃)	—	2	2	no	—
5	C₂×C₃×D₈	—	4	1	no	maximal
6	C₄×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
7	D₄×Q₈	C₂×C₃	1	2	yes	complemented
8	#500	—	3	1	no	—
9	#474	—	3	2	no	—
10	C₈×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
11	C₄×D₈	—	3	1	no	—
12	C₂×C₃×D₄	—	4	2	no	—
13	C₂×SL(2,𝔽₃)	—	4	2	no	—
14	C₂×SL(2,𝔽₃)	D₄	1	1	yes	characteristic
15	C₃×D₈	—	4	4	no	—
16	C₂×C₃×C₈	—	4	1	no	—
17	C₂²×Q₈	—	2	2	no	—
18	C₂×D₈	A₄	1	1	yes	characteristic
19	C₄⋊Q₈	—	3	2	no	—
20	C₂²⋊Q₈	—	6	2	no	—
21	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
22	C₄×D₄	—	3	2	no	—
23	#107	—	3	2	no	—
24	#107	—	3	1	no	—
25	C₄⋊C₈	—	3	1	no	—
26	#102	—	3	2	no	—
27	C₄×C₈	—	3	1	no	—
28	C₂³×C₃	—	8	2	no	—
29	C₃×D₄	—	4	4	no	—
30	C₃×D₄	—	8	2	no	—
31	C₂×C₃×C₄	—	4	1	no	—
32	SL(2,𝔽₃)	D₈	1	1	yes	characteristic, direct product factor
33	C₃×C₈	—	4	2	no	—
34	C₂×Q₈	—	4	2	no	—
35	C₂×Q₈	—	6	2	no	—
36	C₂×Q₈	—	12	2	no	—
37	C₂×Q₈	C₃×D₄	1	1	yes	characteristic
38	C₂×D₄	C₂×A₄	1	2	yes	—
39	C₂²×C₄	—	6	2	no	—
40	D₈	SL(2,𝔽₃)	1	4	yes	direct product factor
41	C₂×C₈	—	3	1	no	—
42	C₂×C₈	C₂×A₄	1	1	yes	characteristic
43	C₄⋊C₄	—	3	1	no	—
44	C₄⋊C₄	—	3	2	no	—
45	C₄⋊C₄	—	6	2	no	—
46	C₂²⋊C₄	—	6	2	no	—
47	C₄²	—	3	1	no	—
48	C₂²×C₃	—	4	1	no	—
49	C₂²×C₃	—	8	4	no	—
50	C₂²×C₃	—	16	2	no	—
51	C₂²×C₃	—	16	2	no	—
52	C₃×C₄	—	4	1	no	—
53	C₃×C₄	—	4	1	no	—
54	C₂³	—	2	2	no	—
55	Q₈	—	3	1	no	—
56	Q₈	—	12	2	no	—
57	Q₈	—	12	2	no	—
58	Q₈	C₃×D₈	1	1	yes	characteristic, complemented
59	D₄	—	2	2	no	—
60	D₄	C₂×SL(2,𝔽₃)	1	4	yes	complemented
61	C₂×C₄	—	3	1	no	—
62	C₂×C₄	—	3	1	no	—
63	C₂×C₄	—	6	2	no	—
64	C₂×C₄	—	12	2	no	—
65	C₂×C₄	—	12	2	no	—
66	C₂×C₄	C₂²×A₄	1	1	yes	characteristic
67	C₈	—	6	1	no	—
68	C₈	C₂×SL(2,𝔽₃)	1	2	yes	complemented
69	C₂×C₃	—	4	1	no	—
70	C₂×C₃	—	4	1	no	—
71	C₂×C₃	—	4	1	no	—
72	C₂×C₃	—	16	4	no	—
73	C₂²	—	2	4	no	—
74	C₂²	—	4	2	no	—
75	C₂²	—	4	2	no	—
76	C₂²	D₄×A₄	1	1	yes	characteristic
77	C₄	—	3	1	no	—
78	C₄	—	3	1	no	—
79	C₄	—	6	1	no	—
80	C₄	—	12	2	no	—
81	C₄	C₂²×SL(2,𝔽₃)	1	1	yes	characteristic
82	C₄	2₋¹⁺⁴⋊C₃	1	1	yes	characteristic
83	C₃	—	4	1	no	Sylow
84	C₂	—	4	4	no	—
85	C₂	D₄×SL(2,𝔽₃)	1	1	yes	characteristic
86	C₂	A₄×D₈	1	1	yes	characteristic
87	C₂	#578⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	D₄×A₄
commutator, cosocle	C₄×Q₈	C₂²×C₃
Frattini	C₂×C₄	C₂²×A₄
Fitting	Q₈×D₈	C₃
hypercenter	C₂×D₈	A₄
hypocenter	Q₈	C₃×D₈
LCS	G ▷ C₄×Q₈ ▷ C₂×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. D₄×SL(2,𝔽₃)	C₂	semidirect
2. C₈×SL(2,𝔽₃)	C₂	semidirect
3. Q₈×D₈	C₃	semidirect
7. D₄×Q₈	C₂×C₃	semidirect
10. C₈×Q₈	C₂×C₃	semidirect
32. SL(2,𝔽₃)	D₈	direct (N × H)
40. D₈	SL(2,𝔽₃)	direct (N × H)
58. Q₈	C₃×D₈	semidirect
60. D₄	C₂×SL(2,𝔽₃)	semidirect
68. C₈	C₂×SL(2,𝔽₃)	semidirect

Maximal subgroups

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

subgroup	index
D₄×SL(2,𝔽₃)	2
C₈×SL(2,𝔽₃)	2
Q₈×D₈	3
C₂×C₃×D₈	4

Subgroups — D8×SL(2,𝔽3) #73267

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₈×SL(2,𝔽₃) #73267