Subgroups — C₂×Q₈×SL(2,𝔽₃) #85599

Order 384.

347 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×SL(2,𝔽₃)	C₂	1	4	yes	maximal, direct product factor
2	C₂×C₄×SL(2,𝔽₃)	C₂	1	3	yes	maximal
3	C₂×Q₈²	C₃	1	1	yes	maximal, characteristic, complemented, Sylow
4	C₂²×C₃×Q₈	—	4	1	no	maximal
5	C₂²×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₄×SL(2,𝔽₃)	C₂²	1	6	yes	—
7	Q₈²	—	3	4	no	—
8	Q₈²	C₂×C₃	1	4	yes	complemented
9	C₂×C₄⋊Q₈	—	3	3	no	—
10	C₂×C₄×Q₈	—	3	1	no	—
11	C₂×C₄×Q₈	C₂×C₃	1	3	yes	—
12	C₂×C₃×Q₈	—	4	8	no	—
13	C₂×C₃×Q₈	—	4	4	no	—
14	C₂²×C₃×C₄	—	4	3	no	—
15	C₂×SL(2,𝔽₃)	Q₈	1	2	yes	direct product factor
16	C₂×SL(2,𝔽₃)	C₂³	1	1	yes	characteristic
17	C₂²×Q₈	—	3	2	no	—
18	C₂²×Q₈	A₄	1	1	yes	characteristic
19	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
20	C₄⋊Q₈	—	3	12	no	—
21	C₄⋊Q₈	—	3	12	no	—
22	C₄×Q₈	—	3	4	no	—
23	C₄×Q₈	—	3	4	no	—
24	C₄×Q₈	—	3	6	no	—
25	C₄×Q₈	C₂²×C₃	1	6	yes	—
26	C₂×C₄⋊C₄	—	3	3	no	—
27	C₂×C₄⋊C₄	—	3	3	no	—
28	C₂×C₄²	—	3	3	no	—
29	C₂³×C₃	—	4	1	no	—
30	C₃×Q₈	—	4	16	no	—
31	C₂×C₃×C₄	—	4	12	no	—
32	C₂×C₃×C₄	—	4	6	no	—
33	SL(2,𝔽₃)	C₂×Q₈	1	1	yes	characteristic, direct product factor
34	C₂×Q₈	—	3	8	no	—
35	C₂×Q₈	—	3	1	no	—
36	C₂×Q₈	—	3	2	no	—
37	C₂×Q₈	—	12	4	no	—
38	C₂×Q₈	SL(2,𝔽₃)	1	8	yes	direct product factor
39	C₂×Q₈	C₃×Q₈	1	2	yes	complemented
40	C₂×Q₈	C₂×A₄	1	4	yes	—
41	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
42	C₂²×C₄	—	3	1	no	—
43	C₂²×C₄	—	3	3	no	—
44	C₂²×C₄	C₂×A₄	1	3	yes	—
45	C₄⋊C₄	—	3	6	no	—
46	C₄⋊C₄	—	3	6	no	—
47	C₄⋊C₄	—	3	12	no	—
48	C₄²	—	3	6	no	—
49	C₄²	—	3	6	no	—
50	C₂²×C₃	—	4	2	no	—
51	C₂²×C₃	—	4	2	no	—
52	C₂²×C₃	—	4	2	no	—
53	C₂²×C₃	—	4	1	no	—
54	C₃×C₄	—	4	12	no	—
55	C₂³	C₂²×A₄	1	1	yes	characteristic
56	Q₈	—	3	2	no	—
57	Q₈	—	3	1	no	—
58	Q₈	—	3	2	no	—
59	Q₈	—	12	8	no	—
60	Q₈	C₂×SL(2,𝔽₃)	1	16	yes	direct product factor
61	Q₈	C₂×C₃×Q₈	1	1	yes	characteristic, complemented
62	C₂×C₄	—	3	2	no	—
63	C₂×C₄	—	3	2	no	—
64	C₂×C₄	—	3	1	no	—
65	C₂×C₄	—	3	1	no	—
66	C₂×C₄	—	3	6	no	—
67	C₂×C₄	—	6	6	no	—
68	C₂×C₄	C₂×SL(2,𝔽₃)	1	12	yes	—
69	C₂×C₄	C₂²×A₄	1	6	yes	—
70	C₂×C₃	—	4	1	no	—
71	C₂×C₃	—	4	1	no	—
72	C₂×C₃	—	4	1	no	—
73	C₂×C₃	—	4	4	no	—
74	C₂²	C₂²×SL(2,𝔽₃)	1	2	yes	—
75	C₂²	Q₈×A₄	1	2	yes	—
76	C₂²	#994	1	2	yes	—
77	C₂²	C₂³×A₄	1	1	yes	characteristic
78	C₄	—	3	1	no	—
79	C₄	—	3	2	no	—
80	C₄	—	3	1	no	—
81	C₄	—	6	6	no	—
82	C₄	C₂²×SL(2,𝔽₃)	1	12	yes	—
83	C₃	—	4	1	no	Sylow
84	C₂	Q₈×SL(2,𝔽₃)	1	4	yes	direct product factor
85	C₂	C₂³×SL(2,𝔽₃)	1	1	yes	characteristic
86	C₂	C₂×Q₈×A₄	1	1	yes	characteristic
87	C₂	C₂×#994	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂²×A₄
commutator, cosocle	C₂×Q₈	C₂³×C₃
Frattini	C₂²	C₂³×A₄
Fitting	C₂×Q₈²	C₃
prime radical	C₂²×SL(2,𝔽₃)	C₂²
hypercenter	C₂²×Q₈	A₄
hypocenter	Q₈	C₂×C₃×Q₈
LCS	G ▷ C₂×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×SL(2,𝔽₃)	C₂	direct (N × H)
3. C₂×Q₈²	C₃	semidirect
8. Q₈²	C₂×C₃	semidirect
15. C₂×SL(2,𝔽₃)	Q₈	direct (N × H)
33. SL(2,𝔽₃)	C₂×Q₈	direct (N × H)
38. C₂×Q₈	SL(2,𝔽₃)	direct (N × H)
39. C₂×Q₈	C₃×Q₈	semidirect
60. Q₈	C₂×SL(2,𝔽₃)	direct (N × H)
61. Q₈	C₂×C₃×Q₈	semidirect
84. C₂	Q₈×SL(2,𝔽₃)	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×SL(2,𝔽₃)	2
C₂×C₄×SL(2,𝔽₃)	2
C₂×Q₈²	3
C₂²×C₃×Q₈	4

Subgroups — C2×Q8×SL(2,𝔽3) #85599

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×SL(2,𝔽₃) #85599