Subgroups — H(3,𝔽₃)×C₄⋊Q₈ #11721920

Order 864.

374 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×H(3,𝔽₃)	C₂	1	2	yes	maximal
2	C₄⋊C₄×H(3,𝔽₃)	C₂	1	4	yes	maximal
3	C₄²×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
4	C₃²×C₄⋊Q₈	C₃	1	4	yes	maximal, complemented
5	Q₈×H(3,𝔽₃)	—	2	4	no	—
6	C₂×C₄×H(3,𝔽₃)	C₂²	1	4	yes	—
7	C₂×C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
8	C₂×C₄×H(3,𝔽₃)	C₂²	1	2	yes	—
9	C₂×C₃²×Q₈	C₂×C₃	1	8	yes	—
10	C₃²×C₄⋊C₄	C₂×C₃	1	16	yes	—
11	C₃²×C₄²	C₂×C₃	1	4	yes	—
12	C₂²×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
13	C₄×H(3,𝔽₃)	—	2	4	no	—
14	C₄×H(3,𝔽₃)	D₄	1	2	yes	—
15	C₄×H(3,𝔽₃)	Q₈	1	4	yes	complemented
16	C₃×C₄⋊Q₈	—	3	4	no	—
17	C₃×C₄⋊Q₈	C₃²	1	1	yes	characteristic
18	C₃²×Q₈	—	2	16	no	—
19	C₂×C₃²×C₄	C₂²×C₃	1	16	yes	—
20	C₂×C₃²×C₄	C₂²×C₃	1	4	yes	—
21	C₂×C₃²×C₄	C₂²×C₃	1	8	yes	—
22	C₂×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
23	C₂×H(3,𝔽₃)	C₂×Q₈	1	2	yes	—
24	C₂×C₃×Q₈	—	3	8	no	—
25	C₂×C₃×Q₈	C₂×C₃²	1	2	yes	—
26	C₃×C₄⋊C₄	—	3	16	no	—
27	C₃×C₄⋊C₄	C₂×C₃²	1	4	yes	—
28	C₃×C₄²	—	3	4	no	—
29	C₃×C₄²	C₂×C₃²	1	1	yes	characteristic
30	C₂²×C₃²	C₂³×C₃	1	4	yes	—
31	C₃²×C₄	—	2	16	no	—
32	C₃²×C₄	C₃×D₄	1	8	yes	—
33	C₃²×C₄	C₃×Q₈	1	16	yes	complemented
34	C₄⋊Q₈	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
35	H(3,𝔽₃)	C₄⋊Q₈	1	1	yes	characteristic, direct product factor, Sylow
36	C₃×Q₈	—	2	4	no	—
37	C₃×Q₈	—	6	16	no	—
38	C₂×C₃×C₄	—	3	8	no	—
39	C₂×C₃×C₄	—	3	4	no	—
40	C₂×C₃×C₄	—	3	16	no	—
41	C₂×C₃×C₄	C₂²×C₃²	1	4	yes	—
42	C₂×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
43	C₂×C₃×C₄	C₂²×C₃²	1	2	yes	—
44	C₂×C₃²	C₂×C₃×D₄	1	4	yes	—
45	C₂×C₃²	C₂×C₃×Q₈	1	8	yes	—
46	C₂×Q₈	C₂×H(3,𝔽₃)	1	2	yes	—
47	C₄⋊C₄	C₂×H(3,𝔽₃)	1	4	yes	—
48	C₄²	C₂×H(3,𝔽₃)	1	1	yes	characteristic
49	C₂²×C₃	—	3	4	no	—
50	C₂²×C₃	C₂³×C₃²	1	1	yes	characteristic
51	C₃×C₄	—	2	4	no	—
52	C₃×C₄	—	3	8	no	—
53	C₃×C₄	—	3	16	no	—
54	C₃×C₄	—	6	16	no	—
55	C₃×C₄	C₃²×D₄	1	2	yes	—
56	C₃×C₄	C₃²×Q₈	1	4	yes	—
57	C₃²	C₃×C₄⋊Q₈	1	4	yes	complemented
58	Q₈	—	2	4	no	—
59	C₂×C₄	C₂²×H(3,𝔽₃)	1	4	yes	—
60	C₂×C₄	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
61	C₂×C₄	C₂²×H(3,𝔽₃)	1	2	yes	—
62	C₂×C₃	—	3	4	no	—
63	C₂×C₃	—	3	8	no	—
64	C₂×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
65	C₂×C₃	C₂×C₃²×Q₈	1	2	yes	—
66	C₂²	C₂³×H(3,𝔽₃)	1	1	yes	characteristic
67	C₄	—	2	4	no	—
68	C₄	D₄×H(3,𝔽₃)	1	2	yes	—
69	C₄	Q₈×H(3,𝔽₃)	1	4	yes	complemented
70	C₃	—	3	4	no	—
71	C₃	C₃²×C₄⋊Q₈	1	1	yes	characteristic
72	C₂	C₂×D₄×H(3,𝔽₃)	1	1	yes	characteristic
73	C₂	C₂×Q₈×H(3,𝔽₃)	1	2	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²×C₃	C₂³×C₃²
prime radical	C₂²×H(3,𝔽₃)	C₂³
LCS	G ▷ C₂²×C₃ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃²×C₄⋊Q₈	C₃	semidirect
15. C₄×H(3,𝔽₃)	Q₈	semidirect
33. C₃²×C₄	C₃×Q₈	semidirect
34. C₄⋊Q₈	H(3,𝔽₃)	direct (N × H)
35. H(3,𝔽₃)	C₄⋊Q₈	direct (N × H)
57. C₃²	C₃×C₄⋊Q₈	semidirect
69. C₄	Q₈×H(3,𝔽₃)	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×H(3,𝔽₃)	2
C₄⋊C₄×H(3,𝔽₃)	2
C₄²×H(3,𝔽₃)	2
C₃²×C₄⋊Q₈	3

Subgroups — H(3,𝔽3)×C4⋊Q8 #11721920

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)×C₄⋊Q₈ #11721920