Subgroups — H(3,𝔽₃)×#100 #11720603

Order 864.

253 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	M₁₆×H(3,𝔽₃)	C₂	1	2	yes	maximal, complemented
3	C₃²×#100	C₃	1	4	yes	maximal, complemented
4	C₂³×H(3,𝔽₃)	C₄	1	2	yes	—
5	D₄×H(3,𝔽₃)	—	2	2	no	—
6	C₂×C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₈×H(3,𝔽₃)	—	2	2	no	—
8	C₂×C₃²×D₄	C₂×C₃	1	4	yes	—
9	C₃²×M₁₆	C₂×C₃	1	8	yes	complemented
10	C₂²×H(3,𝔽₃)	—	2	2	no	—
11	C₂²×H(3,𝔽₃)	—	4	2	no	—
12	C₂²×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
13	C₄×H(3,𝔽₃)	D₄	1	2	yes	—
14	C₃×#100	—	3	4	no	—
15	C₃×#100	C₃²	1	1	yes	characteristic
16	C₂³×C₃²	C₃×C₄	1	8	yes	—
17	C₃²×D₄	—	2	8	no	—
18	C₂×C₃²×C₄	C₂²×C₃	1	4	yes	—
19	C₃²×C₈	—	2	8	no	—
20	C₂×H(3,𝔽₃)	—	2	1	no	—
21	C₂×H(3,𝔽₃)	—	4	2	no	—
22	C₂×H(3,𝔽₃)	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₃×D₄	—	3	4	no	—
24	C₂×C₃×D₄	C₂×C₃²	1	1	yes	characteristic
25	C₃×M₁₆	—	3	8	no	—
26	C₃×M₁₆	C₂×C₃²	1	2	yes	—
27	C₂²×C₃²	—	2	8	no	—
28	C₂²×C₃²	—	4	8	no	—
29	C₂²×C₃²	C₂×C₃×C₄	1	4	yes	—
30	C₃²×C₄	C₃×D₄	1	8	yes	—
31	#100	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
32	H(3,𝔽₃)	#100	1	1	yes	characteristic, direct product factor, Sylow
33	C₂³×C₃	—	3	8	no	—
34	C₂³×C₃	C₃²×C₄	1	2	yes	—
35	C₃×D₄	—	2	2	no	—
36	C₃×D₄	—	6	8	no	—
37	C₂×C₃×C₄	—	3	4	no	—
38	C₂×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
39	C₃×C₈	—	2	2	no	—
40	C₃×C₈	—	6	8	no	—
41	C₂×C₃²	—	2	4	no	—
42	C₂×C₃²	—	4	8	no	—
43	C₂×C₃²	C₃×C₂²⋊C₄	1	4	yes	—
44	C₂×D₄	C₂×H(3,𝔽₃)	1	1	yes	characteristic
45	M₁₆	C₂×H(3,𝔽₃)	1	2	yes	complemented
46	C₂²×C₃	—	2	2	no	—
47	C₂²×C₃	—	3	4	no	—
48	C₂²×C₃	—	4	2	no	—
49	C₂²×C₃	—	6	8	no	—
50	C₂²×C₃	—	12	8	no	—
51	C₂²×C₃	C₂×C₃²×C₄	1	1	yes	characteristic
52	C₃×C₄	—	3	8	no	—
53	C₃×C₄	C₃²×D₄	1	2	yes	—
54	C₃²	C₃×#100	1	4	yes	complemented
55	C₂³	C₄×H(3,𝔽₃)	1	2	yes	—
56	D₄	—	2	2	no	—
57	C₂×C₄	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
58	C₈	—	2	2	no	—
59	C₂×C₃	—	2	1	no	—
60	C₂×C₃	—	3	4	no	—
61	C₂×C₃	—	4	2	no	—
62	C₂×C₃	—	6	4	no	—
63	C₂×C₃	—	12	8	no	—
64	C₂×C₃	C₃²×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²	—	2	2	no	—
66	C₂²	—	4	2	no	—
67	C₂²	C₂×C₄×H(3,𝔽₃)	1	1	yes	characteristic
68	C₄	D₄×H(3,𝔽₃)	1	2	yes	—
69	C₃	—	3	4	no	—
70	C₃	C₃²×#100	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	—	4	2	no	—
73	C₂	C₂²⋊C₄×H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₃²×C₂²⋊C₄
commutator	C₂²×C₃	C₂×C₃²×C₄
Frattini, cosocle	C₂×C₃×C₄	C₂²×C₃²
prime radical	C₂×D₄×H(3,𝔽₃)	C₂
LCS	G ▷ C₂²×C₃ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₄×H(3,𝔽₃) ◁ G

Complements

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

N	H ≅ G/N	type
2. M₁₆×H(3,𝔽₃)	C₂	semidirect
3. C₃²×#100	C₃	semidirect
9. C₃²×M₁₆	C₂×C₃	semidirect
31. #100	H(3,𝔽₃)	direct (N × H)
32. H(3,𝔽₃)	#100	direct (N × H)
45. M₁₆	C₂×H(3,𝔽₃)	semidirect
54. C₃²	C₃×#100	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×H(3,𝔽₃)	2
M₁₆×H(3,𝔽₃)	2
C₃²×#100	3

Subgroups — H(3,𝔽3)×#100 #11720603

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)×#100 #11720603