Subgroups — H(3,𝔽₃)×#368 #13136106

Order 1728.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₆×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	H(3,𝔽₃)×#108	C₂	1	2	yes	maximal
3	C₃²×#368	C₃	1	4	yes	maximal, complemented
4	M₁₆×H(3,𝔽₃)	—	2	2	no	—
5	C₂×C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₁₆×H(3,𝔽₃)	C₄	1	2	yes	—
7	C₂×C₃²×C₁₆	C₂×C₃	1	4	yes	—
8	C₃²×#108	C₂×C₃	1	8	yes	—
9	C₂×C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
10	C₈×H(3,𝔽₃)	—	4	2	no	—
11	C₈×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
12	C₈×H(3,𝔽₃)	Q₈	1	1	yes	characteristic
13	C₃×#368	—	3	4	no	—
14	C₃×#368	C₃²	1	1	yes	characteristic
15	C₃²×M₁₆	—	2	8	no	—
16	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
17	C₃²×C₁₆	C₃×C₄	1	8	yes	—
18	C₂²×H(3,𝔽₃)	Q₁₆	1	1	yes	characteristic
19	C₄×H(3,𝔽₃)	C₄⋊C₄	1	1	yes	characteristic
20	C₄×H(3,𝔽₃)	D₈	1	1	yes	characteristic
21	C₂×C₃×C₁₆	—	3	4	no	—
22	C₂×C₃×C₁₆	C₂×C₃²	1	1	yes	characteristic
23	C₃×#108	—	3	8	no	—
24	C₃×#108	C₂×C₃²	1	2	yes	—
25	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
26	C₃²×C₈	—	4	8	no	—
27	C₃²×C₈	C₂×C₃×C₄	1	4	yes	—
28	C₃²×C₈	C₃×Q₈	1	4	yes	—
29	#368	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
30	C₂×H(3,𝔽₃)	—	2	1	no	—
31	C₂×H(3,𝔽₃)	#107	1	1	yes	characteristic
32	C₃×M₁₆	—	2	2	no	—
33	C₃×M₁₆	—	6	8	no	—
34	C₂×C₃×C₈	—	3	4	no	—
35	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
36	C₃×C₁₆	—	3	8	no	—
37	C₃×C₁₆	C₃²×C₄	1	2	yes	—
38	C₂²×C₃²	C₃×Q₁₆	1	4	yes	—
39	C₃²×C₄	C₃×C₄⋊C₄	1	4	yes	—
40	C₃²×C₄	C₃×D₈	1	4	yes	—
41	C₂×C₁₆	C₂×H(3,𝔽₃)	1	1	yes	characteristic
42	#108	C₂×H(3,𝔽₃)	1	2	yes	—
43	H(3,𝔽₃)	#368	1	1	yes	characteristic, direct product factor, Sylow
44	C₂×C₃×C₄	—	3	4	no	—
45	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
46	C₃×C₈	—	3	4	no	—
47	C₃×C₈	—	3	4	no	—
48	C₃×C₈	—	4	2	no	—
49	C₃×C₈	—	12	8	no	—
50	C₃×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
51	C₃×C₈	C₃²×Q₈	1	1	yes	characteristic
52	C₂×C₃²	—	2	4	no	—
53	C₂×C₃²	C₃×#107	1	4	yes	—
54	M₁₆	—	2	2	no	—
55	C₂×C₈	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
56	C₁₆	C₄×H(3,𝔽₃)	1	2	yes	—
57	C₂²×C₃	—	3	4	no	—
58	C₂²×C₃	C₃²×Q₁₆	1	1	yes	characteristic
59	C₃×C₄	—	3	4	no	—
60	C₃×C₄	—	3	4	no	—
61	C₃×C₄	C₃²×C₄⋊C₄	1	1	yes	characteristic
62	C₃×C₄	C₃²×D₈	1	1	yes	characteristic
63	C₃²	C₃×#368	1	4	yes	complemented
64	C₂×C₄	D₄×H(3,𝔽₃)	1	1	yes	characteristic
65	C₈	—	4	2	no	—
66	C₈	C₂×C₄×H(3,𝔽₃)	1	1	yes	characteristic
67	C₈	Q₈×H(3,𝔽₃)	1	1	yes	characteristic
68	C₂×C₃	—	2	1	no	—
69	C₂×C₃	—	3	4	no	—
70	C₂×C₃	—	6	4	no	—
71	C₂×C₃	C₃²×#107	1	1	yes	characteristic
72	C₂²	Q₁₆×H(3,𝔽₃)	1	1	yes	characteristic
73	C₄	C₄⋊C₄×H(3,𝔽₃)	1	1	yes	characteristic
74	C₄	D₈×H(3,𝔽₃)	1	1	yes	characteristic
75	C₃	—	3	4	no	—
76	C₃	C₃²×#368	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	H(3,𝔽₃)×#107	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	C₃²×D₈
commutator	C₃×C₈	C₂×C₃²×C₄
Frattini, cosocle	C₂×C₃×C₈	C₂²×C₃²
socle	C₂×C₃	C₃²×#107
prime radical	C₂²×H(3,𝔽₃)	Q₁₆
LCS	G ▷ C₃×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₄×H(3,𝔽₃) ◁ C₂×C₈×H(3,𝔽₃) ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃²×#368	C₃	semidirect
29. #368	H(3,𝔽₃)	direct (N × H)
43. H(3,𝔽₃)	#368	direct (N × H)
63. C₃²	C₃×#368	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₆×H(3,𝔽₃)	2
H(3,𝔽₃)×#108	2
C₃²×#368	3

Subgroups — H(3,𝔽3)×#368 #13136106

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)×#368 #13136106