Subgroups — H(3,𝔽₃)×#383 #13144908

Order 1728.

759 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	H(3,𝔽₃)×#95	C₂	1	6	yes	maximal
3	C₃²×#383	C₃	1	4	yes	maximal, complemented
4	C₂²×C₄×H(3,𝔽₃)	C₂²	1	4	yes	—
5	C₂²×C₄×H(3,𝔽₃)	C₂²	1	3	yes	—
6	C₄²×H(3,𝔽₃)	C₄	1	4	yes	complemented
7	C₂×C₃²×C₄²	C₂×C₃	1	4	yes	—
8	C₃²×#95	C₂×C₃	1	24	yes	—
9	C₂³×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
10	C₂×C₄×H(3,𝔽₃)	—	2	6	no	—
11	C₂×C₄×H(3,𝔽₃)	—	2	12	no	—
12	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	6	yes	—
13	C₃×#383	—	3	4	no	—
14	C₃×#383	C₃²	1	1	yes	characteristic
15	C₂²×C₃²×C₄	C₂²×C₃	1	16	yes	—
16	C₂²×C₃²×C₄	C₂²×C₃	1	12	yes	—
17	C₃²×C₄²	C₃×C₄	1	16	yes	complemented
18	C₂²×H(3,𝔽₃)	C₂²×C₄	1	1	yes	characteristic
19	C₂²×H(3,𝔽₃)	#41	1	6	yes	—
20	C₄×H(3,𝔽₃)	—	2	6	no	—
21	C₄×H(3,𝔽₃)	—	4	4	no	—
22	C₂×C₃×C₄²	—	3	4	no	—
23	C₂×C₃×C₄²	C₂×C₃²	1	1	yes	characteristic
24	C₃×#95	—	3	24	no	—
25	C₃×#95	C₂×C₃²	1	6	yes	—
26	C₂³×C₃²	C₂³×C₃	1	4	yes	—
27	C₂×C₃²×C₄	—	2	24	no	—
28	C₂×C₃²×C₄	—	2	48	no	—
29	C₂×C₃²×C₄	C₂×C₃×C₄	1	24	yes	—
30	#383	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
31	C₂×H(3,𝔽₃)	#117	1	3	yes	—
32	C₂×H(3,𝔽₃)	#126	1	4	yes	—
33	C₂²×C₃×C₄	—	3	12	no	—
34	C₂²×C₃×C₄	—	3	16	no	—
35	C₂²×C₃×C₄	C₂²×C₃²	1	4	yes	—
36	C₂²×C₃×C₄	C₂²×C₃²	1	3	yes	—
37	C₃×C₄²	—	3	16	no	—
38	C₃×C₄²	C₃²×C₄	1	4	yes	—
39	C₂²×C₃²	C₂²×C₃×C₄	1	4	yes	—
40	C₂²×C₃²	C₃×#41	1	24	yes	—
41	C₃²×C₄	—	2	24	no	—
42	C₃²×C₄	—	4	16	no	—
43	C₂×C₄²	C₂×H(3,𝔽₃)	1	1	yes	characteristic
44	#95	C₂×H(3,𝔽₃)	1	6	yes	—
45	H(3,𝔽₃)	#383	1	1	yes	characteristic, direct product factor, Sylow
46	C₂³×C₃	—	3	4	no	—
47	C₂³×C₃	C₂³×C₃²	1	1	yes	characteristic
48	C₂×C₃×C₄	—	2	6	no	—
49	C₂×C₃×C₄	—	2	12	no	—
50	C₂×C₃×C₄	—	3	24	no	—
51	C₂×C₃×C₄	—	6	24	no	—
52	C₂×C₃×C₄	—	6	48	no	—
53	C₂×C₃×C₄	C₂×C₃²×C₄	1	6	yes	—
54	C₂×C₃²	C₃×#117	1	12	yes	—
55	C₂×C₃²	C₃×#126	1	16	yes	—
56	C₂²×C₄	C₂²×H(3,𝔽₃)	1	4	yes	—
57	C₂²×C₄	C₂²×H(3,𝔽₃)	1	3	yes	—
58	C₄²	C₄×H(3,𝔽₃)	1	4	yes	complemented
59	C₂²×C₃	—	3	4	no	—
60	C₂²×C₃	—	3	24	no	—
61	C₂²×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
62	C₂²×C₃	C₃²×#41	1	6	yes	—
63	C₃×C₄	—	2	6	no	—
64	C₃×C₄	—	4	4	no	—
65	C₃×C₄	—	6	24	no	—
66	C₃×C₄	—	12	16	no	—
67	C₃²	C₃×#383	1	4	yes	complemented
68	C₂³	C₂³×H(3,𝔽₃)	1	1	yes	characteristic
69	C₂×C₄	—	2	6	no	—
70	C₂×C₄	—	2	12	no	—
71	C₂×C₄	C₂×C₄×H(3,𝔽₃)	1	6	yes	—
72	C₂×C₃	—	3	12	no	—
73	C₂×C₃	—	3	16	no	—
74	C₂×C₃	C₃²×#117	1	3	yes	—
75	C₂×C₃	C₃²×#126	1	4	yes	—
76	C₂²	C₂²×C₄×H(3,𝔽₃)	1	1	yes	characteristic
77	C₂²	#41×H(3,𝔽₃)	1	6	yes	—
78	C₄	—	2	6	no	—
79	C₄	—	4	4	no	—
80	C₃	—	3	4	no	—
81	C₃	C₃²×#383	1	1	yes	characteristic
82	C₂	H(3,𝔽₃)×#117	1	3	yes	—
83	C₂	H(3,𝔽₃)×#126	1	4	yes	—

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³×C₃	C₂³×C₃²
commutator	C₂²×C₃	C₂²×C₃²×C₄
prime radical	C₂³×H(3,𝔽₃)	C₂³
LCS	G ▷ C₂²×C₃ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃²×#383	C₃	semidirect
6. C₄²×H(3,𝔽₃)	C₄	semidirect
17. C₃²×C₄²	C₃×C₄	semidirect
30. #383	H(3,𝔽₃)	direct (N × H)
45. H(3,𝔽₃)	#383	direct (N × H)
58. C₄²	C₄×H(3,𝔽₃)	semidirect
67. C₃²	C₃×#383	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄²×H(3,𝔽₃)	2
H(3,𝔽₃)×#95	2
C₃²×#383	3

Subgroups — H(3,𝔽3)×#383 #13144908

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)×#383 #13144908