Subgroups — H(3,𝔽₃)×#338 #13136016

Order 1728.

473 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×H(3,𝔽₃)	C₂	1	3	yes	maximal
2	C₃²×#338	C₃	1	4	yes	maximal, complemented
3	C₂²×C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
4	M₁₆×H(3,𝔽₃)	—	2	6	no	—
5	C₂×C₈×H(3,𝔽₃)	C₄	1	6	yes	—
6	C₂×C₃²×M₁₆	C₂×C₃	1	12	yes	—
7	C₂³×H(3,𝔽₃)	Q₈	1	1	yes	characteristic
8	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	3	yes	—
9	C₂×C₄×H(3,𝔽₃)	D₄	1	3	yes	—
10	C₈×H(3,𝔽₃)	—	2	6	no	—
11	C₃×#338	—	3	4	no	—
12	C₃×#338	C₃²	1	1	yes	characteristic
13	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
14	C₃²×M₁₆	—	2	24	no	—
15	C₂×C₃²×C₈	C₃×C₄	1	24	yes	—
16	C₂²×H(3,𝔽₃)	—	4	1	no	—
17	C₂²×H(3,𝔽₃)	C₄⋊C₄	1	3	yes	—
18	C₄×H(3,𝔽₃)	C₄²	1	1	yes	characteristic
19	C₄×H(3,𝔽₃)	C₂²⋊C₄	1	3	yes	—
20	C₂×C₃×M₁₆	—	3	12	no	—
21	C₂×C₃×M₁₆	C₂×C₃²	1	3	yes	—
22	C₂³×C₃²	C₃×Q₈	1	4	yes	—
23	C₂×C₃²×C₄	C₂×C₃×C₄	1	12	yes	—
24	C₂×C₃²×C₄	C₃×D₄	1	12	yes	—
25	C₃²×C₈	—	2	24	no	—
26	#338	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
27	C₂×H(3,𝔽₃)	—	2	3	no	—
28	C₂×H(3,𝔽₃)	#95	1	1	yes	characteristic
29	C₂²×C₃×C₄	—	3	4	no	—
30	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
31	C₃×M₁₆	—	2	6	no	—
32	C₃×M₁₆	—	6	24	no	—
33	C₂×C₃×C₈	—	3	24	no	—
34	C₂×C₃×C₈	C₃²×C₄	1	6	yes	—
35	C₂²×C₃²	—	4	4	no	—
36	C₂²×C₃²	C₃×C₄⋊C₄	1	12	yes	—
37	C₃²×C₄	C₃×C₄²	1	4	yes	—
38	C₃²×C₄	C₃×C₂²⋊C₄	1	12	yes	—
39	C₂×M₁₆	C₂×H(3,𝔽₃)	1	3	yes	—
40	H(3,𝔽₃)	#338	1	1	yes	characteristic, direct product factor, Sylow
41	C₂³×C₃	—	3	4	no	—
42	C₂³×C₃	C₃²×Q₈	1	1	yes	characteristic
43	C₂×C₃×C₄	—	3	12	no	—
44	C₂×C₃×C₄	—	3	12	no	—
45	C₂×C₃×C₄	C₂×C₃²×C₄	1	3	yes	—
46	C₂×C₃×C₄	C₃²×D₄	1	3	yes	—
47	C₃×C₈	—	2	6	no	—
48	C₃×C₈	—	6	24	no	—
49	C₂×C₃²	—	2	12	no	—
50	C₂×C₃²	C₃×#95	1	4	yes	—
51	C₂²×C₄	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
52	M₁₆	—	2	6	no	—
53	C₂×C₈	C₄×H(3,𝔽₃)	1	6	yes	—
54	C₂²×C₃	—	3	12	no	—
55	C₂²×C₃	—	4	1	no	—
56	C₂²×C₃	—	12	4	no	—
57	C₂²×C₃	C₃²×C₄⋊C₄	1	3	yes	—
58	C₃×C₄	—	3	4	no	—
59	C₃×C₄	—	3	12	no	—
60	C₃×C₄	C₃²×C₄²	1	1	yes	characteristic
61	C₃×C₄	C₃²×C₂²⋊C₄	1	3	yes	—
62	C₃²	C₃×#338	1	4	yes	complemented
63	C₂³	Q₈×H(3,𝔽₃)	1	1	yes	characteristic
64	C₂×C₄	C₂×C₄×H(3,𝔽₃)	1	3	yes	—
65	C₂×C₄	D₄×H(3,𝔽₃)	1	3	yes	—
66	C₈	—	2	6	no	—
67	C₂×C₃	—	2	3	no	—
68	C₂×C₃	—	3	4	no	—
69	C₂×C₃	—	6	12	no	—
70	C₂×C₃	C₃²×#95	1	1	yes	characteristic
71	C₂²	—	4	1	no	—
72	C₂²	C₄⋊C₄×H(3,𝔽₃)	1	3	yes	—
73	C₄	C₄²×H(3,𝔽₃)	1	1	yes	characteristic
74	C₄	C₂²⋊C₄×H(3,𝔽₃)	1	3	yes	—
75	C₃	—	3	4	no	—
76	C₃	C₃²×#338	1	1	yes	characteristic
77	C₂	—	2	3	no	—
78	C₂	H(3,𝔽₃)×#95	1	1	yes	characteristic

	subgroup	G/H
center, commutator	C₃×C₄	C₃²×C₄²
Frattini, cosocle	C₂²×C₃×C₄	C₂²×C₃²
socle	C₂×C₃	C₃²×#95
prime radical	C₂³×H(3,𝔽₃)	Q₈
LCS	G ▷ C₃×C₄ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃²×#338	C₃	semidirect
26. #338	H(3,𝔽₃)	direct (N × H)
40. H(3,𝔽₃)	#338	direct (N × H)
62. C₃²	C₃×#338	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×H(3,𝔽₃)	2
C₃²×#338	3

Subgroups — H(3,𝔽3)×#338 #13136016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)×#338 #13136016