Subgroups — C₃³×#367 #13146160

Order 1728.

644 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃³×#107	C₂	1	2	yes	maximal
3	C₃²×#367	C₃	1	13	yes	maximal, direct product factor
4	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
5	C₃³×C₄⋊C₄	—	2	2	no	—
6	C₃³×C₁₆	C₄	1	2	yes	complemented
7	C₂×C₃²×C₁₆	C₂×C₃	1	13	yes	—
8	C₃²×#107	C₂×C₃	1	26	yes	—
9	C₂×C₃³×C₄	—	4	2	no	—
10	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
11	C₃³×C₈	C₂×C₄	1	1	yes	characteristic
12	C₃³×C₈	Q₈	1	1	yes	characteristic
13	C₃×#367	C₃²	1	13	yes	direct product factor
14	C₂×C₃²×C₈	C₂²×C₃	1	13	yes	—
15	C₃²×C₄⋊C₄	—	2	26	no	—
16	C₃²×C₁₆	C₃×C₄	1	26	yes	complemented
17	C₂²×C₃³	D₈	1	1	yes	characteristic
18	C₃³×C₄	—	8	2	no	—
19	C₃³×C₄	C₄⋊C₄	1	1	yes	characteristic
20	C₃³×C₄	Q₁₆	1	1	yes	characteristic
21	C₂×C₃×C₁₆	C₂×C₃²	1	13	yes	—
22	C₃×#107	C₂×C₃²	1	26	yes	—
23	C₂×C₃²×C₄	—	4	26	no	—
24	C₂×C₃²×C₄	C₃×D₄	1	13	yes	—
25	C₃²×C₈	C₂×C₃×C₄	1	13	yes	—
26	C₃²×C₈	C₃×Q₈	1	13	yes	—
27	#367	C₃³	1	1	yes	characteristic, direct product factor, Sylow
28	C₂×C₃³	#107	1	1	yes	characteristic
29	C₂×C₃³	SD₃₂	1	2	yes	—
30	C₂×C₃×C₈	C₂²×C₃²	1	13	yes	—
31	C₃×C₄⋊C₄	—	2	26	no	—
32	C₃×C₁₆	C₃²×C₄	1	26	yes	complemented
33	C₂²×C₃²	C₃×D₈	1	13	yes	—
34	C₃²×C₄	—	8	26	no	—
35	C₃²×C₄	C₃×C₄⋊C₄	1	13	yes	—
36	C₃²×C₄	C₃×Q₁₆	1	13	yes	—
37	C₂×C₁₆	C₂×C₃³	1	1	yes	characteristic
38	#107	C₂×C₃³	1	2	yes	—
39	C₃³	#367	1	1	yes	characteristic, direct product factor, Sylow
40	C₂×C₃×C₄	—	4	26	no	—
41	C₂×C₃×C₄	C₃²×D₄	1	13	yes	—
42	C₃×C₈	C₂×C₃²×C₄	1	13	yes	—
43	C₃×C₈	C₃²×Q₈	1	13	yes	—
44	C₂×C₃²	C₃×#107	1	13	yes	—
45	C₂×C₃²	C₃×SD₃₂	1	26	yes	—
46	C₂×C₈	C₂²×C₃³	1	1	yes	characteristic
47	C₄⋊C₄	—	2	2	no	—
48	C₁₆	C₃³×C₄	1	2	yes	complemented
49	C₂²×C₃	C₃²×D₈	1	13	yes	—
50	C₃×C₄	—	8	26	no	—
51	C₃×C₄	C₃²×C₄⋊C₄	1	13	yes	—
52	C₃×C₄	C₃²×Q₁₆	1	13	yes	—
53	C₃²	C₃×#367	1	13	yes	direct product factor
54	C₂×C₄	—	4	2	no	—
55	C₂×C₄	C₃³×D₄	1	1	yes	characteristic
56	C₈	C₂×C₃³×C₄	1	1	yes	characteristic
57	C₈	C₃³×Q₈	1	1	yes	characteristic
58	C₂×C₃	C₃²×#107	1	13	yes	—
59	C₂×C₃	C₃²×SD₃₂	1	26	yes	—
60	C₂²	C₃³×D₈	1	1	yes	characteristic
61	C₄	—	8	2	no	—
62	C₄	C₃³×C₄⋊C₄	1	1	yes	characteristic
63	C₄	C₃³×Q₁₆	1	1	yes	characteristic
64	C₃	C₃²×#367	1	13	yes	direct product factor
65	C₂	C₃³×#107	1	1	yes	characteristic
66	C₂	C₃³×SD₃₂	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₃³	D₈
commutator	C₈	C₂×C₃³×C₄
Frattini, cosocle	C₂×C₈	C₂²×C₃³
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃³ ◁ C₂×C₃³×C₄ ◁ C₂×C₃³×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃²×#367	C₃	direct (N × H)
6. C₃³×C₁₆	C₄	semidirect
13. C₃×#367	C₃²	direct (N × H)
16. C₃²×C₁₆	C₃×C₄	semidirect
27. #367	C₃³	direct (N × H)
32. C₃×C₁₆	C₃²×C₄	semidirect
39. C₃³	#367	direct (N × H)
48. C₁₆	C₃³×C₄	semidirect
53. C₃²	C₃×#367	direct (N × H)
64. C₃	C₃²×#367	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃³×C₁₆	2
C₃³×#107	2
C₃²×#367	3

Subgroups — C33×#367 #13146160

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×#367 #13146160