Subgroups — C₃³×#368 #13146161

Order 1728.

560 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×#108	C₂	1	2	yes	maximal
3	C₃²×#368	C₃	1	13	yes	maximal, direct product factor
4	C₃³×M₁₆	—	2	2	no	—
5	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
6	C₃³×C₁₆	C₄	1	2	yes	—
7	C₂×C₃²×C₁₆	C₂×C₃	1	13	yes	—
8	C₃²×#108	C₂×C₃	1	26	yes	—
9	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
10	C₃³×C₈	—	4	2	no	—
11	C₃³×C₈	C₂×C₄	1	1	yes	characteristic
12	C₃³×C₈	Q₈	1	1	yes	characteristic
13	C₃×#368	C₃²	1	13	yes	direct product factor
14	C₃²×M₁₆	—	2	26	no	—
15	C₂×C₃²×C₈	C₂²×C₃	1	13	yes	—
16	C₃²×C₁₆	C₃×C₄	1	26	yes	—
17	C₂²×C₃³	Q₁₆	1	1	yes	characteristic
18	C₃³×C₄	C₄⋊C₄	1	1	yes	characteristic
19	C₃³×C₄	D₈	1	1	yes	characteristic
20	C₂×C₃×C₁₆	C₂×C₃²	1	13	yes	—
21	C₃×#108	C₂×C₃²	1	26	yes	—
22	C₂×C₃²×C₄	C₃×D₄	1	13	yes	—
23	C₃²×C₈	—	4	26	no	—
24	C₃²×C₈	C₂×C₃×C₄	1	13	yes	—
25	C₃²×C₈	C₃×Q₈	1	13	yes	—
26	#368	C₃³	1	1	yes	characteristic, direct product factor, Sylow
27	C₂×C₃³	—	2	1	no	—
28	C₂×C₃³	#107	1	1	yes	characteristic
29	C₃×M₁₆	—	2	26	no	—
30	C₂×C₃×C₈	C₂²×C₃²	1	13	yes	—
31	C₃×C₁₆	C₃²×C₄	1	26	yes	—
32	C₂²×C₃²	C₃×Q₁₆	1	13	yes	—
33	C₃²×C₄	C₃×C₄⋊C₄	1	13	yes	—
34	C₃²×C₄	C₃×D₈	1	13	yes	—
35	C₂×C₁₆	C₂×C₃³	1	1	yes	characteristic
36	#108	C₂×C₃³	1	2	yes	—
37	C₃³	#368	1	1	yes	characteristic, direct product factor, Sylow
38	C₂×C₃×C₄	C₃²×D₄	1	13	yes	—
39	C₃×C₈	—	4	26	no	—
40	C₃×C₈	C₂×C₃²×C₄	1	13	yes	—
41	C₃×C₈	C₃²×Q₈	1	13	yes	—
42	C₂×C₃²	—	2	13	no	—
43	C₂×C₃²	C₃×#107	1	13	yes	—
44	M₁₆	—	2	2	no	—
45	C₂×C₈	C₂²×C₃³	1	1	yes	characteristic
46	C₁₆	C₃³×C₄	1	2	yes	—
47	C₂²×C₃	C₃²×Q₁₆	1	13	yes	—
48	C₃×C₄	C₃²×C₄⋊C₄	1	13	yes	—
49	C₃×C₄	C₃²×D₈	1	13	yes	—
50	C₃²	C₃×#368	1	13	yes	direct product factor
51	C₂×C₄	C₃³×D₄	1	1	yes	characteristic
52	C₈	—	4	2	no	—
53	C₈	C₂×C₃³×C₄	1	1	yes	characteristic
54	C₈	C₃³×Q₈	1	1	yes	characteristic
55	C₂×C₃	—	2	13	no	—
56	C₂×C₃	C₃²×#107	1	13	yes	—
57	C₂²	C₃³×Q₁₆	1	1	yes	characteristic
58	C₄	C₃³×C₄⋊C₄	1	1	yes	characteristic
59	C₄	C₃³×D₈	1	1	yes	characteristic
60	C₃	C₃²×#368	1	13	yes	direct product factor
61	C₂	—	2	1	no	—
62	C₂	C₃³×#107	1	1	yes	characteristic

	subgroup	G/H
center	C₃³×C₄	D₈
commutator	C₈	C₂×C₃³×C₄
Frattini, cosocle	C₂×C₈	C₂²×C₃³
socle	C₂×C₃³	#107
prime radical	C₂²×C₃³	Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃³×C₄ ◁ 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×#368	C₃	direct (N × H)
13. C₃×#368	C₃²	direct (N × H)
26. #368	C₃³	direct (N × H)
37. C₃³	#368	direct (N × H)
50. C₃²	C₃×#368	direct (N × H)
60. C₃	C₃²×#368	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×#108	2
C₃²×#368	3

Subgroups — C33×#368 #13146161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×#368 #13146161