Subgroups — C₃³×#330 #13146123

Order 1728.

840 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×#125	C₂	1	1	yes	maximal, characteristic
2	C₃³×#97	C₂	1	2	yes	maximal
3	C₃²×#330	C₃	1	13	yes	maximal, direct product factor
4	C₂×C₃³×C₈	—	2	2	no	—
5	C₃³×C₄⋊C₄	—	2	2	no	—
6	C₃³×C₄⋊C₄	C₄	1	2	yes	—
7	C₃³×C₄²	C₂²	1	1	yes	characteristic
8	C₃²×#125	C₂×C₃	1	13	yes	—
9	C₃²×#97	C₂×C₃	1	26	yes	—
10	C₂×C₃³×C₄	—	2	2	no	—
11	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
12	C₂×C₃³×C₄	D₄	1	2	yes	—
13	C₃³×C₈	—	2	4	no	—
14	C₃×#330	C₃²	1	13	yes	direct product factor
15	C₂×C₃²×C₈	—	2	26	no	—
16	C₃²×C₄⋊C₄	—	2	26	no	—
17	C₃²×C₄⋊C₄	C₃×C₄	1	26	yes	—
18	C₃²×C₄²	C₂²×C₃	1	13	yes	—
19	C₂²×C₃³	C₂²⋊C₄	1	1	yes	characteristic
20	C₃³×C₄	—	2	1	no	—
21	C₃³×C₄	—	2	2	no	—
22	C₃³×C₄	—	4	2	no	—
23	C₃×#125	C₂×C₃²	1	13	yes	—
24	C₃×#97	C₂×C₃²	1	26	yes	—
25	C₂×C₃²×C₄	—	2	26	no	—
26	C₂×C₃²×C₄	C₂×C₃×C₄	1	13	yes	—
27	C₂×C₃²×C₄	C₃×D₄	1	26	yes	—
28	C₃²×C₈	—	2	52	no	—
29	#330	C₃³	1	1	yes	characteristic, direct product factor, Sylow
30	C₂×C₃³	#101	1	1	yes	characteristic
31	C₂×C₃³	#104	1	2	yes	—
32	C₂×C₃×C₈	—	2	26	no	—
33	C₃×C₄⋊C₄	—	2	26	no	—
34	C₃×C₄⋊C₄	C₃²×C₄	1	26	yes	—
35	C₃×C₄²	C₂²×C₃²	1	13	yes	—
36	C₂²×C₃²	C₃×C₂²⋊C₄	1	13	yes	—
37	C₃²×C₄	—	2	13	no	—
38	C₃²×C₄	—	2	26	no	—
39	C₃²×C₄	—	4	26	no	—
40	#125	C₂×C₃³	1	1	yes	characteristic
41	#97	C₂×C₃³	1	2	yes	—
42	C₃³	#330	1	1	yes	characteristic, direct product factor, Sylow
43	C₂×C₃×C₄	—	2	26	no	—
44	C₂×C₃×C₄	C₂×C₃²×C₄	1	13	yes	—
45	C₂×C₃×C₄	C₃²×D₄	1	26	yes	—
46	C₃×C₈	—	2	52	no	—
47	C₂×C₃²	C₃×#101	1	13	yes	—
48	C₂×C₃²	C₃×#104	1	26	yes	—
49	C₂×C₈	—	2	2	no	—
50	C₄⋊C₄	—	2	2	no	—
51	C₄⋊C₄	C₃³×C₄	1	2	yes	—
52	C₄²	C₂²×C₃³	1	1	yes	characteristic
53	C₂²×C₃	C₃²×C₂²⋊C₄	1	13	yes	—
54	C₃×C₄	—	2	13	no	—
55	C₃×C₄	—	2	26	no	—
56	C₃×C₄	—	4	26	no	—
57	C₃²	C₃×#330	1	13	yes	direct product factor
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	C₂×C₃³×C₄	1	1	yes	characteristic
60	C₂×C₄	C₃³×D₄	1	2	yes	—
61	C₈	—	2	4	no	—
62	C₂×C₃	C₃²×#101	1	13	yes	—
63	C₂×C₃	C₃²×#104	1	26	yes	—
64	C₂²	C₃³×C₂²⋊C₄	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	—	2	2	no	—
67	C₄	—	4	2	no	—
68	C₃	C₃²×#330	1	13	yes	direct product factor
69	C₂	C₃³×#101	1	1	yes	characteristic
70	C₂	C₃³×#104	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₃³	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₃³×C₄
Frattini, cosocle	C₄²	C₂²×C₃³
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃³ ◁ C₃³×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃²×#330	C₃	direct (N × H)
14. C₃×#330	C₃²	direct (N × H)
29. #330	C₃³	direct (N × H)
42. C₃³	#330	direct (N × H)
57. C₃²	C₃×#330	direct (N × H)
68. C₃	C₃²×#330	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×#125	2
C₃³×#97	2
C₃²×#330	3

Subgroups — C33×#330 #13146123

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×#330 #13146123