Subgroups — C₃³×#331 #13146124

Order 1728.

1176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×#127	C₂	1	1	yes	maximal, characteristic
2	C₃³×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₃²×#331	C₃	1	13	yes	maximal, direct product factor
4	C₂×C₃³×D₄	—	2	2	no	—
5	C₂×C₃³×D₄	C₄	1	2	yes	—
6	C₂×C₃³×C₈	—	2	2	no	—
7	C₃³×C₄²	C₂²	1	1	yes	characteristic
8	C₃²×#127	C₂×C₃	1	13	yes	—
9	C₃²×C₄⋊C₈	C₂×C₃	1	26	yes	complemented
10	C₂³×C₃³	—	2	2	no	—
11	C₃³×D₄	—	4	2	no	—
12	C₃³×D₄	—	4	2	no	—
13	C₃³×D₄	—	4	2	no	—
14	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
15	C₂×C₃³×C₄	D₄	1	2	yes	—
16	C₃³×C₈	—	4	2	no	—
17	C₃×#331	C₃²	1	13	yes	direct product factor
18	C₂×C₃²×D₄	—	2	26	no	—
19	C₂×C₃²×D₄	C₃×C₄	1	26	yes	—
20	C₂×C₃²×C₈	—	2	26	no	—
21	C₃²×C₄²	C₂²×C₃	1	13	yes	—
22	C₂²×C₃³	—	4	2	no	—
23	C₂²×C₃³	—	4	4	no	—
24	C₂²×C₃³	C₂²⋊C₄	1	1	yes	characteristic
25	C₃³×C₄	—	2	1	no	—
26	C₃³×C₄	D₈	1	2	yes	—
27	C₃³×C₄	SD₁₆	1	2	yes	—
28	C₃×#127	C₂×C₃²	1	13	yes	—
29	C₃×C₄⋊C₈	C₂×C₃²	1	26	yes	complemented
30	C₂³×C₃²	—	2	26	no	—
31	C₃²×D₄	—	4	26	no	—
32	C₃²×D₄	—	4	26	no	—
33	C₃²×D₄	—	4	26	no	—
34	C₂×C₃²×C₄	C₂×C₃×C₄	1	13	yes	—
35	C₂×C₃²×C₄	C₃×D₄	1	26	yes	—
36	C₃²×C₈	—	4	26	no	—
37	#331	C₃³	1	1	yes	characteristic, direct product factor, Sylow
38	C₂×C₃³	—	8	2	no	—
39	C₂×C₃³	#100	1	1	yes	characteristic
40	C₂×C₃³	#102	1	2	yes	—
41	C₂×C₃×D₄	—	2	26	no	—
42	C₂×C₃×D₄	C₃²×C₄	1	26	yes	—
43	C₂×C₃×C₈	—	2	26	no	—
44	C₃×C₄²	C₂²×C₃²	1	13	yes	—
45	C₂²×C₃²	—	4	26	no	—
46	C₂²×C₃²	—	4	52	no	—
47	C₂²×C₃²	C₃×C₂²⋊C₄	1	13	yes	—
48	C₃²×C₄	—	2	13	no	—
49	C₃²×C₄	C₃×D₈	1	26	yes	—
50	C₃²×C₄	C₃×SD₁₆	1	26	yes	—
51	#127	C₂×C₃³	1	1	yes	characteristic
52	C₄⋊C₈	C₂×C₃³	1	2	yes	complemented
53	C₃³	#331	1	1	yes	characteristic, direct product factor, Sylow
54	C₂³×C₃	—	2	26	no	—
55	C₃×D₄	—	4	26	no	—
56	C₃×D₄	—	4	26	no	—
57	C₃×D₄	—	4	26	no	—
58	C₂×C₃×C₄	C₂×C₃²×C₄	1	13	yes	—
59	C₂×C₃×C₄	C₃²×D₄	1	26	yes	—
60	C₃×C₈	—	4	26	no	—
61	C₂×C₃²	—	8	26	no	—
62	C₂×C₃²	C₃×#100	1	13	yes	—
63	C₂×C₃²	C₃×#102	1	26	yes	—
64	C₂×D₄	—	2	2	no	—
65	C₂×D₄	C₃³×C₄	1	2	yes	—
66	C₂×C₈	—	2	2	no	—
67	C₄²	C₂²×C₃³	1	1	yes	characteristic
68	C₂²×C₃	—	4	26	no	—
69	C₂²×C₃	—	4	52	no	—
70	C₂²×C₃	C₃²×C₂²⋊C₄	1	13	yes	—
71	C₃×C₄	—	2	13	no	—
72	C₃×C₄	C₃²×D₈	1	26	yes	—
73	C₃×C₄	C₃²×SD₁₆	1	26	yes	—
74	C₃²	C₃×#331	1	13	yes	direct product factor
75	C₂³	—	2	2	no	—
76	D₄	—	4	2	no	—
77	D₄	—	4	2	no	—
78	D₄	—	4	2	no	—
79	C₂×C₄	C₂×C₃³×C₄	1	1	yes	characteristic
80	C₂×C₄	C₃³×D₄	1	2	yes	—
81	C₈	—	4	2	no	—
82	C₂×C₃	—	8	26	no	—
83	C₂×C₃	C₃²×#100	1	13	yes	—
84	C₂×C₃	C₃²×#102	1	26	yes	—
85	C₂²	—	4	2	no	—
86	C₂²	—	4	4	no	—
87	C₂²	C₃³×C₂²⋊C₄	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	C₃³×D₈	1	2	yes	—
90	C₄	C₃³×SD₁₆	1	2	yes	—
91	C₃	C₃²×#331	1	13	yes	direct product factor
92	C₂	—	8	2	no	—
93	C₂	C₃³×#100	1	1	yes	characteristic
94	C₂	C₃³×#102	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃³	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₃³×C₄
Frattini, cosocle	C₄²	C₂²×C₃³
prime radical	C₃³×#127	C₂
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
2. C₃³×C₄⋊C₈	C₂	semidirect
3. C₃²×#331	C₃	direct (N × H)
9. C₃²×C₄⋊C₈	C₂×C₃	semidirect
17. C₃×#331	C₃²	direct (N × H)
29. C₃×C₄⋊C₈	C₂×C₃²	semidirect
37. #331	C₃³	direct (N × H)
52. C₄⋊C₈	C₂×C₃³	semidirect
53. C₃³	#331	direct (N × H)
74. C₃²	C₃×#331	direct (N × H)
91. C₃	C₃²×#331	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×#127	2
C₃³×C₄⋊C₈	2
C₃²×#331	3

Subgroups — C33×#331 #13146124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×#331 #13146124