Subgroups — C₃×C₉×#384 #13144910

Order 1728.

930 conjugacy classes of subgroups.

The subgroup lattice has 126 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×C₄⋊C₄	C₂	1	6	yes	maximal
2	C₂×C₃×C₄²×C₉	C₂	1	1	yes	maximal, characteristic
3	C₃²×#384	C₃	1	1	yes	maximal, characteristic
4	C₉×#384	C₃	1	3	yes	maximal, direct product factor
5	C₂²×C₃×C₄×C₉	C₂²	1	4	yes	—
6	C₂²×C₃×C₄×C₉	C₂²	1	3	yes	—
7	C₃×C₉×C₄⋊C₄	—	2	12	no	—
8	C₃×C₄²×C₉	C₄	1	4	yes	complemented
9	C₂×C₃²×C₄⋊C₄	C₂×C₃	1	6	yes	—
10	C₂×C₃²×C₄²	C₂×C₃	1	1	yes	characteristic
11	C₂×C₉×C₄⋊C₄	C₂×C₃	1	18	yes	—
12	C₂×C₄²×C₉	C₂×C₃	1	3	yes	—
13	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
14	C₂×C₃×C₄×C₉	—	2	12	no	—
15	C₂×C₃×C₄×C₉	C₂×C₄	1	6	yes	—
16	C₂×C₃×C₄×C₉	D₄	1	6	yes	—
17	C₂×C₃×C₄×C₉	Q₈	1	6	yes	—
18	C₃×#384	C₉	1	3	yes	direct product factor
19	C₃×#384	C₃²	1	1	yes	characteristic
20	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
21	C₂²×C₃²×C₄	C₂²×C₃	1	3	yes	—
22	C₃²×C₄⋊C₄	—	2	12	no	—
23	C₃²×C₄²	C₃×C₄	1	4	yes	—
24	C₂²×C₄×C₉	C₂²×C₃	1	12	yes	—
25	C₂²×C₄×C₉	C₂²×C₃	1	9	yes	—
26	C₉×C₄⋊C₄	—	2	36	no	—
27	C₄²×C₉	C₃×C₄	1	12	yes	complemented
28	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₉	C₂×D₄	1	3	yes	—
30	C₂²×C₃×C₉	C₂×Q₈	1	3	yes	—
31	C₃×C₄×C₉	—	4	4	no	—
32	C₃×C₄×C₉	C₄⋊C₄	1	12	yes	complemented
33	C₂×C₃×C₄⋊C₄	C₂×C₉	1	18	yes	—
34	C₂×C₃×C₄⋊C₄	C₂×C₃²	1	6	yes	—
35	C₂×C₃×C₄²	C₂×C₉	1	3	yes	—
36	C₂×C₃×C₄²	C₂×C₃²	1	1	yes	characteristic
37	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
38	C₂×C₃²×C₄	—	2	12	no	—
39	C₂×C₃²×C₄	C₂×C₃×C₄	1	6	yes	—
40	C₂×C₃²×C₄	C₃×D₄	1	6	yes	—
41	C₂×C₃²×C₄	C₃×Q₈	1	6	yes	—
42	C₂³×C₉	C₂³×C₃	1	3	yes	—
43	C₂×C₄×C₉	—	2	36	no	—
44	C₂×C₄×C₉	C₂×C₃×C₄	1	18	yes	—
45	C₂×C₄×C₉	C₃×D₄	1	18	yes	—
46	C₂×C₄×C₉	C₃×Q₈	1	18	yes	—
47	#384	C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
48	C₂×C₃×C₉	C₂×C₄⋊C₄	1	3	yes	—
49	C₂×C₃×C₉	#127	1	1	yes	characteristic
50	C₂×C₃×C₉	C₄⋊Q₈	1	3	yes	—
51	C₂²×C₃×C₄	C₂²×C₉	1	12	yes	—
52	C₂²×C₃×C₄	C₂²×C₉	1	9	yes	—
53	C₂²×C₃×C₄	C₂²×C₃²	1	4	yes	—
54	C₂²×C₃×C₄	C₂²×C₃²	1	3	yes	—
55	C₃×C₄⋊C₄	—	2	36	no	—
56	C₃×C₄⋊C₄	—	2	12	no	—
57	C₃×C₄²	C₄×C₉	1	12	yes	complemented
58	C₃×C₄²	C₃²×C₄	1	4	yes	—
59	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
60	C₂²×C₃²	C₂×C₃×D₄	1	3	yes	—
61	C₂²×C₃²	C₂×C₃×Q₈	1	3	yes	—
62	C₃²×C₄	—	4	4	no	—
63	C₃²×C₄	C₃×C₄⋊C₄	1	12	yes	—
64	C₂²×C₉	C₂²×C₃×C₄	1	3	yes	—
65	C₂²×C₉	C₂×C₃×D₄	1	9	yes	—
66	C₂²×C₉	C₂×C₃×Q₈	1	9	yes	—
67	C₄×C₉	—	4	12	no	—
68	C₄×C₉	C₃×C₄⋊C₄	1	36	yes	complemented
69	C₂×C₄⋊C₄	C₂×C₃×C₉	1	6	yes	—
70	C₂×C₄²	C₂×C₃×C₉	1	1	yes	characteristic
71	C₃×C₉	#384	1	1	yes	characteristic, direct product factor, Sylow
72	C₂³×C₃	C₂³×C₉	1	3	yes	—
73	C₂³×C₃	C₂³×C₃²	1	1	yes	characteristic
74	C₂×C₃×C₄	—	2	36	no	—
75	C₂×C₃×C₄	—	2	12	no	—
76	C₂×C₃×C₄	C₂×C₄×C₉	1	18	yes	—
77	C₂×C₃×C₄	D₄×C₉	1	18	yes	—
78	C₂×C₃×C₄	Q₈×C₉	1	18	yes	—
79	C₂×C₃×C₄	C₂×C₃²×C₄	1	6	yes	—
80	C₂×C₃×C₄	C₃²×D₄	1	6	yes	—
81	C₂×C₃×C₄	C₃²×Q₈	1	6	yes	—
82	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	3	yes	—
83	C₂×C₃²	C₃×#127	1	1	yes	characteristic
84	C₂×C₃²	C₃×C₄⋊Q₈	1	3	yes	—
85	C₂×C₉	C₂×C₃×C₄⋊C₄	1	9	yes	—
86	C₂×C₉	C₃×#127	1	3	yes	—
87	C₂×C₉	C₃×C₄⋊Q₈	1	9	yes	—
88	C₂²×C₄	C₂²×C₃×C₉	1	4	yes	—
89	C₂²×C₄	C₂²×C₃×C₉	1	3	yes	—
90	C₄⋊C₄	—	2	12	no	—
91	C₄²	C₃×C₄×C₉	1	4	yes	complemented
92	C₂²×C₃	C₂²×C₄×C₉	1	3	yes	—
93	C₂²×C₃	C₂×D₄×C₉	1	9	yes	—
94	C₂²×C₃	C₂×Q₈×C₉	1	9	yes	—
95	C₂²×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
96	C₂²×C₃	C₂×C₃²×D₄	1	3	yes	—
97	C₂²×C₃	C₂×C₃²×Q₈	1	3	yes	—
98	C₃×C₄	—	4	12	no	—
99	C₃×C₄	—	4	4	no	—
100	C₃×C₄	C₉×C₄⋊C₄	1	36	yes	complemented
101	C₃×C₄	C₃²×C₄⋊C₄	1	12	yes	—
102	C₃²	C₃×#384	1	1	yes	characteristic
103	C₉	C₃×#384	1	3	yes	direct product factor
104	C₂³	C₂³×C₃×C₉	1	1	yes	characteristic
105	C₂×C₄	—	2	12	no	—
106	C₂×C₄	C₂×C₃×C₄×C₉	1	6	yes	—
107	C₂×C₄	C₃×D₄×C₉	1	6	yes	—
108	C₂×C₄	C₃×Q₈×C₉	1	6	yes	—
109	C₂×C₃	C₂×C₉×C₄⋊C₄	1	9	yes	—
110	C₂×C₃	C₉×#127	1	3	yes	—
111	C₂×C₃	C₉×C₄⋊Q₈	1	9	yes	—
112	C₂×C₃	C₂×C₃²×C₄⋊C₄	1	3	yes	—
113	C₂×C₃	C₃²×#127	1	1	yes	characteristic
114	C₂×C₃	C₃²×C₄⋊Q₈	1	3	yes	—
115	C₂²	C₂²×C₃×C₄×C₉	1	1	yes	characteristic
116	C₂²	C₂×C₃×D₄×C₉	1	3	yes	—
117	C₂²	C₂×C₃×Q₈×C₉	1	3	yes	—
118	C₄	—	4	4	no	—
119	C₄	C₃×C₉×C₄⋊C₄	1	12	yes	complemented
120	C₃	C₉×#384	1	3	yes	direct product factor
121	C₃	C₃²×#384	1	1	yes	characteristic
122	C₂	C₂×C₃×C₉×C₄⋊C₄	1	3	yes	—
123	C₂	C₃×C₉×#127	1	1	yes	characteristic
124	C₂	C₃×C₉×C₄⋊Q₈	1	3	yes	—

	subgroup	G/H
center	C₂³×C₃×C₉	C₂³
commutator	C₂²	C₂²×C₃×C₄×C₉
Frattini, cosocle	C₂³×C₃	C₂³×C₃²
socle, prime radical	C₂³×C₃²	C₂³×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₉×#384	C₃	direct (N × H)
8. C₃×C₄²×C₉	C₄	semidirect
18. C₃×#384	C₉	direct (N × H)
27. C₄²×C₉	C₃×C₄	semidirect
32. C₃×C₄×C₉	C₄⋊C₄	semidirect
47. #384	C₃×C₉	direct (N × H)
57. C₃×C₄²	C₄×C₉	semidirect
68. C₄×C₉	C₃×C₄⋊C₄	semidirect
71. C₃×C₉	#384	direct (N × H)
91. C₄²	C₃×C₄×C₉	semidirect
100. C₃×C₄	C₉×C₄⋊C₄	semidirect
103. C₉	C₃×#384	direct (N × H)
119. C₄	C₃×C₉×C₄⋊C₄	semidirect
120. C₃	C₉×#384	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×C₃×C₉×C₄⋊C₄	2
C₂×C₃×C₄²×C₉	2
C₃²×#384	3
C₉×#384	3

Subgroups — C3×C9×#384 #13144910

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₉×#384 #13144910