Subgroups — C₄×Dic₃×Dic₉ #13139530

Order 1728.

882 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₃⋊Dic₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₉×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×Dic₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃×Dic₉	C₂	1	4	yes	maximal
5	C₄×Dic₃²	—	3	1	no	maximal
6	C₄²×Dic₉	—	3	1	no	maximal
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊Dic₉	C₂²	1	2	yes	—
9	C₂²×C₉×Dic₃	C₂²	1	2	yes	—
10	C₂²×C₃×Dic₉	C₂²	1	2	yes	—
11	C₄×C₃⋊Dic₉	C₄	1	4	yes	complemented
12	C₄×C₉×Dic₃	C₄	1	4	yes	complemented
13	C₃×C₄×Dic₉	C₄	1	4	yes	complemented
14	Dic₃×Dic₉	C₄	1	16	yes	direct product factor
15	C₂×C₄×C₃⋊Dic₃	—	3	1	no	—
16	C₂×C₃×C₄×Dic₃	—	3	1	no	—
17	C₂×C₃×C₄×Dic₃	D₃	1	1	yes	characteristic
18	C₂×Dic₃²	—	3	4	no	—
19	C₂×C₄²×C₉	—	3	1	no	—
20	C₂×C₄×Dic₉	—	3	1	no	—
21	C₂×C₄×Dic₉	—	3	4	no	—
22	C₂×C₄×Dic₉	—	6	1	no	—
23	C₂×C₄×Dic₉	D₃	1	1	yes	characteristic
24	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
25	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
26	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
28	C₂×C₃⋊Dic₉	C₂×C₄	1	8	yes	—
29	C₂×C₃⋊Dic₉	C₂×C₄	1	4	yes	—
30	C₂×C₉×Dic₃	C₂×C₄	1	8	yes	—
31	C₂×C₉×Dic₃	C₂×C₄	1	4	yes	—
32	C₂×C₃×Dic₉	C₂×C₄	1	8	yes	—
33	C₂×C₃×Dic₉	C₂×C₄	1	4	yes	—
34	C₄²×Dic₃	—	9	1	no	—
35	C₄²×Dic₃	—	9	1	no	—
36	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
37	C₂²×C₃⋊Dic₃	—	3	2	no	—
38	C₂²×C₃×Dic₃	—	3	2	no	—
39	C₂²×C₃×Dic₃	C₂×D₃	1	2	yes	—
40	C₄×C₃⋊Dic₃	—	3	4	no	—
41	C₃×C₄×Dic₃	—	3	4	no	—
42	C₃×C₄×Dic₃	Dic₃	1	4	yes	—
43	Dic₃²	—	3	16	no	—
44	C₂²×C₄×C₉	—	2	1	no	—
45	C₂²×C₄×C₉	—	3	2	no	—
46	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
47	C₂²×Dic₉	—	3	2	no	—
48	C₂²×Dic₉	—	6	2	no	—
49	C₂²×Dic₉	C₂×D₃	1	2	yes	—
50	C₄²×C₉	—	3	4	no	—
51	C₄×Dic₉	—	3	4	no	—
52	C₄×Dic₉	—	3	16	no	—
53	C₄×Dic₉	—	6	4	no	—
54	C₄×Dic₉	Dic₃	1	4	yes	direct product factor
55	C₂²×C₃×C₉	C₂²×C₄	1	4	yes	—
56	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
59	C₃×C₄×C₉	C₄²	1	4	yes	complemented
60	C₃⋊Dic₉	C₄²	1	8	yes	complemented
61	C₉×Dic₃	C₄²	1	8	yes	complemented
62	C₃×Dic₉	C₄²	1	8	yes	complemented
63	C₂×C₃×C₄²	—	3	1	no	—
64	C₂×C₃×C₄²	—	9	1	no	—
65	C₂×C₄×Dic₃	—	3	1	no	—
66	C₂×C₄×Dic₃	—	9	4	no	—
67	C₂×C₄×Dic₃	—	9	1	no	—
68	C₂×C₄×Dic₃	—	9	1	no	—
69	C₂×C₄×Dic₃	—	9	4	no	—
70	C₂×C₄×Dic₃	—	18	1	no	—
71	C₂×C₄×Dic₃	D₉	1	1	yes	characteristic
72	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
74	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
75	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
76	C₂×C₃⋊Dic₃	—	3	8	no	—
77	C₂×C₃⋊Dic₃	—	3	4	no	—
78	C₂×C₃×Dic₃	—	3	8	no	—
79	C₂×C₃×Dic₃	—	3	4	no	—
80	C₂×C₃×Dic₃	C₄×D₃	1	8	yes	—
81	C₂×C₃×Dic₃	C₂×Dic₃	1	4	yes	—
82	C₂³×C₉	—	2	1	no	—
83	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
84	C₂×C₄×C₉	—	2	2	no	—
85	C₂×C₄×C₉	—	2	2	no	—
86	C₂×C₄×C₉	—	2	2	no	—
87	C₂×C₄×C₉	—	3	8	no	—
88	C₂×C₄×C₉	—	3	4	no	—
89	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
90	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
91	C₂×C₄×C₉	C₂×Dic₃	1	2	yes	—
92	C₂×Dic₉	—	3	8	no	—
93	C₂×Dic₉	—	3	4	no	—
94	C₂×Dic₉	—	6	4	no	—
95	C₂×Dic₉	—	6	8	no	—
96	C₂×Dic₉	C₄×D₃	1	8	yes	—
97	C₂×Dic₉	C₂×Dic₃	1	4	yes	—
98	C₄³	—	27	1	no	Sylow
99	C₂×C₃×C₉	C₂×C₄²	1	2	yes	—
100	C₂×C₃×C₉	C₂×C₄²	1	2	yes	—
101	C₂×C₃×C₉	C₂×C₄²	1	2	yes	—
102	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
103	C₂²×C₃×C₄	—	2	1	no	—
104	C₂²×C₃×C₄	—	3	2	no	—
105	C₂²×C₃×C₄	—	9	2	no	—
106	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
107	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
108	C₂²×Dic₃	—	3	2	no	—
109	C₂²×Dic₃	—	9	2	no	—
110	C₂²×Dic₃	—	9	2	no	—
111	C₂²×Dic₃	—	18	2	no	—
112	C₂²×Dic₃	C₂×D₉	1	2	yes	—
113	C₃×C₄²	—	3	4	no	—
114	C₃×C₄²	—	9	4	no	—
115	C₄×Dic₃	—	3	4	no	—
116	C₄×Dic₃	—	9	16	no	—
117	C₄×Dic₃	—	9	4	no	—
118	C₄×Dic₃	—	9	4	no	—
119	C₄×Dic₃	—	9	16	no	—
120	C₄×Dic₃	—	18	4	no	—
121	C₄×Dic₃	Dic₉	1	4	yes	direct product factor
122	C₂²×C₃²	C₂×C₄×D₃	1	4	yes	—
123	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
124	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
125	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
126	C₃²×C₄	C₄×Dic₃	1	4	yes	—
127	C₃⋊Dic₃	—	3	8	no	—
128	C₃×Dic₃	—	3	8	no	—
129	C₃×Dic₃	C₄×Dic₃	1	8	yes	—
130	C₂²×C₉	—	2	4	no	—
131	C₂²×C₉	—	2	1	no	—
132	C₂²×C₉	—	2	1	no	—
133	C₂²×C₉	—	2	1	no	—
134	C₂²×C₉	C₂×C₄×D₃	1	4	yes	—
135	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
136	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
137	C₂²×C₉	C₂²×Dic₃	1	1	yes	characteristic
138	C₄×C₉	—	2	4	no	—
139	C₄×C₉	—	3	8	no	—
140	C₄×C₉	C₄×Dic₃	1	4	yes	complemented
141	Dic₉	—	3	8	no	—
142	Dic₉	—	6	8	no	—
143	Dic₉	C₄×Dic₃	1	8	yes	direct product factor
144	C₂×C₄²	—	3	1	no	—
145	C₂×C₄²	—	9	1	no	—
146	C₂×C₄²	—	27	4	no	—
147	C₂×C₄²	—	27	1	no	—
148	C₃×C₉	C₄³	1	1	yes	characteristic, complemented, Sylow
149	C₂³×C₃	—	2	1	no	—
150	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
151	C₂³×C₃	C₂×D₃²	1	1	yes	characteristic
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	3	8	no	—
156	C₂×C₃×C₄	—	3	4	no	—
157	C₂×C₃×C₄	—	9	8	no	—
158	C₂×C₃×C₄	—	9	4	no	—
159	C₂×C₃×C₄	C₄×D₉	1	2	yes	—
160	C₂×C₃×C₄	C₄×D₉	1	2	yes	—
161	C₂×C₃×C₄	C₂×Dic₉	1	2	yes	—
162	C₂×C₃×C₄	D₃×Dic₃	1	2	yes	—
163	C₂×C₃×C₄	D₃×Dic₃	1	2	yes	—
164	C₂×C₃×C₄	#624	1	2	yes	—
165	C₂×Dic₃	—	3	8	no	—
166	C₂×Dic₃	—	3	4	no	—
167	C₂×Dic₃	—	9	8	no	—
168	C₂×Dic₃	—	9	4	no	—
169	C₂×Dic₃	—	9	8	no	—
170	C₂×Dic₃	—	9	4	no	—
171	C₂×Dic₃	—	18	4	no	—
172	C₂×Dic₃	—	18	8	no	—
173	C₂×Dic₃	C₄×D₉	1	8	yes	—
174	C₂×Dic₃	C₂×Dic₉	1	4	yes	—
175	C₂×C₃²	C₄²×D₃	1	2	yes	—
176	C₂×C₃²	C₄²×D₃	1	2	yes	—
177	C₂×C₃²	C₂×C₄×Dic₃	1	2	yes	—
178	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
179	C₂×C₉	—	2	2	no	—
180	C₂×C₉	—	2	2	no	—
181	C₂×C₉	—	2	2	no	—
182	C₂×C₉	—	2	1	no	—
183	C₂×C₉	C₄²×D₃	1	2	yes	—
184	C₂×C₉	C₄²×D₃	1	2	yes	—
185	C₂×C₉	C₂×C₄×Dic₃	1	2	yes	—
186	C₂×C₉	C₂×C₄×Dic₃	1	1	yes	characteristic
187	C₂²×C₄	—	3	2	no	—
188	C₂²×C₄	—	9	2	no	—
189	C₂²×C₄	—	27	2	no	—
190	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
191	C₄²	—	3	4	no	—
192	C₄²	—	9	4	no	—
193	C₄²	—	27	16	no	—
194	C₄²	—	27	4	no	—
195	C₂²×C₃	—	2	4	no	—
196	C₂²×C₃	—	2	1	no	—
197	C₂²×C₃	—	2	1	no	—
198	C₂²×C₃	—	2	1	no	—
199	C₂²×C₃	C₂×C₄×D₉	1	4	yes	—
200	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
201	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
202	C₂²×C₃	C₂²×Dic₉	1	1	yes	characteristic
203	C₂²×C₃	C₄×D₃²	1	4	yes	—
204	C₂²×C₃	C₂×D₃×Dic₃	1	1	yes	characteristic
205	C₂²×C₃	C₂×D₃×Dic₃	1	1	yes	characteristic
206	C₂²×C₃	C₂×#624	1	1	yes	characteristic
207	C₃×C₄	—	2	4	no	—
208	C₃×C₄	—	3	8	no	—
209	C₃×C₄	—	9	8	no	—
210	C₃×C₄	C₄×Dic₉	1	4	yes	complemented
211	C₃×C₄	Dic₃²	1	4	yes	—
212	Dic₃	—	3	8	no	—
213	Dic₃	—	9	8	no	—
214	Dic₃	—	9	8	no	—
215	Dic₃	—	18	8	no	—
216	Dic₃	C₄×Dic₉	1	8	yes	direct product factor
217	C₃²	C₄²×Dic₃	1	1	yes	characteristic
218	C₉	—	2	1	no	—
219	C₉	C₄²×Dic₃	1	1	yes	characteristic, complemented
220	C₂³	C₂×D₃×D₉	1	1	yes	characteristic
221	C₂×C₄	—	3	8	no	—
222	C₂×C₄	—	3	4	no	—
223	C₂×C₄	—	9	8	no	—
224	C₂×C₄	—	9	4	no	—
225	C₂×C₄	—	27	8	no	—
226	C₂×C₄	—	27	4	no	—
227	C₂×C₄	Dic₃×D₉	1	2	yes	—
228	C₂×C₄	(C₃⋊D₉)⋊C₄	1	2	yes	—
229	C₂×C₄	D₃×Dic₉	1	2	yes	—
230	C₂×C₃	—	2	2	no	—
231	C₂×C₃	—	2	2	no	—
232	C₂×C₃	—	2	1	no	—
233	C₂×C₃	—	2	2	no	—
234	C₂×C₃	C₄²×D₉	1	2	yes	—
235	C₂×C₃	C₄²×D₉	1	2	yes	—
236	C₂×C₃	C₂×C₄×Dic₉	1	1	yes	characteristic
237	C₂×C₃	C₂×C₄×Dic₉	1	2	yes	—
238	C₂×C₃	C₄×D₃×Dic₃	1	2	yes	—
239	C₂×C₃	C₄×D₃×Dic₃	1	2	yes	—
240	C₂×C₃	C₄×#624	1	2	yes	—
241	C₂×C₃	C₂×Dic₃²	1	1	yes	characteristic
242	C₂²	C₄×D₃×D₉	1	4	yes	—
243	C₂²	C₂×Dic₃×D₉	1	1	yes	characteristic
244	C₂²	C₂×(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
245	C₂²	C₂×D₃×Dic₉	1	1	yes	characteristic
246	C₄	—	3	8	no	—
247	C₄	—	9	8	no	—
248	C₄	—	27	8	no	—
249	C₄	Dic₃×Dic₉	1	4	yes	direct product factor
250	C₃	—	2	1	no	—
251	C₃	C₄²×Dic₉	1	1	yes	characteristic, complemented
252	C₃	C₄×Dic₃²	1	1	yes	characteristic
253	C₂	C₄×Dic₃×D₉	1	2	yes	—
254	C₂	C₄×(C₃⋊D₉)⋊C₄	1	2	yes	—
255	C₂	C₄×D₃×Dic₉	1	2	yes	—
256	C₂	C₂×Dic₃×Dic₉	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄	D₃×D₉
commutator, hypocenter	C₃×C₉	C₄³
Frattini	C₂³×C₃	C₂×D₃²
Fitting	C₂²×C₃×C₄×C₉	C₂²
socle, prime radical	C₂³×C₃²	C₂²×D₃
cosocle	C₂³×C₃×C₉	C₂³
LCS	G ▷ C₃×C₉
UCS	C₁ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
11. C₄×C₃⋊Dic₉	C₄	semidirect
12. C₄×C₉×Dic₃	C₄	semidirect
13. C₃×C₄×Dic₉	C₄	semidirect
14. Dic₃×Dic₉	C₄	direct (N × H)
54. C₄×Dic₉	Dic₃	direct (N × H)
59. C₃×C₄×C₉	C₄²	semidirect
60. C₃⋊Dic₉	C₄²	semidirect
61. C₉×Dic₃	C₄²	semidirect
62. C₃×Dic₉	C₄²	semidirect
121. C₄×Dic₃	Dic₉	direct (N × H)
140. C₄×C₉	C₄×Dic₃	semidirect
143. Dic₉	C₄×Dic₃	direct (N × H)
148. C₃×C₉	C₄³	semidirect
210. C₃×C₄	C₄×Dic₉	semidirect
216. Dic₃	C₄×Dic₉	direct (N × H)
219. C₉	C₄²×Dic₃	semidirect
249. C₄	Dic₃×Dic₉	direct (N × H)
251. C₃	C₄²×Dic₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₃⋊Dic₉	2
C₂×C₄×C₉×Dic₃	2
C₂×C₃×C₄×Dic₉	2
C₂×Dic₃×Dic₉	2
C₄×Dic₃²	3
C₄²×Dic₉	3

Subgroups — C4×Dic3×Dic9 #13139530

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Dic₃×Dic₉ #13139530