Subgroups — C₄²×D₃×D₉ #13147694

Order 1728.

1602 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₃×D₉	C₂	1	3	yes	maximal
2	C₄²×C₃⋊D₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×D₃×C₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄²×D₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₃×Dic₉	C₂	1	3	yes	maximal, complemented
6	C₄×(C₃⋊D₉)⋊C₄	C₂	1	3	yes	maximal, complemented
7	C₄×Dic₃×D₉	C₂	1	3	yes	maximal, complemented
8	C₄²×D₃²	—	3	1	no	maximal
9	C₂×C₄²×D₉	—	3	1	no	maximal
10	C₂²×D₃×D₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₃⋊D₉	C₂²	1	3	yes	—
12	C₂×C₄×D₃×C₉	C₂²	1	3	yes	—
13	C₂×C₃×C₄×D₉	C₂²	1	3	yes	—
14	C₂×D₃×Dic₉	C₂²	1	3	yes	—
15	C₂×(C₃⋊D₉)⋊C₄	C₂²	1	3	yes	—
16	C₂×Dic₃×D₉	C₂²	1	3	yes	—
17	C₄×D₃×D₉	C₄	1	24	yes	direct product factor
18	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic, complemented
19	C₄×C₃⋊Dic₉	C₂²	1	3	yes	complemented
20	C₄×C₉×Dic₃	C₂²	1	3	yes	complemented
21	C₃×C₄×Dic₉	C₂²	1	3	yes	complemented
22	Dic₃×Dic₉	C₂²	1	6	yes	complemented
23	C₂×C₄×D₃²	—	3	3	no	—
24	C₄²×C₃⋊D₃	—	3	1	no	—
25	C₃×C₄²×D₃	—	3	1	no	—
26	C₃×C₄²×D₃	D₃	1	1	yes	characteristic
27	C₄×#624	—	3	3	no	—
28	C₄×D₃×Dic₃	—	3	3	no	—
29	C₄×D₃×Dic₃	—	3	3	no	—
30	C₂²×C₄×D₉	—	3	3	no	—
31	C₂×C₄²×C₉	—	3	1	no	—
32	C₂×C₄×Dic₉	—	3	3	no	—
33	C₄²×D₉	—	3	3	no	—
34	C₄²×D₉	—	3	1	no	—
35	C₄²×D₉	—	3	3	no	—
36	C₄²×D₉	—	6	1	no	—
37	C₄²×D₉	D₃	1	1	yes	characteristic, direct product factor
38	C₂²×C₃⋊D₉	C₂³	1	1	yes	characteristic
39	C₂²×D₃×C₉	C₂³	1	1	yes	characteristic
40	C₂²×C₃×D₉	C₂³	1	1	yes	characteristic
41	C₂×D₃×D₉	C₂×C₄	1	12	yes	—
42	C₂×C₃×C₄×C₉	C₂³	1	3	yes	—
43	C₂×C₃⋊Dic₉	C₂³	1	3	yes	—
44	C₄×C₃⋊D₉	C₂×C₄	1	12	yes	complemented
45	C₂×C₉×Dic₃	C₂³	1	3	yes	—
46	C₂×C₃×Dic₉	C₂³	1	3	yes	—
47	C₄×D₃×C₉	C₂×C₄	1	12	yes	complemented
48	C₃×C₄×D₉	C₂×C₄	1	12	yes	complemented
49	D₃×Dic₉	C₂×C₄	1	12	yes	complemented
50	(C₃⋊D₉)⋊C₄	C₂×C₄	1	12	yes	complemented
51	Dic₃×D₉	C₂×C₄	1	12	yes	complemented
52	C₂×C₄²×D₃	—	9	1	no	—
53	C₂×C₄²×D₃	—	9	1	no	—
54	C₂²×D₃²	—	3	1	no	—
55	C₂×C₄×C₃⋊D₃	—	3	3	no	—
56	C₂×C₃×C₄×D₃	—	3	3	no	—
57	C₂×C₃×C₄×D₃	C₂×D₃	1	3	yes	—
58	C₂×#624	—	3	3	no	—
59	C₂×D₃×Dic₃	—	3	3	no	—
60	C₂×D₃×Dic₃	—	3	3	no	—
61	C₄×D₃²	—	3	24	no	—
62	C₂³×D₉	—	3	1	no	—
63	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
64	C₄×C₃⋊Dic₃	—	3	3	no	—
65	C₃×C₄×Dic₃	—	3	3	no	—
66	C₃×C₄×Dic₃	C₂×D₃	1	3	yes	—
67	Dic₃²	—	3	6	no	—
68	C₂²×C₄×C₉	—	3	3	no	—
69	C₂²×Dic₉	—	3	3	no	—
70	C₂×C₄×D₉	—	3	3	no	—
71	C₂×C₄×D₉	—	3	3	no	—
72	C₂×C₄×D₉	—	3	3	no	—
73	C₂×C₄×D₉	—	3	24	no	—
74	C₂×C₄×D₉	—	6	3	no	—
75	C₂×C₄×D₉	C₂×D₃	1	3	yes	—
76	C₄²×C₉	—	2	1	no	—
77	C₄²×C₉	—	3	3	no	—
78	C₄²×C₉	C₂×D₃	1	1	yes	characteristic, complemented
79	C₄×Dic₉	—	3	3	no	—
80	C₄×Dic₉	—	3	6	no	—
81	C₄×Dic₉	—	6	3	no	—
82	C₄×Dic₉	C₂×D₃	1	3	yes	complemented
83	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
84	C₂×C₃⋊D₉	C₂²×C₄	1	6	yes	—
85	C₂×D₃×C₉	C₂²×C₄	1	6	yes	—
86	C₂×C₃×D₉	C₂²×C₄	1	6	yes	—
87	D₃×D₉	C₄²	1	16	yes	direct product factor
88	C₃×C₄×C₉	C₂²×C₄	1	6	yes	complemented
89	C₃⋊Dic₉	C₂²×C₄	1	6	yes	complemented
90	C₉×Dic₃	C₂²×C₄	1	6	yes	complemented
91	C₃×Dic₉	C₂²×C₄	1	6	yes	complemented
92	C₂²×C₄×D₃	—	9	3	no	—
93	C₂²×C₄×D₃	—	9	3	no	—
94	C₂×C₃×C₄²	—	3	1	no	—
95	C₂×C₃×C₄²	—	9	1	no	—
96	C₂×C₄×Dic₃	—	9	3	no	—
97	C₂×C₄×Dic₃	—	9	3	no	—
98	C₄²×D₃	—	3	1	no	—
99	C₄²×D₃	—	9	3	no	—
100	C₄²×D₃	—	9	3	no	—
101	C₄²×D₃	—	9	3	no	—
102	C₄²×D₃	—	9	1	no	—
103	C₄²×D₃	—	9	1	no	—
104	C₄²×D₃	—	9	3	no	—
105	C₄²×D₃	—	18	1	no	—
106	C₄²×D₃	D₉	1	1	yes	characteristic, direct product factor
107	C₂²×C₃⋊D₃	—	3	1	no	—
108	C₂²×C₃×D₃	—	3	1	no	—
109	C₂²×C₃×D₃	C₂²×D₃	1	1	yes	characteristic
110	C₂×D₃²	—	3	12	no	—
111	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
112	C₂×C₃⋊Dic₃	—	3	3	no	—
113	C₄×C₃⋊D₃	—	3	12	no	—
114	C₂×C₃×Dic₃	—	3	3	no	—
115	C₂×C₃×Dic₃	C₂²×D₃	1	3	yes	—
116	C₃×C₄×D₃	—	3	12	no	—
117	C₃×C₄×D₃	C₄×D₃	1	12	yes	—
118	#624	—	3	12	no	—
119	D₃×Dic₃	—	3	12	no	—
120	D₃×Dic₃	—	3	12	no	—
121	C₂³×C₉	—	3	1	no	—
122	C₂²×D₉	—	3	1	no	—
123	C₂²×D₉	—	3	12	no	—
124	C₂²×D₉	—	6	1	no	—
125	C₂²×D₉	C₂²×D₃	1	1	yes	characteristic
126	C₂×C₄×C₉	—	2	3	no	—
127	C₂×C₄×C₉	—	3	3	no	—
128	C₂×C₄×C₉	—	3	12	no	—
129	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
130	C₂×Dic₉	—	3	3	no	—
131	C₂×Dic₉	—	3	12	no	—
132	C₂×Dic₉	—	6	3	no	—
133	C₂×Dic₉	C₂²×D₃	1	3	yes	—
134	C₄×D₉	—	3	12	no	—
135	C₄×D₉	—	3	12	no	—
136	C₄×D₉	—	3	12	no	—
137	C₄×D₉	—	6	12	no	—
138	C₄×D₉	C₄×D₃	1	12	yes	direct product factor
139	C₂²×C₄²	—	27	1	no	Sylow
140	C₂×C₃×C₉	C₂³×C₄	1	3	yes	—
141	C₃⋊D₉	C₂×C₄²	1	4	yes	complemented
142	D₃×C₉	C₂×C₄²	1	4	yes	complemented
143	C₃×D₉	C₂×C₄²	1	4	yes	complemented
144	C₂³×D₃	—	9	1	no	—
145	C₂³×D₃	—	9	1	no	—
146	C₂²×C₃×C₄	—	3	3	no	—
147	C₂²×C₃×C₄	—	9	3	no	—
148	C₂²×Dic₃	—	9	3	no	—
149	C₂²×Dic₃	—	9	3	no	—
150	C₂×C₄×D₃	—	3	3	no	—
151	C₂×C₄×D₃	—	9	24	no	—
152	C₂×C₄×D₃	—	9	3	no	—
153	C₂×C₄×D₃	—	9	3	no	—
154	C₂×C₄×D₃	—	9	3	no	—
155	C₂×C₄×D₃	—	9	3	no	—
156	C₂×C₄×D₃	—	9	3	no	—
157	C₂×C₄×D₃	—	9	3	no	—
158	C₂×C₄×D₃	—	9	24	no	—
159	C₂×C₄×D₃	—	18	3	no	—
160	C₂×C₄×D₃	C₂×D₉	1	3	yes	—
161	C₃×C₄²	—	2	1	no	—
162	C₃×C₄²	—	3	3	no	—
163	C₃×C₄²	—	9	3	no	—
164	C₃×C₄²	C₂×D₉	1	1	yes	characteristic, complemented
165	C₃×C₄²	D₃²	1	1	yes	characteristic
166	C₄×Dic₃	—	3	3	no	—
167	C₄×Dic₃	—	9	6	no	—
168	C₄×Dic₃	—	9	3	no	—
169	C₄×Dic₃	—	9	3	no	—
170	C₄×Dic₃	—	9	6	no	—
171	C₄×Dic₃	—	18	3	no	—
172	C₄×Dic₃	C₂×D₉	1	3	yes	complemented
173	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
174	C₂×C₃⋊D₃	—	3	6	no	—
175	C₂×C₃×D₃	—	3	6	no	—
176	C₂×C₃×D₃	C₂×C₄×D₃	1	6	yes	—
177	D₃²	—	3	16	no	—
178	C₃²×C₄	C₂×C₄×D₃	1	6	yes	—
179	C₃⋊Dic₃	—	3	6	no	—
180	C₃×Dic₃	—	3	6	no	—
181	C₃×Dic₃	C₂×C₄×D₃	1	6	yes	—
182	C₂²×C₉	—	2	1	no	—
183	C₂²×C₉	—	3	6	no	—
184	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
185	C₂×D₉	—	3	6	no	—
186	C₂×D₉	—	3	16	no	—
187	C₂×D₉	—	6	6	no	—
188	C₂×D₉	C₂×C₄×D₃	1	6	yes	—
189	C₄×C₉	—	2	6	no	—
190	C₄×C₉	—	3	6	no	—
191	C₄×C₉	C₂×C₄×D₃	1	6	yes	complemented
192	Dic₉	—	3	6	no	—
193	Dic₉	—	6	6	no	—
194	Dic₉	C₂×C₄×D₃	1	6	yes	complemented
195	C₂³×C₄	—	27	3	no	—
196	C₂×C₄²	—	3	1	no	—
197	C₂×C₄²	—	9	1	no	—
198	C₂×C₄²	—	27	3	no	—
199	C₂×C₄²	—	27	3	no	—
200	C₂×C₄²	—	27	3	no	—
201	C₂×C₄²	—	27	1	no	—
202	C₃×C₉	C₂²×C₄²	1	1	yes	characteristic, complemented, Sylow
203	C₂³×C₃	—	3	1	no	—
204	C₂³×C₃	—	9	1	no	—
205	C₂²×D₃	—	3	1	no	—
206	C₂²×D₃	—	9	12	no	—
207	C₂²×D₃	—	9	1	no	—
208	C₂²×D₃	—	9	1	no	—
209	C₂²×D₃	—	9	12	no	—
210	C₂²×D₃	—	18	1	no	—
211	C₂²×D₃	C₂²×D₉	1	1	yes	characteristic
212	C₂×C₃×C₄	—	2	3	no	—
213	C₂×C₃×C₄	—	3	3	no	—
214	C₂×C₃×C₄	—	3	12	no	—
215	C₂×C₃×C₄	—	9	3	no	—
216	C₂×C₃×C₄	—	9	12	no	—
217	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
218	C₂×C₃×C₄	C₂×D₃²	1	3	yes	—
219	C₂×Dic₃	—	3	3	no	—
220	C₂×Dic₃	—	9	12	no	—
221	C₂×Dic₃	—	9	3	no	—
222	C₂×Dic₃	—	9	3	no	—
223	C₂×Dic₃	—	9	12	no	—
224	C₂×Dic₃	—	18	3	no	—
225	C₂×Dic₃	C₂²×D₉	1	3	yes	—
226	C₄×D₃	—	3	12	no	—
227	C₄×D₃	—	9	12	no	—
228	C₄×D₃	—	9	12	no	—
229	C₄×D₃	—	9	12	no	—
230	C₄×D₃	—	9	12	no	—
231	C₄×D₃	—	9	12	no	—
232	C₄×D₃	—	9	12	no	—
233	C₄×D₃	—	18	12	no	—
234	C₄×D₃	C₄×D₉	1	12	yes	direct product factor
235	C₂×C₃²	C₂²×C₄×D₃	1	3	yes	—
236	C₃⋊D₃	—	3	4	no	—
237	C₃×D₃	—	3	4	no	—
238	C₃×D₃	C₄²×D₃	1	4	yes	—
239	C₂×C₉	—	2	3	no	—
240	C₂×C₉	—	3	4	no	—
241	C₂×C₉	C₂²×C₄×D₃	1	3	yes	—
242	D₉	—	3	4	no	—
243	D₉	—	6	4	no	—
244	D₉	C₄²×D₃	1	4	yes	direct product factor
245	C₂⁴	—	27	1	no	—
246	C₂²×C₄	—	3	3	no	—
247	C₂²×C₄	—	9	3	no	—
248	C₂²×C₄	—	27	3	no	—
249	C₂²×C₄	—	27	24	no	—
250	C₂²×C₄	—	27	3	no	—
251	C₂²×C₄	—	27	3	no	—
252	C₂²×C₄	—	27	3	no	—
253	C₄²	—	3	3	no	—
254	C₄²	—	9	3	no	—
255	C₄²	—	27	6	no	—
256	C₄²	—	27	3	no	—
257	C₄²	D₃×D₉	1	1	yes	characteristic, direct product factor
258	C₂²×C₃	—	2	1	no	—
259	C₂²×C₃	—	3	6	no	—
260	C₂²×C₃	—	9	6	no	—
261	C₂²×C₃	C₂³×D₉	1	1	yes	characteristic
262	C₂²×C₃	C₂²×D₃²	1	1	yes	characteristic
263	C₂×D₃	—	3	6	no	—
264	C₂×D₃	—	9	16	no	—
265	C₂×D₃	—	9	6	no	—
266	C₂×D₃	—	9	6	no	—
267	C₂×D₃	—	9	16	no	—
268	C₂×D₃	—	18	6	no	—
269	C₂×D₃	C₂×C₄×D₉	1	6	yes	—
270	C₃×C₄	—	2	6	no	—
271	C₃×C₄	—	3	6	no	—
272	C₃×C₄	—	9	6	no	—
273	C₃×C₄	C₂×C₄×D₉	1	6	yes	complemented
274	C₃×C₄	C₄×D₃²	1	6	yes	—
275	Dic₃	—	3	6	no	—
276	Dic₃	—	9	6	no	—
277	Dic₃	—	9	6	no	—
278	Dic₃	—	18	6	no	—
279	Dic₃	C₂×C₄×D₉	1	6	yes	complemented
280	C₃²	C₂×C₄²×D₃	1	1	yes	characteristic
281	C₉	—	2	1	no	—
282	C₉	C₂×C₄²×D₃	1	1	yes	characteristic, complemented
283	C₂³	—	3	1	no	—
284	C₂³	—	9	1	no	—
285	C₂³	—	27	12	no	—
286	C₂³	—	27	1	no	—
287	C₂×C₄	—	3	3	no	—
288	C₂×C₄	—	3	12	no	—
289	C₂×C₄	—	9	3	no	—
290	C₂×C₄	—	9	12	no	—
291	C₂×C₄	—	27	12	no	—
292	C₂×C₄	—	27	12	no	—
293	C₂×C₄	—	27	12	no	—
294	C₂×C₄	—	27	3	no	—
295	C₂×C₄	—	27	12	no	—
296	C₂×C₄	C₂×D₃×D₉	1	3	yes	—
297	C₂×C₃	—	2	3	no	—
298	C₂×C₃	—	3	4	no	—
299	C₂×C₃	—	9	4	no	—
300	C₂×C₃	C₂²×C₄×D₉	1	3	yes	—
301	C₂×C₃	C₂×C₄×D₃²	1	3	yes	—
302	D₃	—	3	4	no	—
303	D₃	—	9	4	no	—
304	D₃	—	9	4	no	—
305	D₃	—	18	4	no	—
306	D₃	C₄²×D₉	1	4	yes	direct product factor
307	C₂²	—	3	6	no	—
308	C₂²	—	9	6	no	—
309	C₂²	—	27	16	no	—
310	C₂²	—	27	6	no	—
311	C₂²	C₂²×D₃×D₉	1	1	yes	characteristic
312	C₄	—	3	6	no	—
313	C₄	—	9	6	no	—
314	C₄	—	27	6	no	—
315	C₄	C₄×D₃×D₉	1	6	yes	direct product factor
316	C₃	—	2	1	no	—
317	C₃	C₂×C₄²×D₉	1	1	yes	characteristic, complemented
318	C₃	C₄²×D₃²	1	1	yes	characteristic
319	C₂	—	3	4	no	—
320	C₂	—	9	4	no	—
321	C₂	—	27	4	no	—
322	C₂	C₂×C₄×D₃×D₉	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₄²	D₃×D₉
commutator, hypocenter	C₃×C₉	C₂²×C₄²
Frattini	C₂²×C₃	C₂²×D₃²
Fitting	C₃×C₄²×C₉	C₂²
socle	C₂²×C₃²	C₂³×D₃
prime radical	C₂²×D₃×D₉	C₂²
cosocle	C₂²×C₃×C₉	C₂⁴
LCS	G ▷ C₃×C₉
UCS	C₁ ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. C₄²×C₃⋊D₉	C₂	semidirect
3. C₄²×D₃×C₉	C₂	semidirect
4. C₃×C₄²×D₉	C₂	semidirect
5. C₄×D₃×Dic₉	C₂	semidirect
6. C₄×(C₃⋊D₉)⋊C₄	C₂	semidirect
7. C₄×Dic₃×D₉	C₂	semidirect
17. C₄×D₃×D₉	C₄	direct (N × H)
18. C₃×C₄²×C₉	C₂²	semidirect
19. C₄×C₃⋊Dic₉	C₂²	semidirect
20. C₄×C₉×Dic₃	C₂²	semidirect
21. C₃×C₄×Dic₉	C₂²	semidirect
22. Dic₃×Dic₉	C₂²	semidirect
37. C₄²×D₉	D₃	direct (N × H)
44. C₄×C₃⋊D₉	C₂×C₄	semidirect
47. C₄×D₃×C₉	C₂×C₄	semidirect
48. C₃×C₄×D₉	C₂×C₄	semidirect
49. D₃×Dic₉	C₂×C₄	semidirect
50. (C₃⋊D₉)⋊C₄	C₂×C₄	semidirect
51. Dic₃×D₉	C₂×C₄	semidirect
78. C₄²×C₉	C₂×D₃	semidirect
82. C₄×Dic₉	C₂×D₃	semidirect
87. D₃×D₉	C₄²	direct (N × H)
88. C₃×C₄×C₉	C₂²×C₄	semidirect
89. C₃⋊Dic₉	C₂²×C₄	semidirect
90. C₉×Dic₃	C₂²×C₄	semidirect
91. C₃×Dic₉	C₂²×C₄	semidirect
106. C₄²×D₃	D₉	direct (N × H)
138. C₄×D₉	C₄×D₃	direct (N × H)
141. C₃⋊D₉	C₂×C₄²	semidirect
142. D₃×C₉	C₂×C₄²	semidirect
143. C₃×D₉	C₂×C₄²	semidirect
164. C₃×C₄²	C₂×D₉	semidirect
172. C₄×Dic₃	C₂×D₉	semidirect
191. C₄×C₉	C₂×C₄×D₃	semidirect
194. Dic₉	C₂×C₄×D₃	semidirect
202. C₃×C₉	C₂²×C₄²	semidirect
234. C₄×D₃	C₄×D₉	direct (N × H)
244. D₉	C₄²×D₃	direct (N × H)
257. C₄²	D₃×D₉	direct (N × H)
273. C₃×C₄	C₂×C₄×D₉	semidirect
279. Dic₃	C₂×C₄×D₉	semidirect
282. C₉	C₂×C₄²×D₃	semidirect
306. D₃	C₄²×D₉	direct (N × H)
315. C₄	C₄×D₃×D₉	direct (N × H)
317. C₃	C₂×C₄²×D₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₃×D₉	2
C₄²×C₃⋊D₉	2
C₄²×D₃×C₉	2
C₃×C₄²×D₉	2
C₄×D₃×Dic₉	2
C₄×(C₃⋊D₉)⋊C₄	2
C₄×Dic₃×D₉	2
C₄²×D₃²	3
C₂×C₄²×D₉	3

Subgroups — C42×D3×D9 #13147694

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×D₃×D₉ #13147694