Subgroups — C₄×Dic₉⋊Dic₃ #13139568

Order 1728.

690 conjugacy classes of subgroups.

The subgroup lattice has 318 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×Dic₉⋊Dic₃	C₂	1	2	yes	maximal
5	#11720331	C₂	1	2	yes	maximal
6	C₄×Dic₃⋊Dic₃	—	3	1	no	maximal
7	C₄×Dic₉⋊C₄	—	3	1	no	maximal
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃⋊Dic₉	C₂²	1	2	yes	—
10	C₂²×C₉×Dic₃	C₂²	1	2	yes	—
11	C₂²×C₃×Dic₉	C₂²	1	1	yes	characteristic
12	C₂²×C₃×Dic₉	C₂²	1	1	yes	characteristic
13	C₄×C₃⋊Dic₉	—	2	2	no	—
14	C₄×C₉×Dic₃	—	2	2	no	—
15	C₃×C₄×Dic₉	C₄	1	4	yes	complemented
16	Dic₉⋊Dic₃	C₄	1	8	yes	direct product factor
17	C₂×C₄×C₃⋊Dic₃	—	3	1	no	—
18	C₂×C₃×C₄×Dic₃	—	3	1	no	—
19	C₂×C₃×C₄×Dic₃	D₃	1	1	yes	characteristic
20	C₂×Dic₃⋊Dic₃	—	3	2	no	—
21	#63579	—	3	2	no	—
22	C₂×C₄²×C₉	—	3	1	no	—
23	C₂×Dic₉⋊C₄	—	3	2	no	—
24	C₂×C₄×Dic₉	—	3	1	no	—
25	C₂×C₄×Dic₉	—	6	1	no	—
26	C₂×C₄×Dic₉	D₃	1	1	yes	characteristic
27	C₉⋊#95	—	3	2	no	—
28	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
31	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
32	C₂×C₃⋊Dic₉	—	2	2	no	—
33	C₂×C₃⋊Dic₉	—	2	2	no	—
34	C₂×C₃⋊Dic₉	C₂×C₄	1	4	yes	—
35	C₂×C₉×Dic₃	—	2	2	no	—
36	C₂×C₉×Dic₃	—	2	2	no	—
37	C₂×C₉×Dic₃	C₂×C₄	1	4	yes	—
38	C₂×C₃×Dic₉	—	2	2	no	—
39	C₂×C₃×Dic₉	C₂×C₄	1	2	yes	—
40	C₂×C₃×Dic₉	C₂×C₄	1	4	yes	—
41	C₂×C₃×Dic₉	C₂×C₄	1	2	yes	—
42	C₄×C₄⋊Dic₃	—	9	1	no	—
43	C₄×Dic₃⋊C₄	—	9	1	no	—
44	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
45	C₂²×C₃⋊Dic₃	—	3	2	no	—
46	C₂²×C₃×Dic₃	—	3	1	no	—
47	C₂²×C₃×Dic₃	—	3	1	no	—
48	C₂²×C₃×Dic₃	C₂×D₃	1	2	yes	—
49	C₄×C₃⋊Dic₃	—	6	2	no	—
50	C₃×C₄×Dic₃	—	2	2	no	—
51	C₃×C₄×Dic₃	—	3	4	no	—
52	Dic₃⋊Dic₃	—	3	8	no	—
53	C₂²×C₄×C₉	—	2	1	no	—
54	C₂²×C₄×C₉	—	3	2	no	—
55	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
56	C₂²×Dic₉	—	3	2	no	—
57	C₂²×Dic₉	—	6	2	no	—
58	C₂²×Dic₉	C₂×D₃	1	1	yes	characteristic
59	C₂²×Dic₉	C₂×D₃	1	1	yes	characteristic
60	C₄²×C₉	—	6	2	no	—
61	Dic₉⋊C₄	—	3	8	no	—
62	C₄×Dic₉	—	6	4	no	—
63	C₄×Dic₉	—	6	2	no	—
64	C₄×Dic₉	Dic₃	1	4	yes	complemented
65	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₃×C₉	C₂²×C₄	1	2	yes	—
67	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₃×C₉	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₃×C₉	#41	1	2	yes	—
70	C₃×C₄×C₉	C₄⋊C₄	1	4	yes	complemented
71	C₃⋊Dic₉	—	2	4	no	—
72	C₉×Dic₃	—	2	4	no	—
73	C₃×Dic₉	—	2	2	no	—
74	C₃×Dic₉	C₄²	1	4	yes	complemented
75	C₂×C₃×C₄²	—	3	1	no	—
76	C₂×C₃×C₄²	—	9	1	no	—
77	C₂×C₄⋊Dic₃	—	9	2	no	—
78	C₂×Dic₃⋊C₄	—	9	2	no	—
79	C₂×C₄×Dic₃	—	3	1	no	—
80	C₂×C₄×Dic₃	—	9	1	no	—
81	C₂×C₄×Dic₃	—	9	1	no	—
82	C₂×C₄×Dic₃	—	18	1	no	—
83	C₂×C₄×Dic₃	D₉	1	1	yes	characteristic
84	C₃⋊#95	—	9	2	no	—
85	C₃⋊#95	—	9	2	no	—
86	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
87	C₂×C₃²×C₄	Dic₆	1	2	yes	—
88	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
89	C₂×C₃²×C₄	#67	1	2	yes	—
90	C₂×C₃⋊Dic₃	—	3	4	no	—
91	C₂×C₃⋊Dic₃	—	6	2	no	—
92	C₂×C₃⋊Dic₃	—	6	2	no	—
93	C₂×C₃×Dic₃	—	2	2	no	—
94	C₂×C₃×Dic₃	—	2	2	no	—
95	C₂×C₃×Dic₃	—	3	4	no	—
96	C₂×C₃×Dic₃	—	3	2	no	—
97	C₂×C₃×Dic₃	—	3	2	no	—
98	C₂×C₃×Dic₃	—	6	2	no	—
99	C₂×C₃×Dic₃	C₄×D₃	1	4	yes	—
100	C₂³×C₉	—	2	1	no	—
101	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
102	C₂×C₄×C₉	—	2	2	no	—
103	C₂×C₄×C₉	—	2	2	no	—
104	C₂×C₄×C₉	—	2	2	no	—
105	C₂×C₄×C₉	—	3	4	no	—
106	C₂×C₄×C₉	—	6	2	no	—
107	C₂×C₄×C₉	—	6	2	no	—
108	C₂×C₄×C₉	Dic₆	1	2	yes	—
109	C₂×C₄×C₉	D₁₂	1	2	yes	—
110	C₂×C₄×C₉	C₂×Dic₃	1	2	yes	—
111	C₂×Dic₉	—	2	2	no	—
112	C₂×Dic₉	—	3	4	no	—
113	C₂×Dic₉	—	6	4	no	—
114	C₂×Dic₉	—	6	4	no	—
115	C₂×Dic₉	—	6	4	no	—
116	C₂×Dic₉	—	6	2	no	—
117	C₂×Dic₉	—	6	2	no	—
118	C₂×Dic₉	C₄×D₃	1	4	yes	—
119	C₂×Dic₉	C₂×Dic₃	1	2	yes	—
120	C₂×Dic₉	C₂×Dic₃	1	2	yes	—
121	C₄×C₄⋊C₄	—	27	1	no	Sylow
122	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₃×C₉	#117	1	1	yes	characteristic
125	C₂×C₃×C₉	C₄×D₄	1	2	yes	—
126	C₂×C₃×C₉	C₄×Q₈	1	2	yes	—
127	C₂²×C₃×C₄	—	2	1	no	—
128	C₂²×C₃×C₄	—	3	2	no	—
129	C₂²×C₃×C₄	—	9	1	no	—
130	C₂²×C₃×C₄	—	9	1	no	—
131	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
132	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
133	C₂²×Dic₃	—	3	1	no	—
134	C₂²×Dic₃	—	3	1	no	—
135	C₂²×Dic₃	—	9	2	no	—
136	C₂²×Dic₃	—	9	2	no	—
137	C₂²×Dic₃	—	18	2	no	—
138	C₂²×Dic₃	C₂×D₉	1	2	yes	—
139	C₃×C₄²	—	6	2	no	—
140	C₃×C₄²	—	9	4	no	—
141	C₄⋊Dic₃	—	9	8	no	—
142	Dic₃⋊C₄	—	9	8	no	—
143	C₄×Dic₃	—	2	2	no	—
144	C₄×Dic₃	—	3	4	no	—
145	C₄×Dic₃	—	18	4	no	—
146	C₄×Dic₃	—	18	2	no	—
147	C₄×Dic₃	—	18	2	no	—
148	C₂²×C₃²	C₂×Dic₆	1	1	yes	characteristic
149	C₂²×C₃²	C₂×C₄×D₃	1	2	yes	—
150	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
151	C₂²×C₃²	#235	1	2	yes	—
152	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
153	C₃²×C₄	Dic₃⋊C₄	1	4	yes	—
154	C₃⋊Dic₃	—	6	4	no	—
155	C₃×Dic₃	—	2	4	no	—
156	C₃×Dic₃	—	3	4	no	—
157	C₃×Dic₃	—	6	2	no	—
158	C₂²×C₉	—	2	2	no	—
159	C₂²×C₉	—	2	2	no	—
160	C₂²×C₉	—	2	1	no	—
161	C₂²×C₉	—	2	1	no	—
162	C₂²×C₉	—	2	1	no	—
163	C₂²×C₉	C₂×Dic₆	1	1	yes	characteristic
164	C₂²×C₉	C₂×C₄×D₃	1	2	yes	—
165	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
166	C₂²×C₉	#235	1	2	yes	—
167	C₂²×C₉	C₂²×Dic₃	1	1	yes	characteristic
168	C₄×C₉	—	2	4	no	—
169	C₄×C₉	—	6	4	no	—
170	C₄×C₉	C₄⋊Dic₃	1	4	yes	complemented
171	Dic₉	—	2	2	no	—
172	Dic₉	—	6	8	no	—
173	Dic₉	—	6	4	no	—
174	Dic₉	C₄×Dic₃	1	4	yes	complemented
175	C₂×C₄⋊C₄	—	27	2	no	—
176	C₂×C₄²	—	3	1	no	—
177	C₂×C₄²	—	9	1	no	—
178	C₂×C₄²	—	27	1	no	—
179	#95	—	27	2	no	—
180	C₃×C₉	C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
181	C₂³×C₃	—	2	1	no	—
182	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
183	C₂³×C₃	C₂×D₃²	1	1	yes	characteristic
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	2	2	no	—
187	C₂×C₃×C₄	—	3	4	no	—
188	C₂×C₃×C₄	—	6	2	no	—
189	C₂×C₃×C₄	—	6	2	no	—
190	C₂×C₃×C₄	—	9	4	no	—
191	C₂×C₃×C₄	—	9	2	no	—
192	C₂×C₃×C₄	—	9	2	no	—
193	C₂×C₃×C₄	—	18	2	no	—
194	C₂×C₃×C₄	Dic₁₈	1	2	yes	—
195	C₂×C₃×C₄	C₄×D₉	1	2	yes	—
196	C₂×C₃×C₄	#611	1	2	yes	—
197	C₂×C₃×C₄	D₃×Dic₃	1	2	yes	—
198	C₂×C₃×C₄	#626	1	2	yes	—
199	C₂×C₃×C₄	#627	1	2	yes	—
200	C₂×Dic₃	—	2	2	no	—
201	C₂×Dic₃	—	2	2	no	—
202	C₂×Dic₃	—	3	4	no	—
203	C₂×Dic₃	—	3	2	no	—
204	C₂×Dic₃	—	3	2	no	—
205	C₂×Dic₃	—	6	2	no	—
206	C₂×Dic₃	—	9	4	no	—
207	C₂×Dic₃	—	9	4	no	—
208	C₂×Dic₃	—	18	4	no	—
209	C₂×Dic₃	—	18	4	no	—
210	C₂×Dic₃	—	18	4	no	—
211	C₂×Dic₃	—	18	2	no	—
212	C₂×Dic₃	—	18	2	no	—
213	C₂×Dic₃	—	18	2	no	—
214	C₂×Dic₃	—	18	2	no	—
215	C₂×Dic₃	C₄×D₉	1	4	yes	—
216	C₂×C₃²	C₄×Dic₆	1	2	yes	—
217	C₂×C₃²	C₄²×D₃	1	1	yes	characteristic
218	C₂×C₃²	#872	1	1	yes	characteristic
219	C₂×C₃²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
220	C₂×C₃²	C₄×#67	1	2	yes	—
221	C₂×C₉	—	2	1	no	—
222	C₂×C₉	—	2	1	no	—
223	C₂×C₉	—	2	2	no	—
224	C₂×C₉	—	2	2	no	—
225	C₂×C₉	—	2	1	no	—
226	C₂×C₉	C₄×Dic₆	1	2	yes	—
227	C₂×C₉	C₄×D₁₂	1	2	yes	—
228	C₂×C₉	C₂×C₄×Dic₃	1	1	yes	characteristic
229	C₂×C₉	C₂×C₄⋊Dic₃	1	1	yes	characteristic
230	C₂×C₉	#926	1	1	yes	characteristic
231	C₂²×C₄	—	3	2	no	—
232	C₂²×C₄	—	9	1	no	—
233	C₂²×C₄	—	9	1	no	—
234	C₂²×C₄	—	27	2	no	—
235	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
236	C₄⋊C₄	—	27	8	no	—
237	C₄²	—	6	2	no	—
238	C₄²	—	9	4	no	—
239	C₄²	—	54	2	no	—
240	C₂²×C₃	—	2	2	no	—
241	C₂²×C₃	—	2	2	no	—
242	C₂²×C₃	—	2	1	no	—
243	C₂²×C₃	—	2	1	no	—
244	C₂²×C₃	—	2	1	no	—
245	C₂²×C₃	C₂×Dic₁₈	1	1	yes	characteristic
246	C₂²×C₃	C₂×C₄×D₉	1	2	yes	—
247	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
248	C₂²×C₃	#3697	1	2	yes	—
249	C₂²×C₃	C₂×#611	1	1	yes	characteristic
250	C₂²×C₃	#3798	1	2	yes	—
251	C₂²×C₃	C₄×D₃²	1	2	yes	—
252	C₂²×C₃	C₂×D₃×Dic₃	1	1	yes	characteristic
253	C₂²×C₃	C₂×#626	1	1	yes	characteristic
254	C₂²×C₃	C₂×#627	1	1	yes	characteristic
255	C₃×C₄	—	2	4	no	—
256	C₃×C₄	—	6	4	no	—
257	C₃×C₄	—	9	4	no	—
258	C₃×C₄	—	18	2	no	—
259	C₃×C₄	Dic₉⋊C₄	1	4	yes	complemented
260	C₃×C₄	Dic₃⋊Dic₃	1	4	yes	—
261	Dic₃	—	2	4	no	—
262	Dic₃	—	3	4	no	—
263	Dic₃	—	6	2	no	—
264	Dic₃	—	18	8	no	—
265	Dic₃	—	18	4	no	—
266	Dic₃	—	18	4	no	—
267	C₃²	C₄×Dic₃⋊C₄	1	1	yes	characteristic
268	C₉	—	2	1	no	—
269	C₉	C₄×C₄⋊Dic₃	1	1	yes	characteristic, complemented
270	C₂³	C₂×D₃×D₉	1	1	yes	characteristic
271	C₂×C₄	—	3	4	no	—
272	C₂×C₄	—	6	2	no	—
273	C₂×C₄	—	6	2	no	—
274	C₂×C₄	—	9	4	no	—
275	C₂×C₄	—	9	2	no	—
276	C₂×C₄	—	9	2	no	—
277	C₂×C₄	—	18	2	no	—
278	C₂×C₄	—	27	4	no	—
279	C₂×C₄	—	54	2	no	—
280	C₂×C₄	—	54	2	no	—
281	C₂×C₄	#6189	1	2	yes	—
282	C₂×C₄	Dic₃×D₉	1	2	yes	—
283	C₂×C₄	#6195	1	2	yes	—
284	C₂×C₃	—	2	2	no	—
285	C₂×C₃	—	2	1	no	—
286	C₂×C₃	—	2	1	no	—
287	C₂×C₃	—	2	2	no	—
288	C₂×C₃	—	2	1	no	—
289	C₂×C₃	C₄×Dic₁₈	1	2	yes	—
290	C₂×C₃	C₄²×D₉	1	1	yes	characteristic
291	C₂×C₃	#63434	1	1	yes	characteristic
292	C₂×C₃	C₂×Dic₉⋊C₄	1	1	yes	characteristic
293	C₂×C₃	C₄×#611	1	2	yes	—
294	C₂×C₃	#63855	1	1	yes	characteristic
295	C₂×C₃	C₄×D₃×Dic₃	1	1	yes	characteristic
296	C₂×C₃	C₄×#626	1	2	yes	—
297	C₂×C₃	C₄×#627	1	2	yes	—
298	C₂×C₃	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
299	C₂²	#88640	1	2	yes	—
300	C₂²	C₄×D₃×D₉	1	2	yes	—
301	C₂²	C₂×#6189	1	1	yes	characteristic
302	C₂²	C₂×Dic₃×D₉	1	1	yes	characteristic
303	C₂²	C₂×#6195	1	1	yes	characteristic
304	C₄	—	6	4	no	—
305	C₄	—	9	4	no	—
306	C₄	—	18	2	no	—
307	C₄	—	54	4	no	—
308	C₄	Dic₉⋊Dic₃	1	4	yes	direct product factor
309	C₃	—	2	1	no	—
310	C₃	C₄×Dic₉⋊C₄	1	1	yes	characteristic, complemented
311	C₃	C₄×Dic₃⋊Dic₃	1	1	yes	characteristic
312	C₂	#11720943	1	1	yes	characteristic
313	C₂	C₄×#6189	1	2	yes	—
314	C₂	C₄×Dic₃×D₉	1	1	yes	characteristic
315	C₂	C₄×#6195	1	2	yes	—
316	C₂	C₂×Dic₉⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄	D₃×D₉
commutator	C₂×C₃×C₉	C₂×C₄²
Frattini	C₂³×C₃	C₂×D₃²
Fitting	C₂²×C₃×C₄×C₉	C₂²
socle, prime radical	C₂³×C₃²	C₂²×D₃
cosocle	C₂³×C₃×C₉	C₂³
hypocenter	C₃×C₉	C₄×C₄⋊C₄
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
15. C₃×C₄×Dic₉	C₄	semidirect
16. Dic₉⋊Dic₃	C₄	direct (N × H)
64. C₄×Dic₉	Dic₃	semidirect
70. C₃×C₄×C₉	C₄⋊C₄	semidirect
74. C₃×Dic₉	C₄²	semidirect
170. C₄×C₉	C₄⋊Dic₃	semidirect
174. Dic₉	C₄×Dic₃	semidirect
180. C₃×C₉	C₄×C₄⋊C₄	semidirect
259. C₃×C₄	Dic₉⋊C₄	semidirect
269. C₉	C₄×C₄⋊Dic₃	semidirect
308. C₄	Dic₉⋊Dic₃	direct (N × H)
310. C₃	C₄×Dic₉⋊C₄	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×Dic₉⋊Dic₃	2
#11720331	2
C₄×Dic₃⋊Dic₃	3
C₄×Dic₉⋊C₄	3

Subgroups — C4×Dic9⋊Dic3 #13139568

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Dic₉⋊Dic₃ #13139568