Subgroups — #13138159

Order 1728.

398 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₃×D₉	C₂	1	1	yes	maximal, characteristic
2	C₈×C₃⋊Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×(C₂×D₉)⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₉×Dic₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	(C₂×D₉)⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#11720271	C₂	1	1	yes	maximal, characteristic, complemented
7	Dic₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	#10590656	—	3	1	no	maximal
9	#10588517	—	3	1	no	maximal
10	C₂×C₃×C₄×D₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₃×D₉	C₄	1	1	yes	characteristic
12	C₂×Dic₃×D₉	C₄	1	1	yes	characteristic
13	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₄×C₃⋊Dic₉	C₂²	1	1	yes	characteristic
15	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
16	C₄×C₉×Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
19	Dic₃×Dic₉	C₄	1	1	yes	characteristic
20	Dic₃×Dic₉	C₄	1	1	yes	characteristic
21	C₄×D₃×Dic₃	—	3	1	no	—
22	C₈×C₃⋊Dic₃	—	3	1	no	—
23	C₃×(C₂×D₃)⋊C₈	—	3	1	no	—
24	C₃×Dic₃⋊C₈	D₃	1	1	yes	characteristic
25	Dic₃⋊(C₃⋊C₈)	—	3	1	no	—
26	(C₂×D₃)⋊(C₃⋊C₈)	—	3	1	no	—
27	#63553	—	3	1	no	—
28	C₄²×D₉	—	3	1	no	—
29	C₉×C₄⋊C₈	—	3	1	no	—
30	(C₂×D₉)⋊C₈	—	3	1	no	—
31	(C₂×D₉)⋊C₈	D₃	1	1	yes	characteristic
32	C₈×Dic₉	—	3	1	no	—
33	C₈×Dic₉	—	3	1	no	—
34	C₈×Dic₉	—	6	1	no	—
35	C₄⋊(C₉⋊C₈)	—	3	1	no	—
36	C₂²×C₃×D₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
38	C₂×C₃⋊Dic₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃⋊Dic₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₉×Dic₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₉×Dic₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×Dic₉	C₂×C₄	1	1	yes	characteristic
43	C₃×C₄×D₉	—	2	1	no	—
44	C₃×C₄×D₉	—	2	1	no	—
45	Dic₃×D₉	—	2	2	no	—
46	Dic₃×D₉	C₈	1	4	yes	complemented
47	C₃×C₈×C₉	—	2	1	no	—
48	C₃⋊(C₉⋊C₈)	—	2	1	no	—
49	C₉×C₃⋊C₈	—	2	1	no	—
50	C₃×C₉⋊C₈	—	2	1	no	—
51	#4833	—	9	1	no	—
52	#4718	—	9	1	no	—
53	C₂×C₃×C₄×D₃	—	3	1	no	—
54	C₂×D₃×Dic₃	—	3	1	no	—
55	C₂×D₃×Dic₃	—	3	1	no	—
56	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
57	C₄×C₃⋊Dic₃	—	3	1	no	—
58	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
59	C₃×C₄×Dic₃	C₂×D₃	1	1	yes	characteristic
60	C₂×C₃×C₃⋊C₈	—	3	1	no	—
61	C₂×C₃×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
62	Dic₃²	—	3	1	no	—
63	Dic₃²	—	3	1	no	—
64	C₂×C₄×D₉	—	3	1	no	—
65	C₂×C₄×D₉	—	3	1	no	—
66	C₂×C₄×D₉	C₂×D₃	1	1	yes	characteristic
67	C₂×C₈×C₉	—	2	1	no	—
68	C₂×C₈×C₉	—	3	1	no	—
69	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
70	C₄²×C₉	—	3	1	no	—
71	C₄×Dic₉	—	3	1	no	—
72	C₄×Dic₉	—	3	1	no	—
73	C₄×Dic₉	—	3	1	no	—
74	C₄×Dic₉	—	6	1	no	—
75	C₂×C₉⋊C₈	—	3	1	no	—
76	C₂×C₉⋊C₈	—	6	1	no	—
77	C₂×C₉⋊C₈	C₂×D₃	1	1	yes	characteristic
78	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃×D₉	—	2	1	no	—
80	C₂×C₃×D₉	—	2	1	no	—
81	C₂×C₃×D₉	C₂×C₈	1	2	yes	—
82	C₃×C₄×C₉	#41	1	1	yes	characteristic
83	C₃×C₄×C₉	#41	1	1	yes	characteristic
84	C₃⋊Dic₉	C₂×C₈	1	2	yes	—
85	C₃⋊Dic₉	M₁₆	1	2	yes	—
86	C₉×Dic₃	—	2	1	no	—
87	C₉×Dic₃	C₂×C₈	1	2	yes	—
88	C₃×Dic₉	—	2	1	no	—
89	C₂×C₄×Dic₃	—	9	1	no	—
90	C₄²×D₃	—	9	1	no	—
91	C₃×C₄⋊C₈	—	3	1	no	—
92	C₃×C₂²⋊C₈	—	9	1	no	—
93	#830	—	9	1	no	—
94	(C₂×D₃)⋊C₈	—	3	1	no	—
95	(C₂×D₃)⋊C₈	—	9	1	no	—
96	Dic₃⋊C₈	—	9	1	no	—
97	Dic₃⋊C₈	D₉	1	1	yes	characteristic, complemented
98	C₈×Dic₃	—	9	1	no	—
99	C₈×Dic₃	—	9	1	no	—
100	C₈×Dic₃	—	9	1	no	—
101	C₈×Dic₃	—	9	1	no	—
102	C₈×Dic₃	—	18	1	no	—
103	C₄⋊(C₃⋊C₈)	—	9	1	no	—
104	C₂²×C₃×D₃	—	3	1	no	—
105	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
106	C₂×C₃⋊Dic₃	—	3	1	no	—
107	C₂×C₃⋊Dic₃	—	3	1	no	—
108	C₂×C₃×Dic₃	—	3	1	no	—
109	C₂×C₃×Dic₃	C₄×D₃	1	1	yes	characteristic
110	C₂×C₃×Dic₃	C₄×D₃	1	1	yes	characteristic
111	C₃×C₄×D₃	—	6	1	no	—
112	C₃×C₄×D₃	—	6	1	no	—
113	D₃×Dic₃	—	3	4	no	—
114	D₃×Dic₃	—	6	2	no	—
115	C₂²×D₉	C₄×D₃	1	1	yes	characteristic
116	C₃²×C₈	—	2	1	no	—
117	C₃⋊(C₃⋊C₈)	—	6	1	no	—
118	C₃×C₃⋊C₈	—	2	1	no	—
119	C₃×C₃⋊C₈	—	6	1	no	—
120	C₂×C₄×C₉	—	2	1	no	—
121	C₂×C₄×C₉	—	3	1	no	—
122	C₂×C₄×C₉	—	3	1	no	—
123	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
124	C₂×Dic₉	—	3	1	no	—
125	C₂×Dic₉	—	3	1	no	—
126	C₂×Dic₉	—	6	1	no	—
127	C₂×Dic₉	—	6	1	no	—
128	C₂×Dic₉	C₄×D₃	1	1	yes	characteristic
129	C₄×D₉	—	2	1	no	—
130	C₄×D₉	—	2	1	no	—
131	C₄×D₉	—	3	4	no	—
132	C₄×D₉	—	6	2	no	—
133	C₈×C₉	—	2	2	no	—
134	C₈×C₉	—	2	1	no	—
135	C₈×C₉	—	6	1	no	—
136	C₉⋊C₈	—	2	1	no	—
137	C₉⋊C₈	—	6	2	no	—
138	C₉⋊C₈	—	6	1	no	—
139	#431	—	27	1	no	Sylow
140	C₂×C₃×C₉	#117	1	1	yes	characteristic
141	C₂×C₃×C₉	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
143	C₃×D₉	—	2	2	no	—
144	C₂²×C₃×C₄	—	9	1	no	—
145	C₂²×Dic₃	—	9	1	no	—
146	C₂²×Dic₃	—	9	1	no	—
147	C₂×C₄×D₃	—	3	1	no	—
148	C₂×C₄×D₃	—	9	1	no	—
149	C₂×C₄×D₃	—	9	1	no	—
150	C₂×C₃×C₈	—	2	1	no	—
151	C₂×C₃×C₈	—	3	1	no	—
152	C₂×C₃×C₈	—	9	1	no	—
153	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
154	C₂×C₃×C₈	D₃²	1	1	yes	characteristic
155	C₃×C₄²	—	3	1	no	—
156	C₄×Dic₃	—	9	1	no	—
157	C₄×Dic₃	—	9	1	no	—
158	C₄×Dic₃	—	9	1	no	—
159	C₄×Dic₃	—	9	1	no	—
160	C₄×Dic₃	—	9	1	no	—
161	C₄×Dic₃	—	9	1	no	—
162	C₄×Dic₃	—	18	1	no	—
163	C₄×Dic₃	C₂×D₉	1	1	yes	characteristic
164	C₂×C₃⋊C₈	—	3	1	no	—
165	C₂×C₃⋊C₈	—	9	1	no	—
166	C₂×C₃⋊C₈	—	9	1	no	—
167	C₂×C₃⋊C₈	—	18	1	no	—
168	C₂×C₃⋊C₈	C₂×D₉	1	1	yes	characteristic
169	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
170	C₂×C₃×D₃	—	3	2	no	—
171	C₂×C₃×D₃	—	6	1	no	—
172	C₂×C₃×D₃	—	6	1	no	—
173	C₃²×C₄	#237	1	1	yes	characteristic
174	C₃²×C₄	#239	1	1	yes	characteristic
175	C₃⋊Dic₃	—	3	2	no	—
176	C₃⋊Dic₃	—	3	2	no	—
177	C₃×Dic₃	—	2	1	no	—
178	C₃×Dic₃	—	6	1	no	—
179	C₃×Dic₃	D₃×C₈	1	2	yes	—
180	C₂²×C₉	—	2	1	no	—
181	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
182	C₂×D₉	—	2	1	no	—
183	C₂×D₉	—	2	1	no	—
184	C₂×D₉	D₃×C₈	1	2	yes	—
185	C₄×C₉	—	2	1	no	—
186	C₄×C₉	—	2	1	no	—
187	C₄×C₉	—	3	2	no	—
188	C₄×C₉	—	6	1	no	—
189	C₄×C₉	#237	1	1	yes	characteristic
190	C₄×C₉	#237	1	1	yes	characteristic
191	Dic₉	—	2	1	no	—
192	Dic₉	—	3	2	no	—
193	Dic₉	—	3	2	no	—
194	Dic₉	—	6	2	no	—
195	Dic₉	—	6	2	no	—
196	C₂×C₄²	—	27	1	no	—
197	C₄⋊C₈	—	3	1	no	—
198	C₄⋊C₈	—	27	1	no	—
199	C₂²⋊C₈	—	9	1	no	—
200	C₂²⋊C₈	—	27	1	no	—
201	C₄×C₈	—	27	1	no	—
202	C₄×C₈	—	27	1	no	—
203	C₃×C₉	#431	1	1	yes	characteristic, complemented, Sylow
204	C₂³×C₃	—	9	1	no	—
205	C₂²×D₃	—	3	1	no	—
206	C₂×C₃×C₄	—	2	1	no	—
207	C₂×C₃×C₄	—	3	1	no	—
208	C₂×C₃×C₄	—	3	1	no	—
209	C₂×C₃×C₄	—	9	1	no	—
210	C₂×C₃×C₄	—	18	1	no	—
211	C₂×C₃×C₄	—	18	1	no	—
212	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
214	C₂×Dic₃	—	3	1	no	—
215	C₂×Dic₃	—	9	1	no	—
216	C₂×Dic₃	—	9	1	no	—
217	C₂×Dic₃	—	9	1	no	—
218	C₂×Dic₃	—	9	1	no	—
219	C₂×Dic₃	—	9	4	no	—
220	C₂×Dic₃	—	18	1	no	—
221	C₂×Dic₃	—	18	1	no	—
222	C₂×Dic₃	—	18	2	no	—
223	C₂×Dic₃	C₄×D₉	1	1	yes	characteristic
224	C₂×Dic₃	C₄×D₉	1	1	yes	characteristic
225	C₄×D₃	—	6	1	no	—
226	C₄×D₃	—	6	1	no	—
227	C₄×D₃	—	9	4	no	—
228	C₄×D₃	—	18	2	no	—
229	C₃×C₈	—	2	2	no	—
230	C₃×C₈	—	2	1	no	—
231	C₃×C₈	—	2	1	no	—
232	C₃×C₈	—	6	1	no	—
233	C₃×C₈	—	18	1	no	—
234	C₃⋊C₈	—	2	1	no	—
235	C₃⋊C₈	—	6	1	no	—
236	C₃⋊C₈	—	18	2	no	—
237	C₃⋊C₈	—	18	1	no	—
238	C₃⋊C₈	—	18	1	no	—
239	C₂×C₃²	#892	1	1	yes	characteristic
240	C₂×C₃²	C₂×D₃×C₈	1	1	yes	characteristic
241	C₂×C₃²	D₃×M₁₆	1	1	yes	characteristic
242	C₃×D₃	—	6	2	no	—
243	C₂×C₉	—	2	1	no	—
244	C₂×C₉	—	2	1	no	—
245	C₂×C₉	—	2	1	no	—
246	C₂×C₉	#877	1	1	yes	characteristic
247	C₂×C₉	C₂×D₃×C₈	1	1	yes	characteristic
248	C₂×C₉	D₃×M₁₆	1	1	yes	characteristic
249	D₉	—	2	2	no	—
250	C₂²×C₄	—	9	1	no	—
251	C₂²×C₄	—	27	1	no	—
252	C₂²×C₄	—	27	1	no	—
253	C₂×C₈	—	3	1	no	—
254	C₂×C₈	—	9	1	no	—
255	C₂×C₈	—	27	1	no	—
256	C₂×C₈	D₃×D₉	1	1	yes	characteristic
257	C₄²	—	3	1	no	—
258	C₄²	—	27	1	no	—
259	C₄²	—	27	1	no	—
260	C₄²	—	27	1	no	—
261	C₂²×C₃	—	2	1	no	—
262	C₂²×C₃	—	9	2	no	—
263	C₂²×C₃	—	18	1	no	—
264	C₂²×C₃	—	18	1	no	—
265	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
266	C₂²×C₃	C₄×D₃²	1	1	yes	characteristic
267	C₂×D₃	—	3	2	no	—
268	C₂×D₃	—	6	1	no	—
269	C₂×D₃	—	6	1	no	—
270	C₃×C₄	—	2	1	no	—
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	—	3	2	no	—
273	C₃×C₄	—	6	1	no	—
274	C₃×C₄	—	18	1	no	—
275	C₃×C₄	#3699	1	1	yes	characteristic
276	C₃×C₄	#3701	1	1	yes	characteristic
277	C₃×C₄	#3796	1	1	yes	characteristic
278	C₃×C₄	#3805	1	1	yes	characteristic
279	Dic₃	—	2	1	no	—
280	Dic₃	—	6	1	no	—
281	Dic₃	—	9	2	no	—
282	Dic₃	—	9	2	no	—
283	Dic₃	—	9	2	no	—
284	Dic₃	—	9	2	no	—
285	Dic₃	—	18	2	no	—
286	Dic₃	—	18	2	no	—
287	Dic₃	C₈×D₉	1	2	yes	—
288	C₃²	#4833	1	1	yes	characteristic
289	C₉	—	2	1	no	—
290	C₉	#4718	1	1	yes	characteristic, complemented
291	C₂³	—	9	1	no	—
292	C₂×C₄	—	3	1	no	—
293	C₂×C₄	—	3	1	no	—
294	C₂×C₄	—	9	1	no	—
295	C₂×C₄	—	18	1	no	—
296	C₂×C₄	—	18	1	no	—
297	C₂×C₄	—	27	4	no	—
298	C₂×C₄	—	27	1	no	—
299	C₂×C₄	—	27	1	no	—
300	C₂×C₄	—	54	2	no	—
301	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	6	1	no	—
304	C₈	—	18	1	no	—
305	C₈	—	54	1	no	—
306	C₂×C₃	—	2	1	no	—
307	C₂×C₃	—	2	1	no	—
308	C₂×C₃	—	2	1	no	—
309	C₂×C₃	—	18	2	no	—
310	C₂×C₃	#63454	1	1	yes	characteristic
311	C₂×C₃	C₂×C₈×D₉	1	1	yes	characteristic
312	C₂×C₃	M₁₆×D₉	1	1	yes	characteristic
313	C₂×C₃	D₃²×C₈	1	1	yes	characteristic
314	C₂×C₃	#63791	1	1	yes	characteristic
315	C₂×C₃	#63878	1	1	yes	characteristic
316	D₃	—	6	2	no	—
317	C₂²	—	9	2	no	—
318	C₂²	—	18	1	no	—
319	C₂²	—	18	1	no	—
320	C₂²	C₄×D₃×D₉	1	1	yes	characteristic
321	C₄	—	3	2	no	—
322	C₄	—	6	1	no	—
323	C₄	—	18	1	no	—
324	C₄	—	27	2	no	—
325	C₄	—	27	2	no	—
326	C₄	#88634	1	1	yes	characteristic
327	C₄	#88663	1	1	yes	characteristic
328	C₃	—	2	1	no	—
329	C₃	#10588517	1	1	yes	characteristic, complemented
330	C₃	#10590656	1	1	yes	characteristic
331	C₂	—	18	2	no	—
332	C₂	D₃×C₈×D₉	1	1	yes	characteristic
333	C₂	#11720801	1	1	yes	characteristic
334	C₂	#11720975	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃×D₉
commutator	C₂×C₃×C₉	C₂²×C₈
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₂×C₃×C₈×C₉	C₂²
socle	C₂²×C₃²	C₂×C₄×D₃
prime radical	C₂²×C₃×D₉	C₂×C₄
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂×C₈	D₃×D₉
hypocenter	C₃×C₉	#431
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
2. C₈×C₃⋊Dic₉	C₂	semidirect
4. C₉×Dic₃⋊C₈	C₂	semidirect
6. #11720271	C₂	semidirect
7. Dic₃⋊(C₉⋊C₈)	C₂	semidirect
46. Dic₃×D₉	C₈	semidirect
97. Dic₃⋊C₈	D₉	semidirect
203. C₃×C₉	#431	semidirect
290. C₉	#4718	semidirect
329. C₃	#10588517	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₃×D₉	2
C₈×C₃⋊Dic₉	2
C₃×(C₂×D₉)⋊C₈	2
C₉×Dic₃⋊C₈	2
(C₂×D₉)⋊(C₃⋊C₈)	2
#11720271	2
Dic₃⋊(C₉⋊C₈)	2
#10590656	3
#10588517	3