Subgroups — #13138168

Order 1728.

416 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#3663	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₃×D₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#88413	C₂	1	1	yes	maximal, characteristic
4	#11720537	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×Dic₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#11720273	C₂	1	1	yes	maximal, characteristic, complemented
7	Dic₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	#10590663	—	3	1	no	maximal
9	#10588521	—	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	C₃×#3663	—	2	2	no	—
20	Dic₃×Dic₉	C₄	1	1	yes	characteristic
21	Dic₃×Dic₉	C₄	1	1	yes	characteristic
22	#88413	—	2	2	no	—
23	C₂×C₃×ΓL(1,𝔽₂₅)	—	3	1	no	—
24	C₄×D₃×Dic₃	—	3	1	no	—
25	C₂×#3711	—	3	1	no	—
26	#63641	—	3	1	no	—
27	C₃×Dic₃⋊C₈	D₃	1	1	yes	characteristic
28	#63572	—	3	1	no	—
29	Dic₃⋊(C₃⋊C₈)	—	3	1	no	—
30	C₂×#3663	—	3	1	no	—
31	C₂×#3663	D₃	1	1	yes	characteristic
32	C₄²×D₉	—	3	1	no	—
33	C₉×C₄⋊C₈	—	3	1	no	—
34	Dic₉⋊C₈	—	3	1	no	—
35	Dic₉⋊C₈	—	3	1	no	—
36	Dic₉⋊C₈	—	6	1	no	—
37	C₄⋊(C₉⋊C₈)	—	3	1	no	—
38	C₂²×C₃×D₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
40	C₂×C₃⋊Dic₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃⋊Dic₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₉×Dic₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₉×Dic₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃×Dic₉	C₂×C₄	1	1	yes	characteristic
45	C₃×C₄×D₉	D₄	1	2	yes	—
46	C₃×C₄×D₉	Q₈	1	2	yes	—
47	Dic₃×D₉	—	2	2	no	—
48	Dic₃×D₉	—	2	2	no	—
49	C₃×C₈×C₉	—	2	1	no	—
50	C₃⋊(C₉⋊C₈)	—	2	1	no	—
51	C₉×C₃⋊C₈	—	2	1	no	—
52	C₃×C₉⋊C₈	—	2	1	no	—
53	#5118	—	9	1	no	—
54	#4837	—	9	1	no	—
55	C₂×C₃×C₄×D₃	—	3	1	no	—
56	C₂×D₃×Dic₃	—	3	1	no	—
57	C₂×D₃×Dic₃	—	3	1	no	—
58	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
59	C₄×C₃⋊Dic₃	—	3	1	no	—
60	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
61	C₃×C₄×Dic₃	C₂×D₃	1	1	yes	characteristic
62	C₂×C₃×C₃⋊C₈	—	3	1	no	—
63	C₂×C₃×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
64	C₃×ΓL(1,𝔽₂₅)	—	6	2	no	—
65	Dic₃²	—	3	1	no	—
66	Dic₃²	—	3	1	no	—
67	#3711	—	6	2	no	—
68	C₂×C₄×D₉	—	3	1	no	—
69	C₂×C₄×D₉	—	3	1	no	—
70	C₂×C₄×D₉	C₂×D₃	1	1	yes	characteristic
71	C₂×C₈×C₉	—	2	1	no	—
72	C₂×C₈×C₉	—	3	1	no	—
73	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
74	C₄²×C₉	—	3	1	no	—
75	C₄×Dic₉	—	3	1	no	—
76	C₄×Dic₉	—	3	1	no	—
77	C₄×Dic₉	—	3	1	no	—
78	C₄×Dic₉	—	6	1	no	—
79	C₂×C₉⋊C₈	—	3	1	no	—
80	C₂×C₉⋊C₈	—	6	1	no	—
81	C₂×C₉⋊C₈	C₂×D₃	1	1	yes	characteristic
82	#3663	—	2	2	no	—
83	#3663	—	6	2	no	—
84	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₃×D₉	—	2	1	no	—
86	C₂×C₃×D₉	—	2	1	no	—
87	C₂×C₃×D₉	C₄⋊C₄	1	2	yes	—
88	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
89	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
90	C₃⋊Dic₉	—	2	1	no	—
91	C₃⋊Dic₉	M₁₆	1	2	yes	complemented
92	C₉×Dic₃	—	2	1	no	—
93	C₉×Dic₃	M₁₆	1	2	yes	complemented
94	C₃×Dic₉	C₄⋊C₄	1	2	yes	—
95	C₂×C₃×M₁₆	—	9	1	no	—
96	C₂×C₄×Dic₃	—	9	1	no	—
97	C₂×#208	—	9	1	no	—
98	C₂×ΓL(1,𝔽₂₅)	—	3	1	no	—
99	C₂×ΓL(1,𝔽₂₅)	—	9	1	no	—
100	C₄²×D₃	—	9	1	no	—
101	C₃×C₄⋊C₈	—	3	1	no	—
102	Dic₃⋊C₈	—	9	1	no	—
103	Dic₃⋊C₈	—	9	1	no	—
104	Dic₃⋊C₈	—	9	1	no	—
105	Dic₃⋊C₈	—	9	1	no	—
106	Dic₃⋊C₈	—	9	1	no	—
107	Dic₃⋊C₈	—	18	1	no	—
108	Dic₃⋊C₈	D₉	1	1	yes	characteristic, complemented
109	C₄⋊(C₃⋊C₈)	—	9	1	no	—
110	C₂²×C₃×D₃	—	3	1	no	—
111	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
112	C₂×C₃⋊Dic₃	—	3	1	no	—
113	C₂×C₃⋊Dic₃	—	3	1	no	—
114	C₂×C₃×Dic₃	—	3	1	no	—
115	C₂×C₃×Dic₃	C₄×D₃	1	1	yes	characteristic
116	C₂×C₃×Dic₃	C₄×D₃	1	1	yes	characteristic
117	C₃×C₄×D₃	—	3	2	no	—
118	C₃×C₄×D₃	—	3	2	no	—
119	D₃×Dic₃	—	6	2	no	—
120	D₃×Dic₃	—	6	2	no	—
121	C₂²×D₉	C₄×D₃	1	1	yes	characteristic
122	C₃²×C₈	—	2	1	no	—
123	C₃⋊(C₃⋊C₈)	—	6	1	no	—
124	C₃×C₃⋊C₈	—	2	1	no	—
125	C₃×C₃⋊C₈	—	6	1	no	—
126	C₂×C₄×C₉	—	2	1	no	—
127	C₂×C₄×C₉	—	3	1	no	—
128	C₂×C₄×C₉	—	3	1	no	—
129	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
130	C₂×Dic₉	—	3	1	no	—
131	C₂×Dic₉	—	3	1	no	—
132	C₂×Dic₉	—	6	1	no	—
133	C₂×Dic₉	—	6	1	no	—
134	C₂×Dic₉	C₄×D₃	1	1	yes	characteristic
135	C₄×D₉	—	6	2	no	—
136	C₄×D₉	—	6	2	no	—
137	C₄×D₉	Dic₆	1	2	yes	—
138	C₄×D₉	#67	1	2	yes	—
139	C₈×C₉	—	2	1	no	—
140	C₈×C₉	—	4	1	no	—
141	C₈×C₉	—	6	1	no	—
142	C₉⋊C₈	—	2	1	no	—
143	C₉⋊C₈	—	6	1	no	—
144	C₉⋊C₈	—	12	1	no	—
145	#423	—	27	1	no	Sylow
146	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
148	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
149	C₃×D₉	—	2	2	no	—
150	C₂²×C₃×C₄	—	9	1	no	—
151	C₂²×Dic₃	—	9	1	no	—
152	C₂²×Dic₃	—	9	1	no	—
153	C₂×C₄×D₃	—	3	1	no	—
154	C₂×C₄×D₃	—	9	1	no	—
155	C₂×C₄×D₃	—	9	1	no	—
156	C₃×M₁₆	—	18	2	no	—
157	C₂×C₃×C₈	—	2	1	no	—
158	C₂×C₃×C₈	—	3	1	no	—
159	C₂×C₃×C₈	—	9	1	no	—
160	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
161	C₂×C₃×C₈	D₃²	1	1	yes	characteristic
162	C₃×C₄²	—	3	1	no	—
163	C₄×Dic₃	—	9	1	no	—
164	C₄×Dic₃	—	9	1	no	—
165	C₄×Dic₃	—	9	1	no	—
166	C₄×Dic₃	—	9	1	no	—
167	C₄×Dic₃	—	9	1	no	—
168	C₄×Dic₃	—	9	1	no	—
169	C₄×Dic₃	—	18	1	no	—
170	C₄×Dic₃	C₂×D₉	1	1	yes	characteristic
171	#208	—	18	2	no	—
172	C₂×C₃⋊C₈	—	3	1	no	—
173	C₂×C₃⋊C₈	—	9	1	no	—
174	C₂×C₃⋊C₈	—	9	1	no	—
175	C₂×C₃⋊C₈	—	18	1	no	—
176	C₂×C₃⋊C₈	C₂×D₉	1	1	yes	characteristic
177	ΓL(1,𝔽₂₅)	—	6	2	no	—
178	ΓL(1,𝔽₂₅)	—	18	2	no	—
179	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
180	C₂×C₃×D₃	—	3	2	no	—
181	C₂×C₃×D₃	—	6	1	no	—
182	C₂×C₃×D₃	—	6	1	no	—
183	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
184	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
185	C₃⋊Dic₃	—	3	2	no	—
186	C₃⋊Dic₃	—	6	1	no	—
187	C₃×Dic₃	—	2	1	no	—
188	C₃×Dic₃	—	3	2	no	—
189	C₃×Dic₃	ΓL(1,𝔽₂₅)	1	2	yes	—
190	C₂²×C₉	—	2	1	no	—
191	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
192	C₂×D₉	—	2	1	no	—
193	C₂×D₉	—	2	1	no	—
194	C₂×D₉	Dic₃⋊C₄	1	2	yes	—
195	C₄×C₉	—	2	1	no	—
196	C₄×C₉	—	2	1	no	—
197	C₄×C₉	—	3	2	no	—
198	C₄×C₉	—	6	1	no	—
199	C₄×C₉	C₂×Dic₆	1	1	yes	characteristic
200	C₄×C₉	C₂×#67	1	1	yes	characteristic
201	Dic₉	—	3	2	no	—
202	Dic₉	—	6	2	no	—
203	Dic₉	—	6	1	no	—
204	Dic₉	—	12	1	no	—
205	Dic₉	Dic₃⋊C₄	1	2	yes	—
206	C₂×M₁₆	—	9	1	no	—
207	C₂×M₁₆	—	27	1	no	—
208	C₂×C₄²	—	27	1	no	—
209	C₄⋊C₈	—	3	1	no	—
210	C₄⋊C₈	—	27	1	no	—
211	C₄⋊C₈	—	27	1	no	—
212	C₄⋊C₈	—	27	1	no	—
213	C₃×C₉	#423	1	1	yes	characteristic, complemented, Sylow
214	C₂³×C₃	—	9	1	no	—
215	C₂²×D₃	—	3	1	no	—
216	C₂×C₃×C₄	—	2	1	no	—
217	C₂×C₃×C₄	—	3	1	no	—
218	C₂×C₃×C₄	—	3	1	no	—
219	C₂×C₃×C₄	—	9	1	no	—
220	C₂×C₃×C₄	—	9	2	no	—
221	C₂×C₃×C₄	—	9	2	no	—
222	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
223	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
224	C₂×Dic₃	—	3	1	no	—
225	C₂×Dic₃	—	9	1	no	—
226	C₂×Dic₃	—	9	1	no	—
227	C₂×Dic₃	—	9	1	no	—
228	C₂×Dic₃	—	9	1	no	—
229	C₂×Dic₃	—	18	1	no	—
230	C₂×Dic₃	—	18	1	no	—
231	C₂×Dic₃	—	18	2	no	—
232	C₂×Dic₃	—	18	2	no	—
233	C₂×Dic₃	C₄×D₉	1	1	yes	characteristic
234	C₂×Dic₃	C₄×D₉	1	1	yes	characteristic
235	C₄×D₃	—	3	2	no	—
236	C₄×D₃	—	3	2	no	—
237	C₄×D₃	—	18	2	no	—
238	C₄×D₃	—	18	2	no	—
239	C₃×C₈	—	2	1	no	—
240	C₃×C₈	—	2	1	no	—
241	C₃×C₈	—	4	1	no	—
242	C₃×C₈	—	6	1	no	—
243	C₃×C₈	—	18	1	no	—
244	C₃⋊C₈	—	2	1	no	—
245	C₃⋊C₈	—	6	1	no	—
246	C₃⋊C₈	—	18	1	no	—
247	C₃⋊C₈	—	18	1	no	—
248	C₃⋊C₈	—	36	1	no	—
249	C₂×C₃²	D₃×C₄⋊C₄	1	1	yes	characteristic
250	C₂×C₃²	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
251	C₂×C₃²	D₃×M₁₆	1	1	yes	characteristic
252	C₃×D₃	—	6	2	no	—
253	C₂×C₉	—	2	1	no	—
254	C₂×C₉	—	2	1	no	—
255	C₂×C₉	—	2	1	no	—
256	C₂×C₉	D₃×M₁₆	1	1	yes	characteristic
257	C₂×C₉	D₃×M₁₆	1	1	yes	characteristic
258	C₂×C₉	C₂×Dic₃⋊C₄	1	1	yes	characteristic
259	D₉	—	2	2	no	—
260	C₂²×C₄	—	9	1	no	—
261	C₂²×C₄	—	27	1	no	—
262	C₂²×C₄	—	27	1	no	—
263	M₁₆	—	18	2	no	—
264	M₁₆	—	54	2	no	—
265	C₂×C₈	—	3	1	no	—
266	C₂×C₈	—	9	1	no	—
267	C₂×C₈	—	27	1	no	—
268	C₂×C₈	D₃×D₉	1	1	yes	characteristic
269	C₄²	—	3	1	no	—
270	C₄²	—	27	1	no	—
271	C₄²	—	27	1	no	—
272	C₄²	—	27	1	no	—
273	C₂²×C₃	—	2	1	no	—
274	C₂²×C₃	—	9	2	no	—
275	C₂²×C₃	—	18	1	no	—
276	C₂²×C₃	—	18	1	no	—
277	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
278	C₂²×C₃	C₄×D₃²	1	1	yes	characteristic
279	C₂×D₃	—	3	2	no	—
280	C₂×D₃	—	6	1	no	—
281	C₂×D₃	—	6	1	no	—
282	C₃×C₄	—	2	1	no	—
283	C₃×C₄	—	2	1	no	—
284	C₃×C₄	—	3	2	no	—
285	C₃×C₄	—	6	1	no	—
286	C₃×C₄	—	9	2	no	—
287	C₃×C₄	D₄×D₉	1	1	yes	characteristic
288	C₃×C₄	Q₈×D₉	1	1	yes	characteristic
289	C₃×C₄	D₃×Dic₆	1	1	yes	characteristic
290	C₃×C₄	D₃×#67	1	1	yes	characteristic
291	Dic₃	—	2	1	no	—
292	Dic₃	—	3	2	no	—
293	Dic₃	—	9	2	no	—
294	Dic₃	—	9	2	no	—
295	Dic₃	—	18	2	no	—
296	Dic₃	—	18	1	no	—
297	Dic₃	—	18	1	no	—
298	Dic₃	—	36	1	no	—
299	Dic₃	#3663	1	2	yes	complemented
300	C₃²	#4837	1	1	yes	characteristic
301	C₉	—	2	1	no	—
302	C₉	#5118	1	1	yes	characteristic, complemented
303	C₂³	—	9	1	no	—
304	C₂×C₄	—	3	1	no	—
305	C₂×C₄	—	3	1	no	—
306	C₂×C₄	—	9	1	no	—
307	C₂×C₄	—	9	2	no	—
308	C₂×C₄	—	9	2	no	—
309	C₂×C₄	—	27	1	no	—
310	C₂×C₄	—	27	1	no	—
311	C₂×C₄	—	54	2	no	—
312	C₂×C₄	—	54	2	no	—
313	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	6	1	no	—
316	C₈	—	18	1	no	—
317	C₈	—	54	1	no	—
318	C₂×C₃	—	2	1	no	—
319	C₂×C₃	—	2	1	no	—
320	C₂×C₃	—	2	1	no	—
321	C₂×C₃	—	18	2	no	—
322	C₂×C₃	C₄⋊C₄×D₉	1	1	yes	characteristic
323	C₂×C₃	C₂×#3663	1	1	yes	characteristic
324	C₂×C₃	M₁₆×D₉	1	1	yes	characteristic
325	C₂×C₃	D₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
326	C₂×C₃	#63791	1	1	yes	characteristic
327	C₂×C₃	D₃×Dic₃⋊C₄	1	1	yes	characteristic
328	D₃	—	6	2	no	—
329	C₂²	—	9	2	no	—
330	C₂²	—	18	1	no	—
331	C₂²	—	18	1	no	—
332	C₂²	C₄×D₃×D₉	1	1	yes	characteristic
333	C₄	—	3	2	no	—
334	C₄	—	6	1	no	—
335	C₄	—	9	2	no	—
336	C₄	—	27	2	no	—
337	C₄	—	54	1	no	—
338	C₄	D₉×Dic₆	1	1	yes	characteristic
339	C₄	D₉×#67	1	1	yes	characteristic
340	C₃	—	2	1	no	—
341	C₃	#10588521	1	1	yes	characteristic, complemented
342	C₃	#10590663	1	1	yes	characteristic
343	C₂	—	18	2	no	—
344	C₂	D₃×#3663	1	1	yes	characteristic
345	C₂	#11720801	1	1	yes	characteristic
346	C₂	D₉×Dic₃⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃×D₉
commutator	C₂²×C₃×C₉	C₂²×C₄
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	#423
LCS	G ▷ C₂²×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
4. #11720537	C₂	semidirect
5. C₉×Dic₃⋊C₈	C₂	semidirect
6. #11720273	C₂	semidirect
7. Dic₃⋊(C₉⋊C₈)	C₂	semidirect
91. C₃⋊Dic₉	M₁₆	semidirect
93. C₉×Dic₃	M₁₆	semidirect
108. Dic₃⋊C₈	D₉	semidirect
213. C₃×C₉	#423	semidirect
299. Dic₃	#3663	semidirect
302. C₉	#5118	semidirect
341. C₃	#10588521	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#3663	2
C₄×Dic₃×D₉	2
C₂×#88413	2
#11720537	2
C₉×Dic₃⋊C₈	2
#11720273	2
Dic₃⋊(C₉⋊C₈)	2
#10590663	3
#10588521	3