Subgroups — #11786379

Order 1152.

486 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#3663	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×(C₂×D₉)⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×(C₂×D₉)⋊C₄	C₂	1	1	yes	maximal, characteristic
6	#10588228	C₂	1	1	yes	maximal, characteristic, complemented
7	#10588200	C₂	1	1	yes	maximal, characteristic, complemented
8	#70478	—	3	1	no	maximal
9	C₂²×C₄×D₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₂×(C₂×D₉)⋊C₄	C₄	1	1	yes	characteristic
14	C₂×(C₂×D₉)⋊C₄	C₄	1	1	yes	characteristic
15	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
16	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×#3663	—	2	2	no	—
19	C₂×#3663	—	2	2	no	—
20	C₂×#3663	—	2	2	no	—
21	C₉×C₄⋊C₈	—	2	2	no	—
22	C₉⋊#95	C₄	1	1	yes	characteristic
23	C₉⋊#95	C₄	1	1	yes	characteristic
24	(C₂×D₉)⋊C₈	—	2	2	no	—
25	Dic₉⋊C₈	—	2	2	no	—
26	C₂²×ΓL(1,𝔽₂₅)	—	3	1	no	—
27	C₂×C₃×C₄⋊C₈	D₃	1	1	yes	characteristic
28	C₂×(C₂×D₃)⋊C₈	—	3	1	no	—
29	C₂×Dic₃⋊C₈	—	3	1	no	—
30	C₄×(C₂×D₃)⋊C₄	—	3	1	no	—
31	#4550	—	3	1	no	—
32	#4524	—	3	1	no	—
33	C₂³×D₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
37	C₂²×Dic₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×Dic₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×Dic₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×D₉	—	2	1	no	—
41	C₂×C₄×D₉	—	2	1	no	—
42	C₂×C₄×D₉	—	2	1	no	—
43	C₂×C₄×D₉	—	2	1	no	—
44	C₂×C₄×D₉	D₄	1	2	yes	—
45	C₂×C₄×D₉	Q₈	1	2	yes	—
46	C₂×C₈×C₉	—	2	1	no	—
47	C₂×C₈×C₉	—	2	1	no	—
48	C₂×C₈×C₉	—	2	1	no	—
49	C₂×C₈×C₉	—	2	1	no	—
50	C₂×C₈×C₉	—	2	1	no	—
51	C₂×C₈×C₉	D₄	1	2	yes	—
52	C₄²×C₉	—	2	1	no	—
53	C₄²×C₉	—	2	1	no	—
54	(C₂×D₉)⋊C₄	—	2	2	no	—
55	(C₂×D₉)⋊C₄	—	2	2	no	—
56	C₄×Dic₉	—	2	1	no	—
57	C₄×Dic₉	—	2	1	no	—
58	C₂×C₉⋊C₈	—	2	1	no	—
59	C₂×C₉⋊C₈	—	2	1	no	—
60	C₂×C₉⋊C₈	—	2	1	no	—
61	C₂×C₉⋊C₈	—	2	1	no	—
62	C₂×C₉⋊C₈	—	2	1	no	—
63	C₂×C₉⋊C₈	D₄	1	2	yes	—
64	#3663	—	2	8	no	—
65	#1851	—	9	1	no	Sylow
66	C₂²×C₄×D₃	—	3	1	no	—
67	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
68	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
69	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
70	C₂×(C₂×D₃)⋊C₄	—	3	1	no	—
71	C₂×(C₂×D₃)⋊C₄	—	3	1	no	—
72	C₂×C₄×Dic₃	—	3	1	no	—
73	C₂²×C₃⋊C₈	—	3	1	no	—
74	C₂²×C₃⋊C₈	—	3	1	no	—
75	C₂×ΓL(1,𝔽₂₅)	—	6	2	no	—
76	C₂×ΓL(1,𝔽₂₅)	—	6	2	no	—
77	C₂×ΓL(1,𝔽₂₅)	—	6	2	no	—
78	C₃×C₄⋊C₈	—	2	2	no	—
79	C₃⋊#95	—	3	1	no	—
80	C₃⋊#95	—	3	1	no	—
81	(C₂×D₃)⋊C₈	—	6	2	no	—
82	Dic₃⋊C₈	—	6	2	no	—
83	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
84	C₂²×D₉	—	2	1	no	—
85	C₂²×D₉	—	2	1	no	—
86	C₂²×D₉	—	2	1	no	—
87	C₂²×D₉	—	2	1	no	—
88	C₂²×D₉	—	2	1	no	—
89	C₂²×D₉	—	2	1	no	—
90	C₂²×D₉	C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₉	—	2	1	no	—
92	C₂×C₄×C₉	—	2	1	no	—
93	C₂×C₄×C₉	—	2	1	no	—
94	C₂×C₄×C₉	—	2	1	no	—
95	C₂×C₄×C₉	—	2	1	no	—
96	C₂×C₄×C₉	M₁₆	1	2	yes	complemented
97	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₉	#41	1	1	yes	characteristic
102	C₂×C₄×C₉	#41	1	1	yes	characteristic
103	C₂×Dic₉	—	2	1	no	—
104	C₂×Dic₉	—	2	1	no	—
105	C₂×Dic₉	—	2	1	no	—
106	C₂×Dic₉	—	2	1	no	—
107	C₂×Dic₉	—	2	1	no	—
108	C₂×Dic₉	—	2	1	no	—
109	C₂×Dic₉	—	2	1	no	—
110	C₂×Dic₉	C₄⋊C₄	1	2	yes	—
111	C₂×Dic₉	M₁₆	1	2	yes	complemented
112	C₄×D₉	—	2	2	no	—
113	C₄×D₉	—	2	2	no	—
114	C₄×D₉	—	2	2	no	—
115	C₄×D₉	—	2	2	no	—
116	C₈×C₉	—	2	2	no	—
117	C₈×C₉	—	4	1	no	—
118	C₉⋊C₈	—	2	2	no	—
119	C₉⋊C₈	—	4	1	no	—
120	C₂²×M₁₆	—	9	1	no	—
121	C₂×C₄⋊C₈	—	9	1	no	—
122	C₂×C₄⋊C₈	D₉	1	1	yes	characteristic, complemented
123	C₂×C₂²⋊C₈	—	9	1	no	—
124	C₄×C₂²⋊C₄	—	9	1	no	—
125	#336	—	9	1	no	—
126	#336	—	9	1	no	—
127	C₂³×D₃	—	3	1	no	—
128	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
129	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
130	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
131	C₂²×Dic₃	—	3	1	no	—
132	C₂²×Dic₃	—	3	1	no	—
133	C₂²×Dic₃	—	3	1	no	—
134	C₂×C₄×D₃	—	3	2	no	—
135	C₂×C₄×D₃	—	3	2	no	—
136	C₂×C₄×D₃	—	6	1	no	—
137	C₂×C₄×D₃	—	6	1	no	—
138	C₂×C₄×D₃	—	6	1	no	—
139	C₂×C₄×D₃	—	6	1	no	—
140	C₂×C₃×C₈	—	2	1	no	—
141	C₂×C₃×C₈	—	2	1	no	—
142	C₂×C₃×C₈	—	2	1	no	—
143	C₂×C₃×C₈	—	2	1	no	—
144	C₂×C₃×C₈	—	2	1	no	—
145	C₂×C₃×C₈	#67	1	2	yes	—
146	C₃×C₄²	—	2	1	no	—
147	C₃×C₄²	—	2	1	no	—
148	(C₂×D₃)⋊C₄	—	6	2	no	—
149	(C₂×D₃)⋊C₄	—	6	2	no	—
150	C₄×Dic₃	—	6	1	no	—
151	C₄×Dic₃	—	6	1	no	—
152	C₂×C₃⋊C₈	—	3	2	no	—
153	C₂×C₃⋊C₈	—	6	1	no	—
154	C₂×C₃⋊C₈	—	6	1	no	—
155	C₂×C₃⋊C₈	—	6	1	no	—
156	C₂×C₃⋊C₈	—	6	1	no	—
157	C₂×C₃⋊C₈	—	6	1	no	—
158	ΓL(1,𝔽₂₅)	—	6	8	no	—
159	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
161	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
162	C₂²×C₉	C₂×M₁₆	1	1	yes	characteristic
163	C₂²×C₉	C₂×M₁₆	1	1	yes	characteristic
164	C₂²×C₉	#131	1	1	yes	characteristic
165	C₂²×C₉	#131	1	1	yes	characteristic
166	C₂×D₉	—	2	2	no	—
167	C₂×D₉	—	2	2	no	—
168	C₂×D₉	—	2	2	no	—
169	C₂×D₉	—	4	1	no	—
170	C₂×D₉	—	4	1	no	—
171	C₂×D₉	—	4	1	no	—
172	C₂×D₉	—	4	1	no	—
173	C₄×C₉	—	2	2	no	—
174	C₄×C₉	—	4	1	no	—
175	C₄×C₉	#120	1	1	yes	characteristic
176	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
177	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
178	C₄×C₉	#123	1	1	yes	characteristic
179	Dic₉	—	2	2	no	—
180	Dic₉	—	2	2	no	—
181	Dic₉	—	4	1	no	—
182	C₂³×C₄	—	9	1	no	—
183	C₂×M₁₆	—	18	2	no	—
184	C₂×M₁₆	—	18	2	no	—
185	C₂×M₁₆	—	18	2	no	—
186	C₂²×C₈	—	9	1	no	—
187	C₂²×C₈	—	9	1	no	—
188	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
189	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
190	C₂×C₂²⋊C₄	—	9	1	no	—
191	C₂×C₂²⋊C₄	—	9	1	no	—
192	C₂×C₄²	—	9	1	no	—
193	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
194	C₄⋊C₈	—	2	2	no	—
195	C₄⋊C₈	—	18	2	no	—
196	C₂²⋊C₈	—	18	2	no	—
197	#95	—	9	1	no	—
198	#95	—	9	1	no	—
199	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
200	C₂²×D₃	—	3	2	no	—
201	C₂²×D₃	—	6	1	no	—
202	C₂²×D₃	—	6	1	no	—
203	C₂²×D₃	—	6	1	no	—
204	C₂²×D₃	—	6	1	no	—
205	C₂²×D₃	—	6	1	no	—
206	C₂²×D₃	—	6	1	no	—
207	C₂×C₃×C₄	—	2	1	no	—
208	C₂×C₃×C₄	—	2	1	no	—
209	C₂×C₃×C₄	—	2	1	no	—
210	C₂×C₃×C₄	—	2	1	no	—
211	C₂×C₃×C₄	—	2	1	no	—
212	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
213	C₂×C₃×C₄	#235	1	1	yes	characteristic
214	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
215	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
216	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
217	C₂×C₃×C₄	#239	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
219	C₂×Dic₃	—	3	2	no	—
220	C₂×Dic₃	—	3	2	no	—
221	C₂×Dic₃	—	6	1	no	—
222	C₂×Dic₃	—	6	1	no	—
223	C₂×Dic₃	—	6	1	no	—
224	C₂×Dic₃	—	6	1	no	—
225	C₂×Dic₃	—	6	1	no	—
226	C₂×Dic₃	—	6	1	no	—
227	C₂×Dic₃	—	6	1	no	—
228	C₄×D₃	—	6	2	no	—
229	C₄×D₃	—	6	2	no	—
230	C₄×D₃	—	6	2	no	—
231	C₄×D₃	—	6	2	no	—
232	C₃×C₈	—	2	2	no	—
233	C₃×C₈	—	4	1	no	—
234	C₃⋊C₈	—	6	2	no	—
235	C₃⋊C₈	—	12	1	no	—
236	C₂×C₉	#385	1	1	yes	characteristic
237	C₂×C₉	#423	1	1	yes	characteristic
238	C₂×C₉	#424	1	1	yes	characteristic
239	C₂×C₉	#435	1	1	yes	characteristic
240	C₂×C₉	#435	1	1	yes	characteristic
241	C₂×C₉	#436	1	1	yes	characteristic
242	C₂×C₉	#436	1	1	yes	characteristic
243	D₉	—	4	2	no	—
244	C₂⁴	—	9	1	no	—
245	C₂²×C₄	—	9	1	no	—
246	C₂²×C₄	—	9	1	no	—
247	C₂²×C₄	—	9	2	no	—
248	C₂²×C₄	—	9	2	no	—
249	C₂²×C₄	—	9	1	no	—
250	C₂²×C₄	—	18	1	no	—
251	C₂²×C₄	—	18	1	no	—
252	C₂²×C₄	—	18	1	no	—
253	C₂²×C₄	—	18	1	no	—
254	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
255	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
256	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
257	M₁₆	—	18	8	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	9	2	no	—
264	C₂×C₈	—	18	1	no	—
265	C₂×C₈	—	18	1	no	—
266	C₂×C₈	—	18	1	no	—
267	C₂×C₈	—	18	1	no	—
268	C₂×C₈	—	18	1	no	—
269	C₂×C₈	#611	1	2	yes	—
270	C₂²⋊C₄	—	18	2	no	—
271	C₂²⋊C₄	—	18	2	no	—
272	C₄²	—	2	1	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	18	1	no	—
275	C₄²	—	18	1	no	—
276	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
277	C₂²×C₃	#893	1	1	yes	characteristic
278	C₂²×C₃	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
279	C₂²×C₃	#901	1	1	yes	characteristic
280	C₂²×C₃	D₃×M₁₆	1	1	yes	characteristic
281	C₂²×C₃	#907	1	1	yes	characteristic
282	C₂²×C₃	C₄×#67	1	1	yes	characteristic
283	C₂×D₃	—	6	2	no	—
284	C₂×D₃	—	6	2	no	—
285	C₂×D₃	—	6	2	no	—
286	C₂×D₃	—	12	1	no	—
287	C₂×D₃	—	12	1	no	—
288	C₂×D₃	—	12	1	no	—
289	C₂×D₃	—	12	1	no	—
290	C₃×C₄	—	2	2	no	—
291	C₃×C₄	—	4	1	no	—
292	C₃×C₄	#894	1	1	yes	characteristic
293	C₃×C₄	#896	1	1	yes	characteristic
294	C₃×C₄	#937	1	1	yes	characteristic
295	C₃×C₄	#946	1	1	yes	characteristic
296	Dic₃	—	6	2	no	—
297	Dic₃	—	6	2	no	—
298	Dic₃	—	12	1	no	—
299	C₉	#1851	1	1	yes	characteristic, complemented, Sylow
300	C₂³	—	9	2	no	—
301	C₂³	—	18	1	no	—
302	C₂³	—	18	1	no	—
303	C₂³	—	18	1	no	—
304	C₂³	—	18	1	no	—
305	C₂³	—	18	1	no	—
306	C₂³	—	18	1	no	—
307	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	9	2	no	—
314	C₂×C₄	—	9	2	no	—
315	C₂×C₄	—	18	1	no	—
316	C₂×C₄	—	18	1	no	—
317	C₂×C₄	—	18	1	no	—
318	C₂×C₄	—	18	1	no	—
319	C₂×C₄	—	18	2	no	—
320	C₂×C₄	—	18	2	no	—
321	C₂×C₄	—	18	2	no	—
322	C₂×C₄	—	18	2	no	—
323	C₂×C₄	—	18	1	no	—
324	C₂×C₄	—	18	1	no	—
325	C₂×C₄	—	18	1	no	—
326	C₂×C₄	#3663	1	2	yes	complemented
327	C₂×C₄	#3697	1	1	yes	characteristic
328	C₂×C₄	D₄×D₉	1	1	yes	characteristic
329	C₂×C₄	D₄×D₉	1	1	yes	characteristic
330	C₂×C₄	Q₈×D₉	1	1	yes	characteristic
331	C₂×C₄	#3701	1	1	yes	characteristic
332	C₂×C₄	C₂×#611	1	1	yes	characteristic
333	C₈	—	2	2	no	—
334	C₈	—	4	1	no	—
335	C₈	—	18	2	no	—
336	C₈	—	36	1	no	—
337	C₂×C₃	#4836	1	1	yes	characteristic
338	C₂×C₃	#4837	1	1	yes	characteristic
339	C₂×C₃	#4838	1	1	yes	characteristic
340	C₂×C₃	#4839	1	1	yes	characteristic
341	C₂×C₃	#4989	1	1	yes	characteristic
342	C₂×C₃	#5111	1	1	yes	characteristic
343	C₂×C₃	#5128	1	1	yes	characteristic
344	D₃	—	12	2	no	—
345	C₂²	—	18	2	no	—
346	C₂²	—	18	2	no	—
347	C₂²	—	18	2	no	—
348	C₂²	—	36	1	no	—
349	C₂²	—	36	1	no	—
350	C₂²	—	36	1	no	—
351	C₂²	—	36	1	no	—
352	C₂²	C₄⋊C₄×D₉	1	1	yes	characteristic
353	C₂²	#63455	1	1	yes	characteristic
354	C₂²	C₂×#3663	1	1	yes	characteristic
355	C₂²	#63464	1	1	yes	characteristic
356	C₂²	M₁₆×D₉	1	1	yes	characteristic
357	C₂²	#63469	1	1	yes	characteristic
358	C₂²	C₄×#611	1	1	yes	characteristic
359	C₄	—	2	2	no	—
360	C₄	—	4	1	no	—
361	C₄	—	18	2	no	—
362	C₄	—	18	2	no	—
363	C₄	—	36	1	no	—
364	C₄	#63456	1	1	yes	characteristic
365	C₄	#63458	1	1	yes	characteristic
366	C₄	#63499	1	1	yes	characteristic
367	C₄	#63508	1	1	yes	characteristic
368	C₃	#70478	1	1	yes	characteristic
369	C₂	—	36	2	no	—
370	C₂	#10588520	1	1	yes	characteristic
371	C₂	#10588521	1	1	yes	characteristic
372	C₂	#10588522	1	1	yes	characteristic
373	C₂	#10588523	1	1	yes	characteristic
374	C₂	#10588669	1	1	yes	characteristic
375	C₂	#10588792	1	1	yes	characteristic
376	C₂	#10588810	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉
commutator	C₂³×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₉×C₄⋊C₈	C₂
socle	C₂³×C₃	C₂×C₄×D₃
prime radical	C₂³×D₉	C₂×C₄
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×C₄⋊C₈	D₉
hypocenter	C₉	#1851
LCS	G ▷ C₂³×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₉×C₄⋊C₈	C₂	semidirect
4. C₂×Dic₉⋊C₈	C₂	semidirect
6. #10588228	C₂	semidirect
7. #10588200	C₂	semidirect
96. C₂×C₄×C₉	M₁₆	semidirect
111. C₂×Dic₉	M₁₆	semidirect
122. C₂×C₄⋊C₈	D₉	semidirect
299. C₉	#1851	semidirect
326. C₂×C₄	#3663	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#3663	2
C₂×C₉×C₄⋊C₈	2
C₂×(C₂×D₉)⋊C₈	2
C₂×Dic₉⋊C₈	2
C₄×(C₂×D₉)⋊C₄	2
#10588228	2
#10588200	2
#70478	3