Subgroups — #11794041

Order 1152.

486 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#3675	C₂	1	1	yes	maximal, characteristic
2	C₉×#391	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#63395	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#63369	C₂	1	1	yes	maximal, characteristic
5	#10588619	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
7	#10588206	C₂	1	1	yes	maximal, characteristic, complemented
8	#69680	—	3	1	no	maximal
9	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
10	C₂²×D₃₆	C₂²	1	1	yes	characteristic
11	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₂×#3675	—	2	2	no	—
14	C₂×#3675	—	2	2	no	—
15	C₂×#3675	—	2	2	no	—
16	C₂×(C₂×D₉)⋊C₄	—	2	1	no	—
17	C₂×(C₂×D₉)⋊C₄	—	2	1	no	—
18	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
19	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
21	C₉×#95	—	2	1	no	—
22	C₉×#95	—	2	1	no	—
23	#63395	—	2	2	no	—
24	#63369	—	2	2	no	—
25	C₄⋊(C₉⋊C₈)	—	2	2	no	—
26	C₂²×#215	—	3	1	no	—
27	C₃×#391	D₃	1	1	yes	characteristic
28	C₂×#835	—	3	1	no	—
29	C₂×#809	—	3	1	no	—
30	#4939	—	3	1	no	—
31	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
32	#4529	—	3	1	no	—
33	C₂³×D₉	—	2	1	no	—
34	C₂×Q₈×C₉	—	2	1	no	—
35	C₂×Q₈×C₉	—	2	1	no	—
36	C₂×Q₈×C₉	—	2	1	no	—
37	C₂×Q₈×C₉	—	2	1	no	—
38	C₂×Q₈×C₉	—	2	1	no	—
39	C₂×Q₈×C₉	D₄	1	2	yes	—
40	C₂²×C₄×C₉	—	2	1	no	—
41	C₂²×C₄×C₉	—	2	1	no	—
42	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
45	C₂²×Dic₉	—	2	1	no	—
46	C₂×D₃₆	—	2	1	no	—
47	C₂×D₃₆	—	2	1	no	—
48	C₂×D₃₆	—	2	1	no	—
49	C₂×D₃₆	—	2	1	no	—
50	C₂×D₃₆	—	2	1	no	—
51	C₂×D₃₆	D₄	1	2	yes	—
52	C₉×C₄⋊C₄	—	2	1	no	—
53	C₉×C₄⋊C₄	—	2	1	no	—
54	C₄²×C₉	—	2	1	no	—
55	C₄²×C₉	—	2	1	no	—
56	#3675	—	2	8	no	—
57	(C₂×D₉)⋊C₄	—	4	2	no	—
58	(C₂×D₉)⋊C₄	—	4	2	no	—
59	C₄⋊Dic₉	—	2	1	no	—
60	C₄⋊Dic₉	—	2	1	no	—
61	C₂×C₉⋊C₈	—	2	1	no	—
62	C₂×C₉⋊C₈	—	2	1	no	—
63	C₂×C₉⋊C₈	—	2	1	no	—
64	C₂×C₉⋊C₈	—	2	1	no	—
65	C₂×C₉⋊C₈	—	2	1	no	—
66	C₂×C₉⋊C₈	D₄	1	2	yes	—
67	#2055	—	9	1	no	Sylow
68	C₂²×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
69	C₂²×D₁₂	—	3	1	no	—
70	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
71	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
72	C₂×#215	—	6	2	no	—
73	C₂×#215	—	6	2	no	—
74	C₂×#215	—	6	2	no	—
75	C₂×(C₂×D₃)⋊C₄	—	6	1	no	—
76	C₂×(C₂×D₃)⋊C₄	—	6	1	no	—
77	C₂×C₄⋊Dic₃	—	3	1	no	—
78	C₂²×C₃⋊C₈	—	3	1	no	—
79	C₂²×C₃⋊C₈	—	3	1	no	—
80	C₃×#95	—	2	1	no	—
81	C₃×#95	—	2	1	no	—
82	#835	—	6	2	no	—
83	#809	—	6	2	no	—
84	C₄⋊(C₃⋊C₈)	—	6	2	no	—
85	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×D₉	—	2	2	no	—
87	C₂²×D₉	—	4	1	no	—
88	C₂²×D₉	—	4	1	no	—
89	C₂²×D₉	—	4	1	no	—
90	C₂²×D₉	—	4	1	no	—
91	C₂²×D₉	—	4	1	no	—
92	C₂²×D₉	—	4	1	no	—
93	Q₈×C₉	—	2	2	no	—
94	Q₈×C₉	—	2	2	no	—
95	Q₈×C₉	—	4	1	no	—
96	Q₈×C₉	—	4	1	no	—
97	C₂×C₄×C₉	—	2	1	no	—
98	C₂×C₄×C₉	—	2	1	no	—
99	C₂×C₄×C₉	—	2	1	no	—
100	C₂×C₄×C₉	—	2	2	no	—
101	C₂×C₄×C₉	—	2	1	no	—
102	C₂×C₄×C₉	—	2	1	no	—
103	C₂×C₄×C₉	—	4	1	no	—
104	C₂×C₄×C₉	—	4	1	no	—
105	C₂×C₄×C₉	—	4	1	no	—
106	C₂×C₄×C₉	—	4	1	no	—
107	C₂×C₄×C₉	—	4	1	no	—
108	C₂×C₄×C₉	SD₁₆	1	2	yes	complemented
109	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₉	#41	1	1	yes	characteristic
113	C₂×C₄×C₉	#41	1	1	yes	characteristic
114	C₂×C₄×C₉	#41	1	1	yes	characteristic
115	C₂×Dic₉	—	4	1	no	—
116	C₂×Dic₉	—	4	1	no	—
117	C₂×Dic₉	—	4	1	no	—
118	D₃₆	—	2	2	no	—
119	D₃₆	—	2	2	no	—
120	D₃₆	—	4	1	no	—
121	D₃₆	—	4	1	no	—
122	C₉⋊C₈	—	2	2	no	—
123	C₉⋊C₈	—	4	1	no	—
124	C₂²×SD₁₆	—	9	1	no	—
125	C₂×C₄⋊C₈	—	9	1	no	—
126	C₂×#103	—	9	1	no	—
127	C₂×#102	—	9	1	no	—
128	#391	D₉	1	1	yes	characteristic, complemented
129	#390	—	9	1	no	—
130	#340	—	9	1	no	—
131	C₂³×D₃	—	6	1	no	—
132	C₂×C₃×Q₈	—	2	1	no	—
133	C₂×C₃×Q₈	—	2	1	no	—
134	C₂×C₃×Q₈	—	2	1	no	—
135	C₂×C₃×Q₈	—	2	1	no	—
136	C₂×C₃×Q₈	—	2	1	no	—
137	C₂×C₃×Q₈	D₁₂	1	2	yes	—
138	C₂²×C₃×C₄	—	2	1	no	—
139	C₂²×C₃×C₄	—	2	1	no	—
140	C₂²×C₃×C₄	#67	1	1	yes	characteristic
141	C₂²×C₃×C₄	#67	1	1	yes	characteristic
142	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
143	C₂²×Dic₃	—	6	1	no	—
144	C₂×D₁₂	—	3	2	no	—
145	C₂×D₁₂	—	6	1	no	—
146	C₂×D₁₂	—	6	1	no	—
147	C₂×D₁₂	—	6	1	no	—
148	C₂×D₁₂	—	6	1	no	—
149	C₂×D₁₂	—	6	1	no	—
150	C₃×C₄⋊C₄	—	2	1	no	—
151	C₃×C₄⋊C₄	—	2	1	no	—
152	C₃×C₄²	—	2	1	no	—
153	C₃×C₄²	—	2	1	no	—
154	#215	—	6	8	no	—
155	(C₂×D₃)⋊C₄	—	12	2	no	—
156	(C₂×D₃)⋊C₄	—	12	2	no	—
157	C₄⋊Dic₃	—	6	1	no	—
158	C₄⋊Dic₃	—	6	1	no	—
159	C₂×C₃⋊C₈	—	3	2	no	—
160	C₂×C₃⋊C₈	—	6	1	no	—
161	C₂×C₃⋊C₈	—	6	1	no	—
162	C₂×C₃⋊C₈	—	6	1	no	—
163	C₂×C₃⋊C₈	—	6	1	no	—
164	C₂×C₃⋊C₈	—	6	1	no	—
165	C₂²×C₉	#121	1	1	yes	characteristic
166	C₂²×C₉	#121	1	1	yes	characteristic
167	C₂²×C₉	#121	1	1	yes	characteristic
168	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
169	C₂²×C₉	#135	1	1	yes	characteristic
170	C₂²×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
171	C₂²×C₉	#137	1	1	yes	characteristic
172	C₂×D₉	—	4	2	no	—
173	C₂×D₉	—	4	2	no	—
174	C₂×D₉	—	4	2	no	—
175	C₂×D₉	—	8	1	no	—
176	C₂×D₉	—	8	1	no	—
177	C₂×D₉	—	8	1	no	—
178	C₂×D₉	—	8	1	no	—
179	C₄×C₉	—	2	2	no	—
180	C₄×C₉	—	4	2	no	—
181	C₄×C₉	—	4	1	no	—
182	C₄×C₉	—	8	1	no	—
183	C₄×C₉	#120	1	1	yes	characteristic
184	C₄×C₉	#123	1	1	yes	characteristic
185	C₄×C₉	#123	1	1	yes	characteristic
186	C₄×C₉	#124	1	1	yes	characteristic
187	Dic₉	—	8	1	no	—
188	C₂²×Q₈	C₂×D₉	1	1	yes	characteristic
189	C₂²×D₄	—	9	1	no	—
190	C₂×SD₁₆	—	18	2	no	—
191	C₂×SD₁₆	—	18	2	no	—
192	C₂×SD₁₆	—	18	2	no	—
193	C₂²×C₈	—	9	1	no	—
194	C₂²×C₈	—	9	1	no	—
195	C₂×C₄⋊C₄	—	9	1	no	—
196	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
197	C₂×C₂²⋊C₄	—	18	1	no	—
198	C₂×C₂²⋊C₄	—	18	1	no	—
199	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
200	C₄⋊C₈	—	18	2	no	—
201	#103	—	18	2	no	—
202	#102	—	18	2	no	—
203	#95	—	2	1	no	—
204	#95	—	2	1	no	—
205	C₂³×C₃	C₂×#67	1	1	yes	characteristic
206	C₂²×D₃	—	6	2	no	—
207	C₂²×D₃	—	12	1	no	—
208	C₂²×D₃	—	12	1	no	—
209	C₂²×D₃	—	12	1	no	—
210	C₂²×D₃	—	12	1	no	—
211	C₂²×D₃	—	12	1	no	—
212	C₂²×D₃	—	12	1	no	—
213	C₃×Q₈	—	2	2	no	—
214	C₃×Q₈	—	2	2	no	—
215	C₃×Q₈	—	4	1	no	—
216	C₃×Q₈	—	4	1	no	—
217	C₂×C₃×C₄	—	2	1	no	—
218	C₂×C₃×C₄	—	2	1	no	—
219	C₂×C₃×C₄	—	2	2	no	—
220	C₂×C₃×C₄	—	2	1	no	—
221	C₂×C₃×C₄	—	2	1	no	—
222	C₂×C₃×C₄	—	2	1	no	—
223	C₂×C₃×C₄	—	4	1	no	—
224	C₂×C₃×C₄	—	4	1	no	—
225	C₂×C₃×C₄	—	4	1	no	—
226	C₂×C₃×C₄	—	4	1	no	—
227	C₂×C₃×C₄	—	4	1	no	—
228	C₂×C₃×C₄	#215	1	2	yes	—
229	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
230	C₂×C₃×C₄	#235	1	1	yes	characteristic
231	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
232	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
233	C₂×C₃×C₄	#237	1	1	yes	characteristic
234	C₂×C₃×C₄	#237	1	1	yes	characteristic
235	C₂×Dic₃	—	12	1	no	—
236	C₂×Dic₃	—	12	1	no	—
237	C₂×Dic₃	—	12	1	no	—
238	D₁₂	—	6	2	no	—
239	D₁₂	—	6	2	no	—
240	D₁₂	—	12	1	no	—
241	D₁₂	—	12	1	no	—
242	C₃⋊C₈	—	6	2	no	—
243	C₃⋊C₈	—	12	1	no	—
244	C₂×C₉	#394	1	1	yes	characteristic
245	C₂×C₉	#460	1	1	yes	characteristic
246	C₂×C₉	#461	1	1	yes	characteristic
247	C₂×C₉	#463	1	1	yes	characteristic
248	C₂×C₉	#464	1	1	yes	characteristic
249	C₂×C₉	#465	1	1	yes	characteristic
250	C₂×C₉	#468	1	1	yes	characteristic
251	D₉	—	8	2	no	—
252	C₂⁴	—	18	1	no	—
253	C₂×Q₈	—	2	1	no	—
254	C₂×Q₈	—	2	1	no	—
255	C₂×Q₈	—	2	1	no	—
256	C₂×Q₈	—	2	1	no	—
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	D₃₆	1	2	yes	—
259	C₂×D₄	—	9	2	no	—
260	C₂×D₄	—	18	1	no	—
261	C₂×D₄	—	18	1	no	—
262	C₂×D₄	—	18	1	no	—
263	C₂×D₄	—	18	1	no	—
264	C₂×D₄	—	18	1	no	—
265	C₂²×C₄	—	2	1	no	—
266	C₂²×C₄	—	2	1	no	—
267	C₂²×C₄	—	18	1	no	—
268	C₂²×C₄	#611	1	1	yes	characteristic
269	C₂²×C₄	#611	1	1	yes	characteristic
270	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
271	SD₁₆	—	18	8	no	—
272	C₂×C₈	—	9	2	no	—
273	C₂×C₈	—	18	1	no	—
274	C₂×C₈	—	18	1	no	—
275	C₂×C₈	—	18	1	no	—
276	C₂×C₈	—	18	1	no	—
277	C₂×C₈	—	18	1	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	18	1	no	—
281	C₄⋊C₄	—	18	1	no	—
282	C₂²⋊C₄	—	36	2	no	—
283	C₂²⋊C₄	—	36	2	no	—
284	C₄²	—	2	1	no	—
285	C₄²	—	2	1	no	—
286	C₂²×C₃	#930	1	1	yes	characteristic
287	C₂²×C₃	#938	1	1	yes	characteristic
288	C₂²×C₃	#939	1	1	yes	characteristic
289	C₂²×C₃	C₂×#215	1	1	yes	characteristic
290	C₂²×C₃	#942	1	1	yes	characteristic
291	C₂²×C₃	#949	1	1	yes	characteristic
292	C₂²×C₃	#950	1	1	yes	characteristic
293	C₂×D₃	—	12	2	no	—
294	C₂×D₃	—	12	2	no	—
295	C₂×D₃	—	12	2	no	—
296	C₂×D₃	—	24	1	no	—
297	C₂×D₃	—	24	1	no	—
298	C₂×D₃	—	24	1	no	—
299	C₂×D₃	—	24	1	no	—
300	C₃×C₄	—	2	2	no	—
301	C₃×C₄	—	4	2	no	—
302	C₃×C₄	—	4	1	no	—
303	C₃×C₄	—	8	1	no	—
304	C₃×C₄	#882	1	1	yes	characteristic
305	C₃×C₄	#883	1	1	yes	characteristic
306	C₃×C₄	#885	1	1	yes	characteristic
307	C₃×C₄	#886	1	1	yes	characteristic
308	Dic₃	—	24	1	no	—
309	C₉	#2055	1	1	yes	characteristic, complemented, Sylow
310	C₂³	—	18	2	no	—
311	C₂³	—	36	1	no	—
312	C₂³	—	36	1	no	—
313	C₂³	—	36	1	no	—
314	C₂³	—	36	1	no	—
315	C₂³	—	36	1	no	—
316	C₂³	—	36	1	no	—
317	C₂³	C₂×#611	1	1	yes	characteristic
318	Q₈	—	2	2	no	—
319	Q₈	—	2	2	no	—
320	Q₈	—	4	1	no	—
321	Q₈	—	4	1	no	—
322	D₄	—	18	2	no	—
323	D₄	—	18	2	no	—
324	D₄	—	36	1	no	—
325	D₄	—	36	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	36	1	no	—
338	C₂×C₄	—	36	1	no	—
339	C₂×C₄	—	36	1	no	—
340	C₂×C₄	#3675	1	2	yes	complemented
341	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
342	C₂×C₄	#3697	1	1	yes	characteristic
343	C₂×C₄	D₄×D₉	1	1	yes	characteristic
344	C₂×C₄	D₄×D₉	1	1	yes	characteristic
345	C₂×C₄	#3699	1	1	yes	characteristic
346	C₂×C₄	#3699	1	1	yes	characteristic
347	C₈	—	18	2	no	—
348	C₈	—	36	1	no	—
349	C₂×C₃	#4961	1	1	yes	characteristic
350	C₂×C₃	#5027	1	1	yes	characteristic
351	C₂×C₃	#5028	1	1	yes	characteristic
352	C₂×C₃	#5048	1	1	yes	characteristic
353	C₂×C₃	#5049	1	1	yes	characteristic
354	C₂×C₃	#5057	1	1	yes	characteristic
355	C₂×C₃	#5083	1	1	yes	characteristic
356	D₃	—	24	2	no	—
357	C₂²	—	36	2	no	—
358	C₂²	—	36	2	no	—
359	C₂²	—	36	2	no	—
360	C₂²	—	72	1	no	—
361	C₂²	—	72	1	no	—
362	C₂²	—	72	1	no	—
363	C₂²	—	72	1	no	—
364	C₂²	#63492	1	1	yes	characteristic
365	C₂²	#63500	1	1	yes	characteristic
366	C₂²	#63501	1	1	yes	characteristic
367	C₂²	C₂×#3675	1	1	yes	characteristic
368	C₂²	#63505	1	1	yes	characteristic
369	C₂²	#63512	1	1	yes	characteristic
370	C₂²	#63513	1	1	yes	characteristic
371	C₄	—	2	2	no	—
372	C₄	—	4	1	no	—
373	C₄	—	4	2	no	—
374	C₄	—	8	1	no	—
375	C₄	—	72	1	no	—
376	C₄	#63444	1	1	yes	characteristic
377	C₄	#63445	1	1	yes	characteristic
378	C₄	#63447	1	1	yes	characteristic
379	C₄	#63448	1	1	yes	characteristic
380	C₃	#69680	1	1	yes	characteristic
381	C₂	—	72	2	no	—
382	C₂	#10588641	1	1	yes	characteristic
383	C₂	#10588710	1	1	yes	characteristic
384	C₂	#10588711	1	1	yes	characteristic
385	C₂	#10588731	1	1	yes	characteristic
386	C₂	#10588732	1	1	yes	characteristic
387	C₂	#10588738	1	1	yes	characteristic
388	C₂	#10588768	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#611
commutator, cosocle	C₂²×C₄×C₉	C₂³
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₉×#391	C₂
socle	C₂³×C₃	C₂×#67
prime radical	C₂²×D₃₆	C₂²
hypercenter	#391	D₉
hypocenter	C₉	#2055
LCS	G ▷ C₂²×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

Complements

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

N	H ≅ G/N	type
2. C₉×#391	C₂	semidirect
3. C₂×#63395	C₂	semidirect
6. C₂×C₄⋊(C₉⋊C₈)	C₂	semidirect
7. #10588206	C₂	semidirect
108. C₂×C₄×C₉	SD₁₆	semidirect
128. #391	D₉	semidirect
309. C₉	#2055	semidirect
340. C₂×C₄	#3675	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#3675	2
C₉×#391	2
C₂×#63395	2
C₂×#63369	2
#10588619	2
C₂×C₄⋊(C₉⋊C₈)	2
#10588206	2
#69680	3