Subgroups — #11868013

Order 1152.

678 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#131	C₂	1	1	yes	maximal, characteristic, complemented
2	#10589925	C₂	1	2	yes	maximal, complemented
3	C₂×#63467	C₂	1	1	yes	maximal, characteristic, complemented
4	#10588823	C₂	1	2	yes	maximal, complemented
5	#10588822	C₂	1	4	yes	maximal, complemented
6	#10588821	C₂	1	2	yes	maximal, complemented
7	#10588814	C₂	1	2	yes	maximal, complemented
8	C₂×#63372	C₂	1	1	yes	maximal, characteristic, complemented
9	#83805	—	3	1	no	maximal
10	C₂×C₉×#41	C₂²	1	1	yes	characteristic
11	C₂×#3697	C₂²	1	2	yes	—
12	C₉×#131	—	2	1	no	—
13	C₉×#131	—	2	1	no	—
14	C₉×#131	C₂²	1	2	yes	—
15	C₉×#131	C₂²	1	2	yes	—
16	C₂×C₉×M₁₆	—	2	1	no	—
17	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₉	—	2	1	no	—
19	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
20	#63513	—	2	4	no	—
21	#63512	C₂²	1	4	yes	—
22	#63511	C₂²	1	4	yes	—
23	#63505	—	2	2	no	—
24	#63495	—	2	2	no	—
25	C₂×#3667	C₂²	1	2	yes	—
26	#63467	—	2	1	no	—
27	#63467	—	2	1	no	—
28	#63467	—	2	1	no	—
29	#63467	—	2	1	no	—
30	C₂×D₇₂	C₂²	1	1	yes	characteristic, complemented
31	C₂×#3664	C₂²	1	2	yes	complemented
32	C₂×Dic₃₆	C₂²	1	1	yes	characteristic, complemented
33	#63383	C₂²	1	4	yes	complemented
34	#63382	C₂²	1	4	yes	complemented
35	#63372	C₂²	1	2	yes	complemented
36	#63372	C₂²	1	2	yes	complemented
37	C₂×C₃×#131	D₃	1	1	yes	characteristic
38	#5822	—	3	2	no	—
39	C₂×#904	—	3	1	no	—
40	#5143	—	3	2	no	—
41	#5142	—	3	2	no	—
42	#5141	—	3	4	no	—
43	#5132	—	3	2	no	—
44	C₂×#819	—	3	1	no	—
45	C₉×#41	—	2	1	no	—
46	C₉×#41	—	2	1	no	—
47	C₉×#41	D₄	1	2	yes	—
48	C₉×#41	C₂³	1	2	yes	—
49	C₂×Q₈×C₉	D₄	1	1	yes	characteristic
50	C₂×D₄×C₉	—	2	1	no	—
51	C₂×D₄×C₉	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₉	—	2	1	no	—
53	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
54	C₂×#611	—	2	2	no	—
55	#3697	—	2	2	no	—
56	#3697	—	2	2	no	—
57	#3697	—	2	2	no	—
58	#3697	—	2	2	no	—
59	C₂×D₃₆	C₂³	1	2	yes	—
60	C₂×C₄×D₉	—	2	2	no	—
61	C₂×Dic₁₈	C₂³	1	2	yes	—
62	C₉×M₁₆	—	2	1	no	—
63	C₉×M₁₆	—	2	2	no	—
64	C₉×M₁₆	—	2	2	no	—
65	C₉×M₁₆	C₂³	1	2	yes	—
66	C₂×C₈×C₉	—	2	1	no	—
67	C₂×C₈×C₉	—	2	2	no	—
68	C₂×C₈×C₉	—	2	2	no	—
69	C₂×C₈×C₉	D₄	1	1	yes	characteristic
70	C₂×C₈×C₉	D₄	1	1	yes	characteristic
71	C₂×C₈×C₉	D₄	1	1	yes	characteristic
72	C₂×C₈×C₉	D₄	1	1	yes	characteristic
73	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
75	#3675	—	2	4	no	—
76	#3674	—	2	4	no	—
77	#3673	—	2	4	no	—
78	#3672	—	2	4	no	—
79	#3667	—	2	2	no	—
80	#3667	C₂³	1	4	yes	—
81	C₂×C₉⋊C₈	—	2	2	no	—
82	D₇₂	—	2	1	no	—
83	D₇₂	—	2	1	no	—
84	#3664	—	2	2	no	—
85	#3664	—	2	2	no	—
86	Dic₃₆	—	2	1	no	—
87	Dic₃₆	—	2	1	no	—
88	#3067	—	9	1	no	Sylow
89	C₂×C₃×#41	C₂×D₃	1	1	yes	characteristic
90	C₂×#235	—	3	2	no	—
91	C₃×#131	—	2	1	no	—
92	C₃×#131	—	2	1	no	—
93	C₃×#131	C₂×D₃	1	2	yes	—
94	C₃×#131	C₂×D₃	1	2	yes	—
95	C₂×C₃×M₁₆	—	2	1	no	—
96	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
97	C₂²×C₃×C₈	—	2	1	no	—
98	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
99	#951	—	3	4	no	—
100	#950	—	6	4	no	—
101	#949	—	3	4	no	—
102	#942	—	6	2	no	—
103	#932	—	6	2	no	—
104	C₂×#208	—	3	2	no	—
105	C₂×Dic₁₂	—	3	1	no	—
106	#904	—	6	1	no	—
107	#904	—	6	1	no	—
108	#904	—	6	1	no	—
109	#904	—	6	1	no	—
110	C₂×D₂₄	—	3	1	no	—
111	C₂×#204	—	3	2	no	—
112	#824	—	3	4	no	—
113	#823	—	3	4	no	—
114	#819	—	3	2	no	—
115	#819	—	3	2	no	—
116	C₂³×C₉	—	2	1	no	—
117	C₂³×C₉	#41	1	1	yes	characteristic
118	C₂²×D₉	—	2	2	no	—
119	Q₈×C₉	—	2	1	no	—
120	Q₈×C₉	C₂×D₄	1	2	yes	—
121	D₄×C₉	—	2	2	no	—
122	D₄×C₉	—	2	2	no	—
123	D₄×C₉	—	2	1	no	—
124	D₄×C₉	C₂×D₄	1	2	yes	—
125	C₂×C₄×C₉	—	2	1	no	—
126	C₂×C₄×C₉	—	2	2	no	—
127	C₂×C₄×C₉	—	2	2	no	—
128	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₉	#41	1	1	yes	characteristic
133	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
134	#611	—	4	2	no	—
135	#611	—	4	2	no	—
136	C₂×Dic₉	—	2	2	no	—
137	D₃₆	—	2	2	no	—
138	D₃₆	—	2	2	no	—
139	C₄×D₉	—	4	2	no	—
140	C₄×D₉	—	4	2	no	—
141	Dic₁₈	—	2	2	no	—
142	Dic₁₈	—	2	2	no	—
143	C₈×C₉	—	2	2	no	—
144	C₈×C₉	C₂×D₄	1	1	yes	characteristic
145	C₈×C₉	C₂×D₄	1	1	yes	characteristic
146	C₈×C₉	C₂×D₄	1	1	yes	characteristic
147	C₈×C₉	C₂×D₄	1	1	yes	characteristic
148	C₉⋊C₈	—	2	4	no	—
149	#575	—	9	2	no	—
150	C₂×#135	—	9	1	no	—
151	C₂×#131	D₉	1	1	yes	characteristic, complemented
152	#473	—	9	2	no	—
153	#472	—	9	2	no	—
154	#471	—	9	4	no	—
155	C₂×#108	—	9	1	no	—
156	#413	—	9	2	no	—
157	C₃×#41	—	2	1	no	—
158	C₃×#41	—	2	1	no	—
159	C₃×#41	D₁₂	1	2	yes	—
160	C₃×#41	C₂²×D₃	1	2	yes	—
161	C₂×C₃×Q₈	D₁₂	1	1	yes	characteristic
162	C₂×C₃×D₄	—	2	1	no	—
163	C₂×C₃×D₄	D₁₂	1	1	yes	characteristic
164	C₂²×C₃×C₄	—	2	1	no	—
165	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
166	C₂×#67	—	6	2	no	—
167	#235	—	6	2	no	—
168	#235	—	6	2	no	—
169	#235	—	6	2	no	—
170	#235	—	6	2	no	—
171	C₂×D₁₂	—	3	2	no	—
172	C₂×C₄×D₃	—	6	2	no	—
173	C₂×Dic₆	—	3	2	no	—
174	C₃×M₁₆	—	2	1	no	—
175	C₃×M₁₆	—	2	2	no	—
176	C₃×M₁₆	—	2	2	no	—
177	C₃×M₁₆	C₂²×D₃	1	2	yes	—
178	C₂×C₃×C₈	—	2	1	no	—
179	C₂×C₃×C₈	—	2	2	no	—
180	C₂×C₃×C₈	—	2	2	no	—
181	C₂×C₃×C₈	#67	1	1	yes	characteristic
182	C₂×C₃×C₈	#67	1	1	yes	characteristic
183	C₂×C₃×C₈	#67	1	1	yes	characteristic
184	C₂×C₃×C₈	#67	1	1	yes	characteristic
185	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
186	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
187	#216	—	6	4	no	—
188	#215	—	6	4	no	—
189	#214	—	6	4	no	—
190	#213	—	6	4	no	—
191	#208	—	3	4	no	—
192	#208	—	6	2	no	—
193	C₂×C₃⋊C₈	—	6	2	no	—
194	Dic₁₂	—	6	1	no	—
195	Dic₁₂	—	6	1	no	—
196	D₂₄	—	6	1	no	—
197	D₂₄	—	6	1	no	—
198	#204	—	6	2	no	—
199	#204	—	6	2	no	—
200	C₂²×C₉	—	2	2	no	—
201	C₂²×C₉	—	4	1	no	—
202	C₂²×C₉	—	4	2	no	—
203	C₂²×C₉	#121	1	1	yes	characteristic
204	C₂²×C₉	#121	1	1	yes	characteristic
205	C₂²×C₉	C₂×#41	1	1	yes	characteristic
206	C₂×D₉	—	4	2	no	—
207	C₂×D₉	—	8	2	no	—
208	C₄×C₉	—	2	2	no	—
209	C₄×C₉	#121	1	1	yes	characteristic
210	C₄×C₉	#121	1	1	yes	characteristic
211	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
212	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
213	Dic₉	—	4	2	no	—
214	C₂×#41	—	9	2	no	—
215	C₂×#41	C₂×D₉	1	1	yes	characteristic
216	#137	—	9	4	no	—
217	#137	—	18	2	no	—
218	AGL(1,ℤ/8)	—	9	4	no	—
219	AGL(1,ℤ/8)	—	18	2	no	—
220	#135	—	18	4	no	—
221	#135	—	18	1	no	—
222	#135	—	18	1	no	—
223	#135	—	18	1	no	—
224	#135	—	18	1	no	—
225	C₂×Q₁₆	—	9	1	no	—
226	C₂×SD₁₆	—	9	2	no	—
227	C₂×D₈	—	9	1	no	—
228	#131	—	2	1	no	—
229	#131	—	2	1	no	—
230	#131	C₂×D₉	1	2	yes	—
231	#131	C₂×D₉	1	2	yes	—
232	C₂×M₁₆	—	2	1	no	—
233	C₂×M₁₆	—	9	2	no	—
234	C₂×M₁₆	C₂×D₉	1	1	yes	characteristic
235	C₂²×C₈	—	2	1	no	—
236	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
237	#108	—	9	2	no	—
238	#108	—	9	2	no	—
239	#101	—	9	4	no	—
240	#100	—	9	4	no	—
241	C₂³×C₃	—	2	1	no	—
242	C₂³×C₃	#235	1	1	yes	characteristic
243	C₂²×D₃	—	6	2	no	—
244	C₃×Q₈	—	2	1	no	—
245	C₃×Q₈	C₂×D₁₂	1	2	yes	—
246	C₃×D₄	—	2	2	no	—
247	C₃×D₄	—	2	2	no	—
248	C₃×D₄	—	2	1	no	—
249	C₃×D₄	C₂×D₁₂	1	2	yes	—
250	C₂×C₃×C₄	—	2	1	no	—
251	C₂×C₃×C₄	—	2	2	no	—
252	C₂×C₃×C₄	—	2	2	no	—
253	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
254	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
255	C₂×C₃×C₄	#235	1	1	yes	characteristic
256	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
257	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
258	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
259	#67	—	12	2	no	—
260	#67	—	12	2	no	—
261	C₂×Dic₃	—	6	2	no	—
262	D₁₂	—	6	2	no	—
263	D₁₂	—	6	2	no	—
264	C₄×D₃	—	12	2	no	—
265	C₄×D₃	—	12	2	no	—
266	Dic₆	—	6	2	no	—
267	Dic₆	—	6	2	no	—
268	C₃×C₈	—	2	2	no	—
269	C₃×C₈	C₂×#67	1	1	yes	characteristic
270	C₃×C₈	C₂×#67	1	1	yes	characteristic
271	C₃×C₈	C₂×#67	1	1	yes	characteristic
272	C₃×C₈	C₂×#67	1	1	yes	characteristic
273	C₃⋊C₈	—	6	4	no	—
274	C₂×C₉	—	2	1	no	—
275	C₂×C₉	—	2	1	no	—
276	C₂×C₉	—	2	1	no	—
277	C₂×C₉	—	4	2	no	—
278	C₂×C₉	C₂×#121	1	1	yes	characteristic
279	D₉	—	8	2	no	—
280	#41	—	2	1	no	—
281	#41	—	2	1	no	—
282	#41	—	18	2	no	—
283	#41	—	18	2	no	—
284	#41	—	18	2	no	—
285	#41	—	18	2	no	—
286	#41	D₃₆	1	2	yes	—
287	#41	C₂²×D₉	1	2	yes	—
288	C₂×Q₈	—	9	2	no	—
289	C₂×Q₈	D₃₆	1	1	yes	characteristic
290	C₂×D₄	—	2	1	no	—
291	C₂×D₄	—	9	2	no	—
292	C₂×D₄	—	18	2	no	—
293	C₂×D₄	D₃₆	1	1	yes	characteristic
294	C₂²×C₄	—	2	1	no	—
295	C₂²×C₄	—	18	2	no	—
296	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
297	Q₁₆	—	18	4	no	—
298	Q₁₆	—	18	1	no	—
299	Q₁₆	—	18	1	no	—
300	SD₁₆	—	18	4	no	—
301	SD₁₆	—	18	4	no	—
302	SD₁₆	—	18	2	no	—
303	SD₁₆	—	18	2	no	—
304	D₈	—	18	4	no	—
305	D₈	—	18	1	no	—
306	D₈	—	18	1	no	—
307	M₁₆	—	2	1	no	—
308	M₁₆	—	2	2	no	—
309	M₁₆	—	2	2	no	—
310	M₁₆	—	9	4	no	—
311	M₁₆	—	18	2	no	—
312	M₁₆	C₂²×D₉	1	2	yes	—
313	C₂×C₈	—	2	1	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	18	2	no	—
317	C₂×C₈	#611	1	1	yes	characteristic
318	C₂×C₈	#611	1	1	yes	characteristic
319	C₂×C₈	#611	1	1	yes	characteristic
320	C₂×C₈	#611	1	1	yes	characteristic
321	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
322	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
323	C₂²×C₃	—	2	2	no	—
324	C₂²×C₃	—	4	1	no	—
325	C₂²×C₃	—	4	2	no	—
326	C₂²×C₃	#930	1	1	yes	characteristic
327	C₂²×C₃	#930	1	1	yes	characteristic
328	C₂²×C₃	C₂×#235	1	1	yes	characteristic
329	C₂×D₃	—	12	2	no	—
330	C₂×D₃	—	24	2	no	—
331	C₃×C₄	—	2	2	no	—
332	C₃×C₄	#930	1	1	yes	characteristic
333	C₃×C₄	#930	1	1	yes	characteristic
334	C₃×C₄	C₂²×D₁₂	1	1	yes	characteristic
335	C₃×C₄	C₂²×#67	1	1	yes	characteristic
336	Dic₃	—	12	2	no	—
337	C₉	#3067	1	1	yes	characteristic, complemented, Sylow
338	C₂³	—	2	1	no	—
339	C₂³	—	18	2	no	—
340	C₂³	#3697	1	1	yes	characteristic
341	Q₈	—	2	1	no	—
342	Q₈	—	18	2	no	—
343	Q₈	—	18	2	no	—
344	Q₈	C₂×D₃₆	1	2	yes	—
345	D₄	—	2	2	no	—
346	D₄	—	2	2	no	—
347	D₄	—	2	1	no	—
348	D₄	—	18	2	no	—
349	D₄	—	18	2	no	—
350	D₄	—	36	2	no	—
351	D₄	—	36	2	no	—
352	D₄	C₂×D₃₆	1	2	yes	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	18	2	no	—
357	C₂×C₄	—	36	2	no	—
358	C₂×C₄	—	36	2	no	—
359	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
360	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
361	C₂×C₄	#3697	1	1	yes	characteristic
362	C₂×C₄	C₂×#611	1	1	yes	characteristic
363	C₂×C₄	C₂×#611	1	1	yes	characteristic
364	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
365	C₈	—	2	2	no	—
366	C₈	—	18	4	no	—
367	C₈	C₂×#611	1	1	yes	characteristic
368	C₈	C₂×#611	1	1	yes	characteristic
369	C₈	C₂×#611	1	1	yes	characteristic
370	C₈	C₂×#611	1	1	yes	characteristic
371	C₂×C₃	—	2	1	no	—
372	C₂×C₃	—	2	1	no	—
373	C₂×C₃	—	2	1	no	—
374	C₂×C₃	—	4	2	no	—
375	C₂×C₃	C₂×#930	1	1	yes	characteristic
376	D₃	—	24	2	no	—
377	C₂²	—	2	2	no	—
378	C₂²	—	4	1	no	—
379	C₂²	—	4	2	no	—
380	C₂²	—	36	2	no	—
381	C₂²	—	72	2	no	—
382	C₂²	#63492	1	1	yes	characteristic
383	C₂²	#63492	1	1	yes	characteristic
384	C₂²	C₂×#3697	1	1	yes	characteristic
385	C₄	—	2	2	no	—
386	C₄	—	36	2	no	—
387	C₄	#63492	1	1	yes	characteristic
388	C₄	#63492	1	1	yes	characteristic
389	C₄	C₂²×D₃₆	1	1	yes	characteristic
390	C₄	C₂²×#611	1	1	yes	characteristic
391	C₃	#83805	1	1	yes	characteristic
392	C₂	—	2	1	no	—
393	C₂	—	2	1	no	—
394	C₂	—	2	1	no	—
395	C₂	—	4	2	no	—
396	C₂	—	72	2	no	—
397	C₂	C₂×#63492	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#63492
commutator, cosocle	C₂×C₄×C₉	C₂⁴
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂×C₉×#131	C₂
socle	C₂×C₃	C₂×#930
hypercenter	C₂×#131	D₉
hypocenter	C₉	#3067
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. C₂×C₉×#131	C₂	semidirect
2. #10589925	C₂	semidirect
3. C₂×#63467	C₂	semidirect
4. #10588823	C₂	semidirect
5. #10588822	C₂	semidirect
6. #10588821	C₂	semidirect
7. #10588814	C₂	semidirect
8. C₂×#63372	C₂	semidirect
30. C₂×D₇₂	C₂²	semidirect
31. C₂×#3664	C₂²	semidirect
32. C₂×Dic₃₆	C₂²	semidirect
33. #63383	C₂²	semidirect
34. #63382	C₂²	semidirect
35. #63372	C₂²	semidirect
36. #63372	C₂²	semidirect
151. C₂×#131	D₉	semidirect
337. C₉	#3067	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×#131	2
#10589925	2
C₂×#63467	2
#10588823	2
#10588822	2
#10588821	2
#10588814	2
C₂×#63372	2
#83805	3