Subgroups — #11787474

Order 1152.

507 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3674	C₂	1	1	yes	maximal, characteristic
3	C₂×#63395	C₂	1	1	yes	maximal, characteristic
4	C₂×#63368	C₂	1	1	yes	maximal, characteristic
5	#10588609	C₂	1	1	yes	maximal, characteristic
6	#10588206	C₂	1	1	yes	maximal, characteristic
7	#10588200	C₂	1	1	yes	maximal, characteristic
8	#71632	—	3	1	no	maximal
9	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
10	C₂²×Dic₁₈	C₂²	1	1	yes	characteristic
11	C₄×Q₈×C₉	—	2	2	no	—
12	C₄×Q₈×C₉	—	2	2	no	—
13	C₂×C₉×C₄⋊C₄	—	2	1	no	—
14	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₉	—	2	1	no	—
16	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
17	C₂×#3674	—	2	2	no	—
18	C₂×#3674	—	2	2	no	—
19	C₂×#3674	C₄	1	4	yes	complemented
20	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
21	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
22	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
23	#63395	—	2	2	no	—
24	C₉⋊#95	—	2	1	no	—
25	C₉⋊#95	—	2	1	no	—
26	#63368	—	2	2	no	—
27	C₂×C₃×C₄×Q₈	D₃	1	1	yes	characteristic
28	C₂²×#216	—	3	1	no	—
29	C₂×#835	—	3	1	no	—
30	C₂×#810	—	3	1	no	—
31	#4929	—	3	1	no	—
32	#4529	—	3	1	no	—
33	#4524	—	3	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	C₂×C₄	1	2	yes	—
38	C₂×Q₈×C₉	D₄	1	2	yes	—
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	C₂²×Dic₉	—	2	1	no	—
47	C₂×Dic₁₈	—	2	1	no	—
48	C₂×Dic₁₈	—	2	1	no	—
49	C₂×Dic₁₈	—	2	1	no	—
50	C₂×Dic₁₈	—	2	1	no	—
51	C₂×Dic₁₈	—	2	1	no	—
52	C₂×Dic₁₈	C₂×C₄	1	2	yes	—
53	C₉×C₄⋊C₄	—	2	4	no	—
54	C₉×C₄⋊C₄	—	2	1	no	—
55	C₉×C₄⋊C₄	D₄	1	2	yes	—
56	C₄²×C₉	—	2	4	no	—
57	C₄²×C₉	—	2	1	no	—
58	C₄²×C₉	—	2	1	no	—
59	#3674	—	2	8	no	—
60	C₄⋊Dic₉	—	2	1	no	—
61	C₄⋊Dic₉	—	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	—	2	1	no	—
67	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	—
68	#1878	—	9	1	no	Sylow
69	C₂²×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
70	C₂²×Dic₆	—	3	1	no	—
71	C₃×C₄×Q₈	—	2	2	no	—
72	C₃×C₄×Q₈	—	2	2	no	—
73	C₂×C₃×C₄⋊C₄	—	2	1	no	—
74	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
75	C₂×C₃×C₄²	—	2	1	no	—
76	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
77	C₂×#216	—	3	4	no	—
78	C₂×#216	—	6	2	no	—
79	C₂×#216	—	6	2	no	—
80	C₂×C₄⋊Dic₃	—	3	1	no	—
81	C₂²×C₃⋊C₈	—	3	1	no	—
82	C₂²×C₃⋊C₈	—	3	1	no	—
83	#835	—	6	2	no	—
84	C₃⋊#95	—	6	1	no	—
85	C₃⋊#95	—	6	1	no	—
86	#810	—	6	2	no	—
87	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
88	Q₈×C₉	—	2	2	no	—
89	Q₈×C₉	—	2	2	no	—
90	Q₈×C₉	—	2	2	no	—
91	Q₈×C₉	C₂²⋊C₄	1	4	yes	—
92	C₂×C₄×C₉	—	2	2	no	—
93	C₂×C₄×C₉	—	2	1	no	—
94	C₂×C₄×C₉	—	2	2	no	—
95	C₂×C₄×C₉	—	2	2	no	—
96	C₂×C₄×C₉	—	2	1	no	—
97	C₂×C₄×C₉	—	2	1	no	—
98	C₂×C₄×C₉	—	2	2	no	—
99	C₂×C₄×C₉	—	2	1	no	—
100	C₂×C₄×C₉	—	2	1	no	—
101	C₂×C₄×C₉	—	4	1	no	—
102	C₂×C₄×C₉	—	4	1	no	—
103	C₂×C₄×C₉	Q₁₆	1	2	yes	complemented
104	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₉	#41	1	1	yes	characteristic
109	C₂×C₄×C₉	#41	1	1	yes	characteristic
110	C₂×Dic₉	—	2	2	no	—
111	C₂×Dic₉	—	4	1	no	—
112	C₂×Dic₉	—	4	1	no	—
113	C₂×Dic₉	—	4	1	no	—
114	C₂×Dic₉	—	4	1	no	—
115	C₂×Dic₉	—	4	1	no	—
116	Dic₁₈	—	2	2	no	—
117	Dic₁₈	—	2	2	no	—
118	Dic₁₈	—	4	1	no	—
119	Dic₁₈	—	4	1	no	—
120	C₉⋊C₈	—	2	2	no	—
121	C₉⋊C₈	—	4	1	no	—
122	C₂²×Q₁₆	—	9	1	no	—
123	C₂×C₄×Q₈	D₉	1	1	yes	characteristic
124	C₂×#103	—	9	1	no	—
125	C₂×#103	—	9	1	no	—
126	#391	—	9	1	no	—
127	#340	—	9	1	no	—
128	#336	—	9	1	no	—
129	C₂×C₃×Q₈	—	2	1	no	—
130	C₂×C₃×Q₈	—	2	1	no	—
131	C₂×C₃×Q₈	—	2	1	no	—
132	C₂×C₃×Q₈	C₄×D₃	1	2	yes	—
133	C₂×C₃×Q₈	D₁₂	1	2	yes	—
134	C₂×C₃×Q₈	#67	1	2	yes	—
135	C₂²×C₃×C₄	—	2	1	no	—
136	C₂²×C₃×C₄	—	2	1	no	—
137	C₂²×C₃×C₄	#67	1	1	yes	characteristic
138	C₂²×C₃×C₄	#67	1	1	yes	characteristic
139	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
140	C₂²×Dic₃	—	6	1	no	—
141	C₂²×Dic₃	—	6	1	no	—
142	C₂×Dic₆	—	3	2	no	—
143	C₂×Dic₆	—	6	1	no	—
144	C₂×Dic₆	—	6	1	no	—
145	C₂×Dic₆	—	6	1	no	—
146	C₂×Dic₆	—	6	1	no	—
147	C₂×Dic₆	—	6	1	no	—
148	C₃×C₄⋊C₄	—	2	4	no	—
149	C₃×C₄⋊C₄	—	2	1	no	—
150	C₃×C₄⋊C₄	#67	1	2	yes	—
151	C₃×C₄²	—	2	4	no	—
152	C₃×C₄²	—	2	1	no	—
153	C₃×C₄²	—	2	1	no	—
154	#216	—	6	8	no	—
155	C₄⋊Dic₃	—	6	1	no	—
156	C₄⋊Dic₃	—	6	1	no	—
157	C₂×C₃⋊C₈	—	3	2	no	—
158	C₂×C₃⋊C₈	—	6	1	no	—
159	C₂×C₃⋊C₈	—	6	1	no	—
160	C₂×C₃⋊C₈	—	6	1	no	—
161	C₂×C₃⋊C₈	—	6	1	no	—
162	C₂×C₃⋊C₈	—	6	1	no	—
163	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
164	C₂²×C₉	#120	1	1	yes	characteristic
165	C₂²×C₉	#121	1	1	yes	characteristic
166	C₂²×C₉	C₂×Q₁₆	1	1	yes	characteristic
167	C₂²×C₉	#135	1	1	yes	characteristic
168	C₂²×C₉	#137	1	1	yes	characteristic
169	C₂²×C₉	#137	1	1	yes	characteristic
170	C₄×C₉	—	2	4	no	—
171	C₄×C₉	—	2	2	no	—
172	C₄×C₉	—	4	1	no	—
173	C₄×C₉	—	4	2	no	—
174	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
175	C₄×C₉	C₄×D₄	1	1	yes	characteristic
176	C₄×C₉	#121	1	1	yes	characteristic
177	C₄×C₉	#123	1	1	yes	characteristic
178	Dic₉	—	4	2	no	—
179	Dic₉	—	8	1	no	—
180	C₂²×Q₈	—	9	1	no	—
181	C₂²×Q₈	C₂×D₉	1	1	yes	characteristic
182	C₂×Q₁₆	—	9	4	no	—
183	C₂×Q₁₆	—	18	2	no	—
184	C₂×Q₁₆	—	18	2	no	—
185	C₂²×C₈	—	9	1	no	—
186	C₂²×C₈	—	9	1	no	—
187	C₄×Q₈	—	2	2	no	—
188	C₄×Q₈	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	—	9	1	no	—
191	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
194	#103	—	18	2	no	—
195	#103	—	18	2	no	—
196	#95	—	18	1	no	—
197	#95	—	18	1	no	—
198	C₂³×C₃	C₂×#67	1	1	yes	characteristic
199	C₃×Q₈	—	2	2	no	—
200	C₃×Q₈	—	2	2	no	—
201	C₃×Q₈	—	2	2	no	—
202	C₃×Q₈	(C₂×D₃)⋊C₄	1	4	yes	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	2	2	no	—
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	—	2	1	no	—
207	C₂×C₃×C₄	—	2	2	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	2	no	—
212	C₂×C₃×C₄	—	4	1	no	—
213	C₂×C₃×C₄	—	4	1	no	—
214	C₂×C₃×C₄	#216	1	2	yes	—
215	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
217	C₂×C₃×C₄	#235	1	1	yes	characteristic
218	C₂×C₃×C₄	#235	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
221	C₂×Dic₃	—	6	2	no	—
222	C₂×Dic₃	—	12	1	no	—
223	C₂×Dic₃	—	12	1	no	—
224	C₂×Dic₃	—	12	1	no	—
225	C₂×Dic₃	—	12	1	no	—
226	C₂×Dic₃	—	12	1	no	—
227	Dic₆	—	6	2	no	—
228	Dic₆	—	6	2	no	—
229	Dic₆	—	12	1	no	—
230	Dic₆	—	12	1	no	—
231	C₃⋊C₈	—	6	2	no	—
232	C₃⋊C₈	—	12	1	no	—
233	C₂×C₉	#386	1	1	yes	characteristic
234	C₂×C₉	C₄×Q₁₆	1	1	yes	characteristic
235	C₂×C₉	#441	1	1	yes	characteristic
236	C₂×C₉	#451	1	1	yes	characteristic
237	C₂×C₉	#452	1	1	yes	characteristic
238	C₂×C₉	#462	1	1	yes	characteristic
239	C₂×C₉	#464	1	1	yes	characteristic
240	C₂×Q₈	—	2	1	no	—
241	C₂×Q₈	—	2	1	no	—
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	—	9	2	no	—
244	C₂×Q₈	—	18	1	no	—
245	C₂×Q₈	—	18	1	no	—
246	C₂×Q₈	—	18	1	no	—
247	C₂×Q₈	—	18	1	no	—
248	C₂×Q₈	—	18	1	no	—
249	C₂×Q₈	C₄×D₉	1	2	yes	—
250	C₂×Q₈	D₃₆	1	2	yes	—
251	C₂×Q₈	#611	1	2	yes	—
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	18	1	no	—
255	C₂²×C₄	—	18	1	no	—
256	C₂²×C₄	#611	1	1	yes	characteristic
257	C₂²×C₄	#611	1	1	yes	characteristic
258	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
259	Q₁₆	—	18	8	no	—
260	C₂×C₈	—	9	2	no	—
261	C₂×C₈	—	18	1	no	—
262	C₂×C₈	—	18	1	no	—
263	C₂×C₈	—	18	1	no	—
264	C₂×C₈	—	18	1	no	—
265	C₂×C₈	—	18	1	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	18	1	no	—
269	C₄⋊C₄	—	18	1	no	—
270	C₄⋊C₄	#611	1	2	yes	—
271	C₄²	—	2	4	no	—
272	C₄²	—	2	1	no	—
273	C₄²	—	2	1	no	—
274	C₂²×C₃	C₄×#67	1	1	yes	characteristic
275	C₂²×C₃	#930	1	1	yes	characteristic
276	C₂²×C₃	#942	1	1	yes	characteristic
277	C₂²×C₃	C₂×#216	1	1	yes	characteristic
278	C₂²×C₃	#950	1	1	yes	characteristic
279	C₂²×C₃	#951	1	1	yes	characteristic
280	C₂²×C₃	#953	1	1	yes	characteristic
281	C₃×C₄	—	2	4	no	—
282	C₃×C₄	—	2	2	no	—
283	C₃×C₄	—	4	1	no	—
284	C₃×C₄	—	4	2	no	—
285	C₃×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
286	C₃×C₄	C₄×#67	1	1	yes	characteristic
287	C₃×C₄	#929	1	1	yes	characteristic
288	C₃×C₄	#930	1	1	yes	characteristic
289	Dic₃	—	12	2	no	—
290	Dic₃	—	24	1	no	—
291	C₉	#1878	1	1	yes	characteristic, complemented, Sylow
292	C₂³	C₂×#611	1	1	yes	characteristic
293	Q₈	—	2	2	no	—
294	Q₈	—	2	2	no	—
295	Q₈	—	2	2	no	—
296	Q₈	—	18	2	no	—
297	Q₈	—	18	2	no	—
298	Q₈	—	36	1	no	—
299	Q₈	—	36	1	no	—
300	Q₈	(C₂×D₉)⋊C₄	1	4	yes	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	18	2	no	—
313	C₂×C₄	—	36	1	no	—
314	C₂×C₄	—	36	1	no	—
315	C₂×C₄	—	36	1	no	—
316	C₂×C₄	—	36	1	no	—
317	C₂×C₄	—	36	1	no	—
318	C₂×C₄	#3674	1	2	yes	complemented
319	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
320	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
321	C₂×C₄	#3697	1	1	yes	characteristic
322	C₂×C₄	#3697	1	1	yes	characteristic
323	C₂×C₄	C₂×#611	1	1	yes	characteristic
324	C₂×C₄	C₂×#611	1	1	yes	characteristic
325	C₈	—	18	2	no	—
326	C₈	—	36	1	no	—
327	C₂×C₃	#5028	1	1	yes	characteristic
328	C₂×C₃	C₄×#216	1	1	yes	characteristic
329	C₂×C₃	#5030	1	1	yes	characteristic
330	C₂×C₃	#5031	1	1	yes	characteristic
331	C₂×C₃	#5213	1	1	yes	characteristic
332	C₂×C₃	#5228	1	1	yes	characteristic
333	C₂×C₃	#5239	1	1	yes	characteristic
334	C₂²	C₄×#611	1	1	yes	characteristic
335	C₂²	#63492	1	1	yes	characteristic
336	C₂²	C₂×#3674	1	1	yes	characteristic
337	C₂²	#63505	1	1	yes	characteristic
338	C₂²	#63511	1	1	yes	characteristic
339	C₂²	#63513	1	1	yes	characteristic
340	C₂²	#63515	1	1	yes	characteristic
341	C₄	—	2	4	no	—
342	C₄	—	2	2	no	—
343	C₄	—	4	1	no	—
344	C₄	—	4	2	no	—
345	C₄	—	36	2	no	—
346	C₄	—	72	1	no	—
347	C₄	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
348	C₄	C₄×#611	1	1	yes	characteristic
349	C₄	#63491	1	1	yes	characteristic
350	C₄	#63492	1	1	yes	characteristic
351	C₃	#71632	1	1	yes	characteristic
352	C₂	C₄×#3674	1	1	yes	characteristic
353	C₂	#10588706	1	1	yes	characteristic
354	C₂	#10588707	1	1	yes	characteristic
355	C₂	#10588711	1	1	yes	characteristic
356	C₂	#10588893	1	1	yes	characteristic
357	C₂	#10588907	1	1	yes	characteristic
358	C₂	#10588918	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#611
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₄×Q₈×C₉	C₂
socle, prime radical	C₂³×C₃	C₂×#67
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×C₄×Q₈	D₉
hypocenter	C₉	#1878
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
19. C₂×#3674	C₄	semidirect
103. C₂×C₄×C₉	Q₁₆	semidirect
291. C₉	#1878	semidirect
318. C₂×C₄	#3674	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₈×C₉	2
C₂²×#3674	2
C₂×#63395	2
C₂×#63368	2
#10588609	2
#10588206	2
#10588200	2
#71632	3