Subgroups — #11861318

Order 1152.

546 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10589680	C₂	1	1	yes	maximal, characteristic
2	C₈×Q₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#10588701	C₂	1	1	yes	maximal, characteristic, complemented
4	#10588523	C₂	1	2	yes	maximal
5	#10588521	C₂	1	1	yes	maximal, characteristic
6	#10588510	C₂	1	2	yes	maximal, complemented
7	#10588509	C₂	1	1	yes	maximal, characteristic, complemented
8	#10588371	C₂	1	2	yes	maximal
9	C₄×C₈×D₉	C₂	1	1	yes	maximal, characteristic
10	C₈×Dic₁₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Dic₁₈	C₂	1	2	yes	maximal, complemented
12	#79917	—	3	1	no	maximal
13	C₄×Q₈×C₉	C₂²	1	1	yes	characteristic
14	#63508	C₄	1	1	yes	characteristic
15	#63506	C₄	1	1	yes	characteristic
16	C₂×#3663	C₂²	1	2	yes	—
17	C₂×C₈×D₉	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₉	C₂²	1	1	yes	characteristic
19	#63459	C₄	1	1	yes	characteristic
20	#63458	C₄	1	2	yes	—
21	#63451	C₄	1	2	yes	—
22	#63450	C₄	1	1	yes	characteristic
23	#63434	C₂²	1	2	yes	—
24	C₄²×D₉	C₂²	1	1	yes	characteristic
25	C₄×Dic₁₈	C₂²	1	1	yes	characteristic
26	C₄×Dic₁₈	C₂²	1	2	yes	—
27	C₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₉×C₄⋊C₈	C₂²	1	2	yes	—
29	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
30	C₄×C₈×C₉	C₂²	1	2	yes	—
31	#63375	C₂²	1	2	yes	—
32	Dic₉⋊C₈	C₂²	1	2	yes	—
33	Dic₉⋊C₈	C₂²	1	2	yes	—
34	Dic₉⋊C₈	C₂²	1	2	yes	—
35	C₈×Dic₉	C₂²	1	1	yes	characteristic
36	C₈×Dic₉	C₂²	1	1	yes	characteristic
37	C₈×Dic₉	C₂²	1	2	yes	—
38	C₄⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
39	C₄⋊(C₉⋊C₈)	C₂²	1	2	yes	—
40	#63362	C₂²	1	2	yes	—
41	C₄×C₉⋊C₈	C₂²	1	1	yes	characteristic
42	#5578	—	3	1	no	—
43	C₃×C₈×Q₈	D₃	1	1	yes	characteristic
44	#5024	—	3	1	no	—
45	#4839	—	3	2	no	—
46	#4837	—	3	1	no	—
47	#4831	—	3	1	no	—
48	#4828	—	3	2	no	—
49	#4689	—	3	2	no	—
50	C₄×D₃×C₈	—	3	1	no	—
51	C₈×Dic₆	—	3	1	no	—
52	C₈×Dic₆	—	3	2	no	—
53	C₂×Q₈×C₉	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₉	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₉	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×D₉	C₂³	1	1	yes	characteristic
57	C₂×Dic₁₈	C₂×C₄	1	1	yes	characteristic
58	C₂×Dic₁₈	C₂×C₄	1	2	yes	—
59	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₉	C₂³	1	2	yes	—
61	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
62	C₉×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
63	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
64	C₄²×C₉	C₂³	1	1	yes	characteristic
65	C₄²×C₉	C₂³	1	2	yes	—
66	(C₂×D₉)⋊C₄	C₂×C₄	1	2	yes	—
67	(C₂×D₉)⋊C₄	C₂×C₄	1	2	yes	—
68	C₄⋊Dic₉	C₂×C₄	1	1	yes	characteristic
69	C₄⋊Dic₉	C₂×C₄	1	2	yes	—
70	Dic₉⋊C₄	C₂×C₄	1	2	yes	—
71	Dic₉⋊C₄	C₂×C₄	1	2	yes	—
72	Dic₉⋊C₄	C₂×C₄	1	2	yes	—
73	Dic₉⋊C₄	C₂×C₄	1	2	yes	—
74	Dic₉⋊C₄	C₂×C₄	1	2	yes	—
75	C₄×Dic₉	C₂×C₄	1	1	yes	characteristic
76	C₄×Dic₉	C₂×C₄	1	1	yes	characteristic
77	C₄×Dic₉	C₂³	1	1	yes	characteristic
78	C₄×Dic₉	C₂³	1	2	yes	—
79	C₂×C₉⋊C₈	C₂³	1	1	yes	characteristic
80	C₂×C₉⋊C₈	C₂³	1	2	yes	—
81	C₂×C₉⋊C₈	C₂³	1	1	yes	characteristic
82	#3663	—	2	4	no	—
83	C₈×D₉	—	2	2	no	—
84	C₈×D₉	Q₈	1	4	yes	complemented
85	#2960	—	9	1	no	Sylow
86	C₃×C₄×Q₈	C₂×D₃	1	1	yes	characteristic
87	#946	—	3	1	no	—
88	#944	—	3	1	no	—
89	C₂×ΓL(1,𝔽₂₅)	—	3	2	no	—
90	C₂×D₃×C₈	—	3	1	no	—
91	C₂×D₃×C₈	—	3	1	no	—
92	#897	—	3	1	no	—
93	#896	—	3	2	no	—
94	#889	—	3	2	no	—
95	#888	—	3	1	no	—
96	#872	—	3	2	no	—
97	C₄²×D₃	—	3	1	no	—
98	C₄×Dic₆	—	3	2	no	—
99	C₄×Dic₆	—	3	1	no	—
100	C₃×C₄⋊C₈	C₂×D₃	1	2	yes	—
101	C₃×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
102	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
103	C₃×C₄×C₈	C₂×D₃	1	2	yes	—
104	#815	—	3	2	no	—
105	Dic₃⋊C₈	—	3	2	no	—
106	Dic₃⋊C₈	—	3	2	no	—
107	Dic₃⋊C₈	—	3	2	no	—
108	C₈×Dic₃	—	3	1	no	—
109	C₈×Dic₃	—	3	2	no	—
110	C₈×Dic₃	—	3	1	no	—
111	C₄⋊(C₃⋊C₈)	—	3	1	no	—
112	C₄⋊(C₃⋊C₈)	—	3	2	no	—
113	#803	—	3	2	no	—
114	C₄×C₃⋊C₈	—	3	1	no	—
115	C₂²×D₉	C₂²×C₄	1	1	yes	characteristic
116	Q₈×C₉	—	2	2	no	—
117	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
119	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
120	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
122	C₂×Dic₉	C₂²×C₄	1	1	yes	characteristic
123	C₂×Dic₉	C₂²×C₄	1	2	yes	—
124	C₂×Dic₉	C₂²×C₄	1	2	yes	—
125	C₂×Dic₉	C₂²×C₄	1	1	yes	characteristic
126	C₂×Dic₉	C₂²×C₄	1	1	yes	characteristic
127	C₄×D₉	—	2	2	no	—
128	C₄×D₉	—	2	2	no	—
129	C₄×D₉	C₂×Q₈	1	2	yes	—
130	C₄×D₉	#41	1	2	yes	—
131	Dic₁₈	—	2	2	no	—
132	Dic₁₈	—	2	4	no	—
133	C₈×C₉	—	2	1	no	—
134	C₈×C₉	—	2	2	no	—
135	C₈×C₉	C₂×Q₈	1	2	yes	—
136	C₉⋊C₈	—	2	1	no	—
137	C₉⋊C₈	—	2	2	no	—
138	C₉⋊C₈	C₂×Q₈	1	2	yes	—
139	#533	—	9	1	no	—
140	#446	—	9	1	no	—
141	#446	—	9	2	no	—
142	#446	—	9	1	no	—
143	C₈×Q₈	—	9	2	no	—
144	C₈×Q₈	—	9	1	no	—
145	C₈×Q₈	D₉	1	1	yes	characteristic, complemented
146	#424	—	9	2	no	—
147	#423	—	9	1	no	—
148	#405	—	9	2	no	—
149	C₂×C₄×C₈	—	9	1	no	—
150	C₂×C₃×Q₈	C₄×D₃	1	1	yes	characteristic
151	C₂×C₄×D₃	—	3	1	no	—
152	C₂×C₄×D₃	—	3	1	no	—
153	C₂×C₄×D₃	—	3	1	no	—
154	C₂×Dic₆	—	3	2	no	—
155	C₂×Dic₆	—	3	1	no	—
156	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
157	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
158	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
159	C₃×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
160	C₃×C₄⋊C₄	C₄×D₃	1	2	yes	—
161	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
162	C₃×C₄²	C₂²×D₃	1	2	yes	—
163	(C₂×D₃)⋊C₄	—	3	2	no	—
164	(C₂×D₃)⋊C₄	—	3	2	no	—
165	C₄⋊Dic₃	—	3	1	no	—
166	C₄⋊Dic₃	—	3	2	no	—
167	Dic₃⋊C₄	—	3	2	no	—
168	Dic₃⋊C₄	—	3	2	no	—
169	Dic₃⋊C₄	—	3	2	no	—
170	Dic₃⋊C₄	—	3	2	no	—
171	Dic₃⋊C₄	—	3	2	no	—
172	C₄×Dic₃	—	3	1	no	—
173	C₄×Dic₃	—	3	1	no	—
174	C₄×Dic₃	—	3	1	no	—
175	C₄×Dic₃	—	3	2	no	—
176	C₂×C₃⋊C₈	—	3	1	no	—
177	C₂×C₃⋊C₈	—	3	1	no	—
178	C₂×C₃⋊C₈	—	3	2	no	—
179	ΓL(1,𝔽₂₅)	—	6	4	no	—
180	D₃×C₈	—	3	4	no	—
181	D₃×C₈	—	6	2	no	—
182	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
183	C₂×D₉	—	2	1	no	—
184	C₂×D₉	—	2	1	no	—
185	C₂×D₉	C₄×Q₈	1	2	yes	—
186	C₄×C₉	—	2	2	no	—
187	C₄×C₉	—	2	1	no	—
188	C₄×C₉	—	2	2	no	—
189	C₄×C₉	#131	1	2	yes	—
190	C₄×C₉	C₂²×Q₈	1	1	yes	characteristic
191	C₄×C₉	C₂×#41	1	1	yes	characteristic
192	Dic₉	—	2	2	no	—
193	Dic₉	—	2	2	no	—
194	Dic₉	—	2	1	no	—
195	Dic₉	C₄×Q₈	1	2	yes	—
196	Dic₉	#131	1	2	yes	—
197	C₂×M₁₆	—	9	2	no	—
198	C₂²×C₈	—	9	1	no	—
199	C₂²×C₈	—	9	1	no	—
200	C₄⋊Q₈	—	9	1	no	—
201	C₄⋊Q₈	—	9	1	no	—
202	#125	—	9	2	no	—
203	C₂²⋊Q₈	—	9	1	no	—
204	C₂²⋊Q₈	—	9	2	no	—
205	C₂²⋊Q₈	—	9	1	no	—
206	C₄×Q₈	—	9	2	no	—
207	C₄×Q₈	—	9	1	no	—
208	C₄×Q₈	C₂×D₉	1	1	yes	characteristic
209	#117	—	9	2	no	—
210	C₂×C₄²	—	9	1	no	—
211	C₄⋊C₈	—	9	2	no	—
212	C₄⋊C₈	—	9	2	no	—
213	C₄⋊C₈	—	9	2	no	—
214	C₄⋊C₈	—	9	1	no	—
215	C₄⋊C₈	—	9	2	no	—
216	C₄⋊C₈	C₂×D₉	1	1	yes	characteristic
217	C₄⋊C₈	C₂×D₉	1	2	yes	—
218	#97	—	9	2	no	—
219	#97	—	9	2	no	—
220	C₄×C₈	—	9	1	no	—
221	C₄×C₈	—	9	1	no	—
222	C₄×C₈	—	9	2	no	—
223	C₄×C₈	—	9	1	no	—
224	C₄×C₈	C₂×D₉	1	1	yes	characteristic
225	C₄×C₈	C₂×D₉	1	2	yes	—
226	C₂²×D₃	—	3	1	no	—
227	C₃×Q₈	—	2	2	no	—
228	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
230	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
231	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
233	C₂×Dic₃	—	3	1	no	—
234	C₂×Dic₃	—	3	1	no	—
235	C₂×Dic₃	—	3	1	no	—
236	C₂×Dic₃	—	3	2	no	—
237	C₂×Dic₃	—	3	2	no	—
238	C₄×D₃	—	3	2	no	—
239	C₄×D₃	—	3	2	no	—
240	C₄×D₃	—	6	2	no	—
241	C₄×D₃	—	6	2	no	—
242	Dic₆	—	6	4	no	—
243	Dic₆	—	6	2	no	—
244	C₃×C₈	—	2	1	no	—
245	C₃×C₈	—	2	2	no	—
246	C₃×C₈	D₃×Q₈	1	2	yes	—
247	C₃⋊C₈	—	3	2	no	—
248	C₃⋊C₈	—	6	1	no	—
249	C₃⋊C₈	—	6	2	no	—
250	C₂×C₉	C₂×C₄×Q₈	1	1	yes	characteristic
251	C₂×C₉	C₂×#131	1	1	yes	characteristic
252	C₂×C₉	C₂×#131	1	1	yes	characteristic
253	D₉	—	2	2	no	—
254	C₂×Q₈	—	9	2	no	—
255	C₂×Q₈	—	9	1	no	—
256	C₂×Q₈	C₄×D₉	1	1	yes	characteristic
257	C₂²×C₄	—	9	1	no	—
258	C₂²×C₄	—	9	1	no	—
259	C₂²×C₄	—	9	1	no	—
260	M₁₆	—	18	4	no	—
261	C₂×C₈	—	9	1	no	—
262	C₂×C₈	—	9	4	no	—
263	C₂×C₈	—	9	1	no	—
264	C₂×C₈	—	9	2	no	—
265	C₂×C₈	—	18	2	no	—
266	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
267	C₂×C₈	C₂²×D₉	1	2	yes	—
268	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
269	C₄⋊C₄	—	9	2	no	—
270	C₄⋊C₄	—	9	2	no	—
271	C₄⋊C₄	—	9	2	no	—
272	C₄⋊C₄	—	9	2	no	—
273	C₄⋊C₄	—	9	2	no	—
274	C₄⋊C₄	—	9	2	no	—
275	C₄⋊C₄	—	9	1	no	—
276	C₄⋊C₄	C₄×D₉	1	2	yes	—
277	C₄⋊C₄	C₄×D₉	1	1	yes	characteristic
278	C₂²⋊C₄	—	9	2	no	—
279	C₂²⋊C₄	—	9	2	no	—
280	C₄²	—	9	1	no	—
281	C₄²	—	9	1	no	—
282	C₄²	—	9	1	no	—
283	C₄²	—	9	2	no	—
284	C₄²	C₂²×D₉	1	1	yes	characteristic
285	C₄²	C₂²×D₉	1	2	yes	—
286	C₂²×C₃	C₂²×C₄×D₃	1	1	yes	characteristic
287	C₂×D₃	—	3	2	no	—
288	C₂×D₃	—	6	1	no	—
289	C₂×D₃	—	6	1	no	—
290	C₃×C₄	—	2	2	no	—
291	C₃×C₄	—	2	2	no	—
292	C₃×C₄	—	2	1	no	—
293	C₃×C₄	#901	1	2	yes	—
294	C₃×C₄	C₂×D₃×Q₈	1	1	yes	characteristic
295	C₃×C₄	D₃×#41	1	1	yes	characteristic
296	Dic₃	—	3	2	no	—
297	Dic₃	—	3	2	no	—
298	Dic₃	—	6	2	no	—
299	Dic₃	—	6	1	no	—
300	Dic₃	—	6	2	no	—
301	C₉	#2960	1	1	yes	characteristic, complemented, Sylow
302	C₂³	—	9	1	no	—
303	Q₈	—	2	2	no	—
304	Q₈	—	18	4	no	—
305	Q₈	—	18	2	no	—
306	C₂×C₄	—	9	1	no	—
307	C₂×C₄	—	9	1	no	—
308	C₂×C₄	—	9	2	no	—
309	C₂×C₄	—	9	2	no	—
310	C₂×C₄	—	9	1	no	—
311	C₂×C₄	—	9	2	no	—
312	C₂×C₄	—	9	2	no	—
313	C₂×C₄	—	18	2	no	—
314	C₂×C₄	—	18	2	no	—
315	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
316	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
317	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
318	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
319	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
320	C₈	—	2	1	no	—
321	C₈	—	2	2	no	—
322	C₈	—	9	2	no	—
323	C₈	—	18	1	no	—
324	C₈	—	18	2	no	—
325	C₈	Q₈×D₉	1	2	yes	—
326	C₂×C₃	C₄×D₃×Q₈	1	1	yes	characteristic
327	C₂×C₃	C₂×#901	1	1	yes	characteristic
328	C₂×C₃	D₃×#131	1	1	yes	characteristic
329	D₃	—	6	2	no	—
330	C₂²	—	9	2	no	—
331	C₂²	—	18	1	no	—
332	C₂²	—	18	1	no	—
333	C₂²	C₂²×C₄×D₉	1	1	yes	characteristic
334	C₄	—	2	2	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	2	no	—
337	C₄	—	9	2	no	—
338	C₄	—	9	2	no	—
339	C₄	—	18	1	no	—
340	C₄	—	18	2	no	—
341	C₄	—	18	2	no	—
342	C₄	#63464	1	2	yes	—
343	C₄	C₂×Q₈×D₉	1	1	yes	characteristic
344	C₄	#41×D₉	1	1	yes	characteristic
345	C₃	#79917	1	1	yes	characteristic
346	C₂	—	18	2	no	—
347	C₂	C₄×Q₈×D₉	1	1	yes	characteristic
348	C₂	C₂×#63464	1	1	yes	characteristic
349	C₂	D₉×#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₉
commutator	C₂²×C₉	C₂³×C₄
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₈×Q₈×C₉	C₂
socle	C₂²×C₃	C₂²×C₄×D₃
prime radical	C₂²×D₉	C₂²×C₄
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	C₈×Q₈	D₉
hypocenter	C₉	#2960
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. C₈×Q₈×C₉	C₂	semidirect
3. #10588701	C₂	semidirect
6. #10588510	C₂	semidirect
7. #10588509	C₂	semidirect
10. C₈×Dic₁₈	C₂	semidirect
11. C₈×Dic₁₈	C₂	semidirect
84. C₈×D₉	Q₈	semidirect
145. C₈×Q₈	D₉	semidirect
301. C₉	#2960	semidirect

Maximal subgroups

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

subgroup	index
#10589680	2
C₈×Q₈×C₉	2
#10588701	2
#10588523	2
#10588521	2
#10588510	2
#10588509	2
#10588371	2
C₄×C₈×D₉	2
C₈×Dic₁₈	2
#79917	3