Subgroups — #11861498

Order 1152.

582 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	C₄×Q₈×D₉	C₂	1	1	yes	maximal, characteristic
2	C₉×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	#10588701	C₂	1	1	yes	maximal, characteristic, complemented
4	#10588521	C₂	1	1	yes	maximal, characteristic
5	#10588521	C₂	1	2	yes	maximal
6	#10588510	C₂	1	2	yes	maximal, complemented
7	#10588508	C₂	1	1	yes	maximal, characteristic, complemented
8	#10588388	C₂	1	2	yes	maximal
9	#10588381	C₂	1	2	yes	maximal, complemented
10	C₄×#3663	C₂	1	1	yes	maximal, characteristic
11	C₈×Dic₁₈	C₂	1	1	yes	maximal, characteristic, complemented
12	#79962	—	3	1	no	maximal
13	C₂×Q₈×D₉	C₄	1	1	yes	characteristic
14	C₄×Q₈×C₉	C₂²	1	1	yes	characteristic
15	Q₈×Dic₉	C₄	1	1	yes	characteristic
16	C₂×#3663	C₂²	1	1	yes	characteristic
17	C₂×#3663	C₂²	1	2	yes	—
18	C₂×#3663	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×D₉	C₄	1	1	yes	characteristic
20	C₄⋊C₄×D₉	C₄	1	2	yes	—
21	#63449	C₄	1	2	yes	—
22	#63449	C₄	1	1	yes	characteristic
23	C₄²×D₉	C₂²	1	1	yes	characteristic
24	C₄²×D₉	C₂²	1	2	yes	—
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	C₉×#97	C₂²	1	2	yes	—
30	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
31	#63375	C₂²	1	1	yes	characteristic
32	#63375	C₂²	1	1	yes	characteristic
33	Dic₉⋊C₈	C₂²	1	2	yes	—
34	Dic₉⋊C₈	C₂²	1	2	yes	—
35	Dic₉⋊C₈	C₂²	1	2	yes	—
36	C₈×Dic₉	C₂²	1	2	yes	—
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	C₄×D₃×Q₈	—	3	1	no	—
43	C₃×#446	D₃	1	1	yes	characteristic
44	#5024	—	3	1	no	—
45	#4837	—	3	1	no	—
46	#4837	—	3	2	no	—
47	#4830	—	3	1	no	—
48	#4828	—	3	2	no	—
49	#4707	—	3	2	no	—
50	#4701	—	3	2	no	—
51	C₄×ΓL(1,𝔽₂₅)	—	3	1	no	—
52	C₈×Dic₆	—	3	1	no	—
53	C₂×Q₈×C₉	C₂×C₄	1	1	yes	characteristic
54	Q₈×D₉	—	2	4	no	—
55	C₂×C₄×D₉	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×D₉	C₂×C₄	1	2	yes	—
57	C₂×C₄×D₉	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×D₉	C₂×C₄	1	2	yes	—
59	C₂×C₄×D₉	C₂³	1	1	yes	characteristic
60	C₂×Dic₁₈	C₂×C₄	1	2	yes	—
61	C₂×Dic₁₈	C₂×C₄	1	1	yes	characteristic
62	C₂×C₈×C₉	C₂³	1	2	yes	—
63	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
65	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
66	C₉×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
67	C₄²×C₉	C₂³	1	1	yes	characteristic
68	C₄²×C₉	C₂³	1	2	yes	—
69	C₄⋊Dic₉	C₂×C₄	1	2	yes	—
70	C₄⋊Dic₉	C₂×C₄	1	1	yes	characteristic
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	C₄×Dic₉	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	C₂×C₄	1	2	yes	—
78	C₄×Dic₉	C₂³	1	2	yes	—
79	C₄×Dic₉	C₂³	1	1	yes	characteristic
80	C₂×C₉⋊C₈	C₂³	1	1	yes	characteristic
81	C₂×C₉⋊C₈	C₂³	1	2	yes	—
82	C₂×C₉⋊C₈	C₂³	1	1	yes	characteristic
83	#3663	—	2	2	no	—
84	#3663	—	2	4	no	—
85	#3663	Q₈	1	4	yes	complemented
86	Q₈×M₁₆	—	9	1	no	Sylow
87	C₂×D₃×Q₈	—	3	1	no	—
88	C₃×C₄×Q₈	C₂×D₃	1	1	yes	characteristic
89	Q₈×Dic₃	—	3	1	no	—
90	C₂×ΓL(1,𝔽₂₅)	—	3	1	no	—
91	C₂×ΓL(1,𝔽₂₅)	—	3	1	no	—
92	C₂×ΓL(1,𝔽₂₅)	—	3	2	no	—
93	D₃×C₄⋊C₄	—	3	2	no	—
94	D₃×C₄⋊C₄	—	3	1	no	—
95	#887	—	3	1	no	—
96	#887	—	3	2	no	—
97	C₄²×D₃	—	3	1	no	—
98	C₄²×D₃	—	3	2	no	—
99	C₄×Dic₆	—	3	1	no	—
100	C₄×Dic₆	—	3	2	no	—
101	C₃×C₄⋊C₈	C₂×D₃	1	2	yes	—
102	C₃×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
103	C₃×#97	C₂×D₃	1	2	yes	—
104	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
105	#815	—	3	1	no	—
106	#815	—	3	1	no	—
107	Dic₃⋊C₈	—	3	2	no	—
108	Dic₃⋊C₈	—	3	2	no	—
109	Dic₃⋊C₈	—	3	2	no	—
110	C₈×Dic₃	—	3	2	no	—
111	C₈×Dic₃	—	3	2	no	—
112	C₄⋊(C₃⋊C₈)	—	3	1	no	—
113	C₄⋊(C₃⋊C₈)	—	3	2	no	—
114	#803	—	3	2	no	—
115	C₄×C₃⋊C₈	—	3	1	no	—
116	C₂²×D₉	C₂²×C₄	1	1	yes	characteristic
117	Q₈×C₉	—	2	2	no	—
118	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
120	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
121	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
123	C₂×Dic₉	C₂²×C₄	1	2	yes	—
124	C₂×Dic₉	C₂²×C₄	1	1	yes	characteristic
125	C₂×Dic₉	C₂²×C₄	1	1	yes	characteristic
126	C₂×Dic₉	C₂²×C₄	1	1	yes	characteristic
127	C₂×Dic₉	C₂²×C₄	1	2	yes	—
128	C₄×D₉	—	2	2	no	—
129	C₄×D₉	—	2	4	no	—
130	C₄×D₉	—	4	1	no	—
131	C₄×D₉	—	4	2	no	—
132	C₄×D₉	C₂×Q₈	1	2	yes	—
133	C₄×D₉	#41	1	2	yes	—
134	Dic₁₈	—	2	4	no	—
135	Dic₁₈	M₁₆	1	4	yes	complemented
136	C₈×C₉	—	2	2	no	—
137	C₈×C₉	—	2	1	no	—
138	C₈×C₉	C₂×Q₈	1	2	yes	complemented
139	C₉⋊C₈	—	2	1	no	—
140	C₉⋊C₈	—	2	2	no	—
141	C₉⋊C₈	C₂×Q₈	1	2	yes	complemented
142	C₂×C₄×Q₈	—	9	1	no	—
143	#446	—	9	1	no	—
144	#446	—	9	2	no	—
145	#446	—	9	2	no	—
146	#446	D₉	1	1	yes	characteristic, complemented
147	C₈×Q₈	—	9	1	no	—
148	C₈×Q₈	—	9	1	no	—
149	#423	—	9	2	no	—
150	#423	—	9	1	no	—
151	C₄×M₁₆	—	9	2	no	—
152	C₄×M₁₆	—	9	1	no	—
153	C₂×C₃×Q₈	C₄×D₃	1	1	yes	characteristic
154	D₃×Q₈	—	6	4	no	—
155	C₂×C₄×D₃	—	3	1	no	—
156	C₂×C₄×D₃	—	3	2	no	—
157	C₂×C₄×D₃	—	3	1	no	—
158	C₂×C₄×D₃	—	3	1	no	—
159	C₂×C₄×D₃	—	3	2	no	—
160	C₂×Dic₆	—	3	1	no	—
161	C₂×Dic₆	—	3	2	no	—
162	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
163	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
164	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
165	C₃×C₄⋊C₄	C₄×D₃	1	2	yes	—
166	C₃×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
167	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
168	C₃×C₄²	C₂²×D₃	1	2	yes	—
169	C₄⋊Dic₃	—	3	2	no	—
170	C₄⋊Dic₃	—	3	1	no	—
171	Dic₃⋊C₄	—	3	2	no	—
172	Dic₃⋊C₄	—	3	2	no	—
173	Dic₃⋊C₄	—	3	2	no	—
174	C₄×Dic₃	—	3	2	no	—
175	C₄×Dic₃	—	3	1	no	—
176	C₄×Dic₃	—	3	2	no	—
177	C₄×Dic₃	—	3	1	no	—
178	C₄×Dic₃	—	3	2	no	—
179	C₄×Dic₃	—	3	1	no	—
180	C₂×C₃⋊C₈	—	3	1	no	—
181	C₂×C₃⋊C₈	—	3	1	no	—
182	C₂×C₃⋊C₈	—	3	2	no	—
183	ΓL(1,𝔽₂₅)	—	3	4	no	—
184	ΓL(1,𝔽₂₅)	—	6	2	no	—
185	ΓL(1,𝔽₂₅)	—	6	4	no	—
186	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
187	C₂×D₉	—	2	1	no	—
188	C₂×D₉	—	2	1	no	—
189	C₂×D₉	C₄×Q₈	1	2	yes	—
190	C₄×C₉	—	2	2	no	—
191	C₄×C₉	—	2	1	no	—
192	C₄×C₉	—	2	2	no	—
193	C₄×C₉	C₂×M₁₆	1	2	yes	—
194	C₄×C₉	C₂²×Q₈	1	1	yes	characteristic
195	C₄×C₉	C₂×#41	1	1	yes	characteristic
196	Dic₉	—	2	1	no	—
197	Dic₉	—	2	2	no	—
198	Dic₉	—	2	1	no	—
199	Dic₉	C₄×Q₈	1	2	yes	—
200	Dic₉	C₂×M₁₆	1	4	yes	—
201	C₂²×Q₈	—	9	1	no	—
202	C₂×M₁₆	—	9	1	no	—
203	C₂×M₁₆	—	9	2	no	—
204	C₂×M₁₆	—	9	1	no	—
205	C₄×Q₈	—	9	1	no	—
206	C₄×Q₈	—	9	1	no	—
207	C₄×Q₈	—	9	2	no	—
208	C₄×Q₈	—	9	1	no	—
209	C₄×Q₈	—	9	2	no	—
210	C₄×Q₈	C₂×D₉	1	1	yes	characteristic
211	C₂×C₄⋊C₄	—	9	2	no	—
212	C₂×C₄⋊C₄	—	9	1	no	—
213	C₂×C₄²	—	9	2	no	—
214	C₂×C₄²	—	9	1	no	—
215	C₄⋊C₈	—	9	2	no	—
216	C₄⋊C₈	—	9	2	no	—
217	C₄⋊C₈	—	9	2	no	—
218	C₄⋊C₈	—	9	1	no	—
219	C₄⋊C₈	—	9	2	no	—
220	C₄⋊C₈	C₂×D₉	1	1	yes	characteristic
221	C₄⋊C₈	C₂×D₉	1	2	yes	—
222	#97	—	9	1	no	—
223	#97	—	9	1	no	—
224	#97	—	9	2	no	—
225	#97	C₂×D₉	1	2	yes	—
226	C₄×C₈	—	9	2	no	—
227	C₄×C₈	—	9	2	no	—
228	C₄×C₈	—	9	1	no	—
229	C₄×C₈	C₂×D₉	1	1	yes	characteristic
230	C₂²×D₃	—	3	1	no	—
231	C₃×Q₈	—	2	2	no	—
232	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
233	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
234	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
235	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
236	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
237	C₂×Dic₃	—	3	1	no	—
238	C₂×Dic₃	—	3	2	no	—
239	C₂×Dic₃	—	3	2	no	—
240	C₂×Dic₃	—	3	1	no	—
241	C₂×Dic₃	—	3	1	no	—
242	C₄×D₃	—	3	2	no	—
243	C₄×D₃	—	3	2	no	—
244	C₄×D₃	—	6	4	no	—
245	C₄×D₃	—	6	2	no	—
246	C₄×D₃	—	12	1	no	—
247	C₄×D₃	—	12	2	no	—
248	Dic₆	—	3	4	no	—
249	Dic₆	—	6	4	no	—
250	C₃×C₈	—	2	2	no	—
251	C₃×C₈	—	2	1	no	—
252	C₃×C₈	D₃×Q₈	1	2	yes	—
253	C₃⋊C₈	—	3	2	no	—
254	C₃⋊C₈	—	6	1	no	—
255	C₃⋊C₈	—	6	2	no	—
256	C₂×C₉	C₂×C₄×Q₈	1	1	yes	characteristic
257	C₂×C₉	C₂²×M₁₆	1	1	yes	characteristic
258	C₂×C₉	#568	1	1	yes	characteristic
259	D₉	—	2	2	no	—
260	C₂×Q₈	—	9	1	no	—
261	C₂×Q₈	—	9	2	no	—
262	C₂×Q₈	—	18	4	no	—
263	C₂×Q₈	C₄×D₉	1	1	yes	characteristic
264	C₂²×C₄	—	9	1	no	—
265	C₂²×C₄	—	9	2	no	—
266	C₂²×C₄	—	9	1	no	—
267	C₂²×C₄	—	9	2	no	—
268	C₂²×C₄	—	9	1	no	—
269	M₁₆	—	9	4	no	—
270	M₁₆	—	18	4	no	—
271	M₁₆	—	18	2	no	—
272	C₂×C₈	—	9	1	no	—
273	C₂×C₈	—	9	2	no	—
274	C₂×C₈	—	9	1	no	—
275	C₂×C₈	C₂²×D₉	1	2	yes	—
276	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
277	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
278	C₄⋊C₄	—	9	2	no	—
279	C₄⋊C₄	—	9	2	no	—
280	C₄⋊C₄	—	9	2	no	—
281	C₄⋊C₄	—	9	2	no	—
282	C₄⋊C₄	—	9	1	no	—
283	C₄⋊C₄	C₄×D₉	1	2	yes	—
284	C₄⋊C₄	C₄×D₉	1	1	yes	characteristic
285	C₄²	—	9	2	no	—
286	C₄²	—	9	2	no	—
287	C₄²	—	9	1	no	—
288	C₄²	—	9	1	no	—
289	C₄²	—	9	2	no	—
290	C₄²	—	9	1	no	—
291	C₄²	C₂²×D₉	1	1	yes	characteristic
292	C₄²	C₂²×D₉	1	2	yes	—
293	C₂²×C₃	C₂²×C₄×D₃	1	1	yes	characteristic
294	C₂×D₃	—	3	2	no	—
295	C₂×D₃	—	6	1	no	—
296	C₂×D₃	—	6	1	no	—
297	C₃×C₄	—	2	2	no	—
298	C₃×C₄	—	2	1	no	—
299	C₃×C₄	—	2	2	no	—
300	C₃×C₄	D₃×M₁₆	1	2	yes	—
301	C₃×C₄	C₂×D₃×Q₈	1	1	yes	characteristic
302	C₃×C₄	D₃×#41	1	1	yes	characteristic
303	Dic₃	—	3	4	no	—
304	Dic₃	—	3	2	no	—
305	Dic₃	—	6	1	no	—
306	Dic₃	—	6	2	no	—
307	Dic₃	—	6	1	no	—
308	C₉	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
309	C₂³	—	9	1	no	—
310	Q₈	—	2	2	no	—
311	Q₈	—	9	4	no	—
312	Q₈	—	18	4	no	—
313	C₂×C₄	—	9	1	no	—
314	C₂×C₄	—	9	2	no	—
315	C₂×C₄	—	9	2	no	—
316	C₂×C₄	—	9	2	no	—
317	C₂×C₄	—	9	2	no	—
318	C₂×C₄	—	9	1	no	—
319	C₂×C₄	—	9	1	no	—
320	C₂×C₄	—	18	4	no	—
321	C₂×C₄	—	18	2	no	—
322	C₂×C₄	—	36	1	no	—
323	C₂×C₄	—	36	2	no	—
324	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
325	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
326	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
327	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
328	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
329	C₈	—	2	2	no	—
330	C₈	—	2	1	no	—
331	C₈	—	9	2	no	—
332	C₈	—	18	2	no	—
333	C₈	—	18	1	no	—
334	C₈	Q₈×D₉	1	2	yes	complemented
335	C₂×C₃	C₄×D₃×Q₈	1	1	yes	characteristic
336	C₂×C₃	C₂×D₃×M₁₆	1	1	yes	characteristic
337	C₂×C₃	#5750	1	1	yes	characteristic
338	D₃	—	6	2	no	—
339	C₂²	—	9	2	no	—
340	C₂²	—	18	1	no	—
341	C₂²	—	18	1	no	—
342	C₂²	C₂²×C₄×D₉	1	1	yes	characteristic
343	C₄	—	2	2	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	2	no	—
346	C₄	—	9	4	no	—
347	C₄	—	9	2	no	—
348	C₄	—	18	1	no	—
349	C₄	—	18	2	no	—
350	C₄	—	18	1	no	—
351	C₄	M₁₆×D₉	1	2	yes	—
352	C₄	C₂×Q₈×D₉	1	1	yes	characteristic
353	C₄	#41×D₉	1	1	yes	characteristic
354	C₃	#79962	1	1	yes	characteristic
355	C₂	—	18	2	no	—
356	C₂	C₄×Q₈×D₉	1	1	yes	characteristic
357	C₂	C₂×M₁₆×D₉	1	1	yes	characteristic
358	C₂	#10589852	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₉
commutator	C₂²×C₉	C₂³×C₄
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₉×#446	C₂
socle	C₂²×C₃	C₂²×C₄×D₃
prime radical	C₂²×D₉	C₂²×C₄
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	#446	D₉
hypocenter	C₉	Q₈×M₁₆
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. C₉×#446	C₂	semidirect
3. #10588701	C₂	semidirect
6. #10588510	C₂	semidirect
7. #10588508	C₂	semidirect
9. #10588381	C₂	semidirect
11. C₈×Dic₁₈	C₂	semidirect
85. #3663	Q₈	semidirect
135. Dic₁₈	M₁₆	semidirect
138. C₈×C₉	C₂×Q₈	semidirect
141. C₉⋊C₈	C₂×Q₈	semidirect
146. #446	D₉	semidirect
308. C₉	Q₈×M₁₆	semidirect
334. C₈	Q₈×D₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×D₉	2
C₉×#446	2
#10588701	2
#10588521	2
#10588510	2
#10588508	2
#10588388	2
#10588381	2
C₄×#3663	2
C₈×Dic₁₈	2
#79962	3