Subgroups — #11859085

Order 1152.

588 conjugacy classes of subgroups.

The subgroup lattice has 327 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₉×#41	C₂	1	1	yes	maximal, characteristic
2	#10588912	C₂	1	2	yes	maximal
3	#10588678	C₂	1	4	yes	maximal, complemented
4	#10588675	C₂	1	2	yes	maximal, complemented
5	#10588606	C₂	1	2	yes	maximal, complemented
6	#10588605	C₂	1	2	yes	maximal, complemented
7	#10588604	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
9	#82515	—	3	1	no	maximal
10	C₂×C₉×#41	C₂²	1	1	yes	characteristic
11	C₄×Q₈×C₉	C₄	1	1	yes	characteristic
12	C₄×Q₈×C₉	C₄	1	1	yes	characteristic
13	C₄×D₄×C₉	C₄	1	2	yes	—
14	C₄×D₄×C₉	C₄	1	1	yes	characteristic
15	C₄×D₄×C₉	C₄	1	2	yes	—
16	C₄×D₄×C₉	C₄	1	1	yes	characteristic
17	C₉×#117	C₂²	1	1	yes	characteristic
18	C₉×#117	C₂²	1	2	yes	—
19	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₉	C₂²	1	2	yes	—
21	C₂×#3667	C₂²	1	2	yes	—
22	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
23	#63389	C₂²	1	4	yes	—
24	#63389	C₂²	1	4	yes	—
25	C₄⋊(C₉⋊C₈)	C₂²	1	2	yes	—
26	C₄⋊(C₉⋊C₈)	C₂²	1	2	yes	—
27	C₄⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
28	C₄⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
29	C₄⋊(C₉⋊C₈)	C₂²	1	2	yes	—
30	#63362	C₂²	1	4	yes	—
31	C₄×C₉⋊C₈	C₂²	1	4	yes	—
32	C₃×C₄×#41	D₃	1	1	yes	characteristic
33	#5234	—	3	2	no	—
34	#5000	—	3	4	no	—
35	#4997	—	3	2	no	—
36	#4926	—	3	2	no	—
37	#4925	—	3	2	no	—
38	#4924	—	3	1	no	—
39	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
40	C₉×#41	—	2	2	no	—
41	C₉×#41	—	2	2	no	—
42	C₂×Q₈×C₉	C₂×C₄	1	1	yes	characteristic
43	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
44	C₂×D₄×C₉	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
46	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
48	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₉	C₂³	1	2	yes	—
50	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
51	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
52	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
53	C₉×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
54	C₉×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
55	C₉×C₂²⋊C₄	C₂×C₄	1	2	yes	—
56	C₉×C₂²⋊C₄	C₂×C₄	1	2	yes	—
57	C₉×C₂²⋊C₄	C₂×C₄	1	2	yes	—
58	C₄²×C₉	C₂×C₄	1	2	yes	—
59	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
60	C₄²×C₉	C₂×C₄	1	2	yes	—
61	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
62	C₄²×C₉	C₂³	1	1	yes	characteristic
63	C₄²×C₉	C₂³	1	2	yes	—
64	C₄²×C₉	C₂³	1	1	yes	characteristic
65	#3667	—	2	4	no	—
66	C₂×C₉⋊C₈	—	2	4	no	—
67	C₂×C₉⋊C₈	C₂³	1	2	yes	—
68	C₂×C₉⋊C₈	C₂³	1	4	yes	—
69	C₂×C₉⋊C₈	C₂³	1	2	yes	—
70	#2925	—	9	1	no	Sylow
71	C₂×C₃×#41	C₂×D₃	1	1	yes	characteristic
72	C₃×C₄×Q₈	Dic₃	1	1	yes	characteristic
73	C₃×C₄×Q₈	Dic₃	1	1	yes	characteristic
74	C₃×C₄×D₄	Dic₃	1	2	yes	—
75	C₃×C₄×D₄	Dic₃	1	1	yes	characteristic
76	C₃×C₄×D₄	Dic₃	1	2	yes	—
77	C₃×C₄×D₄	Dic₃	1	1	yes	characteristic
78	C₃×#117	C₂×D₃	1	1	yes	characteristic
79	C₃×#117	C₂×D₃	1	2	yes	—
80	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
81	C₂×C₃×C₄²	C₂×D₃	1	2	yes	—
82	C₂×#208	—	3	2	no	—
83	C₂²×C₃⋊C₈	—	3	2	no	—
84	#830	—	3	4	no	—
85	#830	—	3	4	no	—
86	C₄⋊(C₃⋊C₈)	—	3	1	no	—
87	C₄⋊(C₃⋊C₈)	—	3	1	no	—
88	C₄⋊(C₃⋊C₈)	—	3	2	no	—
89	C₄⋊(C₃⋊C₈)	—	3	2	no	—
90	C₄⋊(C₃⋊C₈)	—	3	2	no	—
91	#803	—	3	4	no	—
92	C₄×C₃⋊C₈	—	3	4	no	—
93	C₂³×C₉	C₂²×C₄	1	2	yes	—
94	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
95	Q₈×C₉	—	2	2	no	—
96	D₄×C₉	—	2	4	no	—
97	D₄×C₉	—	2	2	no	—
98	C₂×C₄×C₉	—	2	2	no	—
99	C₂×C₄×C₉	—	2	4	no	—
100	C₂×C₄×C₉	—	2	2	no	—
101	C₂×C₄×C₉	—	2	2	no	—
102	C₂×C₄×C₉	—	4	1	no	—
103	C₂×C₄×C₉	—	4	2	no	—
104	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
106	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
109	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₉	#41	1	2	yes	—
113	C₂×C₄×C₉	#41	1	2	yes	—
114	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
115	C₉⋊C₈	—	2	2	no	—
116	C₉⋊C₈	—	2	2	no	—
117	C₉⋊C₈	—	2	4	no	—
118	C₄×#41	D₉	1	1	yes	characteristic
119	#433	—	9	4	no	—
120	#432	—	9	2	no	—
121	#431	—	9	2	no	—
122	#423	—	9	1	no	—
123	C₂×C₄⋊C₈	—	9	1	no	—
124	#408	—	9	2	no	—
125	#405	—	9	2	no	—
126	C₃×#41	—	2	2	no	—
127	C₃×#41	—	2	2	no	—
128	C₂×C₃×Q₈	C₂×Dic₃	1	1	yes	characteristic
129	C₂×C₃×D₄	C₂×Dic₃	1	2	yes	—
130	C₂×C₃×D₄	C₂×Dic₃	1	1	yes	characteristic
131	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
132	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
133	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
134	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
135	C₂²×C₃×C₄	C₂²×D₃	1	2	yes	—
136	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
137	C₃×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
138	C₃×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
139	C₃×C₄⋊C₄	C₂×Dic₃	1	1	yes	characteristic
140	C₃×C₄⋊C₄	C₂×Dic₃	1	1	yes	characteristic
141	C₃×C₂²⋊C₄	C₂×Dic₃	1	2	yes	—
142	C₃×C₂²⋊C₄	C₂×Dic₃	1	2	yes	—
143	C₃×C₂²⋊C₄	C₂×Dic₃	1	2	yes	—
144	C₃×C₄²	C₂×Dic₃	1	2	yes	—
145	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
146	C₃×C₄²	C₂×Dic₃	1	2	yes	—
147	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
148	C₃×C₄²	C₂²×D₃	1	2	yes	—
149	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
150	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
151	#208	—	6	4	no	—
152	C₂×C₃⋊C₈	—	3	2	no	—
153	C₂×C₃⋊C₈	—	3	4	no	—
154	C₂×C₃⋊C₈	—	3	2	no	—
155	C₂×C₃⋊C₈	—	6	4	no	—
156	C₂²×C₉	—	2	1	no	—
157	C₂²×C₉	—	2	1	no	—
158	C₂²×C₉	—	2	2	no	—
159	C₂²×C₉	—	2	2	no	—
160	C₂²×C₉	—	2	2	no	—
161	C₂²×C₉	#117	1	2	yes	—
162	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
163	C₄×C₉	—	2	2	no	—
164	C₄×C₉	—	2	1	no	—
165	C₄×C₉	—	2	2	no	—
166	C₄×C₉	—	2	1	no	—
167	C₄×C₉	—	2	1	no	—
168	C₄×C₉	—	2	2	no	—
169	C₄×C₉	#117	1	2	yes	—
170	C₄×C₉	#131	1	2	yes	—
171	C₄×C₉	C₂×#41	1	1	yes	characteristic
172	C₄×C₉	C₂×#41	1	1	yes	characteristic
173	C₂×#41	C₂×D₉	1	1	yes	characteristic
174	C₂×M₁₆	—	9	2	no	—
175	C₂²×C₈	—	9	2	no	—
176	C₄×Q₈	Dic₉	1	1	yes	characteristic
177	C₄×Q₈	Dic₉	1	1	yes	characteristic
178	C₄×D₄	Dic₉	1	2	yes	—
179	C₄×D₄	Dic₉	1	1	yes	characteristic
180	C₄×D₄	Dic₉	1	2	yes	—
181	C₄×D₄	Dic₉	1	1	yes	characteristic
182	#117	C₂×D₉	1	1	yes	characteristic
183	#117	C₂×D₉	1	2	yes	—
184	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
185	C₂×C₄²	C₂×D₉	1	2	yes	—
186	C₄⋊C₈	—	9	1	no	—
187	C₄⋊C₈	—	9	1	no	—
188	C₄⋊C₈	—	9	2	no	—
189	C₄⋊C₈	—	9	2	no	—
190	C₄⋊C₈	—	9	2	no	—
191	C₂²⋊C₈	—	9	4	no	—
192	C₂²⋊C₈	—	9	4	no	—
193	#97	—	9	4	no	—
194	C₄×C₈	—	9	4	no	—
195	C₂³×C₃	C₂²×Dic₃	1	2	yes	—
196	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
197	C₃×Q₈	—	2	2	no	—
198	C₃×D₄	—	2	4	no	—
199	C₃×D₄	—	2	2	no	—
200	C₂×C₃×C₄	—	2	2	no	—
201	C₂×C₃×C₄	—	2	4	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	4	1	no	—
205	C₂×C₃×C₄	—	4	2	no	—
206	C₂×C₃×C₄	#235	1	2	yes	—
207	C₂×C₃×C₄	#235	1	2	yes	—
208	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂²×Dic₃	1	2	yes	—
210	C₂×C₃×C₄	C₂²×Dic₃	1	2	yes	—
211	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂²×Dic₃	1	2	yes	—
213	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
217	C₃⋊C₈	—	6	2	no	—
218	C₃⋊C₈	—	6	2	no	—
219	C₃⋊C₈	—	6	4	no	—
220	C₂×C₉	—	2	2	no	—
221	C₂×C₉	—	4	2	no	—
222	C₂×C₉	C₂×#117	1	1	yes	characteristic
223	C₂×C₉	C₂×#131	1	1	yes	characteristic
224	C₂×C₉	#568	1	1	yes	characteristic
225	#41	—	2	2	no	—
226	#41	—	2	2	no	—
227	C₂×Q₈	C₂×Dic₉	1	1	yes	characteristic
228	C₂×D₄	C₂×Dic₉	1	2	yes	—
229	C₂×D₄	C₂×Dic₉	1	1	yes	characteristic
230	C₂²×C₄	C₂×Dic₉	1	2	yes	—
231	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
232	C₂²×C₄	C₂×Dic₉	1	2	yes	—
233	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
234	C₂²×C₄	C₂²×D₉	1	2	yes	—
235	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
236	M₁₆	—	18	4	no	—
237	C₂×C₈	—	9	2	no	—
238	C₂×C₈	—	9	4	no	—
239	C₂×C₈	—	9	2	no	—
240	C₂×C₈	—	18	4	no	—
241	C₄⋊C₄	C₂×Dic₉	1	2	yes	—
242	C₄⋊C₄	C₂×Dic₉	1	2	yes	—
243	C₄⋊C₄	C₂×Dic₉	1	1	yes	characteristic
244	C₄⋊C₄	C₂×Dic₉	1	1	yes	characteristic
245	C₂²⋊C₄	C₂×Dic₉	1	2	yes	—
246	C₂²⋊C₄	C₂×Dic₉	1	2	yes	—
247	C₂²⋊C₄	C₂×Dic₉	1	2	yes	—
248	C₄²	C₂×Dic₉	1	2	yes	—
249	C₄²	C₂×Dic₉	1	1	yes	characteristic
250	C₄²	C₂×Dic₉	1	2	yes	—
251	C₄²	C₂×Dic₉	1	1	yes	characteristic
252	C₄²	C₂²×D₉	1	2	yes	—
253	C₄²	C₂²×D₉	1	1	yes	characteristic
254	C₄²	C₂²×D₉	1	1	yes	characteristic
255	C₂²×C₃	—	2	1	no	—
256	C₂²×C₃	—	2	1	no	—
257	C₂²×C₃	—	2	2	no	—
258	C₂²×C₃	—	2	2	no	—
259	C₂²×C₃	—	2	2	no	—
260	C₂²×C₃	#926	1	2	yes	—
261	C₂²×C₃	C₂³×Dic₃	1	1	yes	characteristic
262	C₃×C₄	—	2	2	no	—
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	—	2	2	no	—
265	C₃×C₄	—	2	1	no	—
266	C₃×C₄	—	2	1	no	—
267	C₃×C₄	—	2	2	no	—
268	C₃×C₄	#926	1	2	yes	—
269	C₃×C₄	#948	1	2	yes	—
270	C₃×C₄	C₂×#235	1	1	yes	characteristic
271	C₃×C₄	C₂×#235	1	1	yes	characteristic
272	C₉	#2925	1	1	yes	characteristic, complemented, Sylow
273	C₂³	C₂²×Dic₉	1	2	yes	—
274	C₂³	C₂²×Dic₉	1	1	yes	characteristic
275	Q₈	—	2	2	no	—
276	D₄	—	2	4	no	—
277	D₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	#3697	1	2	yes	—
285	C₂×C₄	#3697	1	2	yes	—
286	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
287	C₂×C₄	C₂²×Dic₉	1	2	yes	—
288	C₂×C₄	C₂²×Dic₉	1	2	yes	—
289	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
290	C₂×C₄	C₂²×Dic₉	1	2	yes	—
291	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
292	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
293	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
294	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
295	C₈	—	18	2	no	—
296	C₈	—	18	2	no	—
297	C₈	—	18	4	no	—
298	C₂×C₃	—	2	2	no	—
299	C₂×C₃	—	4	2	no	—
300	C₂×C₃	C₂×#926	1	1	yes	characteristic
301	C₂×C₃	C₂×#948	1	1	yes	characteristic
302	C₂×C₃	#5819	1	1	yes	characteristic
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	#63488	1	2	yes	—
309	C₂²	C₂³×Dic₉	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	2	no	—
316	C₄	#63488	1	2	yes	—
317	C₄	#63510	1	2	yes	—
318	C₄	C₂×#3697	1	1	yes	characteristic
319	C₄	C₂×#3697	1	1	yes	characteristic
320	C₃	#82515	1	1	yes	characteristic
321	C₂	—	2	2	no	—
322	C₂	—	4	2	no	—
323	C₂	C₂×#63488	1	1	yes	characteristic
324	C₂	C₂×#63510	1	1	yes	characteristic
325	C₂	#10589921	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₉
commutator	C₂²×C₉	C₂³×C₄
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₄×C₉×#41	C₂
socle	C₂²×C₃	C₂³×Dic₃
prime radical	C₂×C₃×#41	C₂×D₃
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	C₄×#41	D₉
hypocenter	C₉	#2925
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
3. #10588678	C₂	semidirect
4. #10588675	C₂	semidirect
5. #10588606	C₂	semidirect
6. #10588605	C₂	semidirect
7. #10588604	C₂	semidirect
8. C₂×C₄⋊(C₉⋊C₈)	C₂	semidirect
272. C₉	#2925	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₉×#41	2
#10588912	2
#10588678	2
#10588675	2
#10588606	2
#10588605	2
#10588604	2
C₂×C₄⋊(C₉⋊C₈)	2
#82515	3