Subgroups — #11786446

Order 1152.

540 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₄×D₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#106	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×(C₂×D₉)⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#10588667	C₂	1	1	yes	maximal, characteristic
5	C₂×#63367	C₂	1	1	yes	maximal, characteristic, complemented
6	#10588229	C₂	1	1	yes	maximal, characteristic, complemented
7	#10588206	C₂	1	1	yes	maximal, characteristic, complemented
8	#70579	—	3	1	no	maximal
9	C₂²×C₄×D₉	—	2	1	no	—
10	C₂²×C₄×D₉	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×(C₂×D₉)⋊C₄	—	2	1	no	—
15	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₉⋊C₄	—	2	1	no	—
18	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×D₉	—	2	2	no	—
20	C₄⋊C₄×D₉	—	2	2	no	—
21	C₉×#106	—	2	2	no	—
22	C₉⋊#95	—	2	1	no	—
23	(C₂×D₉)⋊C₈	C₄	1	4	yes	complemented
24	#63367	—	2	2	no	—
25	C₂×D₃×C₄⋊C₄	—	3	1	no	—
26	C₂×C₃×#106	D₃	1	1	yes	characteristic
27	C₂×(C₂×D₃)⋊C₈	—	3	1	no	—
28	#4987	—	3	1	no	—
29	C₂×#808	—	3	1	no	—
30	#4551	—	3	1	no	—
31	#4529	—	3	1	no	—
32	C₂³×D₉	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₉	—	2	1	no	—
34	C₂²×C₄×C₉	—	2	1	no	—
35	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
36	C₂²×Dic₉	—	2	1	no	—
37	C₂²×Dic₉	—	2	1	no	—
38	C₂²×Dic₉	D₄	1	1	yes	characteristic
39	C₂×C₄×D₉	—	2	1	no	—
40	C₂×C₄×D₉	—	2	1	no	—
41	C₂×C₄×D₉	—	2	1	no	—
42	C₂×C₄×D₉	—	2	1	no	—
43	C₂×C₄×D₉	—	2	4	no	—
44	C₂×C₄×D₉	—	4	2	no	—
45	C₂×C₄×D₉	—	4	2	no	—
46	C₂×C₄×D₉	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₉	Q₈	1	2	yes	—
48	C₂×C₈×C₉	—	2	1	no	—
49	C₂×C₈×C₉	—	2	1	no	—
50	C₂×C₈×C₉	C₂×C₄	1	2	yes	complemented
51	C₉×C₄⋊C₄	—	2	1	no	—
52	C₉×C₄⋊C₄	—	2	1	no	—
53	C₉×C₄⋊C₄	—	2	1	no	—
54	C₉×C₄⋊C₄	D₄	1	2	yes	—
55	(C₂×D₉)⋊C₄	—	4	2	no	—
56	C₄⋊Dic₉	—	2	1	no	—
57	C₄⋊Dic₉	—	2	1	no	—
58	C₄⋊Dic₉	—	2	1	no	—
59	C₄⋊Dic₉	D₄	1	2	yes	—
60	Dic₉⋊C₄	—	2	4	no	—
61	C₂×C₉⋊C₈	—	2	1	no	—
62	C₂×C₉⋊C₈	—	2	1	no	—
63	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	complemented
64	#1853	—	9	1	no	Sylow
65	C₂²×C₄×D₃	—	3	1	no	—
66	C₂²×C₄×D₃	—	6	1	no	—
67	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
68	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
69	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
70	C₂×(C₂×D₃)⋊C₄	—	6	1	no	—
71	C₂×C₄⋊Dic₃	—	3	1	no	—
72	C₂×C₄⋊Dic₃	—	3	1	no	—
73	C₂×Dic₃⋊C₄	—	6	1	no	—
74	C₂²×C₃⋊C₈	—	3	1	no	—
75	D₃×C₄⋊C₄	—	6	2	no	—
76	D₃×C₄⋊C₄	—	6	2	no	—
77	C₃×#106	—	2	2	no	—
78	C₃⋊#95	—	6	1	no	—
79	(C₂×D₃)⋊C₈	—	3	4	no	—
80	#808	—	6	2	no	—
81	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×D₉	—	2	1	no	—
83	C₂²×D₉	—	2	1	no	—
84	C₂²×D₉	—	2	1	no	—
85	C₂²×D₉	—	2	1	no	—
86	C₂²×D₉	C₄⋊C₄	1	2	yes	—
87	C₂²×D₉	SD₁₆	1	2	yes	—
88	C₂²×D₉	SD₁₆	1	2	yes	—
89	C₂×C₄×C₉	—	2	2	no	—
90	C₂×C₄×C₉	—	4	1	no	—
91	C₂×C₄×C₉	—	4	1	no	—
92	C₂×C₄×C₉	—	4	1	no	—
93	C₂×C₄×C₉	—	4	1	no	—
94	C₂×C₄×C₉	—	4	1	no	—
95	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₉	#41	1	1	yes	characteristic
100	C₂×C₄×C₉	#41	1	1	yes	characteristic
101	C₂×Dic₉	—	2	1	no	—
102	C₂×Dic₉	—	2	1	no	—
103	C₂×Dic₉	—	2	2	no	—
104	C₂×Dic₉	—	4	1	no	—
105	C₂×Dic₉	—	4	1	no	—
106	C₂×Dic₉	—	4	1	no	—
107	C₂×Dic₉	—	4	1	no	—
108	C₂×Dic₉	—	4	1	no	—
109	C₂×Dic₉	C₄⋊C₄	1	2	yes	—
110	C₄×D₉	—	2	2	no	—
111	C₄×D₉	—	2	2	no	—
112	C₄×D₉	—	2	2	no	—
113	C₄×D₉	—	2	2	no	—
114	C₄×D₉	—	4	8	no	—
115	C₈×C₉	—	2	2	no	—
116	C₉⋊C₈	—	2	2	no	—
117	C₂²×C₄⋊C₄	—	9	1	no	—
118	C₂×#106	—	9	1	no	—
119	C₂×#106	D₉	1	1	yes	characteristic, complemented
120	C₂×C₂²⋊C₈	—	9	1	no	—
121	#380	—	9	1	no	—
122	#340	—	9	1	no	—
123	#340	—	9	1	no	—
124	C₂³×D₃	—	3	1	no	—
125	C₂²×C₃×C₄	—	2	1	no	—
126	C₂²×C₃×C₄	—	2	1	no	—
127	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
128	C₂²×Dic₃	—	3	1	no	—
129	C₂²×Dic₃	—	6	1	no	—
130	C₂²×Dic₃	—	6	1	no	—
131	C₂×C₄×D₃	—	3	2	no	—
132	C₂×C₄×D₃	—	3	2	no	—
133	C₂×C₄×D₃	—	6	1	no	—
134	C₂×C₄×D₃	—	6	1	no	—
135	C₂×C₄×D₃	—	6	1	no	—
136	C₂×C₄×D₃	—	6	1	no	—
137	C₂×C₄×D₃	—	6	4	no	—
138	C₂×C₄×D₃	—	12	2	no	—
139	C₂×C₄×D₃	—	12	2	no	—
140	C₂×C₃×C₈	—	2	1	no	—
141	C₂×C₃×C₈	—	2	1	no	—
142	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
143	C₃×C₄⋊C₄	—	2	1	no	—
144	C₃×C₄⋊C₄	—	2	1	no	—
145	C₃×C₄⋊C₄	—	2	1	no	—
146	C₃×C₄⋊C₄	#67	1	2	yes	—
147	(C₂×D₃)⋊C₄	—	12	2	no	—
148	C₄⋊Dic₃	—	3	2	no	—
149	C₄⋊Dic₃	—	6	1	no	—
150	C₄⋊Dic₃	—	6	1	no	—
151	C₄⋊Dic₃	—	6	1	no	—
152	Dic₃⋊C₄	—	6	4	no	—
153	C₂×C₃⋊C₈	—	3	2	no	—
154	C₂×C₃⋊C₈	—	6	1	no	—
155	C₂×C₃⋊C₈	—	6	1	no	—
156	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂²×C₉	#120	1	1	yes	characteristic
158	C₂²×C₉	#123	1	1	yes	characteristic
159	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
160	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
161	C₂²×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₂²×C₉	#137	1	1	yes	characteristic
163	C₂×D₉	—	2	2	no	—
164	C₂×D₉	—	2	2	no	—
165	C₂×D₉	—	2	2	no	—
166	C₂×D₉	—	2	2	no	—
167	C₂×D₉	—	2	2	no	—
168	C₂×D₉	—	2	2	no	—
169	C₂×D₉	#106	1	4	yes	complemented
170	C₄×C₉	—	4	2	no	—
171	C₄×C₉	—	8	1	no	—
172	C₄×C₉	C₄×D₄	1	1	yes	characteristic
173	C₄×C₉	C₄×D₄	1	1	yes	characteristic
174	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
175	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
176	Dic₉	—	2	2	no	—
177	Dic₉	—	4	2	no	—
178	Dic₉	—	8	1	no	—
179	C₂³×C₄	—	9	1	no	—
180	C₂³×C₄	—	18	1	no	—
181	C₂²×C₈	—	9	1	no	—
182	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
183	C₂×C₄⋊C₄	—	9	1	no	—
184	C₂×C₄⋊C₄	—	9	1	no	—
185	C₂×C₄⋊C₄	—	18	2	no	—
186	C₂×C₄⋊C₄	—	18	1	no	—
187	C₂×C₄⋊C₄	—	18	2	no	—
188	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
189	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
190	C₂×C₂²⋊C₄	—	18	1	no	—
191	#106	—	2	2	no	—
192	#106	—	18	2	no	—
193	C₂²⋊C₈	—	9	4	no	—
194	#95	—	18	1	no	—
195	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
196	C₂²×D₃	—	3	2	no	—
197	C₂²×D₃	—	3	2	no	—
198	C₂²×D₃	—	3	2	no	—
199	C₂²×D₃	—	6	1	no	—
200	C₂²×D₃	—	6	1	no	—
201	C₂²×D₃	—	6	1	no	—
202	C₂²×D₃	—	6	1	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	4	1	no	—
205	C₂×C₃×C₄	—	4	1	no	—
206	C₂×C₃×C₄	—	4	1	no	—
207	C₂×C₃×C₄	—	4	1	no	—
208	C₂×C₃×C₄	—	4	1	no	—
209	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
210	C₂×C₃×C₄	#235	1	1	yes	characteristic
211	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
212	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
213	C₂×C₃×C₄	#239	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
215	C₂×Dic₃	—	3	2	no	—
216	C₂×Dic₃	—	6	1	no	—
217	C₂×Dic₃	—	6	1	no	—
218	C₂×Dic₃	—	6	2	no	—
219	C₂×Dic₃	—	12	1	no	—
220	C₂×Dic₃	—	12	1	no	—
221	C₂×Dic₃	—	12	1	no	—
222	C₂×Dic₃	—	12	1	no	—
223	C₂×Dic₃	—	12	1	no	—
224	C₄×D₃	—	6	2	no	—
225	C₄×D₃	—	6	2	no	—
226	C₄×D₃	—	6	2	no	—
227	C₄×D₃	—	6	2	no	—
228	C₄×D₃	—	12	8	no	—
229	C₃×C₈	—	2	2	no	—
230	C₃⋊C₈	—	6	2	no	—
231	C₂×C₉	#385	1	1	yes	characteristic
232	C₂×C₉	C₂×#106	1	1	yes	characteristic
233	C₂×C₉	#428	1	1	yes	characteristic
234	C₂×C₉	#448	1	1	yes	characteristic
235	C₂×C₉	#450	1	1	yes	characteristic
236	C₂×C₉	#481	1	1	yes	characteristic
237	C₂×C₉	#483	1	1	yes	characteristic
238	D₉	—	2	4	no	—
239	C₂⁴	—	9	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	9	1	no	—
243	C₂²×C₄	—	9	2	no	—
244	C₂²×C₄	—	9	2	no	—
245	C₂²×C₄	—	18	4	no	—
246	C₂²×C₄	—	18	1	no	—
247	C₂²×C₄	—	18	1	no	—
248	C₂²×C₄	—	18	1	no	—
249	C₂²×C₄	—	18	1	no	—
250	C₂²×C₄	—	18	1	no	—
251	C₂²×C₄	—	18	1	no	—
252	C₂²×C₄	—	36	2	no	—
253	C₂²×C₄	—	36	2	no	—
254	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	9	2	no	—
258	C₂×C₈	—	18	1	no	—
259	C₂×C₈	—	18	1	no	—
260	C₂×C₈	C₄×D₉	1	2	yes	complemented
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	9	2	no	—
265	C₄⋊C₄	—	18	4	no	—
266	C₄⋊C₄	—	18	1	no	—
267	C₄⋊C₄	—	18	1	no	—
268	C₄⋊C₄	—	18	1	no	—
269	C₄⋊C₄	#611	1	2	yes	—
270	C₂²⋊C₄	—	36	2	no	—
271	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
272	C₂²×C₃	#894	1	1	yes	characteristic
273	C₂²×C₃	D₃×SD₁₆	1	1	yes	characteristic
274	C₂²×C₃	D₃×SD₁₆	1	1	yes	characteristic
275	C₂²×C₃	#914	1	1	yes	characteristic
276	C₂²×C₃	#915	1	1	yes	characteristic
277	C₂²×C₃	#937	1	1	yes	characteristic
278	C₂×D₃	—	3	4	no	—
279	C₂×D₃	—	6	2	no	—
280	C₂×D₃	—	6	2	no	—
281	C₂×D₃	—	6	2	no	—
282	C₂×D₃	—	6	2	no	—
283	C₂×D₃	—	6	2	no	—
284	C₂×D₃	—	6	2	no	—
285	C₃×C₄	—	4	2	no	—
286	C₃×C₄	—	8	1	no	—
287	C₃×C₄	#893	1	1	yes	characteristic
288	C₃×C₄	#896	1	1	yes	characteristic
289	C₃×C₄	C₄×#67	1	1	yes	characteristic
290	C₃×C₄	#946	1	1	yes	characteristic
291	Dic₃	—	6	2	no	—
292	Dic₃	—	12	2	no	—
293	Dic₃	—	24	1	no	—
294	C₉	#1853	1	1	yes	characteristic, complemented, Sylow
295	C₂³	—	9	2	no	—
296	C₂³	—	9	2	no	—
297	C₂³	—	9	2	no	—
298	C₂³	—	18	1	no	—
299	C₂³	—	18	1	no	—
300	C₂³	—	18	1	no	—
301	C₂³	—	18	1	no	—
302	C₂³	D₄×D₉	1	1	yes	characteristic
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	9	2	no	—
310	C₂×C₄	—	18	1	no	—
311	C₂×C₄	—	18	1	no	—
312	C₂×C₄	—	18	2	no	—
313	C₂×C₄	—	18	2	no	—
314	C₂×C₄	—	18	2	no	—
315	C₂×C₄	—	18	2	no	—
316	C₂×C₄	—	18	2	no	—
317	C₂×C₄	—	36	8	no	—
318	C₂×C₄	—	36	1	no	—
319	C₂×C₄	—	36	1	no	—
320	C₂×C₄	—	36	1	no	—
321	C₂×C₄	—	36	1	no	—
322	C₂×C₄	—	36	1	no	—
323	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
324	C₂×C₄	#3697	1	1	yes	characteristic
325	C₂×C₄	D₄×D₉	1	1	yes	characteristic
326	C₂×C₄	Q₈×D₉	1	1	yes	characteristic
327	C₂×C₄	#3701	1	1	yes	characteristic
328	C₂×C₄	C₂×#611	1	1	yes	characteristic
329	C₈	—	2	2	no	—
330	C₈	—	18	2	no	—
331	C₂×C₃	D₃×#106	1	1	yes	characteristic
332	C₂×C₃	#4861	1	1	yes	characteristic
333	C₂×C₃	#4862	1	1	yes	characteristic
334	C₂×C₃	#4863	1	1	yes	characteristic
335	C₂×C₃	#4989	1	1	yes	characteristic
336	C₂×C₃	#5169	1	1	yes	characteristic
337	C₂×C₃	#5170	1	1	yes	characteristic
338	D₃	—	6	4	no	—
339	C₂²	—	9	4	no	—
340	C₂²	—	18	2	no	—
341	C₂²	—	18	2	no	—
342	C₂²	—	18	2	no	—
343	C₂²	—	18	2	no	—
344	C₂²	—	18	2	no	—
345	C₂²	—	18	2	no	—
346	C₂²	C₄⋊C₄×D₉	1	1	yes	characteristic
347	C₂²	#63456	1	1	yes	characteristic
348	C₂²	SD₁₆×D₉	1	1	yes	characteristic
349	C₂²	SD₁₆×D₉	1	1	yes	characteristic
350	C₂²	#63476	1	1	yes	characteristic
351	C₂²	#63477	1	1	yes	characteristic
352	C₂²	#63499	1	1	yes	characteristic
353	C₄	—	4	2	no	—
354	C₄	—	8	1	no	—
355	C₄	—	18	2	no	—
356	C₄	—	36	2	no	—
357	C₄	—	72	1	no	—
358	C₄	#63455	1	1	yes	characteristic
359	C₄	#63458	1	1	yes	characteristic
360	C₄	C₄×#611	1	1	yes	characteristic
361	C₄	#63508	1	1	yes	characteristic
362	C₃	#70579	1	1	yes	characteristic
363	C₂	—	18	4	no	—
364	C₂	D₉×#106	1	1	yes	characteristic
365	C₂	#10588541	1	1	yes	characteristic
366	C₂	#10588542	1	1	yes	characteristic
367	C₂	#10588543	1	1	yes	characteristic
368	C₂	#10588669	1	1	yes	characteristic
369	C₂	#10588849	1	1	yes	characteristic
370	C₂	#10588850	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×D₉
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₉×#106	C₂
socle	C₂³×C₃	D₃×D₄
prime radical	C₂³×D₉	D₄
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×#106	D₉
hypocenter	C₉	#1853
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

Complements

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

N	H ≅ G/N	type
2. C₂×C₉×#106	C₂	semidirect
5. C₂×#63367	C₂	semidirect
6. #10588229	C₂	semidirect
7. #10588206	C₂	semidirect
23. (C₂×D₉)⋊C₈	C₄	semidirect
50. C₂×C₈×C₉	C₂×C₄	semidirect
63. C₂×C₉⋊C₈	C₂×C₄	semidirect
119. C₂×#106	D₉	semidirect
169. C₂×D₉	#106	semidirect
260. C₂×C₈	C₄×D₉	semidirect
294. C₉	#1853	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₄×D₉	2
C₂×C₉×#106	2
C₂×(C₂×D₉)⋊C₈	2
#10588667	2
C₂×#63367	2
#10588229	2
#10588206	2
#70579	3