Subgroups — #11790076

Order 1152.

593 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#803	C₂	1	1	yes	maximal, characteristic
4	C₂×#63572	C₂	1	2	yes	maximal
5	#10589156	C₂	1	2	yes	maximal
6	#70469	—	3	1	no	maximal
7	#69836	—	3	1	no	maximal
8	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₃⋊Dic₃)	C₄	1	2	yes	—
10	C₂×C₄×C₃⋊Dic₃	C₂²	1	2	yes	—
11	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
12	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	#63658	C₄	1	1	yes	characteristic
15	#63658	C₄	1	1	yes	characteristic
16	C₃×C₄⋊(C₃⋊C₈)	—	2	2	no	—
17	C₃×#803	—	2	2	no	—
18	#63572	—	2	4	no	—
19	C₂×C₃×C₄⋊C₈	—	3	1	no	—
20	C₂×C₃×#97	—	3	1	no	—
21	C₂×Dic₃⋊C₈	—	3	2	no	—
22	C₂×Dic₃⋊C₈	—	3	2	no	—
23	C₄×C₄⋊Dic₃	—	3	1	no	—
24	C₄×C₄⋊Dic₃	—	3	1	no	—
25	C₄×C₄⋊Dic₃	—	6	1	no	—
26	C₂×C₄⋊(C₃⋊C₈)	D₃	1	1	yes	characteristic
27	C₂×#803	D₃	1	1	yes	characteristic
28	#4550	—	3	2	no	—
29	#4550	—	3	2	no	—
30	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
33	C₂²×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
34	C₂²×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
35	C₃²×C₄²	D₄	1	2	yes	—
36	C₃²×C₄²	Q₈	1	2	yes	—
37	C₄⋊(C₃⋊Dic₃)	—	2	4	no	—
38	C₄×C₃⋊Dic₃	—	2	2	no	—
39	C₄×C₃⋊Dic₃	—	2	2	no	—
40	C₂×C₃×C₃⋊C₈	—	2	1	no	—
41	C₂×C₃×C₃⋊C₈	—	2	1	no	—
42	C₂×C₃×C₃⋊C₈	—	2	1	no	—
43	C₂×C₃×C₃⋊C₈	—	2	1	no	—
44	C₂×C₃×C₃⋊C₈	—	2	2	no	—
45	C₂×C₃×C₃⋊C₈	—	2	2	no	—
46	C₂×C₃×C₃⋊C₈	—	2	2	no	—
47	C₂×C₃×C₃⋊C₈	D₄	1	2	yes	—
48	C₂×C₃×C₃⋊C₈	Q₈	1	2	yes	—
49	#1940	—	9	1	no	Sylow
50	C₂²×C₃×C₈	—	3	1	no	—
51	C₂²×C₃×C₈	—	3	1	no	—
52	C₂²×C₃×C₈	—	3	2	no	—
53	C₂×C₃×C₄²	—	2	1	no	—
54	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
55	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
56	C₂×C₄⋊Dic₃	—	3	2	no	—
57	C₂×C₄⋊Dic₃	—	3	2	no	—
58	C₂×C₄⋊Dic₃	—	6	2	no	—
59	C₂×C₄×Dic₃	—	3	2	no	—
60	C₂×C₄×Dic₃	—	3	2	no	—
61	C₂×C₄×Dic₃	—	6	2	no	—
62	C₂²×C₃⋊C₈	C₂×D₃	1	2	yes	—
63	C₂²×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
64	C₂²×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
65	C₃×C₄⋊C₈	—	6	2	no	—
66	C₃×#97	—	6	2	no	—
67	C₃⋊#95	—	3	1	no	—
68	C₃⋊#95	—	3	1	no	—
69	C₃⋊#95	—	3	1	no	—
70	C₃⋊#95	—	3	1	no	—
71	C₃⋊#95	—	6	1	no	—
72	C₃⋊#95	—	6	1	no	—
73	Dic₃⋊C₈	—	6	4	no	—
74	Dic₃⋊C₈	—	6	4	no	—
75	C₄⋊(C₃⋊C₈)	—	2	2	no	—
76	#803	—	2	2	no	—
77	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃²×C₄	—	2	2	no	—
79	C₂×C₃²×C₄	—	2	2	no	—
80	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
81	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
82	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₃²×C₄	#41	1	1	yes	characteristic
87	C₂×C₃²×C₄	#41	1	1	yes	characteristic
88	C₂×C₃⋊Dic₃	—	2	2	no	—
89	C₂×C₃⋊Dic₃	—	2	2	no	—
90	C₂×C₃⋊Dic₃	—	2	2	no	—
91	C₂×C₃⋊Dic₃	—	2	2	no	—
92	C₂×C₃⋊Dic₃	—	2	2	no	—
93	C₂×C₃⋊Dic₃	M₁₆	1	4	yes	—
94	C₃×C₃⋊C₈	—	2	2	no	—
95	C₃×C₃⋊C₈	—	2	2	no	—
96	C₃×C₃⋊C₈	—	4	2	no	—
97	C₂×C₄⋊C₈	—	3	1	no	—
98	C₂×C₄⋊C₈	—	9	2	no	—
99	C₂×#97	—	3	1	no	—
100	C₄×C₄⋊C₄	—	9	1	no	—
101	#336	—	9	2	no	—
102	C₂²×C₃×C₄	—	2	1	no	—
103	C₂²×C₃×C₄	—	2	1	no	—
104	C₂²×C₃×C₄	—	2	1	no	—
105	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
107	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
111	C₂²×Dic₃	—	3	2	no	—
112	C₂²×Dic₃	—	3	2	no	—
113	C₂²×Dic₃	—	3	2	no	—
114	C₂²×Dic₃	—	3	2	no	—
115	C₂²×Dic₃	—	6	2	no	—
116	C₂²×Dic₃	—	6	2	no	—
117	C₂×C₃×C₈	—	3	2	no	—
118	C₂×C₃×C₈	—	3	2	no	—
119	C₂×C₃×C₈	—	6	1	no	—
120	C₂×C₃×C₈	—	6	1	no	—
121	C₂×C₃×C₈	—	6	1	no	—
122	C₂×C₃×C₈	—	6	1	no	—
123	C₂×C₃×C₈	—	6	2	no	—
124	C₂×C₃×C₈	—	6	2	no	—
125	C₂×C₃×C₈	—	6	2	no	—
126	C₃×C₄²	—	2	2	no	—
127	C₃×C₄²	—	2	2	no	—
128	C₃×C₄²	Dic₆	1	2	yes	—
129	C₃×C₄²	Dic₆	1	2	yes	—
130	C₃×C₄²	#67	1	2	yes	—
131	C₃×C₄²	#67	1	2	yes	—
132	C₄⋊Dic₃	—	6	4	no	—
133	C₄⋊Dic₃	—	6	4	no	—
134	C₄⋊Dic₃	—	6	8	no	—
135	C₄×Dic₃	—	6	2	no	—
136	C₄×Dic₃	—	6	2	no	—
137	C₄×Dic₃	—	6	2	no	—
138	C₄×Dic₃	—	6	2	no	—
139	C₄×Dic₃	—	12	2	no	—
140	C₄×Dic₃	—	12	2	no	—
141	C₂×C₃⋊C₈	—	2	1	no	—
142	C₂×C₃⋊C₈	—	2	1	no	—
143	C₂×C₃⋊C₈	—	2	1	no	—
144	C₂×C₃⋊C₈	—	2	1	no	—
145	C₂×C₃⋊C₈	—	2	2	no	—
146	C₂×C₃⋊C₈	—	2	2	no	—
147	C₂×C₃⋊C₈	—	2	2	no	—
148	C₂×C₃⋊C₈	Dic₆	1	2	yes	—
149	C₂×C₃⋊C₈	D₁₂	1	2	yes	—
150	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
152	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
153	C₂²×C₃²	C₂×M₁₆	1	2	yes	—
154	C₂²×C₃²	#131	1	2	yes	—
155	C₃²×C₄	—	2	2	no	—
156	C₃²×C₄	—	2	2	no	—
157	C₃²×C₄	#121	1	1	yes	characteristic
158	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
159	C₃²×C₄	#125	1	1	yes	characteristic
160	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
161	C₃⋊Dic₃	—	2	4	no	—
162	C₃⋊Dic₃	—	4	2	no	—
163	C₂²×C₈	—	3	1	no	—
164	C₂²×C₈	—	3	1	no	—
165	C₂²×C₈	—	3	2	no	—
166	C₂×C₄⋊C₄	—	9	2	no	—
167	C₂×C₄²	—	9	2	no	—
168	C₂×C₄²	D₃²	1	1	yes	characteristic
169	C₄⋊C₈	—	6	2	no	—
170	C₄⋊C₈	—	18	4	no	—
171	#97	—	6	2	no	—
172	#95	—	9	1	no	—
173	#95	—	9	1	no	—
174	C₂³×C₃	—	2	1	no	—
175	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
176	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
177	C₂×C₃×C₄	—	2	4	no	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	2	2	no	—
181	C₂×C₃×C₄	—	2	2	no	—
182	C₂×C₃×C₄	—	2	2	no	—
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	—	2	1	no	—
185	C₂×C₃×C₄	—	2	1	no	—
186	C₂×C₃×C₄	—	2	1	no	—
187	C₂×C₃×C₄	—	2	1	no	—
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	2	1	no	—
190	C₂×C₃×C₄	—	4	2	no	—
191	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
192	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
193	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
194	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
195	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
199	C₂×C₃×C₄	#235	1	1	yes	characteristic
200	C₂×C₃×C₄	#235	1	1	yes	characteristic
201	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
202	C₂×C₃×C₄	#237	1	1	yes	characteristic
203	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
204	C₂×C₃×C₄	#239	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
207	C₂×Dic₃	—	3	4	no	—
208	C₂×Dic₃	—	3	4	no	—
209	C₂×Dic₃	—	6	4	no	—
210	C₂×Dic₃	—	6	2	no	—
211	C₂×Dic₃	—	6	2	no	—
212	C₂×Dic₃	—	6	2	no	—
213	C₂×Dic₃	—	6	2	no	—
214	C₂×Dic₃	—	6	4	no	—
215	C₂×Dic₃	—	6	2	no	—
216	C₂×Dic₃	—	6	2	no	—
217	C₂×Dic₃	—	6	2	no	—
218	C₂×Dic₃	—	6	2	no	—
219	C₂×Dic₃	—	6	2	no	—
220	C₂×Dic₃	—	6	2	no	—
221	C₂×Dic₃	—	12	2	no	—
222	C₂×Dic₃	—	12	2	no	—
223	C₂×Dic₃	—	12	2	no	—
224	C₂×Dic₃	—	12	2	no	—
225	C₃×C₈	—	6	2	no	—
226	C₃×C₈	—	6	2	no	—
227	C₃×C₈	—	12	2	no	—
228	C₃⋊C₈	—	2	2	no	—
229	C₃⋊C₈	—	2	2	no	—
230	C₃⋊C₈	—	4	2	no	—
231	C₂×C₃²	#389	1	1	yes	characteristic
232	C₂×C₃²	#423	1	1	yes	characteristic
233	C₂×C₃²	#424	1	1	yes	characteristic
234	C₂×C₃²	#435	1	2	yes	—
235	C₂×C₃²	#446	1	2	yes	—
236	C₂²×C₄	—	9	2	no	—
237	C₂²×C₄	—	9	2	no	—
238	C₂²×C₄	#624	1	1	yes	characteristic
239	C₂²×C₄	#624	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
241	C₂×C₈	—	3	2	no	—
242	C₂×C₈	—	3	2	no	—
243	C₂×C₈	—	6	1	no	—
244	C₂×C₈	—	6	1	no	—
245	C₂×C₈	—	6	1	no	—
246	C₂×C₈	—	6	1	no	—
247	C₂×C₈	—	6	2	no	—
248	C₂×C₈	—	6	2	no	—
249	C₂×C₈	—	6	2	no	—
250	C₄⋊C₄	—	18	4	no	—
251	C₄²	—	18	2	no	—
252	C₄²	—	18	2	no	—
253	C₄²	#625	1	2	yes	—
254	C₄²	#627	1	2	yes	—
255	C₂²×C₃	—	2	2	no	—
256	C₂²×C₃	—	2	2	no	—
257	C₂²×C₃	—	2	1	no	—
258	C₂²×C₃	—	2	1	no	—
259	C₂²×C₃	—	2	1	no	—
260	C₂²×C₃	C₄×Dic₆	1	1	yes	characteristic
261	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
262	C₂²×C₃	#887	1	1	yes	characteristic
263	C₂²×C₃	#893	1	1	yes	characteristic
264	C₂²×C₃	#901	1	2	yes	—
265	C₂²×C₃	D₃×M₁₆	1	2	yes	—
266	C₂²×C₃	D₃×M₁₆	1	2	yes	—
267	C₂²×C₃	#907	1	2	yes	—
268	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
269	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
270	C₃×C₄	—	2	4	no	—
271	C₃×C₄	—	2	2	no	—
272	C₃×C₄	—	2	2	no	—
273	C₃×C₄	—	2	2	no	—
274	C₃×C₄	—	2	2	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	4	2	no	—
280	C₃×C₄	#869	1	1	yes	characteristic
281	C₃×C₄	#870	1	1	yes	characteristic
282	C₃×C₄	#888	1	1	yes	characteristic
283	C₃×C₄	#890	1	1	yes	characteristic
284	C₃×C₄	#924	1	1	yes	characteristic
285	C₃×C₄	#930	1	1	yes	characteristic
286	C₃×C₄	#938	1	1	yes	characteristic
287	C₃×C₄	#946	1	1	yes	characteristic
288	Dic₃	—	6	4	no	—
289	Dic₃	—	6	4	no	—
290	Dic₃	—	12	4	no	—
291	Dic₃	—	12	4	no	—
292	Dic₃	—	12	2	no	—
293	Dic₃	—	12	2	no	—
294	C₃²	#1940	1	1	yes	characteristic, complemented, Sylow
295	C₂³	C₂×#624	1	1	yes	characteristic
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	9	4	no	—
299	C₂×C₄	—	18	2	no	—
300	C₂×C₄	—	18	2	no	—
301	C₂×C₄	—	18	2	no	—
302	C₂×C₄	—	18	2	no	—
303	C₂×C₄	—	18	2	no	—
304	C₂×C₄	#3724	1	2	yes	—
305	C₂×C₄	#3724	1	2	yes	—
306	C₂×C₄	D₃×Dic₆	1	1	yes	characteristic
307	C₂×C₄	#3795	1	1	yes	characteristic
308	C₂×C₄	#3799	1	1	yes	characteristic
309	C₂×C₄	D₃×D₁₂	1	1	yes	characteristic
310	C₂×C₄	C₂×#625	1	1	yes	characteristic
311	C₂×C₄	C₂×#627	1	1	yes	characteristic
312	C₈	—	6	2	no	—
313	C₈	—	6	2	no	—
314	C₈	—	12	2	no	—
315	C₂×C₃	—	2	2	no	—
316	C₂×C₃	—	2	2	no	—
317	C₂×C₃	—	2	1	no	—
318	C₂×C₃	—	2	1	no	—
319	C₂×C₃	—	2	1	no	—
320	C₂×C₃	#4701	1	2	yes	—
321	C₂×C₃	#4706	1	2	yes	—
322	C₂×C₃	#4828	1	2	yes	—
323	C₂×C₃	#4836	1	2	yes	—
324	C₂×C₃	#4928	1	1	yes	characteristic
325	C₂×C₃	#4972	1	1	yes	characteristic
326	C₂×C₃	#5102	1	1	yes	characteristic
327	C₂×C₃	#5118	1	1	yes	characteristic
328	C₂×C₃	#5118	1	1	yes	characteristic
329	C₂×C₃	#5119	1	1	yes	characteristic
330	C₂²	#63817	1	2	yes	—
331	C₂²	#63818	1	2	yes	—
332	C₂²	#63844	1	1	yes	characteristic
333	C₂²	#63894	1	1	yes	characteristic
334	C₂²	C₂×#3724	1	1	yes	characteristic
335	C₄	—	2	2	no	—
336	C₄	—	2	2	no	—
337	C₄	—	18	4	no	—
338	C₄	—	36	2	no	—
339	C₄	#63856	1	1	yes	characteristic
340	C₄	#63869	1	1	yes	characteristic
341	C₄	#63908	1	1	yes	characteristic
342	C₄	#63918	1	1	yes	characteristic
343	C₃	—	2	1	no	—
344	C₃	#69836	1	1	yes	characteristic, complemented
345	C₃	#70469	1	1	yes	characteristic, complemented
346	C₂	#10590255	1	2	yes	—
347	C₂	#10590328	1	2	yes	—
348	C₂	#10591119	1	1	yes	characteristic
349	C₂	#10591120	1	1	yes	characteristic
350	C₂	#10591212	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₃²
commutator, socle, prime radical	C₂³×C₃²	C₂²×C₄
Fitting	C₂×C₃²×C₄²	C₂²
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×C₄²	D₃²
hypocenter	C₃²	#1940
LCS	G ▷ C₂³×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
294. C₃²	#1940	semidirect
344. C₃	#69836	semidirect
345. C₃	#70469	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₃⋊Dic₃)	2
C₂×C₃×C₄⋊(C₃⋊C₈)	2
C₂×C₃×#803	2
C₂×#63572	2
#10589156	2
#70469	3
#69836	3