Subgroups — #11774775

Order 1152.

575 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10595941	C₂	1	1	yes	maximal, characteristic
2	C₃×#4924	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#10589085	C₂	1	4	yes	maximal, complemented
5	#70501	—	3	1	no	maximal
6	#69738	—	3	1	no	maximal
7	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
8	#64135	—	2	1	no	—
9	#64133	—	2	1	no	—
10	#64133	C₄	1	2	yes	—
11	#64079	—	2	1	no	—
12	#64078	C₄	1	1	yes	characteristic
13	#64078	C₄	1	1	yes	characteristic
14	C₄×#634	—	2	1	no	—
15	C₄×#634	C₂²	1	2	yes	—
16	C₂×C₃×#208	—	2	1	no	—
17	C₂²×C₃×C₃⋊C₈	—	2	1	no	—
18	C₃×C₄⋊(C₃⋊C₈)	—	2	1	no	—
19	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
20	C₃×C₄×C₃⋊C₈	—	2	1	no	—
21	C₃×C₄×C₃⋊C₈	C₂²	1	2	yes	—
22	#5470	—	3	1	no	—
23	#5470	—	3	1	no	—
24	#5470	—	6	1	no	—
25	C₃×#423	—	3	1	no	—
26	C₂×C₃×C₄×C₈	—	3	1	no	—
27	#4924	D₃	1	1	yes	characteristic
28	C₂×C₄×C₃⋊C₈	D₃	1	1	yes	characteristic
29	#4484	—	3	4	no	—
30	#4456	—	3	4	no	—
31	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
33	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
34	C₂×#634	—	2	1	no	—
35	C₂×#634	C₂×C₄	1	2	yes	—
36	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
37	C₃²×C₄²	D₄	1	1	yes	characteristic
38	C₃²×C₄²	D₄	1	1	yes	characteristic
39	C₃²×C₄²	C₂³	1	1	yes	characteristic
40	#3757	—	2	2	no	—
41	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
42	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
43	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
44	C₄⋊(C₃⋊Dic₃)	C₂×C₄	1	2	yes	—
45	#3750	—	2	2	no	—
46	#3750	—	2	2	no	—
47	C₄×C₃⋊Dic₃	—	2	2	no	—
48	C₃×#208	—	4	2	no	—
49	C₂×C₃×C₃⋊C₈	—	2	4	no	—
50	C₂×C₃×C₃⋊C₈	—	2	2	no	—
51	C₂×C₃×C₃⋊C₈	—	2	2	no	—
52	#1493	—	9	1	no	Sylow
53	C₂×C₃×M₁₆	—	6	1	no	—
54	C₂²×C₃×C₈	—	6	1	no	—
55	C₂×C₃×C₄²	—	2	1	no	—
56	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
57	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
58	#926	—	6	2	no	—
59	#926	—	6	1	no	—
60	#926	—	6	1	no	—
61	#924	—	3	2	no	—
62	#924	—	3	2	no	—
63	#924	—	6	2	no	—
64	#924	—	6	2	no	—
65	#924	—	6	1	no	—
66	#924	—	6	1	no	—
67	C₂×#208	—	2	1	no	—
68	C₂²×C₃⋊C₈	—	2	1	no	—
69	#870	—	6	2	no	—
70	#870	—	6	1	no	—
71	#870	—	6	1	no	—
72	#869	—	3	1	no	—
73	#869	—	3	1	no	—
74	#869	—	3	1	no	—
75	#869	—	3	1	no	—
76	#869	—	6	1	no	—
77	#869	—	6	1	no	—
78	C₄×Dic₆	—	3	2	no	—
79	C₄×Dic₆	—	3	2	no	—
80	C₄×Dic₆	—	6	2	no	—
81	C₄×Dic₆	—	6	2	no	—
82	C₄×Dic₆	—	6	1	no	—
83	C₄×Dic₆	—	6	1	no	—
84	C₃×C₄⋊C₈	—	3	2	no	—
85	C₃×C₄⋊C₈	—	6	1	no	—
86	C₃×C₄×C₈	—	3	2	no	—
87	C₃×C₄×C₈	—	6	1	no	—
88	C₄⋊(C₃⋊C₈)	—	2	1	no	—
89	C₄⋊(C₃⋊C₈)	C₂×D₃	1	2	yes	—
90	C₄×C₃⋊C₈	—	2	1	no	—
91	C₄×C₃⋊C₈	C₂×D₃	1	2	yes	—
92	C₂³×C₃²	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₃²×C₄	—	2	2	no	—
94	C₂×C₃²×C₄	—	2	2	no	—
95	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
99	C₂×C₃²×C₄	Q₁₆	1	2	yes	—
100	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
102	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
103	C₂×C₃⋊Dic₃	—	2	2	no	—
104	C₂×C₃⋊Dic₃	—	2	2	no	—
105	#634	—	2	4	no	—
106	#634	—	4	2	no	—
107	C₃×C₃⋊C₈	—	2	4	no	—
108	C₃×C₃⋊C₈	—	4	2	no	—
109	#533	—	9	1	no	—
110	#423	—	3	1	no	—
111	C₂×C₄×C₈	—	3	1	no	—
112	Q₈⋊C₈	—	9	4	no	—
113	C₂²×C₃×C₄	—	2	1	no	—
114	C₂²×C₃×C₄	—	2	1	no	—
115	C₂²×C₃×C₄	—	2	1	no	—
116	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
117	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
118	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
119	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
120	C₂²×C₃×C₄	#67	1	1	yes	characteristic
121	C₂²×C₃×C₄	#67	1	1	yes	characteristic
122	C₂×Dic₆	—	3	2	no	—
123	C₂×Dic₆	—	3	2	no	—
124	C₂×Dic₆	—	6	2	no	—
125	C₂×Dic₆	—	6	2	no	—
126	C₂×Dic₆	—	6	1	no	—
127	C₂×Dic₆	—	6	1	no	—
128	C₃×M₁₆	—	12	2	no	—
129	C₂×C₃×C₈	—	6	4	no	—
130	C₂×C₃×C₈	—	6	2	no	—
131	C₂×C₃×C₈	—	6	2	no	—
132	C₃×C₄²	—	2	1	no	—
133	C₃×C₄²	—	2	1	no	—
134	C₃×C₄²	—	2	1	no	—
135	C₃×C₄²	—	2	1	no	—
136	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
137	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
138	C₃×C₄²	D₁₂	1	1	yes	characteristic
139	C₃×C₄²	D₁₂	1	1	yes	characteristic
140	C₃×C₄²	#67	1	1	yes	characteristic
141	C₃×C₄²	#67	1	1	yes	characteristic
142	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
143	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
144	#217	—	6	4	no	—
145	#217	—	6	2	no	—
146	#217	—	6	2	no	—
147	C₄⋊Dic₃	—	3	2	no	—
148	C₄⋊Dic₃	—	3	2	no	—
149	C₄⋊Dic₃	—	6	2	no	—
150	C₄⋊Dic₃	—	6	2	no	—
151	C₄⋊Dic₃	—	6	2	no	—
152	C₄⋊Dic₃	—	6	2	no	—
153	C₄⋊Dic₃	—	6	1	no	—
154	C₄⋊Dic₃	—	6	1	no	—
155	C₄⋊Dic₃	—	6	1	no	—
156	C₄⋊Dic₃	—	6	1	no	—
157	C₄⋊Dic₃	—	6	1	no	—
158	C₄⋊Dic₃	—	6	1	no	—
159	Dic₃⋊C₄	—	6	4	no	—
160	Dic₃⋊C₄	—	6	4	no	—
161	Dic₃⋊C₄	—	6	2	no	—
162	Dic₃⋊C₄	—	6	2	no	—
163	Dic₃⋊C₄	—	6	2	no	—
164	Dic₃⋊C₄	—	6	2	no	—
165	C₄×Dic₃	—	6	4	no	—
166	C₄×Dic₃	—	6	2	no	—
167	C₄×Dic₃	—	6	2	no	—
168	#208	—	4	2	no	—
169	C₂×C₃⋊C₈	—	2	4	no	—
170	C₂×C₃⋊C₈	—	2	2	no	—
171	C₂×C₃⋊C₈	—	2	2	no	—
172	C₂²×C₃²	—	2	1	no	—
173	C₂²×C₃²	—	2	1	no	—
174	C₂²×C₃²	#103	1	2	yes	—
175	C₂²×C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic
176	C₃²×C₄	—	2	1	no	—
177	C₃²×C₄	—	2	1	no	—
178	C₃²×C₄	#103	1	2	yes	—
179	C₃²×C₄	#104	1	2	yes	—
180	C₃²×C₄	#131	1	2	yes	—
181	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
182	C₃²×C₄	C₂×Q₁₆	1	1	yes	characteristic
183	C₃⋊Dic₃	—	4	2	no	—
184	C₃⋊Dic₃	—	4	2	no	—
185	C₂×M₁₆	—	6	1	no	—
186	C₂²×C₈	—	6	1	no	—
187	C₄⋊Q₈	—	9	1	no	—
188	C₄⋊Q₈	—	9	1	no	—
189	#125	—	18	1	no	—
190	C₂²⋊Q₈	—	9	2	no	—
191	C₂²⋊Q₈	—	18	1	no	—
192	C₄×Q₈	—	9	2	no	—
193	C₄×Q₈	—	18	1	no	—
194	#117	—	18	1	no	—
195	C₂×C₄²	D₃²	1	1	yes	characteristic
196	C₄⋊C₈	—	3	2	no	—
197	C₄⋊C₈	—	6	1	no	—
198	C₄×C₈	—	3	2	no	—
199	C₄×C₈	—	6	1	no	—
200	C₂³×C₃	—	2	1	no	—
201	C₂³×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
202	C₂³×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	2	2	no	—
205	C₂×C₃×C₄	—	2	2	no	—
206	C₂×C₃×C₄	—	2	2	no	—
207	C₂×C₃×C₄	—	2	2	no	—
208	C₂×C₃×C₄	—	2	2	no	—
209	C₂×C₃×C₄	—	2	1	no	—
210	C₂×C₃×C₄	—	2	1	no	—
211	C₂×C₃×C₄	—	2	1	no	—
212	C₂×C₃×C₄	—	2	1	no	—
213	C₂×C₃×C₄	—	2	1	no	—
214	C₂×C₃×C₄	—	2	1	no	—
215	C₂×C₃×C₄	—	4	2	no	—
216	C₂×C₃×C₄	—	4	2	no	—
217	C₂×C₃×C₄	#204	1	2	yes	—
218	C₂×C₃×C₄	Dic₁₂	1	2	yes	—
219	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
220	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
221	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
222	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
223	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
224	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
225	C₂×C₃×C₄	#214	1	2	yes	—
226	C₂×C₃×C₄	#216	1	2	yes	—
227	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
233	C₂×Dic₃	—	6	4	no	—
234	C₂×Dic₃	—	6	4	no	—
235	C₂×Dic₃	—	6	2	no	—
236	C₂×Dic₃	—	6	2	no	—
237	C₂×Dic₃	—	6	2	no	—
238	C₂×Dic₃	—	6	2	no	—
239	Dic₆	—	6	4	no	—
240	Dic₆	—	6	4	no	—
241	Dic₆	—	12	4	no	—
242	Dic₆	—	12	4	no	—
243	Dic₆	—	12	2	no	—
244	Dic₆	—	12	2	no	—
245	C₃×C₈	—	6	4	no	—
246	C₃×C₈	—	12	2	no	—
247	C₃⋊C₈	—	2	4	no	—
248	C₃⋊C₈	—	4	2	no	—
249	C₂×C₃²	—	2	2	no	—
250	C₂×C₃²	#408	1	1	yes	characteristic
251	C₂×C₃²	C₂×#103	1	1	yes	characteristic
252	C₂×C₃²	C₂×#104	1	1	yes	characteristic
253	C₂×Q₈	—	9	2	no	—
254	C₂×Q₈	—	18	1	no	—
255	C₂²×C₄	#624	1	1	yes	characteristic
256	C₂²×C₄	#626	1	1	yes	characteristic
257	C₂²×C₄	#626	1	1	yes	characteristic
258	M₁₆	—	12	2	no	—
259	C₂×C₈	—	6	4	no	—
260	C₂×C₈	—	6	2	no	—
261	C₂×C₈	—	6	2	no	—
262	C₄⋊C₄	—	9	2	no	—
263	C₄⋊C₄	—	18	1	no	—
264	C₄⋊C₄	—	18	1	no	—
265	C₄⋊C₄	—	18	2	no	—
266	C₄⋊C₄	—	18	1	no	—
267	C₄⋊C₄	—	18	2	no	—
268	C₂²⋊C₄	—	18	2	no	—
269	C₄²	—	18	2	no	—
270	C₄²	#624	1	1	yes	characteristic
271	C₄²	#626	1	1	yes	characteristic
272	C₄²	#626	1	1	yes	characteristic
273	C₄²	C₂×D₃²	1	1	yes	characteristic
274	C₂²×C₃	—	2	1	no	—
275	C₂²×C₃	—	2	1	no	—
276	C₂²×C₃	—	2	1	no	—
277	C₂²×C₃	—	2	1	no	—
278	C₂²×C₃	—	2	2	no	—
279	C₂²×C₃	—	2	1	no	—
280	C₂²×C₃	—	4	1	no	—
281	C₂²×C₃	—	4	1	no	—
282	C₂²×C₃	#810	1	2	yes	—
283	C₂²×C₃	#816	1	2	yes	—
284	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
285	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
286	C₃×C₄	—	2	1	no	—
287	C₃×C₄	—	2	1	no	—
288	C₃×C₄	—	2	1	no	—
289	C₃×C₄	—	2	1	no	—
290	C₃×C₄	—	2	2	no	—
291	C₃×C₄	—	2	2	no	—
292	C₃×C₄	—	2	2	no	—
293	C₃×C₄	—	2	1	no	—
294	C₃×C₄	—	2	1	no	—
295	C₃×C₄	—	4	1	no	—
296	C₃×C₄	—	4	1	no	—
297	C₃×C₄	#805	1	2	yes	—
298	C₃×C₄	#810	1	2	yes	—
299	C₃×C₄	#816	1	2	yes	—
300	C₃×C₄	#825	1	2	yes	—
301	C₃×C₄	#901	1	2	yes	—
302	C₃×C₄	C₂×#204	1	1	yes	characteristic
303	C₃×C₄	C₂×Dic₁₂	1	1	yes	characteristic
304	C₃×C₄	#907	1	2	yes	—
305	C₃×C₄	C₂×#214	1	1	yes	characteristic
306	C₃×C₄	C₂×#216	1	1	yes	characteristic
307	Dic₃	—	12	4	no	—
308	Dic₃	—	12	4	no	—
309	Dic₃	—	12	2	no	—
310	Dic₃	—	12	2	no	—
311	Dic₃	—	12	2	no	—
312	Dic₃	—	12	2	no	—
313	C₃²	#1493	1	1	yes	characteristic, complemented, Sylow
314	C₂³	#3722	1	1	yes	characteristic
315	Q₈	—	18	4	no	—
316	Q₈	—	36	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	18	2	no	—
320	C₂×C₄	—	18	2	no	—
321	C₂×C₄	#3716	1	2	yes	—
322	C₂×C₄	#3719	1	2	yes	—
323	C₂×C₄	#3722	1	1	yes	characteristic
324	C₂×C₄	#3722	1	1	yes	characteristic
325	C₂×C₄	#3722	1	1	yes	characteristic
326	C₂×C₄	C₂×#624	1	1	yes	characteristic
327	C₂×C₄	C₂×#626	1	1	yes	characteristic
328	C₂×C₄	C₂×#626	1	1	yes	characteristic
329	C₈	—	6	4	no	—
330	C₈	—	12	2	no	—
331	C₂×C₃	—	2	2	no	—
332	C₂×C₃	—	2	2	no	—
333	C₂×C₃	—	2	1	no	—
334	C₂×C₃	—	2	1	no	—
335	C₂×C₃	—	2	1	no	—
336	C₂×C₃	—	4	2	no	—
337	C₂×C₃	C₂×#805	1	1	yes	characteristic
338	C₂×C₃	C₂×#810	1	1	yes	characteristic
339	C₂×C₃	C₂×#816	1	1	yes	characteristic
340	C₂×C₃	#5110	1	1	yes	characteristic
341	C₂×C₃	#5129	1	1	yes	characteristic
342	C₂×C₃	C₂×#825	1	1	yes	characteristic
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	#63567	1	2	yes	—
346	C₂²	C₂×#3722	1	1	yes	characteristic
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	36	2	no	—
350	C₄	—	36	2	no	—
351	C₄	#63567	1	2	yes	—
352	C₄	#63570	1	2	yes	—
353	C₄	#63817	1	2	yes	—
354	C₄	C₂×#3716	1	1	yes	characteristic
355	C₄	C₂×#3719	1	1	yes	characteristic
356	C₃	—	2	1	no	—
357	C₃	#69738	1	1	yes	characteristic, complemented
358	C₃	#70501	1	1	yes	characteristic, complemented
359	C₂	—	2	2	no	—
360	C₂	#10591073	1	1	yes	characteristic
361	C₂	C₂×#63567	1	1	yes	characteristic
362	C₂	C₂×#63570	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#626
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₄²	C₂×D₃²
Fitting	C₂×C₃²×C₄²	C₂²
socle	C₂²×C₃²	C₂×C₂²⋊C₄
prime radical	C₂³×C₃²	C₂²⋊C₄
cosocle	C₃²×C₄²	C₂³
hypercenter	C₂×C₄²	D₃²
hypocenter	C₃²	#1493
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
4. #10589085	C₂	semidirect
313. C₃²	#1493	semidirect
357. C₃	#69738	semidirect
358. C₃	#70501	semidirect

Maximal subgroups

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

subgroup	index
#10595941	2
C₃×#4924	2
C₂×C₃×C₄×C₃⋊C₈	2
#10589085	2
#70501	3
#69738	3