Subgroups — #10948302

Order 768.

590 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×#131	C₂	1	1	yes	maximal, characteristic
2	#72004	C₂	1	2	yes	maximal
3	#69883	C₂	1	4	yes	maximal, complemented
4	#69881	C₂	1	2	yes	maximal, complemented
5	#69745	C₂	1	2	yes	maximal, complemented
6	#69744	C₂	1	2	yes	maximal, complemented
7	#69743	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
9	#33423	—	3	1	no	maximal, Sylow
10	C₂×C₃×#131	C₄	1	1	yes	characteristic
11	C₂×C₃×#131	C₂²	1	1	yes	characteristic
12	C₃×C₄×#41	C₄	1	1	yes	characteristic
13	C₃×#405	C₄	1	1	yes	characteristic
14	C₃×#405	C₄	1	2	yes	—
15	C₃×#405	C₂²	1	1	yes	characteristic
16	C₃×#405	C₂²	1	2	yes	—
17	C₃×C₄×M₁₆	C₄	1	2	yes	—
18	C₃×C₄×M₁₆	C₄	1	1	yes	characteristic
19	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
21	C₂×#812	C₂²	1	2	yes	—
22	C₂²×C₃⋊C₁₆	C₂²	1	2	yes	—
23	#4549	C₂²	1	4	yes	—
24	#4549	C₂²	1	4	yes	—
25	C₄⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
26	C₄⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
27	C₄⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
28	C₄⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
29	C₄⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
30	#4461	C₂²	1	4	yes	—
31	C₄×C₃⋊C₁₆	C₂²	1	4	yes	—
32	C₄×#131	D₃	1	1	yes	characteristic
33	#2164	—	3	4	no	—
34	#2163	—	3	2	no	—
35	#2162	—	3	2	no	—
36	#2150	—	3	1	no	—
37	C₂×C₄⋊C₁₆	—	3	1	no	—
38	#2113	—	3	2	no	—
39	#2108	—	3	2	no	—
40	C₂×C₃×#41	C₂×C₄	1	1	yes	characteristic
41	C₃×#131	—	2	4	no	—
42	C₃×#131	—	2	2	no	—
43	C₃×#131	—	2	2	no	—
44	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
45	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
47	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
50	C₂²×C₃×C₈	C₂³	1	2	yes	—
51	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
52	C₃×C₄×Q₈	C₈	1	1	yes	characteristic
53	C₃×C₄×Q₈	C₈	1	1	yes	characteristic
54	C₃×C₄×D₄	C₈	1	2	yes	—
55	C₃×C₄×D₄	C₈	1	1	yes	characteristic
56	C₃×C₄×D₄	C₈	1	2	yes	—
57	C₃×C₄×D₄	C₈	1	1	yes	characteristic
58	C₃×#117	C₂×C₄	1	2	yes	—
59	C₃×#117	C₂×C₄	1	1	yes	characteristic
60	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
61	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
62	C₃×#97	C₂×C₄	1	2	yes	—
63	C₃×#97	C₂×C₄	1	2	yes	—
64	C₃×#97	C₂×C₄	1	2	yes	—
65	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
66	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
67	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
68	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
69	C₃×C₄×C₈	C₂³	1	2	yes	—
70	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
71	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
72	#812	—	2	4	no	—
73	C₂×C₃⋊C₁₆	—	2	4	no	—
74	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
75	C₂×C₃⋊C₁₆	C₂³	1	4	yes	—
76	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
77	C₂×#131	Dic₃	1	1	yes	characteristic
78	C₂×#131	C₂×D₃	1	1	yes	characteristic
79	C₄×#41	Dic₃	1	1	yes	characteristic
80	C₂×M₃₂	—	3	2	no	—
81	C₂²×C₁₆	—	3	2	no	—
82	#405	Dic₃	1	1	yes	characteristic
83	#405	Dic₃	1	2	yes	—
84	#405	C₂×D₃	1	1	yes	characteristic
85	#405	C₂×D₃	1	2	yes	—
86	C₄×M₁₆	Dic₃	1	2	yes	—
87	C₄×M₁₆	Dic₃	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂×D₃	1	2	yes	—
90	C₄⋊C₁₆	—	3	1	no	—
91	C₄⋊C₁₆	—	3	1	no	—
92	C₄⋊C₁₆	—	3	2	no	—
93	C₄⋊C₁₆	—	3	2	no	—
94	C₄⋊C₁₆	—	3	2	no	—
95	C₂²⋊C₁₆	—	3	4	no	—
96	C₂²⋊C₁₆	—	3	4	no	—
97	#346	—	3	4	no	—
98	C₄×C₁₆	—	3	4	no	—
99	C₃×#41	—	2	2	no	—
100	C₃×#41	—	2	2	no	—
101	C₂×C₃×Q₈	C₂×C₈	1	1	yes	characteristic
102	C₂×C₃×D₄	C₂×C₈	1	2	yes	—
103	C₂×C₃×D₄	C₂×C₈	1	1	yes	characteristic
104	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
105	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
107	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
109	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
110	C₃×M₁₆	—	2	2	no	—
111	C₃×M₁₆	—	2	4	no	—
112	C₃×M₁₆	—	2	4	no	—
113	C₃×M₁₆	M₁₆	1	4	yes	—
114	C₂×C₃×C₈	—	2	2	no	—
115	C₂×C₃×C₈	—	2	4	no	—
116	C₂×C₃×C₈	—	2	2	no	—
117	C₂×C₃×C₈	—	2	2	no	—
118	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
120	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
123	C₂×C₃×C₈	#41	1	2	yes	—
124	C₂×C₃×C₈	#41	1	2	yes	—
125	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
126	C₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
127	C₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
128	C₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
129	C₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
130	C₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
131	C₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
132	C₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
133	C₃×C₄²	C₂×C₈	1	2	yes	—
134	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
135	C₃×C₄²	C₂×C₈	1	2	yes	—
136	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
137	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
138	C₃×C₄²	C₂²×C₄	1	2	yes	—
139	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
140	C₃⋊C₁₆	—	2	2	no	—
141	C₃⋊C₁₆	—	2	2	no	—
142	C₃⋊C₁₆	—	2	4	no	—
143	C₂×#41	C₂×Dic₃	1	1	yes	characteristic
144	#131	—	2	4	no	—
145	#131	—	2	2	no	—
146	#131	—	2	2	no	—
147	C₂×M₁₆	C₂×Dic₃	1	2	yes	—
148	C₂×M₁₆	C₂×Dic₃	1	1	yes	characteristic
149	C₂×M₁₆	C₂×Dic₃	1	2	yes	—
150	C₂×M₁₆	C₂×Dic₃	1	1	yes	characteristic
151	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
152	C₂²×C₈	C₂×Dic₃	1	2	yes	—
153	C₂²×C₈	C₂²×D₃	1	2	yes	—
154	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
155	C₄×Q₈	C₃⋊C₈	1	1	yes	characteristic
156	C₄×Q₈	C₃⋊C₈	1	1	yes	characteristic
157	C₄×D₄	C₃⋊C₈	1	2	yes	—
158	C₄×D₄	C₃⋊C₈	1	1	yes	characteristic
159	C₄×D₄	C₃⋊C₈	1	2	yes	—
160	C₄×D₄	C₃⋊C₈	1	1	yes	characteristic
161	#117	C₂×Dic₃	1	2	yes	—
162	#117	C₂×Dic₃	1	1	yes	characteristic
163	C₂×C₄²	C₂×Dic₃	1	2	yes	—
164	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
165	M₃₂	—	6	4	no	—
166	C₂×C₁₆	—	3	2	no	—
167	C₂×C₁₆	—	3	4	no	—
168	C₂×C₁₆	—	3	2	no	—
169	C₂×C₁₆	—	6	4	no	—
170	#97	C₂×Dic₃	1	2	yes	—
171	#97	C₂×Dic₃	1	2	yes	—
172	#97	C₂×Dic₃	1	2	yes	—
173	C₄×C₈	C₂×Dic₃	1	2	yes	—
174	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
175	C₄×C₈	C₂×Dic₃	1	2	yes	—
176	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
177	C₄×C₈	C₂²×D₃	1	2	yes	—
178	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
179	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
180	C₂³×C₃	C₂²×C₈	1	2	yes	—
181	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
182	C₃×Q₈	—	2	2	no	—
183	C₃×D₄	—	2	4	no	—
184	C₃×D₄	—	2	2	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	2	2	no	—
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	2	4	no	—
189	C₂×C₃×C₄	—	4	1	no	—
190	C₂×C₃×C₄	—	4	2	no	—
191	C₂×C₃×C₄	#117	1	2	yes	—
192	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
194	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
195	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
197	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
201	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
202	C₃×C₈	—	2	2	no	—
203	C₃×C₈	—	2	1	no	—
204	C₃×C₈	—	2	1	no	—
205	C₃×C₈	#117	1	2	yes	—
206	C₃×C₈	C₂×M₁₆	1	4	yes	—
207	C₃×C₈	C₂×#41	1	1	yes	characteristic
208	C₃×C₈	C₂×#41	1	1	yes	characteristic
209	#41	—	2	2	no	—
210	#41	—	2	2	no	—
211	C₂×Q₈	C₂×C₃⋊C₈	1	1	yes	characteristic
212	C₂×D₄	C₂×C₃⋊C₈	1	2	yes	—
213	C₂×D₄	C₂×C₃⋊C₈	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₃⋊C₈	1	2	yes	—
215	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₃⋊C₈	1	2	yes	—
217	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
218	C₂²×C₄	C₂²×Dic₃	1	2	yes	—
219	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
220	M₁₆	—	2	2	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	#208	1	4	yes	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	#235	1	2	yes	—
229	C₂×C₈	#235	1	2	yes	—
230	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
231	C₂×C₈	C₂²×Dic₃	1	2	yes	—
232	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
233	C₂×C₈	C₂²×Dic₃	1	2	yes	—
234	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
235	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
236	C₄⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
237	C₄⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
238	C₄⋊C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
239	C₄⋊C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
240	C₂²⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
241	C₂²⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
242	C₂²⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
243	C₄²	C₂×C₃⋊C₈	1	2	yes	—
244	C₄²	C₂×C₃⋊C₈	1	1	yes	characteristic
245	C₄²	C₂×C₃⋊C₈	1	2	yes	—
246	C₄²	C₂×C₃⋊C₈	1	1	yes	characteristic
247	C₄²	C₂²×Dic₃	1	1	yes	characteristic
248	C₄²	C₂²×Dic₃	1	2	yes	—
249	C₄²	C₂²×Dic₃	1	1	yes	characteristic
250	C₁₆	—	6	2	no	—
251	C₁₆	—	6	2	no	—
252	C₁₆	—	6	4	no	—
253	C₂²×C₃	—	2	1	no	—
254	C₂²×C₃	—	2	1	no	—
255	C₂²×C₃	—	2	2	no	—
256	C₂²×C₃	—	2	2	no	—
257	C₂²×C₃	—	2	2	no	—
258	C₂²×C₃	#431	1	2	yes	—
259	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	2	2	no	—
262	C₃×C₄	—	2	1	no	—
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	—	2	2	no	—
265	C₃×C₄	—	2	2	no	—
266	C₃×C₄	#431	1	2	yes	—
267	C₃×C₄	#504	1	2	yes	—
268	C₃×C₄	C₂×#117	1	1	yes	characteristic
269	C₃×C₄	C₂²×M₁₆	1	1	yes	characteristic
270	C₂³	C₂²×C₃⋊C₈	1	2	yes	—
271	C₂³	C₂²×C₃⋊C₈	1	1	yes	characteristic
272	Q₈	—	2	2	no	—
273	D₄	—	2	4	no	—
274	D₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
282	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
283	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
284	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
285	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
286	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
287	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
288	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
289	C₂×C₄	C₂×#208	1	2	yes	—
290	C₂×C₄	#926	1	2	yes	—
291	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	—	2	2	no	—
295	C₈	C₂×#208	1	4	yes	—
296	C₈	#926	1	2	yes	—
297	C₈	C₂×#235	1	1	yes	characteristic
298	C₈	C₂×#235	1	1	yes	characteristic
299	C₂×C₃	—	2	2	no	—
300	C₂×C₃	—	4	2	no	—
301	C₂×C₃	C₂×#431	1	1	yes	characteristic
302	C₂×C₃	C₂×#504	1	1	yes	characteristic
303	C₂×C₃	#3407	1	1	yes	characteristic
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	#4925	1	2	yes	—
310	C₂²	C₂³×C₃⋊C₈	1	1	yes	characteristic
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	—	2	2	no	—
317	C₄	#4925	1	2	yes	—
318	C₄	#5140	1	2	yes	—
319	C₄	C₂²×#208	1	1	yes	characteristic
320	C₄	C₂×#926	1	1	yes	characteristic
321	C₃	#33423	1	1	yes	characteristic, complemented, Sylow
322	C₂	—	2	2	no	—
323	C₂	—	4	2	no	—
324	C₂	C₂×#4925	1	1	yes	characteristic
325	C₂	C₂×#5140	1	1	yes	characteristic
326	C₂	#83780	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₈
Fitting	C₃×C₄×#131	C₂
prime radical	C₂×C₃×#41	C₂×C₄
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	C₄×#131	D₃
hypocenter	C₃	#33423
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
3. #69883	C₂	semidirect
4. #69881	C₂	semidirect
5. #69745	C₂	semidirect
6. #69744	C₂	semidirect
7. #69743	C₂	semidirect
8. C₂×C₄⋊(C₃⋊C₁₆)	C₂	semidirect
321. C₃	#33423	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×#131	2
#72004	2
#69883	2
#69881	2
#69745	2
#69744	2
#69743	2
C₂×C₄⋊(C₃⋊C₁₆)	2
#33423	3