Subgroups — #10948284

Order 768.

582 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#78549	C₂	1	1	yes	maximal, characteristic
2	C₃×#2164	C₂	1	1	yes	maximal, characteristic, complemented
3	#69883	C₂	1	1	yes	maximal, characteristic, complemented
4	#69285	C₂	1	2	yes	maximal, complemented
5	#69275	C₂	1	2	yes	maximal, complemented
6	#69107	C₂	1	2	yes	maximal
7	#69093	C₂	1	2	yes	maximal, complemented
8	#69074	C₂	1	1	yes	maximal, characteristic, complemented
9	D₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
10	#69053	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#798	C₂	1	1	yes	maximal, characteristic, complemented
12	#33421	—	3	1	no	maximal, Sylow
13	C₂×#907	C₄	1	1	yes	characteristic
14	C₂×#901	C₂²	1	1	yes	characteristic
15	#5522	C₄	1	1	yes	characteristic
16	C₃×#405	C₂²	1	1	yes	characteristic
17	Dic₃×M₁₆	C₄	1	1	yes	characteristic
18	C₂×C₈×Dic₃	C₂²	1	1	yes	characteristic
19	#4997	C₄	1	1	yes	characteristic
20	C₂×#798	C₂²	1	2	yes	—
21	C₂×D₃×C₁₆	C₂²	1	2	yes	—
22	#4708	C₄	1	1	yes	characteristic
23	#4708	C₄	1	1	yes	characteristic
24	#4707	C₄	1	1	yes	characteristic
25	#4707	C₄	1	1	yes	characteristic
26	#4689	C₂²	1	1	yes	characteristic
27	#4689	C₂²	1	1	yes	characteristic
28	C₄×D₃×C₈	C₂²	1	1	yes	characteristic
29	C₄×D₃×C₈	C₂²	1	1	yes	characteristic
30	C₃×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
32	C₃×#346	C₂²	1	1	yes	characteristic
33	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
34	#4549	C₂²	1	2	yes	—
35	(C₂×D₃)⋊C₁₆	C₂²	1	2	yes	—
36	(C₂×D₃)⋊C₁₆	C₂²	1	2	yes	—
37	#4502	C₂²	1	2	yes	—
38	#4502	C₂²	1	2	yes	—
39	Dic₃⋊C₁₆	C₂²	1	2	yes	—
40	Dic₃⋊C₁₆	C₂²	1	2	yes	—
41	C₄⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
42	#4461	C₂²	1	1	yes	characteristic
43	C₄×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
44	C₄×#131	—	3	1	no	—
45	#2164	—	3	1	no	—
46	#2164	—	3	1	no	—
47	#2164	—	3	1	no	—
48	#2164	D₃	1	1	yes	characteristic, complemented
49	#2163	—	3	2	no	—
50	#2163	—	3	2	no	—
51	#2151	—	3	2	no	—
52	#2113	—	3	2	no	—
53	C₄×M₃₂	—	3	1	no	—
54	C₂×#346	—	3	1	no	—
55	C₂×#235	C₂×C₄	1	1	yes	characteristic
56	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
58	C₃×#117	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
60	C₂×#208	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
62	#907	—	2	4	no	—
63	#901	—	2	4	no	—
64	C₂×ΓL(1,𝔽₂₅)	C₂×C₄	1	1	yes	characteristic
65	C₂×ΓL(1,𝔽₂₅)	C₂×C₄	1	1	yes	characteristic
66	C₂×ΓL(1,𝔽₂₅)	C₂×C₄	1	1	yes	characteristic
67	C₂×ΓL(1,𝔽₂₅)	C₂×C₄	1	1	yes	characteristic
68	C₂×D₃×C₈	C₂×C₄	1	1	yes	characteristic
69	C₂×D₃×C₈	C₂×C₄	1	1	yes	characteristic
70	C₂×D₃×C₈	C₂³	1	1	yes	characteristic
71	C₂×D₃×C₈	C₂³	1	1	yes	characteristic
72	#893	C₈	1	2	yes	—
73	#887	C₈	1	2	yes	—
74	#67⋊C₄	C₈	1	2	yes	—
75	#67⋊C₄	C₈	1	2	yes	—
76	#872	C₂×C₄	1	1	yes	characteristic
77	#872	C₂×C₄	1	1	yes	characteristic
78	C₄²×D₃	C₂×C₄	1	1	yes	characteristic
79	C₄²×D₃	C₂×C₄	1	1	yes	characteristic
80	C₂×C₃×C₁₆	C₂³	1	2	yes	—
81	C₂×C₃×C₁₆	C₂³	1	2	yes	—
82	C₃×#97	C₂×C₄	1	1	yes	characteristic
83	C₃×#97	C₂×C₄	1	1	yes	characteristic
84	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
85	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
86	#815	C₂×C₄	1	1	yes	characteristic
87	#815	C₂×C₄	1	1	yes	characteristic
88	C₈×Dic₃	C₂×C₄	1	1	yes	characteristic
89	C₈×Dic₃	C₂×C₄	1	1	yes	characteristic
90	C₈×Dic₃	C₂×C₄	1	1	yes	characteristic
91	C₈×Dic₃	C₂×C₄	1	1	yes	characteristic
92	C₈×Dic₃	C₂³	1	1	yes	characteristic
93	C₈×Dic₃	C₂³	1	1	yes	characteristic
94	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
95	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
96	#803	C₂×C₄	1	1	yes	characteristic
97	#803	C₂×C₄	1	1	yes	characteristic
98	C₄×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
99	C₄×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
100	#798	—	2	4	no	—
101	D₃×C₁₆	—	2	4	no	—
102	C₂×#131	—	3	1	no	—
103	C₂×#131	—	3	1	no	—
104	C₄×#41	—	3	1	no	—
105	C₂×M₃₂	—	3	2	no	—
106	C₂²×C₁₆	—	3	2	no	—
107	#405	—	3	1	no	—
108	#405	—	3	1	no	—
109	#405	—	3	1	no	—
110	#405	—	3	1	no	—
111	#405	—	3	1	no	—
112	#405	C₂×D₃	1	1	yes	characteristic
113	C₄×M₁₆	—	3	1	no	—
114	C₄×M₁₆	—	3	1	no	—
115	C₄×M₁₆	—	3	1	no	—
116	C₂×C₄×C₈	—	3	1	no	—
117	C₂×C₄×C₈	—	3	1	no	—
118	C₂×C₄×C₈	—	3	1	no	—
119	C₄⋊C₁₆	—	3	2	no	—
120	C₄⋊C₁₆	—	3	2	no	—
121	C₄⋊C₁₆	—	3	2	no	—
122	C₄⋊C₁₆	C₂×D₃	1	2	yes	—
123	C₂²⋊C₁₆	—	3	2	no	—
124	C₂²⋊C₁₆	—	3	2	no	—
125	C₂²⋊C₁₆	—	3	2	no	—
126	C₂²⋊C₁₆	C₂×D₃	1	2	yes	—
127	#346	—	3	2	no	—
128	#346	—	3	2	no	—
129	#346	—	3	1	no	—
130	#346	C₂×D₃	1	1	yes	characteristic
131	C₄×C₁₆	—	3	1	no	—
132	C₄×C₁₆	C₂×D₃	1	1	yes	characteristic
133	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
134	C₂×#67	C₂×C₈	1	1	yes	characteristic
135	C₂×#67	C₂×C₈	1	1	yes	characteristic
136	C₂²×Dic₃	C₂×C₈	1	2	yes	—
137	#235	—	2	2	no	—
138	#235	—	2	2	no	—
139	C₂×D₁₂	C₂×C₈	1	1	yes	characteristic
140	C₂×C₄×D₃	C₂×C₈	1	2	yes	—
141	C₂×C₄×D₃	C₂×C₈	1	2	yes	—
142	C₂×C₄×D₃	C₂²×C₄	1	1	yes	characteristic
143	C₂×C₄×D₃	C₂²×C₄	1	1	yes	characteristic
144	C₂×Dic₆	C₂×C₈	1	1	yes	characteristic
145	C₃×M₁₆	—	2	2	no	—
146	C₂×C₃×C₈	—	2	2	no	—
147	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
148	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
149	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
150	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
151	C₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
152	C₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
153	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
154	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
155	(C₂×D₃)⋊C₄	C₂×C₈	1	2	yes	—
156	(C₂×D₃)⋊C₄	C₂×C₈	1	2	yes	—
157	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
158	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
159	C₄×Dic₃	C₂×C₈	1	2	yes	—
160	C₄×Dic₃	C₂×C₈	1	2	yes	—
161	C₄×Dic₃	C₂×C₈	1	2	yes	—
162	C₄×Dic₃	C₂²×C₄	1	1	yes	characteristic
163	C₄×Dic₃	C₂²×C₄	1	1	yes	characteristic
164	#208	—	2	2	no	—
165	C₂×C₃⋊C₈	—	2	2	no	—
166	C₂×C₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
167	C₂×C₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
168	C₂×C₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
169	C₂×C₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
170	ΓL(1,𝔽₂₅)	—	2	2	no	—
171	ΓL(1,𝔽₂₅)	—	2	2	no	—
172	ΓL(1,𝔽₂₅)	—	2	2	no	—
173	ΓL(1,𝔽₂₅)	—	2	2	no	—
174	D₃×C₈	—	2	2	no	—
175	D₃×C₈	—	2	2	no	—
176	D₃×C₈	M₁₆	1	4	yes	—
177	D₃×C₈	#41	1	4	yes	—
178	C₃×C₁₆	—	2	2	no	—
179	C₃×C₁₆	—	2	2	no	—
180	C₃⋊C₁₆	—	2	2	no	—
181	C₃⋊C₁₆	—	2	2	no	—
182	C₂×#41	—	3	1	no	—
183	#131	—	6	4	no	—
184	#131	—	6	4	no	—
185	C₂×M₁₆	—	3	1	no	—
186	C₂×M₁₆	—	3	1	no	—
187	C₂×M₁₆	—	3	1	no	—
188	C₂×M₁₆	—	3	1	no	—
189	C₂×M₁₆	—	3	1	no	—
190	C₂×M₁₆	C₄×D₃	1	1	yes	characteristic
191	C₂²×C₈	—	3	1	no	—
192	C₂²×C₈	—	3	1	no	—
193	C₂²×C₈	—	3	1	no	—
194	C₂²×C₈	—	3	1	no	—
195	C₂²×C₈	—	3	1	no	—
196	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
197	C₄×Q₈	—	3	2	no	—
198	C₄×D₄	—	3	2	no	—
199	C₄×D₄	—	3	2	no	—
200	C₄×D₄	—	3	2	no	—
201	#117	—	3	1	no	—
202	#117	—	3	1	no	—
203	#117	C₄×D₃	1	1	yes	characteristic
204	C₂×C₄²	—	3	1	no	—
205	C₂×C₄²	—	3	1	no	—
206	C₂×C₄²	—	3	1	no	—
207	M₃₂	—	6	4	no	—
208	C₂×C₁₆	—	3	2	no	—
209	C₂×C₁₆	—	3	2	no	—
210	C₂×C₁₆	—	6	4	no	—
211	C₂×C₁₆	C₂²×D₃	1	2	yes	—
212	C₂×C₁₆	C₂²×D₃	1	2	yes	—
213	#97	—	3	1	no	—
214	#97	—	3	1	no	—
215	#97	—	3	1	no	—
216	#97	—	3	1	no	—
217	#97	C₄×D₃	1	1	yes	characteristic
218	#97	C₄×D₃	1	1	yes	characteristic
219	C₄×C₈	—	3	1	no	—
220	C₄×C₈	—	3	1	no	—
221	C₄×C₈	—	3	1	no	—
222	C₄×C₈	—	3	1	no	—
223	C₄×C₈	—	3	1	no	—
224	C₄×C₈	—	3	1	no	—
225	C₄×C₈	—	3	1	no	—
226	C₄×C₈	—	3	1	no	—
227	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
228	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
229	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
230	C₂²×D₃	C₂²×C₈	1	1	yes	characteristic
231	C₂²×D₃	C₂²×C₈	1	1	yes	characteristic
232	C₂×C₃×C₄	—	2	1	no	—
233	C₂×C₃×C₄	—	2	1	no	—
234	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
235	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
236	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
237	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
238	#67	—	2	2	no	—
239	#67	—	2	2	no	—
240	C₂×Dic₃	—	4	2	no	—
241	C₂×Dic₃	C₂²×C₈	1	2	yes	—
242	C₂×Dic₃	C₂²×C₈	1	1	yes	characteristic
243	C₂×Dic₃	C₂²×C₈	1	2	yes	—
244	C₂×Dic₃	C₂²×C₈	1	1	yes	characteristic
245	D₁₂	—	2	2	no	—
246	C₄×D₃	—	2	1	no	—
247	C₄×D₃	—	2	1	no	—
248	C₄×D₃	—	4	2	no	—
249	C₄×D₃	—	4	2	no	—
250	C₄×D₃	#117	1	2	yes	—
251	C₄×D₃	C₂×M₁₆	1	2	yes	—
252	Dic₆	—	2	2	no	—
253	C₃×C₈	—	2	1	no	—
254	C₃×C₈	—	2	1	no	—
255	C₃×C₈	C₂×M₁₆	1	2	yes	—
256	C₃×C₈	C₂×#41	1	2	yes	—
257	C₃⋊C₈	—	2	1	no	—
258	C₃⋊C₈	—	2	1	no	—
259	C₃⋊C₈	#117	1	2	yes	—
260	C₃⋊C₈	C₂×M₁₆	1	2	yes	—
261	#41	—	6	2	no	—
262	#41	—	6	2	no	—
263	C₂×Q₈	—	3	1	no	—
264	C₂×D₄	—	3	1	no	—
265	C₂×D₄	—	3	1	no	—
266	C₂×D₄	—	3	1	no	—
267	C₂²×C₄	—	3	1	no	—
268	C₂²×C₄	—	3	2	no	—
269	C₂²×C₄	—	3	1	no	—
270	C₂²×C₄	—	3	2	no	—
271	C₂²×C₄	—	3	2	no	—
272	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
273	M₁₆	—	2	2	no	—
274	M₁₆	—	6	2	no	—
275	M₁₆	—	6	2	no	—
276	M₁₆	—	6	2	no	—
277	M₁₆	—	6	2	no	—
278	M₁₆	—	6	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	3	1	no	—
281	C₂×C₈	—	3	1	no	—
282	C₂×C₈	—	3	4	no	—
283	C₂×C₈	—	3	4	no	—
284	C₂×C₈	—	3	1	no	—
285	C₂×C₈	—	3	1	no	—
286	C₂×C₈	—	6	2	no	—
287	C₂×C₈	—	6	2	no	—
288	C₂×C₈	—	6	2	no	—
289	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
290	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
292	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
293	C₄⋊C₄	—	3	2	no	—
294	C₄⋊C₄	—	3	2	no	—
295	C₄⋊C₄	D₃×C₈	1	2	yes	—
296	C₂²⋊C₄	—	3	2	no	—
297	C₂²⋊C₄	—	3	2	no	—
298	C₂²⋊C₄	D₃×C₈	1	2	yes	—
299	C₄²	—	3	2	no	—
300	C₄²	—	3	2	no	—
301	C₄²	—	3	2	no	—
302	C₄²	—	3	1	no	—
303	C₄²	—	3	1	no	—
304	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
305	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
306	C₁₆	—	2	2	no	—
307	C₁₆	—	2	2	no	—
308	C₁₆	—	6	2	no	—
309	C₁₆	—	6	2	no	—
310	C₂²×C₃	—	2	2	no	—
311	C₂²×C₃	—	2	1	no	—
312	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
313	C₂×D₃	—	2	2	no	—
314	C₂×D₃	—	2	2	no	—
315	C₂×D₃	—	2	1	no	—
316	C₂×D₃	#431	1	2	yes	—
317	C₃×C₄	—	2	2	no	—
318	C₃×C₄	—	2	2	no	—
319	C₃×C₄	—	2	1	no	—
320	C₃×C₄	C₂×#117	1	1	yes	characteristic
321	C₃×C₄	C₂²×M₁₆	1	1	yes	characteristic
322	Dic₃	—	2	2	no	—
323	Dic₃	—	2	2	no	—
324	Dic₃	—	2	1	no	—
325	Dic₃	#431	1	2	yes	—
326	C₂³	—	3	1	no	—
327	C₂³	—	3	1	no	—
328	C₂³	C₂×D₃×C₈	1	1	yes	characteristic
329	Q₈	—	6	2	no	—
330	D₄	—	6	2	no	—
331	D₄	—	6	2	no	—
332	D₄	—	6	2	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	3	1	no	—
336	C₂×C₄	—	3	2	no	—
337	C₂×C₄	—	3	2	no	—
338	C₂×C₄	—	3	2	no	—
339	C₂×C₄	—	3	2	no	—
340	C₂×C₄	—	3	1	no	—
341	C₂×C₄	—	6	1	no	—
342	C₂×C₄	—	6	1	no	—
343	C₂×C₄	—	12	2	no	—
344	C₂×C₄	—	12	2	no	—
345	C₂×C₄	—	12	2	no	—
346	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
347	C₂×C₄	C₂×D₃×C₈	1	2	yes	—
348	C₂×C₄	C₂×D₃×C₈	1	2	yes	—
349	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	—	3	2	no	—
353	C₈	—	3	2	no	—
354	C₈	—	6	1	no	—
355	C₈	—	6	1	no	—
356	C₈	D₃×M₁₆	1	2	yes	—
357	C₈	D₃×#41	1	2	yes	—
358	C₂×C₃	—	4	1	no	—
359	C₂×C₃	C₂×#431	1	1	yes	characteristic
360	C₂×C₃	#3407	1	2	yes	—
361	D₃	—	2	2	no	—
362	D₃	—	4	1	no	—
363	C₂²	—	2	2	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	3	2	no	—
366	C₂²	—	6	2	no	—
367	C₂²	—	6	2	no	—
368	C₂²	—	6	1	no	—
369	C₂²	C₂²×D₃×C₈	1	1	yes	characteristic
370	C₄	—	2	2	no	—
371	C₄	—	2	2	no	—
372	C₄	—	2	1	no	—
373	C₄	—	3	2	no	—
374	C₄	—	6	2	no	—
375	C₄	—	6	2	no	—
376	C₄	—	6	1	no	—
377	C₄	D₃×#117	1	1	yes	characteristic
378	C₄	C₂×D₃×M₁₆	1	1	yes	characteristic
379	C₃	#33421	1	1	yes	characteristic, complemented, Sylow
380	C₂	—	4	1	no	—
381	C₂	—	6	2	no	—
382	C₂	—	12	1	no	—
383	C₂	D₃×#431	1	1	yes	characteristic
384	C₂	#82012	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₈
Fitting	C₃×#2164	C₂
prime radical	C₂×#235	C₂×C₄
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	#2164	D₃
hypocenter	C₃	#33421
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ #2164

Complements

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

N	H ≅ G/N	type
2. C₃×#2164	C₂	semidirect
3. #69883	C₂	semidirect
4. #69285	C₂	semidirect
5. #69275	C₂	semidirect
7. #69093	C₂	semidirect
8. #69074	C₂	semidirect
9. D₃×#346	C₂	semidirect
10. #69053	C₂	semidirect
11. C₄×#798	C₂	semidirect
48. #2164	D₃	semidirect
379. C₃	#33421	semidirect

Maximal subgroups

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

subgroup	index
#78549	2
C₃×#2164	2
#69883	2
#69285	2
#69275	2
#69107	2
#69093	2
#69074	2
D₃×#346	2
#69053	2
C₄×#798	2
#33421	3