Subgroups — #10693396

Order 768.

444 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#69787	C₂	1	1	yes	maximal, characteristic
2	C₄×#821	C₂	1	1	yes	maximal, characteristic
3	#69757	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#4459	C₂	1	1	yes	maximal, characteristic
5	C₂×#4457	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#1273	C₂	1	1	yes	maximal, characteristic, complemented
7	#67655	C₂	1	1	yes	maximal, characteristic, complemented
8	#9546	—	3	1	no	maximal, Sylow
9	C₂×C₄×D₁₂	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×#821	C₄	1	1	yes	characteristic
14	C₂×#821	C₄	1	1	yes	characteristic
15	C₂×(C₂×D₃)⋊C₈	—	2	1	no	—
16	C₂×(C₂×D₃)⋊C₈	—	2	1	no	—
17	C₂×#815	—	2	1	no	—
18	C₄×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
21	#4551	C₄	1	1	yes	characteristic
22	#4551	C₄	1	1	yes	characteristic
23	#4550	—	2	1	no	—
24	#4459	—	2	2	no	—
25	#4457	—	2	2	no	—
26	#1954	—	3	1	no	—
27	#1770	—	3	1	no	—
28	C₄×#102	—	3	1	no	—
29	C₂×#334	—	3	1	no	—
30	C₂×D₄⋊C₈	—	3	1	no	—
31	#1279	—	3	1	no	—
32	#1273	D₃	1	1	yes	characteristic, complemented
33	C₂²×D₁₂	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×D₃	—	2	1	no	—
35	C₂²×C₃×C₈	—	2	1	no	—
36	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
39	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
40	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
41	C₂×(C₂×D₃)⋊C₄	—	2	1	no	—
42	C₂×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×Dic₃	—	2	1	no	—
46	C₂²×C₃⋊C₈	—	2	1	no	—
47	C₂²×C₃⋊C₈	—	2	1	no	—
48	C₄×D₁₂	—	2	1	no	—
49	C₄×D₁₂	—	2	1	no	—
50	C₄×D₁₂	—	2	1	no	—
51	C₄×D₁₂	—	2	1	no	—
52	C₃×C₄×C₈	—	2	1	no	—
53	C₃×C₄×C₈	—	2	1	no	—
54	C₃×C₄×C₈	—	2	1	no	—
55	C₃×C₄×C₈	—	2	1	no	—
56	C₃×C₄×C₈	—	2	1	no	—
57	C₃×C₄×C₈	—	2	1	no	—
58	C₃⋊#95	—	2	1	no	—
59	#821	—	2	2	no	—
60	#821	—	2	2	no	—
61	(C₂×D₃)⋊C₈	—	4	2	no	—
62	(C₂×D₃)⋊C₈	—	4	2	no	—
63	#815	—	4	2	no	—
64	C₄⋊(C₃⋊C₈)	—	2	1	no	—
65	C₄⋊(C₃⋊C₈)	—	2	1	no	—
66	C₄⋊(C₃⋊C₈)	—	2	1	no	—
67	C₄⋊(C₃⋊C₈)	D₄	1	2	yes	—
68	C₂×C₄×D₄	—	3	1	no	—
69	C₂×C₄⋊C₈	—	3	1	no	—
70	C₂×C₄⋊C₈	—	3	1	no	—
71	C₂×#102	—	3	1	no	—
72	C₂×#102	—	3	1	no	—
73	C₂×C₂²⋊C₈	—	6	1	no	—
74	C₂×C₂²⋊C₈	—	6	1	no	—
75	C₂×#97	—	6	1	no	—
76	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
77	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
78	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
79	C₄×C₄⋊C₄	—	3	1	no	—
80	#340	—	3	1	no	—
81	#340	—	3	1	no	—
82	#336	—	6	1	no	—
83	#334	—	6	2	no	—
84	D₄⋊C₈	—	6	2	no	—
85	C₂³×D₃	—	2	1	no	—
86	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₃×C₄	#41	1	1	yes	characteristic
89	C₂²×Dic₃	—	2	1	no	—
90	C₂²×Dic₃	—	2	1	no	—
91	C₂²×Dic₃	—	2	1	no	—
92	C₂×D₁₂	—	2	1	no	—
93	C₂×D₁₂	—	2	1	no	—
94	C₂×D₁₂	—	2	1	no	—
95	C₂×D₁₂	—	2	1	no	—
96	C₂×D₁₂	—	2	1	no	—
97	C₂×D₁₂	M₁₆	1	2	yes	—
98	C₂×C₄×D₃	—	4	1	no	—
99	C₂×C₄×D₃	—	4	1	no	—
100	C₂×C₄×D₃	—	4	1	no	—
101	C₂×C₄×D₃	—	4	1	no	—
102	C₂×C₄×D₃	—	4	1	no	—
103	C₂×C₄×D₃	—	4	1	no	—
104	C₂×C₃×C₈	—	2	1	no	—
105	C₂×C₃×C₈	—	2	1	no	—
106	C₂×C₃×C₈	—	2	1	no	—
107	C₂×C₃×C₈	—	2	1	no	—
108	C₂×C₃×C₈	—	2	2	no	—
109	C₂×C₃×C₈	—	2	1	no	—
110	C₂×C₃×C₈	—	2	1	no	—
111	C₂×C₃×C₈	—	2	1	no	—
112	C₂×C₃×C₈	—	2	1	no	—
113	C₂×C₃×C₈	—	2	1	no	—
114	C₂×C₃×C₈	—	2	1	no	—
115	C₂×C₃×C₈	—	4	1	no	—
116	C₂×C₃×C₈	—	4	1	no	—
117	C₂×C₃×C₈	M₁₆	1	2	yes	—
118	C₂×C₃×C₈	SD₁₆	1	2	yes	—
119	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
120	C₃×C₄²	#41	1	1	yes	characteristic
121	C₃×C₄²	#41	1	1	yes	characteristic
122	C₃×C₄²	#41	1	1	yes	characteristic
123	(C₂×D₃)⋊C₄	—	4	1	no	—
124	(C₂×D₃)⋊C₄	—	4	1	no	—
125	C₄⋊Dic₃	—	2	1	no	—
126	C₄⋊Dic₃	—	2	1	no	—
127	C₄⋊Dic₃	—	2	1	no	—
128	C₄⋊Dic₃	—	2	1	no	—
129	C₄⋊Dic₃	—	2	1	no	—
130	C₄⋊Dic₃	—	2	1	no	—
131	C₄×Dic₃	—	4	1	no	—
132	C₄×Dic₃	—	4	1	no	—
133	C₂×C₃⋊C₈	—	2	2	no	—
134	C₂×C₃⋊C₈	—	4	1	no	—
135	C₂×C₃⋊C₈	—	4	1	no	—
136	C₂×C₃⋊C₈	—	4	1	no	—
137	C₂×C₃⋊C₈	—	4	1	no	—
138	C₂×C₃⋊C₈	—	4	1	no	—
139	C₂²×D₄	—	3	1	no	—
140	C₂³×C₄	—	6	1	no	—
141	C₂²×C₈	—	2	1	no	—
142	C₂²×C₈	—	6	1	no	—
143	C₂²×C₈	—	6	1	no	—
144	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
145	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
146	C₂²×C₈	D₁₂	1	1	yes	characteristic
147	C₂²×C₈	D₁₂	1	1	yes	characteristic
148	C₄×D₄	—	6	1	no	—
149	C₄×D₄	—	6	1	no	—
150	C₄×D₄	—	6	1	no	—
151	C₄×D₄	—	6	1	no	—
152	C₂×C₄⋊C₄	—	3	1	no	—
153	C₂×C₄⋊C₄	—	3	1	no	—
154	C₂×C₄⋊C₄	—	3	1	no	—
155	C₂×C₂²⋊C₄	—	6	1	no	—
156	C₂×C₄²	—	6	1	no	—
157	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
158	C₄⋊C₈	—	3	2	no	—
159	C₄⋊C₈	—	6	1	no	—
160	C₄⋊C₈	—	6	1	no	—
161	C₄⋊C₈	—	6	1	no	—
162	#102	—	6	2	no	—
163	#102	—	6	2	no	—
164	C₂²⋊C₈	—	12	2	no	—
165	C₂²⋊C₈	—	12	2	no	—
166	#97	—	12	2	no	—
167	C₄×C₈	—	2	1	no	—
168	C₄×C₈	—	2	1	no	—
169	C₄×C₈	—	2	1	no	—
170	C₄×C₈	—	2	1	no	—
171	C₄×C₈	—	2	1	no	—
172	C₄×C₈	—	2	1	no	—
173	#95	—	6	1	no	—
174	C₂³×C₃	C₄×D₄	1	1	yes	characteristic
175	C₂²×D₃	—	2	2	no	—
176	C₂²×D₃	—	4	1	no	—
177	C₂²×D₃	—	4	1	no	—
178	C₂²×D₃	—	4	1	no	—
179	C₂²×D₃	—	4	1	no	—
180	C₂²×D₃	—	4	1	no	—
181	C₂²×D₃	—	4	1	no	—
182	C₂×C₃×C₄	—	2	1	no	—
183	C₂×C₃×C₄	—	2	1	no	—
184	C₂×C₃×C₄	#117	1	1	yes	characteristic
185	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
186	C₂×C₃×C₄	#121	1	1	yes	characteristic
187	C₂×C₃×C₄	#123	1	1	yes	characteristic
188	C₂×C₃×C₄	#124	1	1	yes	characteristic
189	C₂×C₃×C₄	#126	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
192	C₂×C₃×C₄	#131	1	1	yes	characteristic
193	C₂×C₃×C₄	#131	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
195	C₂×C₃×C₄	#135	1	1	yes	characteristic
196	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
197	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
198	C₂×Dic₃	—	4	1	no	—
199	C₂×Dic₃	—	4	1	no	—
200	C₂×Dic₃	—	4	1	no	—
201	C₂×Dic₃	—	4	1	no	—
202	C₂×Dic₃	—	4	1	no	—
203	C₂×Dic₃	—	4	1	no	—
204	C₂×Dic₃	—	4	1	no	—
205	C₂×Dic₃	—	4	1	no	—
206	C₂×Dic₃	—	4	1	no	—
207	D₁₂	—	2	2	no	—
208	D₁₂	—	2	2	no	—
209	D₁₂	—	4	1	no	—
210	D₁₂	—	4	1	no	—
211	C₄×D₃	—	8	1	no	—
212	C₄×D₃	—	8	1	no	—
213	C₄×D₃	—	8	1	no	—
214	C₄×D₃	—	8	1	no	—
215	C₃×C₈	—	2	2	no	—
216	C₃×C₈	—	2	2	no	—
217	C₃×C₈	—	4	2	no	—
218	C₃×C₈	—	4	1	no	—
219	C₃×C₈	—	4	1	no	—
220	C₃⋊C₈	—	4	2	no	—
221	C₃⋊C₈	—	8	1	no	—
222	C₂⁴	—	6	1	no	—
223	C₂×D₄	—	3	2	no	—
224	C₂×D₄	—	6	1	no	—
225	C₂×D₄	—	6	1	no	—
226	C₂×D₄	—	6	1	no	—
227	C₂×D₄	—	6	1	no	—
228	C₂×D₄	—	6	1	no	—
229	C₂²×C₄	—	6	1	no	—
230	C₂²×C₄	—	6	1	no	—
231	C₂²×C₄	—	6	1	no	—
232	C₂²×C₄	—	12	1	no	—
233	C₂²×C₄	—	12	1	no	—
234	C₂²×C₄	—	12	1	no	—
235	C₂²×C₄	—	12	1	no	—
236	C₂²×C₄	—	12	1	no	—
237	C₂²×C₄	—	12	1	no	—
238	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
240	C₂²×C₄	#235	1	1	yes	characteristic
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	6	2	no	—
255	C₂×C₈	—	12	1	no	—
256	C₂×C₈	—	12	1	no	—
257	C₂×C₈	—	12	1	no	—
258	C₂×C₈	—	12	1	no	—
259	C₂×C₈	—	12	1	no	—
260	C₂×C₈	ΓL(1,𝔽₂₅)	1	2	yes	—
261	C₂×C₈	#204	1	2	yes	—
262	C₄⋊C₄	—	6	1	no	—
263	C₄⋊C₄	—	6	1	no	—
264	C₄⋊C₄	—	6	1	no	—
265	C₄⋊C₄	—	6	1	no	—
266	C₄⋊C₄	—	6	1	no	—
267	C₄⋊C₄	—	6	1	no	—
268	C₂²⋊C₄	—	12	1	no	—
269	C₂²⋊C₄	—	12	1	no	—
270	C₄²	—	12	1	no	—
271	C₄²	—	12	1	no	—
272	C₄²	D₃×D₄	1	1	yes	characteristic
273	C₄²	#237	1	1	yes	characteristic
274	C₄²	#237	1	1	yes	characteristic
275	C₄²	#239	1	1	yes	characteristic
276	C₂²×C₃	#388	1	1	yes	characteristic
277	C₂²×C₃	#435	1	1	yes	characteristic
278	C₂²×C₃	#436	1	1	yes	characteristic
279	C₂²×C₃	C₄×SD₁₆	1	1	yes	characteristic
280	C₂²×C₃	#442	1	1	yes	characteristic
281	C₂²×C₃	#443	1	1	yes	characteristic
282	C₂²×C₃	#444	1	1	yes	characteristic
283	C₂×D₃	—	4	2	no	—
284	C₂×D₃	—	4	2	no	—
285	C₂×D₃	—	4	2	no	—
286	C₂×D₃	—	8	1	no	—
287	C₂×D₃	—	8	1	no	—
288	C₂×D₃	—	8	1	no	—
289	C₂×D₃	—	8	1	no	—
290	C₃×C₄	—	2	1	no	—
291	C₃×C₄	—	2	1	no	—
292	C₃×C₄	#432	1	1	yes	characteristic
293	C₃×C₄	#433	1	1	yes	characteristic
294	C₃×C₄	#435	1	1	yes	characteristic
295	C₃×C₄	#435	1	1	yes	characteristic
296	C₃×C₄	#460	1	1	yes	characteristic
297	C₃×C₄	#463	1	1	yes	characteristic
298	C₃×C₄	#481	1	1	yes	characteristic
299	C₃×C₄	#482	1	1	yes	characteristic
300	Dic₃	—	8	1	no	—
301	Dic₃	—	8	1	no	—
302	Dic₃	—	8	1	no	—
303	C₂³	—	6	2	no	—
304	C₂³	—	12	1	no	—
305	C₂³	—	12	1	no	—
306	C₂³	—	12	1	no	—
307	C₂³	—	12	1	no	—
308	C₂³	—	12	1	no	—
309	C₂³	—	12	1	no	—
310	C₂³	C₄×D₁₂	1	1	yes	characteristic
311	D₄	—	6	2	no	—
312	D₄	—	6	2	no	—
313	D₄	—	12	1	no	—
314	D₄	—	12	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	12	1	no	—
318	C₂×C₄	—	12	1	no	—
319	C₂×C₄	—	12	1	no	—
320	C₂×C₄	—	12	1	no	—
321	C₂×C₄	—	12	1	no	—
322	C₂×C₄	—	12	1	no	—
323	C₂×C₄	—	12	1	no	—
324	C₂×C₄	—	12	1	no	—
325	C₂×C₄	—	12	1	no	—
326	C₂×C₄	—	24	1	no	—
327	C₂×C₄	—	24	1	no	—
328	C₂×C₄	—	24	1	no	—
329	C₂×C₄	—	24	1	no	—
330	C₂×C₄	#67⋊C₄	1	1	yes	characteristic
331	C₂×C₄	#885	1	1	yes	characteristic
332	C₂×C₄	#886	1	1	yes	characteristic
333	C₂×C₄	#892	1	1	yes	characteristic
334	C₂×C₄	#895	1	1	yes	characteristic
335	C₂×C₄	#898	1	1	yes	characteristic
336	C₂×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
337	C₂×C₄	#901	1	1	yes	characteristic
338	C₂×C₄	C₂×#204	1	1	yes	characteristic
339	C₂×C₄	#904	1	1	yes	characteristic
340	C₂×C₄	D₃×M₁₆	1	1	yes	characteristic
341	C₂×C₄	#907	1	1	yes	characteristic
342	C₂×C₄	#908	1	1	yes	characteristic
343	C₂×C₄	#908	1	1	yes	characteristic
344	C₈	—	2	2	no	—
345	C₈	—	2	2	no	—
346	C₈	—	4	2	no	—
347	C₈	—	4	1	no	—
348	C₈	—	4	1	no	—
349	C₈	—	12	2	no	—
350	C₈	—	24	1	no	—
351	C₂×C₃	#1582	1	1	yes	characteristic
352	C₂×C₃	#1586	1	1	yes	characteristic
353	C₂×C₃	#1590	1	1	yes	characteristic
354	C₂×C₃	#1594	1	1	yes	characteristic
355	C₂×C₃	#1924	1	1	yes	characteristic
356	C₂×C₃	#1926	1	1	yes	characteristic
357	C₂×C₃	#1929	1	1	yes	characteristic
358	D₃	—	8	2	no	—
359	C₂²	—	12	2	no	—
360	C₂²	—	12	2	no	—
361	C₂²	—	12	2	no	—
362	C₂²	—	24	1	no	—
363	C₂²	—	24	1	no	—
364	C₂²	—	24	1	no	—
365	C₂²	—	24	1	no	—
366	C₂²	#4670	1	1	yes	characteristic
367	C₂²	#4688	1	1	yes	characteristic
368	C₂²	C₄×#204	1	1	yes	characteristic
369	C₂²	#4700	1	1	yes	characteristic
370	C₂²	#4706	1	1	yes	characteristic
371	C₂²	#4711	1	1	yes	characteristic
372	C₂²	#4717	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	24	1	no	—
376	C₄	—	24	1	no	—
377	C₄	—	24	1	no	—
378	C₄	#4728	1	1	yes	characteristic
379	C₄	#4730	1	1	yes	characteristic
380	C₄	#4735	1	1	yes	characteristic
381	C₄	#4737	1	1	yes	characteristic
382	C₄	#4835	1	1	yes	characteristic
383	C₄	#4840	1	1	yes	characteristic
384	C₄	#4844	1	1	yes	characteristic
385	C₄	#4848	1	1	yes	characteristic
386	C₃	#9546	1	1	yes	characteristic, complemented, Sylow
387	C₂	—	24	2	no	—
388	C₂	#68055	1	1	yes	characteristic
389	C₂	#68079	1	1	yes	characteristic
390	C₂	#68081	1	1	yes	characteristic
391	C₂	#68083	1	1	yes	characteristic
392	C₂	#68616	1	1	yes	characteristic
393	C₂	#68626	1	1	yes	characteristic
394	C₂	#68646	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₂
commutator	C₂²×C₃×C₄	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1273	C₂
socle	C₂³×C₃	C₄×D₄
prime radical	C₂²×D₁₂	C₂×C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#1273	D₃
hypocenter	C₃	#9546
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1273

Complements

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

N	H ≅ G/N	type
3. #69757	C₂	semidirect
5. C₂×#4457	C₂	semidirect
6. C₃×#1273	C₂	semidirect
7. #67655	C₂	semidirect
32. #1273	D₃	semidirect
386. C₃	#9546	semidirect

Maximal subgroups

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

subgroup	index
#69787	2
C₄×#821	2
#69757	2
C₂×#4459	2
C₂×#4457	2
C₃×#1273	2
#67655	2
#9546	3