Finite Group Databasebeta

← #22571890

Subgroups — #22571890

Order 8576.

526 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14524063C211yesmaximal, characteristic, complemented
2C22×#11761613C211yesmaximal, characteristic, complemented
3#16864042C211yesmaximal, characteristic, complemented
4#16863778C211yesmaximal, characteristic, complemented
5#16863635C211yesmaximal, characteristic, complemented
6#16863630C211yesmaximal, characteristic, complemented
7#16863611C211yesmaximal, characteristic, complemented
8#16863608C211yesmaximal, characteristic, complemented
9#16863547C211yesmaximal, characteristic, complemented
10#16863546C211yesmaximal, characteristic, complemented
11#16863541C211yesmaximal, characteristic, complemented
12#16863539C211yesmaximal, characteristic, complemented
13C4×#11761613C211yesmaximal, characteristic, complemented
14D4×C67⋊C8C211yesmaximal, characteristic, complemented
15#546×C67C211yesmaximal, characteristic, complemented
16C2×#11761628C2211yescharacteristic, complemented
17C22×D268C2211yescharacteristic
18#14524063—24no—
19C2×#11761618C2211yescharacteristic, complemented
20C2×#11761616C2211yescharacteristic, complemented
21C2×#11761616C2211yescharacteristic, complemented
22C2×#11761615C2211yescharacteristic, complemented
23C2×#11761615C2211yescharacteristic, complemented
24C2×#11761613—24no—
25C2×#11761613C2211yescharacteristic
26C2×#11761613C2211yescharacteristic
27C2×#11761613C2211yescharacteristic
28C2×#11761613C2211yescharacteristic
29C22×C67⋊C8C2211yescharacteristic
30C22×C67⋊C8C2211yescharacteristic, complemented
31#14524031C2211yescharacteristic, complemented
32#14524028—21no—
33#14524022C2211yescharacteristic
34#14524017C2211yescharacteristic, complemented
35#14524016—21no—
36#14524011—21no—
37C4×D268C2211yescharacteristic, complemented
38C22⋊C4×D67—21no—
39#14523976C2211yescharacteristic, complemented
40C4⋊(C67⋊C8)C2211yescharacteristic, complemented
41#14523967C2211yescharacteristic, complemented
42#14523966C2211yescharacteristic, complemented
43#14523964C2211yescharacteristic, complemented
44#14523964C2211yescharacteristic, complemented
45#14523963C2211yescharacteristic, complemented
46#14523963C2211yescharacteristic, complemented
47C22⋊(C67⋊C8)C2211yescharacteristic
48C22⋊(C67⋊C8)C2211yescharacteristic, complemented
49C4×C67⋊C8C2211yescharacteristic, complemented
50C2×#41×C67C2211yescharacteristic, complemented
51C22×D4×C67C2211yescharacteristic
52#124×C67C2211yescharacteristic, complemented
53#123×C67—21no—
54C22⋊Q8×C67C2211yescharacteristic
55#121×C67—21no—
56#121×C67C2211yescharacteristic, complemented
57#120×C67—21no—
58C4×D4×C67C2211yescharacteristic
59C4×D4×C67C2211yescharacteristic, complemented
60C2×C22⋊C4×C67—21no—
61C23×D67—21no—
62#11761628—22no—
63#11761628—41no—
64C2×Dic134C2311yescharacteristic
65C2×D268—22no—
66C2×D268—21no—
67C2×D268—41no—
68C2×D268C2311yescharacteristic
69C2×D268C2311yescharacteristic
70C2×C22⋊D67—21no—
71C2×C22⋊D67—21no—
72C2×C4×D67—21no—
73C2×C4×D67—21no—
74#11761618—22no—
75#11761616—22no—
76#11761616—41no—
77#11761615—22no—
78#11761615—41no—
79#11761613—22no—
80#11761613—22no—
81#11761613—22no—
82#11761613D414yescomplemented
83C2×C67⋊C8—22no—
84C2×C67⋊C8—22no—
85C2×C67⋊C8C2311yescharacteristic
86C2×C67⋊C8C2311yescharacteristic
87C2×C67⋊C8C2311yescharacteristic
88C2×C67⋊C8C2311yescharacteristic
89C4⋊Dic67C2311yescharacteristic
90Dic67⋊C4—21no—
91C22⋊Dic67—21no—
92(C2×D67)⋊C4—21no—
93(C2×D67)⋊C4—21no—
94(C2×D67)⋊C4—21no—
95(C2×D67)⋊C4—21no—
96C24×C67—21no—
97#41×C67—22no—
98#41×C67—41no—
99C2×Q8×C67C2311yescharacteristic
100C2×D4×C67—22no—
101C2×D4×C67—21no—
102C2×D4×C67—21no—
103C2×D4×C67—21no—
104C2×D4×C67—41no—
105C2×D4×C67D411yescharacteristic
106C2×D4×C67C2311yescharacteristic
107C2×D4×C67C2311yescharacteristic
108C22×C4×C67—21no—
109C22×C4×C67—21no—
110C22×C4×C67C2311yescharacteristic
111C22×C4×C67C2311yescharacteristic
112C4⋊C4×C67—21no—
113C4⋊C4×C67D411yescharacteristic
114C4⋊C4×C67C2311yescharacteristic
115C22⋊C4×C67—21no—
116C22⋊C4×C67—21no—
117C22⋊C4×C67—21no—
118C22⋊C4×C67—21no—
119C22⋊C4×C67—21no—
120C22⋊C4×C67D411yescharacteristic
121C22⋊C4×C67D411yescharacteristic
122C42×C67C2311yescharacteristic
123C23×C67—21no—
124C23×C67—21no—
125C23×C67—42no—
126C23×C67—42no—
127C23×C67—81no—
128C23×C67C2×D411yescharacteristic
129C23×C67C2×D411yescharacteristic
130C22×D67—21no—
131C22×D67—21no—
132C22×D67—42no—
133C22×D67—42no—
134C22×D67—81no—
135Q8×C67—21no—
136Q8×C67—21no—
137D4×C67—22no—
138D4×C67—21no—
139D4×C67—21no—
140D4×C67—21no—
141D4×C67—42no—
142D4×C67—42no—
143D4×C67—81no—
144D4×C67C2×D412yes—
145C2×C4×C67—21no—
146C2×C4×C67—21no—
147C2×C4×C67—21no—
148C2×C4×C67—21no—
149C2×C4×C67—21no—
150C2×C4×C67—21no—
151C2×C4×C67—42no—
152C2×C4×C67—81no—
153C2×C4×C67C2×D411yescharacteristic
154C2×C4×C67C2×D411yescharacteristic
155C2×C4×C67C2×D411yescharacteristic
156C2×C4×C67C2×D411yescharacteristic
157C2×C4×C67C2411yescharacteristic
158C22⋊D67—42no—
159C22⋊D67—81no—
160C2×Dic67—21no—
161C2×Dic67—21no—
162D268—21no—
163D268—21no—
164D268—21no—
165D268—42no—
166D268C2×D412yes—
167C4×D67—42no—
168C4×D67—81no—
169Dic134—21no—
170Dic134—21no—
171C67⋊C8—21no—
172C67⋊C8—21no—
173C67⋊C8—21no—
174C67⋊C8C2×D412yes—
175C22×C67—21no—
176C22×C67—21no—
177C22×C67—21no—
178C22×C67—22no—
179C22×C67—21no—
180C22×C67—21no—
181C22×C67—41no—
182C22×C67—41no—
183C22×C67—41no—
184C22×C67—41no—
185C22×C67—41no—
186C22×C67—84no—
187C22×C67#13512yescomplemented
188C22×C67C22×D411yescharacteristic
189C2×D67—22no—
190C2×D67—41no—
191C2×D67—41no—
192C2×D67—41no—
193C2×D67—41no—
194C2×D67—41no—
195C2×D67—84no—
196C4×C67—21no—
197C4×C67—21no—
198C4×C67—21no—
199C4×C67—21no—
200C4×C67—41no—
201C4×C67—41no—
202C4×C67C22×D411yescharacteristic
203C4×C67H(5,𝔽2)11yescharacteristic
204Dic67—41no—
205Dic67—41no—
206C2×C67—22no—
207C2×C67—41no—
208C2×C67—42no—
209C2×C67—81no—
210C2×C67D4211yescharacteristic
211C2×C67C2×#13511yescharacteristic
212C2×C67#57611yescharacteristic
213D67—42no—
214D67—81no—
215#3280—671nomaximal, Sylow
216C67#328011yescharacteristic, complemented, Sylow
217C2×#135—671no—
218C22×D8—671no—
219#546—671no—
220#546D6711yescharacteristic, complemented
221#495—671no—
222#467—671no—
223#466—671no—
224#463—671no—
225#463—671no—
226#450—671no—
227#450—671no—
228#449—671no—
229#449—671no—
230C4×D8—671no—
231C8×D4—671no—
232C2×#41—671no—
233C2×#41C2×D6711yescharacteristic, complemented
234C22×D4—671no—
235C22×D4C2×D6711yescharacteristic
236#135—1344no—
237C2×Q16—671no—
238C2×SD16—671no—
239C2×SD16—671no—
240C2×SD16—671no—
241C2×SD16—671no—
242C2×D8—671no—
243C2×D8—671no—
244C2×D8—671no—
245C2×D8—671no—
246C2×D8—1344no—
247C22×C8—671no—
248C22×C8—671no—
249#124—671no—
250#124C2×D6711yescharacteristic, complemented
251#123—21no—
252#123—1341no—
253C22⋊Q8—671no—
254C22⋊Q8C2×D6711yescharacteristic
255#121—21no—
256#121—671no—
257#121—1341no—
258#121C2×D6711yescharacteristic, complemented
259#120—21no—
260#120—1341no—
261C4×D4—671no—
262C4×D4C2×D6711yescharacteristic
263C4×D4C2×D6711yescharacteristic, complemented
264C2×C22⋊C4—21no—
265C2×C22⋊C4—1341no—
266#107—671no—
267C4⋊C8—671no—
268#103—671no—
269#103—671no—
270#102—671no—
271#102—671no—
272#102—671no—
273#102—671no—
274C22⋊C8—671no—
275C22⋊C8—671no—
276C4×C8—671no—
277C24—21no—
278C24—1341no—
279#41—22no—
280#41—41no—
281#41—2681no—
282#41—1342no—
283C2×Q8—671no—
284C2×Q8C22×D6711yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—2681no—
291C2×D4—671no—
292C2×D4—671no—
293C2×D4—1342no—
294C2×D4—1341no—
295C2×D4—1341no—
296C2×D4—1341no—
297C2×D4C22⋊D6711yescharacteristic
298C2×D4C22×D6711yescharacteristic
299C2×D4C22×D6711yescharacteristic
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—1341no—
303C22×C4—1341no—
304C22×C4C22×D6711yescharacteristic
305C22×C4C22×D6711yescharacteristic
306Q16—1342no—
307SD16—2681no—
308SD16—2681no—
309SD16—1342no—
310SD16—1342no—
311D8—674no—
312D8—1342no—
313D8—1342no—
314D8—1342no—
315C2×C8—671no—
316C2×C8—671no—
317C2×C8—671no—
318C2×C8—671no—
319C2×C8—1342no—
320C2×C8—1342no—
321C4⋊C4—21no—
322C4⋊C4—671no—
323C4⋊C4—1341no—
324C4⋊C4C22⋊D6711yescharacteristic
325C4⋊C4C22×D6711yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—1341no—
332C22⋊C4—1341no—
333C22⋊C4—1341no—
334C22⋊C4—1341no—
335C22⋊C4—1341no—
336C22⋊C4C22⋊D6711yescharacteristic
337C22⋊C4C22⋊D6711yescharacteristic
338C42C22×D6711yescharacteristic
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—2682no—
345C23—2682no—
346C23—5361no—
347C23—1341no—
348C23—1341no—
349C23C2×C22⋊D6711yescharacteristic
350C23C2×C22⋊D6711yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—1341no—
354Q8—1341no—
355D4—22no—
356D4—21no—
357D4—21no—
358D4—21no—
359D4—42no—
360D4—42no—
361D4—81no—
362D4—2682no—
363D4—2682no—
364D4—5361no—
365D4—672no—
366D4—1341no—
367D4—1341no—
368D4—1341no—
369D4D4×D6712yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—2682no—
379C2×C4—5361no—
380C2×C4—1341no—
381C2×C4—1341no—
382C2×C4C2×C22⋊D6711yescharacteristic
383C2×C4C2×C22⋊D6711yescharacteristic
384C2×C4C2×C22⋊D6711yescharacteristic
385C2×C4C2×C22⋊D6711yescharacteristic
386C2×C4C23×D6711yescharacteristic
387C8—672no—
388C8—1341no—
389C8—1341no—
390C8—1341no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—21no—
396C22—21no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—84no—
403C22—2681no—
404C22—2681no—
405C22—2681no—
406C22—2681no—
407C22—2681no—
408C22—5364no—
409C22—1342no—
410C22#1452406312yescomplemented
411C22C22×C22⋊D6711yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—41no—
417C4—41no—
418C4—2681no—
419C4—2681no—
420C4C2×D4×D6711yescharacteristic
421C4#1452408811yescharacteristic
422C2—22no—
423C2—41no—
424C2—42no—
425C2—81no—
426C2—2682no—
427C2—5361no—
428C2D4×C22⋊D6711yescharacteristic
429C2C2×#1452406311yescharacteristic
430C2#1686419911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#546×C67C2
socleC22×C67C22×D4
hypercenter#546D67
hypocenterC67#3280
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#14524063C2semidirect
2. C22×#11761613C2semidirect
3. #16864042C2semidirect
4. #16863778C2semidirect
5. #16863635C2semidirect
6. #16863630C2semidirect
7. #16863611C2semidirect
8. #16863608C2semidirect
9. #16863547C2semidirect
10. #16863546C2semidirect
11. #16863541C2semidirect
12. #16863539C2semidirect
13. C4×#11761613C2semidirect
14. D4×C67⋊C8C2semidirect
15. #546×C67C2semidirect
16. C2×#11761628C22semidirect
19. C2×#11761618C22semidirect
20. C2×#11761616C22semidirect
21. C2×#11761616C22semidirect
22. C2×#11761615C22semidirect
23. C2×#11761615C22semidirect
30. C22×C67⋊C8C22semidirect
31. #14524031C22semidirect
34. #14524017C22semidirect
37. C4×D268C22semidirect
39. #14523976C22semidirect
40. C4⋊(C67⋊C8)C22semidirect
41. #14523967C22semidirect
42. #14523966C22semidirect
43. #14523964C22semidirect
44. #14523964C22semidirect
45. #14523963C22semidirect
46. #14523963C22semidirect
48. C22⋊(C67⋊C8)C22semidirect
49. C4×C67⋊C8C22semidirect
50. C2×#41×C67C22semidirect
52. #124×C67C22semidirect
56. #121×C67C22semidirect
59. C4×D4×C67C22semidirect
82. #11761613D4semidirect
187. C22×C67#135semidirect
216. C67#3280semidirect
220. #546D67semidirect
233. C2×#41C2×D67semidirect
250. #124C2×D67semidirect
258. #121C2×D67semidirect
263. C4×D4C2×D67semidirect
410. C22#14524063semidirect