Finite Group Databasebeta

← #22571685

Subgroups — #22571685

Order 8576.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14524069C211yesmaximal, characteristic, complemented
2C22×#11761613C211yesmaximal, characteristic, complemented
3#16864042C211yesmaximal, characteristic, complemented
4#16863783C211yesmaximal, characteristic, complemented
5#16863647C211yesmaximal, characteristic, complemented
6#16863623C211yesmaximal, characteristic, complemented
7#16863611C211yesmaximal, characteristic, complemented
8#16863588C211yesmaximal, characteristic, complemented
9#16863547C211yesmaximal, characteristic, complemented
10#16863539C211yesmaximal, characteristic, complemented
11#16863530C211yesmaximal, characteristic, complemented
12#16863529C211yesmaximal, characteristic, complemented
13#16863506C211yesmaximal, characteristic, complemented
14#16863468C211yesmaximal, characteristic, complemented
15D42×C67C211yesmaximal, characteristic, complemented
16C2×#11761628C2211yescharacteristic, complemented
17C22×D268C2211yescharacteristic, complemented
18#14524069—24no—
19C2×#11761616C2211yescharacteristic, complemented
20C2×#11761616C2211yescharacteristic, complemented
21C2×#11761616C2211yescharacteristic, complemented
22C2×#11761613—24no—
23C2×#11761613C2211yescharacteristic
24C2×#11761613C2211yescharacteristic
25C2×#11761613C2211yescharacteristic, complemented
26C2×#11761613C2211yescharacteristic, complemented
27C2×#11761613C2211yescharacteristic, complemented
28C2×#11761613C2211yescharacteristic, complemented
29C2×#11761612C2211yescharacteristic, complemented
30C22×C67⋊C8C2211yescharacteristic, complemented
31#14524031C2211yescharacteristic, complemented
32#14524028—21no—
33#14524022C2211yescharacteristic, complemented
34#14524017C2211yescharacteristic, complemented
35#14524016—21no—
36#14524011—21no—
37C4×D268C2211yescharacteristic, complemented
38C22⋊C4×D67—21no—
39#14523974C2211yescharacteristic, complemented
40C4⋊(C67⋊C8)C2211yescharacteristic, complemented
41#14523966C2211yescharacteristic, complemented
42#14523964C2211yescharacteristic, complemented
43#14523964C2211yescharacteristic, complemented
44#14523964C2211yescharacteristic, complemented
45#14523963C2211yescharacteristic, complemented
46#14523963C2211yescharacteristic, complemented
47C22⋊(C67⋊C8)C2211yescharacteristic, complemented
48C22⋊(C67⋊C8)C2211yescharacteristic, complemented
49#14523954C2211yescharacteristic, complemented
50C22×D4×C67—21no—
51C22×D4×C67C2211yescharacteristic
52C22×D4×C67C2211yescharacteristic, complemented
53#127×C67C2211yescharacteristic, complemented
54#121×C67—21no—
55#121×C67C2211yescharacteristic
56#121×C67C2211yescharacteristic, complemented
57#120×C67—21no—
58#120×C67—21no—
59C4×D4×C67C2211yescharacteristic
60C4×D4×C67C2211yescharacteristic, complemented
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#11761616—22no—
75#11761616—22no—
76#11761616—41no—
77#11761613—22no—
78#11761613—22no—
79#11761613—22no—
80#11761613—22no—
81#11761613—41no—
82#11761613D414yescomplemented
83#11761612—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—
97C24×C67—21no—
98C2×D4×C67—22no—
99C2×D4×C67—22no—
100C2×D4×C67—21no—
101C2×D4×C67—21no—
102C2×D4×C67—21no—
103C2×D4×C67—21no—
104C2×D4×C67—21no—
105C2×D4×C67—21no—
106C2×D4×C67—41no—
107C2×D4×C67—41no—
108C2×D4×C67—44no—
109C2×D4×C67D411yescharacteristic
110C2×D4×C67C2311yescharacteristic
111C2×D4×C67C2311yescharacteristic
112C2×D4×C67C2311yescharacteristic
113C22×C4×C67—21no—
114C22×C4×C67C2311yescharacteristic
115C22×C4×C67C2311yescharacteristic
116C4⋊C4×C67D411yescharacteristic
117C4⋊C4×C67C2311yescharacteristic
118C22⋊C4×C67—21no—
119C22⋊C4×C67—21no—
120C22⋊C4×C67—21no—
121C22⋊C4×C67D411yescharacteristic
122C22⋊C4×C67D411yescharacteristic
123C42×C67C2311yescharacteristic
124C23×C67—21no—
125C23×C67—21no—
126C23×C67—21no—
127C23×C67—42no—
128C23×C67—42no—
129C23×C67—42no—
130C23×C67—42no—
131C23×C67—81no—
132C23×C67—81no—
133C23×C67C2×D411yescharacteristic
134C23×C67C2×D411yescharacteristic
135C22×D67—21no—
136C22×D67—21no—
137C22×D67—42no—
138C22×D67—42no—
139C22×D67—81no—
140D4×C67—22no—
141D4×C67—21no—
142D4×C67—21no—
143D4×C67—21no—
144D4×C67—21no—
145D4×C67—21no—
146D4×C67—42no—
147D4×C67—42no—
148D4×C67—42no—
149D4×C67—42no—
150D4×C67—42no—
151D4×C67—81no—
152D4×C67C2×D412yes—
153C2×C4×C67—21no—
154C2×C4×C67—21no—
155C2×C4×C67—21no—
156C2×C4×C67—21no—
157C2×C4×C67—21no—
158C2×C4×C67—42no—
159C2×C4×C67C2×D411yescharacteristic
160C2×C4×C67C2×D411yescharacteristic
161C2×C4×C67C2×D411yescharacteristic
162C2×C4×C67C2×D411yescharacteristic
163C2×C4×C67C2411yescharacteristic
164C22⋊D67—42no—
165C22⋊D67—81no—
166C2×Dic67—21no—
167C2×Dic67—21no—
168D268—21no—
169D268—21no—
170D268—21no—
171D268—42no—
172D268C2×D412yescomplemented
173C4×D67—42no—
174C4×D67—81no—
175Dic134—21no—
176Dic134—21no—
177C67⋊C8—21no—
178C67⋊C8—21no—
179C67⋊C8—21no—
180C67⋊C8C2×D412yescomplemented
181C22×C67—21no—
182C22×C67—21no—
183C22×C67—21no—
184C22×C67—22no—
185C22×C67—21no—
186C22×C67—21no—
187C22×C67—41no—
188C22×C67—41no—
189C22×C67—41no—
190C22×C67—41no—
191C22×C67—41no—
192C22×C67—41no—
193C22×C67—41no—
194C22×C67—41no—
195C22×C67—84no—
196C22×C67—84no—
197C22×C67AGL(1,ℤ/8)12yescomplemented
198C22×C67C22×D411yescharacteristic
199C2×D67—22no—
200C2×D67—41no—
201C2×D67—41no—
202C2×D67—41no—
203C2×D67—41no—
204C2×D67—41no—
205C2×D67—84no—
206C4×C67—21no—
207C4×C67—21no—
208C4×C67—21no—
209C4×C67—21no—
210C4×C67—41no—
211C4×C67C22×D411yescharacteristic
212C4×C67H(5,𝔽2)11yescharacteristic
213Dic67—41no—
214Dic67—41no—
215C2×C67—22no—
216C2×C67—41no—
217C2×C67—42no—
218C2×C67—81no—
219C2×C67—81no—
220C2×C67D4211yescharacteristic
221C2×C67C2×AGL(1,ℤ/8)11yescharacteristic
222C2×C67#57611yescharacteristic
223D67—42no—
224D67—81no—
225#3264—671nomaximal, Sylow
226C67#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)—671no—
228C22×D8—671no—
229#546—671no—
230D42D6711yescharacteristic, complemented
231#496—671no—
232#469—671no—
233#465—671no—
234#463—671no—
235#459—671no—
236#450—671no—
237#449—671no—
238#447—671no—
239#447—671no—
240#442—671no—
241#435—671no—
242C2×#41—671no—
243C22×D4—21no—
244C22×D4—671no—
245C22×D4C2×D6711yescharacteristic
246C22×D4C2×D6711yescharacteristic, complemented
247AGL(1,ℤ/8)—1344no—
248C2×SD16—671no—
249C2×SD16—671no—
250C2×SD16—671no—
251C2×D8—671no—
252C2×D8—671no—
253C2×D8—671no—
254C2×D8—671no—
255C2×D8—671no—
256C2×D8—671no—
257C2×D8—1344no—
258C2×M16—671no—
259C22×C8—671no—
260#127C2×D6711yescharacteristic, complemented
261#124—671no—
262#123—1341no—
263C22⋊Q8—671no—
264#121—21no—
265#121—671no—
266#121—1341no—
267#121C2×D6711yescharacteristic
268#121C2×D6711yescharacteristic, complemented
269#120—21no—
270#120—21no—
271#120—1341no—
272C4×D4—671no—
273C4×D4C2×D6711yescharacteristic
274C4×D4C2×D6711yescharacteristic, complemented
275C2×C22⋊C4—1341no—
276#106—671no—
277C4⋊C8—671no—
278#103—671no—
279#102—671no—
280#102—671no—
281#102—671no—
282#102—671no—
283#102—671no—
284C22⋊C8—671no—
285C22⋊C8—671no—
286#97—671no—
287C24—21no—
288C24—21no—
289C24—1341no—
290#41—2681no—
291#41—1342no—
292C2×Q8—671no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—44no—
304C2×D4—2681no—
305C2×D4—671no—
306C2×D4—671no—
307C2×D4—1342no—
308C2×D4—1341no—
309C2×D4—1341no—
310C2×D4—1341no—
311C2×D4C22⋊D6711yescharacteristic
312C2×D4C22×D6711yescharacteristic
313C2×D4C22×D6711yescharacteristic
314C2×D4C22×D6711yescharacteristic
315C22×C4—21no—
316C22×C4—1341no—
317C22×C4—1341no—
318C22×C4C22×D6711yescharacteristic
319C22×C4C22×D6711yescharacteristic
320SD16—2681no—
321SD16—1342no—
322SD16—1342no—
323D8—2681no—
324D8—674no—
325D8—1342no—
326D8—1342no—
327D8—1342no—
328D8—1342no—
329M16—1342no—
330C2×C8—671no—
331C2×C8—671no—
332C2×C8—671no—
333C2×C8—671no—
334C2×C8—1342no—
335C4⋊C4—671no—
336C4⋊C4—1341no—
337C4⋊C4C22⋊D6711yescharacteristic
338C4⋊C4C22×D6711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—1341no—
343C22⋊C4—1341no—
344C22⋊C4—1341no—
345C22⋊C4—1341no—
346C22⋊C4—1341no—
347C22⋊C4C22⋊D6711yescharacteristic
348C22⋊C4C22⋊D6711yescharacteristic
349C42C22×D6711yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—81no—
358C23—81no—
359C23—2682no—
360C23—2682no—
361C23—5361no—
362C23—1341no—
363C23—1341no—
364C23C2×C22⋊D6711yescharacteristic
365C23C2×C22⋊D6711yescharacteristic
366Q8—1341no—
367Q8—1341no—
368D4—22no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—42no—
376D4—42no—
377D4—42no—
378D4—42no—
379D4—81no—
380D4—2682no—
381D4—2682no—
382D4—5361no—
383D4—672no—
384D4—1341no—
385D4—1341no—
386D4—1341no—
387D4D4×D6712yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—42no—
394C2×C4—2682no—
395C2×C4—5361no—
396C2×C4—1341no—
397C2×C4—1341no—
398C2×C4C2×C22⋊D6711yescharacteristic
399C2×C4C2×C22⋊D6711yescharacteristic
400C2×C4C2×C22⋊D6711yescharacteristic
401C2×C4C2×C22⋊D6711yescharacteristic
402C2×C4C23×D6711yescharacteristic
403C8—672no—
404C8—1341no—
405C8—1341no—
406C8—1341no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—21no—
412C22—21no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—84no—
422C22—84no—
423C22—2681no—
424C22—2681no—
425C22—2681no—
426C22—2681no—
427C22—2681no—
428C22—5364no—
429C22—1342no—
430C22#1452406912yescomplemented
431C22C22×C22⋊D6711yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—41no—
437C4—2681no—
438C4—2681no—
439C4C2×D4×D6711yescharacteristic
440C4#1452408811yescharacteristic
441C2—22no—
442C2—41no—
443C2—42no—
444C2—81no—
445C2—81no—
446C2—2682no—
447C2—5361no—
448C2D4×C22⋊D6711yescharacteristic
449C2C2×#1452406911yescharacteristic
450C2#1686419911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
FittingD42×C67C2
socleC22×C67C22×D4
hypercenterD42D67
hypocenterC67#3264
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#14524069C2semidirect
2. C22×#11761613C2semidirect
3. #16864042C2semidirect
4. #16863783C2semidirect
5. #16863647C2semidirect
6. #16863623C2semidirect
7. #16863611C2semidirect
8. #16863588C2semidirect
9. #16863547C2semidirect
10. #16863539C2semidirect
11. #16863530C2semidirect
12. #16863529C2semidirect
13. #16863506C2semidirect
14. #16863468C2semidirect
15. D42×C67C2semidirect
16. C2×#11761628C22semidirect
17. C22×D268C22semidirect
19. C2×#11761616C22semidirect
20. C2×#11761616C22semidirect
21. C2×#11761616C22semidirect
25. C2×#11761613C22semidirect
26. C2×#11761613C22semidirect
27. C2×#11761613C22semidirect
28. C2×#11761613C22semidirect
29. C2×#11761612C22semidirect
30. C22×C67⋊C8C22semidirect
31. #14524031C22semidirect
33. #14524022C22semidirect
34. #14524017C22semidirect
37. C4×D268C22semidirect
39. #14523974C22semidirect
40. C4⋊(C67⋊C8)C22semidirect
41. #14523966C22semidirect
42. #14523964C22semidirect
43. #14523964C22semidirect
44. #14523964C22semidirect
45. #14523963C22semidirect
46. #14523963C22semidirect
47. C22⋊(C67⋊C8)C22semidirect
48. C22⋊(C67⋊C8)C22semidirect
49. #14523954C22semidirect
52. C22×D4×C67C22semidirect
53. #127×C67C22semidirect
56. #121×C67C22semidirect
60. C4×D4×C67C22semidirect
82. #11761613D4semidirect
172. D268C2×D4semidirect
180. C67⋊C8C2×D4semidirect
197. C22×C67AGL(1,ℤ/8)semidirect
226. C67#3264semidirect
230. D42D67semidirect
246. C22×D4C2×D67semidirect
260. #127C2×D67semidirect
268. #121C2×D67semidirect
274. C4×D4C2×D67semidirect
430. C22#14524069semidirect