Finite Group Databasebeta

← #22571705

Subgroups — #22571705

Order 8576.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14524068C211yesmaximal, characteristic, complemented
2C22×#11761615C211yesmaximal, characteristic, complemented
3D4×D268C211yesmaximal, characteristic, complemented
4#16863786C211yesmaximal, characteristic, complemented
5#16863644C211yesmaximal, characteristic, complemented
6#16863630C211yesmaximal, characteristic, complemented
7#16863608C211yesmaximal, characteristic, complemented
8#16863596C211yesmaximal, characteristic, complemented
9#16863546C211yesmaximal, characteristic, complemented
10#16863541C211yesmaximal, characteristic, complemented
11#16863535C211yesmaximal, characteristic, complemented
12#16863529C211yesmaximal, characteristic, complemented
13#16863494C211yesmaximal, characteristic, complemented
14#16863468C211yesmaximal, characteristic, complemented
15#548×C67C211yesmaximal, characteristic, complemented
16C22×D268C2211yescharacteristic
17C22×D268C2211yescharacteristic
18C2×D4×D67—21no—
19#14524068—24no—
20C2×#11761615—24no—
21C2×#11761615C2211yescharacteristic
22C2×#11761615C2211yescharacteristic
23C2×#11761615C2211yescharacteristic, complemented
24C2×#11761615C2211yescharacteristic, complemented
25C2×#11761615C2211yescharacteristic, complemented
26C2×#11761615C2211yescharacteristic, complemented
27C2×#11761613C2211yescharacteristic
28C2×#11761613C2211yescharacteristic, complemented
29C2×#11761613C2211yescharacteristic, complemented
30C2×#11761612C2211yescharacteristic, complemented
31C22×C67⋊C8C2211yescharacteristic, complemented
32#14524040C2211yescharacteristic
33#14524018—21no—
34#14524017C2211yescharacteristic
35#14524017C2211yescharacteristic
36#14524013—21no—
37#14524013—21no—
38C4×D268C2211yescharacteristic
39#14523976C2211yescharacteristic, complemented
40C4⋊(C67⋊C8)C2211yescharacteristic, complemented
41#14523967C2211yescharacteristic, complemented
42#14523967C2211yescharacteristic, complemented
43#14523964C2211yescharacteristic, complemented
44#14523963C2211yescharacteristic
45#14523963C2211yescharacteristic, complemented
46#14523963C2211yescharacteristic, complemented
47C22⋊(C67⋊C8)C2211yescharacteristic, complemented
48C22⋊(C67⋊C8)C2211yescharacteristic, complemented
49#14523954C2211yescharacteristic, complemented
50C2×#41×C67C2211yescharacteristic, complemented
51C22×Q8×C67C2211yescharacteristic, complemented
52#124×C67—21no—
53#124×C67C2211yescharacteristic, complemented
54C22⋊Q8×C67—21no—
55C22⋊Q8×C67C2211yescharacteristic, complemented
56#121×C67—21no—
57#121×C67C2211yescharacteristic, complemented
58C4×Q8×C67C2211yescharacteristic, complemented
59C4×D4×C67—21no—
60C4×D4×C67C2211yescharacteristic
61C23×D67—21no—
62C23×D67—21no—
63C2×D268—22no—
64C2×D268—22no—
65C2×D268—21no—
66C2×D268—21no—
67C2×D268—21no—
68C2×D268—21no—
69C2×D268—41no—
70C2×D268—41no—
71C2×D268C2311yescharacteristic
72C2×D268C2311yescharacteristic
73C2×D268C2311yescharacteristic
74C2×C22⋊D67—21no—
75C2×C22⋊D67—21no—
76D4×D67—44no—
77C2×C4×D67—21no—
78#11761615—22no—
79#11761615—22no—
80#11761615—22no—
81#11761615—22no—
82#11761615—41no—
83#11761615D414yescomplemented
84#11761613—22no—
85#11761613—22no—
86#11761613—41no—
87#11761612—22no—
88C2×C67⋊C8—22no—
89C2×C67⋊C8C2311yescharacteristic
90C2×C67⋊C8C2311yescharacteristic
91C2×C67⋊C8C2311yescharacteristic
92C2×C67⋊C8C2311yescharacteristic
93C4⋊Dic67C2311yescharacteristic
94(C2×D67)⋊C4—21no—
95(C2×D67)⋊C4—21no—
96(C2×D67)⋊C4—21no—
97#41×C67—22no—
98#41×C67—41no—
99C2×Q8×C67—21no—
100C2×Q8×C67—22no—
101C2×Q8×C67—41no—
102C2×Q8×C67C2311yescharacteristic
103C2×Q8×C67C2311yescharacteristic
104C2×D4×C67—21no—
105C2×D4×C67—21no—
106C2×D4×C67D411yescharacteristic
107C2×D4×C67C2311yescharacteristic
108C22×C4×C67—21no—
109C22×C4×C67—21no—
110C22×C4×C67C2311yescharacteristic
111C22×C4×C67C2311yescharacteristic
112C4⋊C4×C67—21no—
113C4⋊C4×C67—21no—
114C4⋊C4×C67D411yescharacteristic
115C4⋊C4×C67C2311yescharacteristic
116C22⋊C4×C67—21no—
117C22⋊C4×C67—21no—
118C22⋊C4×C67—21no—
119C22⋊C4×C67—21no—
120C22⋊C4×C67D411yescharacteristic
121C22⋊C4×C67D411yescharacteristic
122C42×C67—21no—
123C42×C67C2311yescharacteristic
124C23×C67—21no—
125C23×C67C2×D411yescharacteristic
126C23×C67C2×D411yescharacteristic
127C22×D67—21no—
128C22×D67—21no—
129C22×D67—21no—
130C22×D67—42no—
131C22×D67—42no—
132C22×D67—42no—
133C22×D67—42no—
134C22×D67—81no—
135C22×D67—81no—
136Q8×C67—21no—
137Q8×C67—21no—
138Q8×C67—21no—
139Q8×C67—42no—
140Q8×C67C2×D412yescomplemented
141D4×C67—22no—
142D4×C67—21no—
143D4×C67—21no—
144D4×C67—42no—
145D4×C67—42no—
146C2×C4×C67—21no—
147C2×C4×C67—21no—
148C2×C4×C67—21no—
149C2×C4×C67—21no—
150C2×C4×C67—21no—
151C2×C4×C67—21no—
152C2×C4×C67—21no—
153C2×C4×C67—42no—
154C2×C4×C67—42no—
155C2×C4×C67C2×D411yescharacteristic
156C2×C4×C67C2×D411yescharacteristic
157C2×C4×C67C2×D411yescharacteristic
158C2×C4×C67C2×D411yescharacteristic
159C2×C4×C67C2411yescharacteristic
160C22⋊D67—42no—
161C22⋊D67—42no—
162C2×Dic67—21no—
163D268—21no—
164D268—21no—
165D268—21no—
166D268—21no—
167D268—21no—
168D268—42no—
169D268—42no—
170D268—42no—
171D268—81no—
172D268C2×D412yes—
173C4×D67—42no—
174C67⋊C8—21no—
175C67⋊C8—21no—
176C67⋊C8—21no—
177C67⋊C8C2×D412yescomplemented
178C22×C67—21no—
179C22×C67—21no—
180C22×C67—21no—
181C22×C67—21no—
182C22×C67—21no—
183C22×C67—41no—
184C22×C67—41no—
185C22×C67—41no—
186C22×C67AGL(1,ℤ/8)12yescomplemented
187C22×C67C22×D411yescharacteristic
188C2×D67—22no—
189C2×D67—41no—
190C2×D67—41no—
191C2×D67—41no—
192C2×D67—41no—
193C2×D67—41no—
194C2×D67—41no—
195C2×D67—41no—
196C2×D67—41no—
197C2×D67—84no—
198C2×D67—84no—
199C4×C67—21no—
200C4×C67—21no—
201C4×C67—22no—
202C4×C67—21no—
203C4×C67—21no—
204C4×C67—41no—
205C4×C67—41no—
206C4×C67C22×D411yescharacteristic
207C4×C67H(5,𝔽2)11yescharacteristic
208Dic67—41no—
209C2×C67—22no—
210C2×C67—41no—
211C2×C67—81no—
212C2×C67D4211yescharacteristic
213C2×C67C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C67#57711yescharacteristic
215D67—42no—
216D67—81no—
217D67—81no—
218#3265—671nomaximal, Sylow
219C67#326511yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—671no—
221C22×SD16—671no—
222#548D6711yescharacteristic, complemented
223D42—671no—
224#496—671no—
225#468—671no—
226#466—671no—
227#463—671no—
228#460—671no—
229#450—671no—
230#449—671no—
231#448—671no—
232#447—671no—
233#440—671no—
234#435—671no—
235C2×#41C2×D6711yescharacteristic, complemented
236C22×Q8C2×D6711yescharacteristic, complemented
237C22×D4—671no—
238C22×D4—671no—
239C22×D4—1341no—
240AGL(1,ℤ/8)—1344no—
241C2×SD16—671no—
242C2×SD16—671no—
243C2×SD16—671no—
244C2×SD16—671no—
245C2×SD16—671no—
246C2×SD16—671no—
247C2×SD16—1344no—
248C2×D8—671no—
249C2×D8—671no—
250C2×D8—671no—
251C2×M16—671no—
252C22×C8—671no—
253#127—671no—
254#124—21no—
255#124C2×D6711yescharacteristic, complemented
256C22⋊Q8—21no—
257C22⋊Q8C2×D6711yescharacteristic, complemented
258#121—21no—
259#121—671no—
260#121—671no—
261#121—1341no—
262#121C2×D6711yescharacteristic, complemented
263#120—1341no—
264#120—1341no—
265C4×Q8C2×D6711yescharacteristic, complemented
266C4×D4—21no—
267C4×D4—671no—
268C4×D4C2×D6711yescharacteristic
269#107—671no—
270C4⋊C8—671no—
271#103—671no—
272#103—671no—
273#102—671no—
274#102—671no—
275#102—671no—
276#102—671no—
277C22⋊C8—671no—
278C22⋊C8—671no—
279#97—671no—
280C24—1341no—
281C24—1341no—
282#41—22no—
283#41—41no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8C22×D6711yescharacteristic
288C2×Q8C22×D6711yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—2681no—
292C2×D4—2681no—
293C2×D4—2684no—
294C2×D4—671no—
295C2×D4—671no—
296C2×D4—671no—
297C2×D4—1342no—
298C2×D4—1342no—
299C2×D4—1341no—
300C2×D4—1341no—
301C2×D4—1341no—
302C2×D4—1341no—
303C2×D4—1341no—
304C2×D4—1341no—
305C2×D4C22⋊D6711yescharacteristic
306C2×D4C22×D6711yescharacteristic
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—1341no—
310C22×C4C22×D6711yescharacteristic
311C22×C4C22×D6711yescharacteristic
312SD16—2681no—
313SD16—674no—
314SD16—1342no—
315SD16—1342no—
316SD16—1342no—
317SD16—1342no—
318D8—2681no—
319D8—1342no—
320D8—1342no—
321M16—1342no—
322C2×C8—671no—
323C2×C8—671no—
324C2×C8—671no—
325C2×C8—671no—
326C2×C8—1342no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—671no—
330C4⋊C4C22⋊D6711yescharacteristic
331C4⋊C4C22×D6711yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—1341no—
337C22⋊C4—1341no—
338C22⋊C4—1341no—
339C22⋊C4C22⋊D6711yescharacteristic
340C22⋊C4C22⋊D6711yescharacteristic
341C42—21no—
342C42C22×D6711yescharacteristic
343C23—21no—
344C23—2682no—
345C23—2682no—
346C23—2682no—
347C23—2682no—
348C23—5361no—
349C23—5361no—
350C23—1341no—
351C23—1341no—
352C23—1341no—
353C23C2×C22⋊D6711yescharacteristic
354C23C2×C22⋊D6711yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—21no—
358Q8—42no—
359Q8D4×D6712yescomplemented
360D4—22no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—42no—
365D4—2682no—
366D4—2682no—
367D4—2682no—
368D4—2682no—
369D4—2682no—
370D4—5361no—
371D4—672no—
372D4—1341no—
373D4—1341no—
374D4—1341no—
375D4—1341no—
376D4—1341no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—2682no—
387C2×C4—1341no—
388C2×C4C2×C22⋊D6711yescharacteristic
389C2×C4C2×C22⋊D6711yescharacteristic
390C2×C4C2×C22⋊D6711yescharacteristic
391C2×C4C2×C22⋊D6711yescharacteristic
392C2×C4C23×D6711yescharacteristic
393C8—672no—
394C8—1341no—
395C8—1341no—
396C8—1341no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—2681no—
406C22—2681no—
407C22—2681no—
408C22—2681no—
409C22—2681no—
410C22—2681no—
411C22—2681no—
412C22—2681no—
413C22—5364no—
414C22—5364no—
415C22—1342no—
416C22#1452406812yescomplemented
417C22C22×C22⋊D6711yescharacteristic
418C4—21no—
419C4—21no—
420C4—22no—
421C4—21no—
422C4—21no—
423C4—41no—
424C4—41no—
425C4—2681no—
426C4C2×D4×D6711yescharacteristic
427C4#1452408811yescharacteristic
428C2—22no—
429C2—41no—
430C2—81no—
431C2—2682no—
432C2—5361no—
433C2—5361no—
434C2D4×C22⋊D6711yescharacteristic
435C2C2×#1452406811yescharacteristic
436C2#1686420411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#548×C67C2
socleC22×C67C22×D4
hypercenter#548D67
hypocenterC67#3265
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#14524068C2semidirect
2. C22×#11761615C2semidirect
3. D4×D268C2semidirect
4. #16863786C2semidirect
5. #16863644C2semidirect
6. #16863630C2semidirect
7. #16863608C2semidirect
8. #16863596C2semidirect
9. #16863546C2semidirect
10. #16863541C2semidirect
11. #16863535C2semidirect
12. #16863529C2semidirect
13. #16863494C2semidirect
14. #16863468C2semidirect
15. #548×C67C2semidirect
23. C2×#11761615C22semidirect
24. C2×#11761615C22semidirect
25. C2×#11761615C22semidirect
26. C2×#11761615C22semidirect
28. C2×#11761613C22semidirect
29. C2×#11761613C22semidirect
30. C2×#11761612C22semidirect
31. C22×C67⋊C8C22semidirect
39. #14523976C22semidirect
40. C4⋊(C67⋊C8)C22semidirect
41. #14523967C22semidirect
42. #14523967C22semidirect
43. #14523964C22semidirect
45. #14523963C22semidirect
46. #14523963C22semidirect
47. C22⋊(C67⋊C8)C22semidirect
48. C22⋊(C67⋊C8)C22semidirect
49. #14523954C22semidirect
50. C2×#41×C67C22semidirect
51. C22×Q8×C67C22semidirect
53. #124×C67C22semidirect
55. C22⋊Q8×C67C22semidirect
57. #121×C67C22semidirect
58. C4×Q8×C67C22semidirect
83. #11761615D4semidirect
140. Q8×C67C2×D4semidirect
177. C67⋊C8C2×D4semidirect
186. C22×C67AGL(1,ℤ/8)semidirect
219. C67#3265semidirect
222. #548D67semidirect
235. C2×#41C2×D67semidirect
236. C22×Q8C2×D67semidirect
255. #124C2×D67semidirect
257. C22⋊Q8C2×D67semidirect
262. #121C2×D67semidirect
265. C4×Q8C2×D67semidirect
359. Q8D4×D67semidirect
416. C22#14524068semidirect