Finite Group Databasebeta

← #22569825

Subgroups — #22569825

Order 8576.

548 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14524067C211yesmaximal, characteristic, complemented
2C22×#11761611C211yesmaximal, characteristic, complemented
3#16863906C211yesmaximal, characteristic, complemented
4#121×D67C211yesmaximal, characteristic, complemented
5#16863648C211yesmaximal, characteristic, complemented
6#16863646C211yesmaximal, characteristic, complemented
7#16863643C211yesmaximal, characteristic, complemented
8#16863641C211yesmaximal, characteristic, complemented
9#16863630C211yesmaximal, characteristic, complemented
10#16863625C211yesmaximal, characteristic, complemented
11#16863623C211yesmaximal, characteristic, complemented
12#16863437C211yesmaximal, characteristic, complemented
13#16863399C211yesmaximal, characteristic, complemented
14#16863394C211yesmaximal, characteristic, complemented
15#466×C67C211yesmaximal, characteristic, complemented
16C2×#11761627C2211yescharacteristic
17C2×D4×D67—21no—
18C2×D4×D67C2211yescharacteristic
19C22×C4×D67C2211yescharacteristic
20#14524067—24no—
21C2×#11761617C2211yescharacteristic, complemented
22C2×#11761616C2211yescharacteristic, complemented
23C2×#11761613C2211yescharacteristic, complemented
24C2×#11761611—24no—
25C2×#11761611C2211yescharacteristic
26C2×#11761611C2211yescharacteristic
27C2×#11761611C2211yescharacteristic, complemented
28C2×#11761611C2211yescharacteristic, complemented
29C22×C67⋊C8C2211yescharacteristic, complemented
30#14524025—21no—
31#14524023C2211yescharacteristic, complemented
32#14524022C2211yescharacteristic, complemented
33#14524018C2211yescharacteristic
34#14524017C2211yescharacteristic
35#14524016—21no—
36#14524015—21no—
37#14524014C2211yescharacteristic
38#14524014C2211yescharacteristic, complemented
39#14524011—21no—
40D4×Dic67—21no—
41(C22⋊D67)⋊C4—21no—
42#14524001C2211yescharacteristic
43C4⋊C4×D67C2211yescharacteristic
44C22⋊C4×D67—21no—
45#14523976C2211yescharacteristic, complemented
46#14523975C2211yescharacteristic, complemented
47#14523974C2211yescharacteristic, complemented
48#14523968C2211yescharacteristic, complemented
49#14523966C2211yescharacteristic, complemented
50#14523965C2211yescharacteristic, complemented
51#14523964C2211yescharacteristic, complemented
52#14523964C2211yescharacteristic, complemented
53#14523963C2211yescharacteristic, complemented
54C2×D8×C67C2211yescharacteristic, complemented
55C22×C8×C67C2211yescharacteristic, complemented
56#121×C67C2211yescharacteristic
57#121×C67C2211yescharacteristic, complemented
58#107×C67C2211yescharacteristic, complemented
59#102×C67C2211yescharacteristic, complemented
60#102×C67C2211yescharacteristic, complemented
61C23×D67—21no—
62C23×D67D411yescharacteristic
63#11761627—22no—
64#11761627—22no—
65C2×Dic134C2311yescharacteristic
66C2×D268—21no—
67C2×D268C2311yescharacteristic
68C2×C22⋊D67—21no—
69C2×C22⋊D67—21no—
70C2×C22⋊D67—21no—
71C2×C22⋊D67—21no—
72C2×C22⋊D67—21no—
73D4×D67—22no—
74D4×D67—22no—
75D4×D67—44no—
76C22×Dic67—21no—
77C22×Dic67D411yescharacteristic
78C2×C4×D67—22no—
79C2×C4×D67—22no—
80C2×C4×D67—21no—
81C2×C4×D67D411yescharacteristic
82C2×C4×D67D411yescharacteristic
83C2×C4×D67C2311yescharacteristic
84C2×C4×D67C2311yescharacteristic
85#11761617—22no—
86#11761616—22no—
87#11761613—22no—
88#11761611—22no—
89#11761611—22no—
90#11761611—22no—
91#11761611D414yescomplemented
92C2×C67⋊C8—22no—
93C2×C67⋊C8C2311yescharacteristic
94C2×C67⋊C8C2311yescharacteristic
95C4⋊Dic67—21no—
96C4⋊Dic67C2311yescharacteristic
97C4⋊Dic67C2311yescharacteristic
98Dic67⋊C4—21no—
99Dic67⋊C4—21no—
100C22⋊Dic67—21no—
101C22⋊Dic67—21no—
102C22⋊Dic67—21no—
103(C2×D67)⋊C4—21no—
104(C2×D67)⋊C4—21no—
105(C2×D67)⋊C4—21no—
106(C2×D67)⋊C4—21no—
107C4×Dic67—21no—
108C2×D4×C67—21no—
109C2×D4×C67—21no—
110C2×D4×C67C2311yescharacteristic
111C2×D4×C67C2311yescharacteristic
112C22×C4×C67C2311yescharacteristic
113D8×C67—22no—
114C2×C8×C67—22no—
115C2×C8×C67C2311yescharacteristic
116C2×C8×C67C2311yescharacteristic
117C4⋊C4×C67C2311yescharacteristic
118C4⋊C4×C67C2311yescharacteristic
119C22⋊C4×C67—21no—
120C22⋊C4×C67—21no—
121C23×C67—21no—
122C23×C67—21no—
123C23×C67C2×D411yescharacteristic
124C22×D67—22no—
125C22×D67—22no—
126C22×D67—21no—
127C22×D67—41no—
128C22×D67—42no—
129C22×D67—42no—
130C22×D67—42no—
131C22×D67C2×D411yescharacteristic
132C22×D67C2×D411yescharacteristic
133D4×C67—21no—
134D4×C67—21no—
135D4×C67—21no—
136D4×C67—21no—
137D4×C67—42no—
138D4×C67—42no—
139C2×C4×C67—21no—
140C2×C4×C67—21no—
141C2×C4×C67—21no—
142C2×C4×C67—21no—
143C2×C4×C67C2×D411yescharacteristic
144C2×C4×C67C2411yescharacteristic
145C22⋊D67—42no—
146C22⋊D67—42no—
147C22⋊D67—42no—
148C22⋊D67—42no—
149C22⋊D67—42no—
150C2×Dic67—22no—
151C2×Dic67—21no—
152C2×Dic67—21no—
153C2×Dic67—21no—
154C2×Dic67—42no—
155C2×Dic67C2×D411yescharacteristic
156C2×Dic67C2×D411yescharacteristic
157D268—21no—
158D268—21no—
159D268—42no—
160C4×D67—22no—
161C4×D67—21no—
162C4×D67—21no—
163C4×D67—22no—
164C4×D67—42no—
165C4×D67C2×D412yes—
166C4×D67#4112yes—
167Dic134—21no—
168Dic134—21no—
169C8×C67—21no—
170C8×C67C2×D412yescomplemented
171C67⋊C8—21no—
172C67⋊C8C2×D412yescomplemented
173C22×C67—21no—
174C22×C67—21no—
175C22×C67—41no—
176C22×C67—41no—
177C22×C67—41no—
178C22×C67—41no—
179C22×C67—41no—
180C22×C67—41no—
181C22×C67AGL(1,ℤ/8)12yescomplemented
182C22×C67C22×D411yescharacteristic
183C2×D67—21no—
184C2×D67—21no—
185C2×D67—21no—
186C2×D67—21no—
187C2×D67—21no—
188C2×D67—44no—
189C2×D67—41no—
190C2×D67—41no—
191C2×D67—41no—
192C2×D67—84no—
193C2×D67#12112yes—
194C4×C67—21no—
195C4×C67—41no—
196C4×C67—41no—
197C4×C67C22×D411yescharacteristic
198C4×C67C2×#4111yescharacteristic
199Dic67—21no—
200Dic67—41no—
201Dic67—41no—
202Dic67—41no—
203Dic67#12112yes—
204C2×C67—22no—
205C2×C67—81no—
206C2×C67—81no—
207C2×C67C2×#12111yescharacteristic
208C2×C67C2×AGL(1,ℤ/8)11yescharacteristic
209C2×C67#57511yescharacteristic
210D67—22no—
211D67—41no—
212D67—81no—
213#3055—671nomaximal, Sylow
214C67#305511yescharacteristic, complemented, Sylow
215C2×AGL(1,ℤ/8)—671no—
216C22×M16—671no—
217#525—671no—
218C2×#121—671no—
219#469—671no—
220#469—671no—
221#468—671no—
222#468—671no—
223#466—671no—
224#466D6711yescharacteristic, complemented
225#465—671no—
226#465—671no—
227#428—671no—
228#418—671no—
229#417—671no—
230C2×#41—671no—
231C22×D4—671no—
232C22×D4—1341no—
233C23×C4—671no—
234AGL(1,ℤ/8)—1344no—
235C2×SD16—671no—
236C2×SD16—671no—
237C2×D8—671no—
238C2×D8C2×D6711yescharacteristic, complemented
239C2×M16—671no—
240C2×M16—671no—
241C2×M16—671no—
242C2×M16—671no—
243C2×M16—1344no—
244C22×C8—671no—
245C22×C8C2×D6711yescharacteristic, complemented
246#123—1341no—
247C22⋊Q8—671no—
248C22⋊Q8—671no—
249#121—671no—
250#121—671no—
251#121—671no—
252#121—671no—
253#121—1341no—
254#121—1341no—
255#121C2×D6711yescharacteristic
256#121C2×D6711yescharacteristic, complemented
257#120—1341no—
258C4×D4—1341no—
259C4×D4—1341no—
260#117—671no—
261C2×C4⋊C4—671no—
262C2×C22⋊C4—1341no—
263#107—671no—
264#107C2×D6711yescharacteristic, complemented
265#106—671no—
266#106—671no—
267#103—671no—
268#103—671no—
269#102—671no—
270#102—671no—
271#102—671no—
272#102—671no—
273#102C2×D6711yescharacteristic, complemented
274#102C2×D6711yescharacteristic, complemented
275C24—671no—
276C24—1341no—
277#41—1342no—
278#41—1342no—
279C2×Q8—671no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—2684no—
283C2×D4—671no—
284C2×D4—1341no—
285C2×D4—1342no—
286C2×D4—1342no—
287C2×D4—1341no—
288C2×D4—1341no—
289C2×D4—1341no—
290C2×D4—1341no—
291C2×D4—1341no—
292C2×D4C22×D6711yescharacteristic
293C2×D4C22×D6711yescharacteristic
294C22×C4—671no—
295C22×C4—671no—
296C22×C4—671no—
297C22×C4—671no—
298C22×C4—671no—
299C22×C4—1341no—
300C22×C4—1342no—
301C22×C4—1342no—
302C22×C4—1341no—
303C22×C4C22×D6711yescharacteristic
304SD16—1342no—
305SD16—1342no—
306D8—22no—
307D8—1342no—
308M16—674no—
309M16—1342no—
310M16—1342no—
311M16—1342no—
312C2×C8—22no—
313C2×C8—671no—
314C2×C8—671no—
315C2×C8—1342no—
316C2×C8C22×D6711yescharacteristic
317C2×C8C22×D6711yescharacteristic
318C4⋊C4—671no—
319C4⋊C4—671no—
320C4⋊C4—1341no—
321C4⋊C4—1341no—
322C4⋊C4—1341no—
323C4⋊C4C22×D6711yescharacteristic
324C4⋊C4C22×D6711yescharacteristic
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—1341no—
328C22⋊C4—1341no—
329C22⋊C4—1341no—
330C22⋊C4—1341no—
331C22⋊C4—1341no—
332C22⋊C4—1341no—
333C22⋊C4—1341no—
334C42—1341no—
335C23—21no—
336C23—21no—
337C23—2681no—
338C23—2682no—
339C23—2682no—
340C23—2682no—
341C23—671no—
342C23—671no—
343C23—1342no—
344C23—1341no—
345C23—1342no—
346C23D4×D6711yescharacteristic
347Q8—1341no—
348Q8—1341no—
349D4—21no—
350D4—21no—
351D4—21no—
352D4—21no—
353D4—42no—
354D4—42no—
355D4—2682no—
356D4—2682no—
357D4—2682no—
358D4—2682no—
359D4—2682no—
360D4—2682no—
361D4—1341no—
362D4—1341no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—2682no—
368C2×C4—2682no—
369C2×C4—671no—
370C2×C4—672no—
371C2×C4—672no—
372C2×C4—671no—
373C2×C4—1342no—
374C2×C4—1342no—
375C2×C4—1341no—
376C2×C4—1341no—
377C2×C4—1341no—
378C2×C4—1342no—
379C2×C4—1341no—
380C2×C4—1341no—
381C2×C4D4×D6711yescharacteristic
382C2×C4C23×D6711yescharacteristic
383C8—21no—
384C8—672no—
385C8—1341no—
386C8D4×D6712yescomplemented
387C22—21no—
388C22—21no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—2684no—
396C22—2681no—
397C22—2681no—
398C22—2681no—
399C22—5364no—
400C22—672no—
401C22—1341no—
402C22—1341no—
403C22—1341no—
404C22—1341no—
405C22—1341no—
406C22#1452406712yescomplemented
407C22C2×D4×D6711yescharacteristic
408C4—21no—
409C4—41no—
410C4—41no—
411C4—2681no—
412C4—2681no—
413C4—2681no—
414C4—672no—
415C4—1341no—
416C4C2×D4×D6711yescharacteristic
417C4#41×D6711yescharacteristic
418C2—22no—
419C2—81no—
420C2—81no—
421C2—2681no—
422C2—5361no—
423C2—1342no—
424C2#121×D6711yescharacteristic
425C2C2×#1452406711yescharacteristic
426C2#1686419311yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#466×C67C2
socleC22×C67C22×D4
hypercenter#466D67
hypocenterC67#3055
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#14524067C2semidirect
2. C22×#11761611C2semidirect
3. #16863906C2semidirect
4. #121×D67C2semidirect
5. #16863648C2semidirect
6. #16863646C2semidirect
7. #16863643C2semidirect
8. #16863641C2semidirect
9. #16863630C2semidirect
10. #16863625C2semidirect
11. #16863623C2semidirect
12. #16863437C2semidirect
13. #16863399C2semidirect
14. #16863394C2semidirect
15. #466×C67C2semidirect
21. C2×#11761617C22semidirect
22. C2×#11761616C22semidirect
23. C2×#11761613C22semidirect
27. C2×#11761611C22semidirect
28. C2×#11761611C22semidirect
29. C22×C67⋊C8C22semidirect
31. #14524023C22semidirect
32. #14524022C22semidirect
38. #14524014C22semidirect
45. #14523976C22semidirect
46. #14523975C22semidirect
47. #14523974C22semidirect
48. #14523968C22semidirect
49. #14523966C22semidirect
50. #14523965C22semidirect
51. #14523964C22semidirect
52. #14523964C22semidirect
53. #14523963C22semidirect
54. C2×D8×C67C22semidirect
55. C22×C8×C67C22semidirect
57. #121×C67C22semidirect
58. #107×C67C22semidirect
59. #102×C67C22semidirect
60. #102×C67C22semidirect
91. #11761611D4semidirect
170. C8×C67C2×D4semidirect
172. C67⋊C8C2×D4semidirect
181. C22×C67AGL(1,ℤ/8)semidirect
214. C67#3055semidirect
224. #466D67semidirect
238. C2×D8C2×D67semidirect
245. C22×C8C2×D67semidirect
256. #121C2×D67semidirect
264. #107C2×D67semidirect
273. #102C2×D67semidirect
274. #102C2×D67semidirect
386. C8D4×D67semidirect
406. C22#14524067semidirect