Finite Group Databasebeta

← #22569821

Subgroups — #22569821

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×#14524070C211yesmaximal, characteristic, complemented
2C22×#11761611C211yesmaximal, characteristic, complemented
3#16863909C211yesmaximal, characteristic, complemented
4#121×D67C211yesmaximal, characteristic, complemented
5#16863650C211yesmaximal, characteristic, complemented
6#16863648C211yesmaximal, characteristic, complemented
7#16863645C211yesmaximal, characteristic, complemented
8#16863639C211yesmaximal, characteristic, complemented
9#16863632C211yesmaximal, characteristic, complemented
10#16863630C211yesmaximal, characteristic, complemented
11#16863627C211yesmaximal, characteristic, complemented
12#16863435C211yesmaximal, characteristic, complemented
13#16863399C211yesmaximal, characteristic, complemented
14#16863396C211yesmaximal, characteristic, complemented
15#465×C67C211yesmaximal, characteristic, complemented
16C2×#11761629C2211yescharacteristic
17C2×D4×D67—21no—
18C2×D4×D67C2211yescharacteristic
19C22×C4×D67C2211yescharacteristic
20#14524070—24no—
21C2×#11761615C2211yescharacteristic, complemented
22C2×D536C2211yescharacteristic, complemented
23C2×#11761613C2211yescharacteristic, complemented
24C2×#11761611—24no—
25C2×#11761611C2211yescharacteristic
26C2×#11761611C2211yescharacteristic
27C2×#11761611C2211yescharacteristic, complemented
28C2×#11761611C2211yescharacteristic, complemented
29C22×C67⋊C8C2211yescharacteristic, complemented
30#14524028—21no—
31#14524019C2211yescharacteristic, complemented
32#14524018C2211yescharacteristic
33#14524018C2211yescharacteristic, complemented
34#14524017C2211yescharacteristic
35#14524017C2211yescharacteristic, complemented
36#14524016—21no—
37#14524015—21no—
38#14524014C2211yescharacteristic
39#14524013—21no—
40#14524005—21no—
41(C22⋊D67)⋊C4—21no—
42#14524001C2211yescharacteristic
43C4⋊C4×D67C2211yescharacteristic
44C22⋊C4×D67—21no—
45#14523977C2211yescharacteristic, complemented
46#14523976C2211yescharacteristic, complemented
47#14523974C2211yescharacteristic, complemented
48#14523967C2211yescharacteristic, complemented
49#14523965C2211yescharacteristic, complemented
50#14523965C2211yescharacteristic, complemented
51#14523964C2211yescharacteristic, complemented
52#14523963C2211yescharacteristic, complemented
53#14523963C2211yescharacteristic, complemented
54C2×SD16×C67C2211yescharacteristic, complemented
55C22×C8×C67C2211yescharacteristic, complemented
56C22⋊Q8×C67C2211yescharacteristic, complemented
57#121×C67C2211yescharacteristic
58#106×C67C2211yescharacteristic, complemented
59#103×C67C2211yescharacteristic, complemented
60#102×C67C2211yescharacteristic, complemented
61C23×D67—21no—
62C23×D67D411yescharacteristic
63#11761629—22no—
64#11761629—22no—
65C2×D268—21no—
66C2×D268—21no—
67C2×D268—21no—
68C2×D268C2311yescharacteristic
69C2×D268C2311yescharacteristic
70C2×C22⋊D67—21no—
71C2×C22⋊D67—21no—
72C2×C22⋊D67—21no—
73C2×C22⋊D67—21no—
74D4×D67—22no—
75D4×D67—22no—
76D4×D67—44no—
77C22×Dic67D411yescharacteristic
78C2×C4×D67—22no—
79C2×C4×D67—22no—
80C2×C4×D67—21no—
81C2×C4×D67—21no—
82C2×C4×D67D411yescharacteristic
83C2×C4×D67D411yescharacteristic
84C2×C4×D67C2311yescharacteristic
85C2×C4×D67C2311yescharacteristic
86#11761615—22no—
87D536—22no—
88#11761613—22no—
89#11761611—22no—
90#11761611—22no—
91#11761611—22no—
92#11761611D414yescomplemented
93C2×C67⋊C8—22no—
94C2×C67⋊C8C2311yescharacteristic
95C2×C67⋊C8C2311yescharacteristic
96C4⋊Dic67C2311yescharacteristic
97C4⋊Dic67C2311yescharacteristic
98Dic67⋊C4—21no—
99Dic67⋊C4—21no—
100C22⋊Dic67—21no—
101(C2×D67)⋊C4—21no—
102(C2×D67)⋊C4—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×Q8×C67C2311yescharacteristic
109C2×D4×C67—21no—
110C2×D4×C67C2311yescharacteristic
111C22×C4×C67C2311yescharacteristic
112SD16×C67—22no—
113C2×C8×C67—22no—
114C2×C8×C67C2311yescharacteristic
115C2×C8×C67C2311yescharacteristic
116C4⋊C4×C67—21no—
117C4⋊C4×C67C2311yescharacteristic
118C4⋊C4×C67C2311yescharacteristic
119C22⋊C4×C67—21no—
120C22⋊C4×C67—21no—
121C23×C67—21no—
122C23×C67C2×D411yescharacteristic
123C22×D67—22no—
124C22×D67—22no—
125C22×D67—21no—
126C22×D67—21no—
127C22×D67—41no—
128C22×D67—42no—
129C22×D67—42no—
130C22×D67—42no—
131C22×D67C2×D411yescharacteristic
132C22×D67C2×D411yescharacteristic
133Q8×C67—21no—
134Q8×C67—21no—
135D4×C67—21no—
136D4×C67—21no—
137D4×C67—42no—
138C2×C4×C67—21no—
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—
149C2×Dic67—22no—
150C2×Dic67—21no—
151C2×Dic67—21no—
152C2×Dic67C2×D411yescharacteristic
153C2×Dic67C2×D411yescharacteristic
154D268—21no—
155D268—21no—
156D268—21no—
157D268—21no—
158D268—42no—
159D268—42no—
160D268—42no—
161C4×D67—22no—
162C4×D67—21no—
163C4×D67—21no—
164C4×D67—22no—
165C4×D67—42no—
166C4×D67—42no—
167C4×D67C2×D412yes—
168C4×D67#4112yes—
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×C67AGL(1,ℤ/8)12yescomplemented
179C22×C67C22×D411yescharacteristic
180C2×D67—21no—
181C2×D67—21no—
182C2×D67—21no—
183C2×D67—21no—
184C2×D67—21no—
185C2×D67—44no—
186C2×D67—41no—
187C2×D67—41no—
188C2×D67—41no—
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×C67—41no—
198C4×C67C22×D411yescharacteristic
199C4×C67C2×#4111yescharacteristic
200Dic67—21no—
201Dic67—41no—
202Dic67—41no—
203Dic67#12112yes—
204C2×C67—22no—
205C2×C67—81no—
206C2×C67C2×#12111yescharacteristic
207C2×C67C2×AGL(1,ℤ/8)11yescharacteristic
208C2×C67#57511yescharacteristic
209D67—22no—
210D67—41no—
211D67—81no—
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#466—671no—
225#465—671no—
226#465D6711yescharacteristic, complemented
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×SD16C2×D6711yescharacteristic, complemented
237C2×D8—671no—
238C2×D8—671no—
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⋊Q8C2×D6711yescharacteristic, complemented
249#121—671no—
250#121—671no—
251#121—671no—
252#121—671no—
253#121—671no—
254#121—1341no—
255#121—1341no—
256#121C2×D6711yescharacteristic
257#120—1341no—
258C4×D4—1341no—
259C4×D4—1341no—
260#117—671no—
261C2×C4⋊C4—671no—
262C2×C22⋊C4—1341no—
263#107—671no—
264#107—671no—
265#106—671no—
266#106C2×D6711yescharacteristic, complemented
267#103—671no—
268#103C2×D6711yescharacteristic, complemented
269#102—671no—
270#102—671no—
271#102—671no—
272#102—671no—
273#102—671no—
274#102C2×D6711yescharacteristic, complemented
275C24—671no—
276C24—1341no—
277#41—1342no—
278#41—1342no—
279C2×Q8C22×D6711yescharacteristic
280C2×D4—21no—
281C2×D4—2684no—
282C2×D4—671no—
283C2×D4—671no—
284C2×D4—1341no—
285C2×D4—1341no—
286C2×D4—1341no—
287C2×D4—1342no—
288C2×D4—1342no—
289C2×D4—1341no—
290C2×D4—1341no—
291C2×D4—1341no—
292C2×D4—1341no—
293C2×D4C22×D6711yescharacteristic
294C22×C4—671no—
295C22×C4—671no—
296C22×C4—671no—
297C22×C4—671no—
298C22×C4—671no—
299C22×C4—1341no—
300C22×C4—1341no—
301C22×C4—1342no—
302C22×C4—1342no—
303C22×C4C22×D6711yescharacteristic
304SD16—22no—
305SD16—1342no—
306D8—1342no—
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—21no—
319C4⋊C4—671no—
320C4⋊C4—671no—
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—2681no—
337C23—2682no—
338C23—2682no—
339C23—2682no—
340C23—671no—
341C23—671no—
342C23—1342no—
343C23—1341no—
344C23—1341no—
345C23—1342no—
346C23D4×D6711yescharacteristic
347Q8—21no—
348Q8—21no—
349D4—21no—
350D4—21no—
351D4—42no—
352D4—2682no—
353D4—2682no—
354D4—2682no—
355D4—2682no—
356D4—2682no—
357D4—2682no—
358D4—2682no—
359D4—1341no—
360D4—1341no—
361D4—1341no—
362D4—1341no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—2682no—
369C2×C4—2682no—
370C2×C4—671no—
371C2×C4—672no—
372C2×C4—672no—
373C2×C4—671no—
374C2×C4—1342no—
375C2×C4—1342no—
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—2684no—
393C22—2681no—
394C22—2681no—
395C22—2681no—
396C22—2681no—
397C22—2681no—
398C22—2681no—
399C22—5364no—
400C22—672no—
401C22—1341no—
402C22—1341no—
403C22—1341no—
404C22—1341no—
405C22—1341no—
406C22#1452407012yescomplemented
407C22C2×D4×D6711yescharacteristic
408C4—21no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—2681no—
413C4—2681no—
414C4—672no—
415C4—1341no—
416C4C2×D4×D6711yescharacteristic
417C4#41×D6711yescharacteristic
418C2—22no—
419C2—81no—
420C2—2681no—
421C2—5361no—
422C2—5361no—
423C2—1342no—
424C2#121×D6711yescharacteristic
425C2C2×#1452407011yescharacteristic
426C2#1686419311yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#465×C67C2
socleC22×C67C22×D4
hypercenter#465D67
hypocenterC67#3055
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#14524070C2semidirect
2. C22×#11761611C2semidirect
3. #16863909C2semidirect
4. #121×D67C2semidirect
5. #16863650C2semidirect
6. #16863648C2semidirect
7. #16863645C2semidirect
8. #16863639C2semidirect
9. #16863632C2semidirect
10. #16863630C2semidirect
11. #16863627C2semidirect
12. #16863435C2semidirect
13. #16863399C2semidirect
14. #16863396C2semidirect
15. #465×C67C2semidirect
21. C2×#11761615C22semidirect
22. C2×D536C22semidirect
23. C2×#11761613C22semidirect
27. C2×#11761611C22semidirect
28. C2×#11761611C22semidirect
29. C22×C67⋊C8C22semidirect
31. #14524019C22semidirect
33. #14524018C22semidirect
35. #14524017C22semidirect
45. #14523977C22semidirect
46. #14523976C22semidirect
47. #14523974C22semidirect
48. #14523967C22semidirect
49. #14523965C22semidirect
50. #14523965C22semidirect
51. #14523964C22semidirect
52. #14523963C22semidirect
53. #14523963C22semidirect
54. C2×SD16×C67C22semidirect
55. C22×C8×C67C22semidirect
56. C22⋊Q8×C67C22semidirect
58. #106×C67C22semidirect
59. #103×C67C22semidirect
60. #102×C67C22semidirect
92. #11761611D4semidirect
170. C8×C67C2×D4semidirect
172. C67⋊C8C2×D4semidirect
178. C22×C67AGL(1,ℤ/8)semidirect
214. C67#3055semidirect
226. #465D67semidirect
236. C2×SD16C2×D67semidirect
245. C22×C8C2×D67semidirect
248. C22⋊Q8C2×D67semidirect
266. #106C2×D67semidirect
268. #103C2×D67semidirect
274. #102C2×D67semidirect
386. C8D4×D67semidirect
406. C22#14524070semidirect