Finite Group Databasebeta

← #22571735

Subgroups — #22571735

Order 8576.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14524068C211yesmaximal, characteristic, complemented
2C2×#14524063C211yesmaximal, characteristic, complemented
3#16864042C211yesmaximal, characteristic, complemented
4#16863788C211yesmaximal, characteristic, complemented
5#16863652C211yesmaximal, characteristic, complemented
6#16863627C211yesmaximal, characteristic, complemented
7#16863611C211yesmaximal, characteristic, complemented
8#16863598C211yesmaximal, characteristic, complemented
9#16863558C211yesmaximal, characteristic, complemented
10#16863556C211yesmaximal, characteristic, complemented
11#16863546C211yesmaximal, characteristic, complemented
12#16863541C211yesmaximal, characteristic, complemented
13#16863506C211yesmaximal, characteristic, complemented
14#16863468C211yesmaximal, characteristic, complemented
15#547×C67C211yesmaximal, characteristic, complemented
16C2×#11761628C2211yescharacteristic
17C22×D268C2211yescharacteristic
18#14524068—24no—
19#14524063—24no—
20C2×#11761618C2211yescharacteristic, complemented
21C2×#11761618C2211yescharacteristic, complemented
22C2×#11761616C2211yescharacteristic, complemented
23C2×#11761615C2211yescharacteristic
24C2×#11761615C2211yescharacteristic
25C2×#11761615C2211yescharacteristic
26C2×#11761615C2211yescharacteristic
27C2×#11761613C2211yescharacteristic
28C2×#11761613C2211yescharacteristic
29C2×#11761612C2211yescharacteristic
30C22×C67⋊C8C2211yescharacteristic, complemented
31#14524031C2211yescharacteristic
32#14524028—21no—
33#14524022C2211yescharacteristic
34#14524017C2211yescharacteristic
35#14524016—21no—
36#14524011—21no—
37C4×D268C2211yescharacteristic
38C22⋊C4×D67—21no—
39#14523974C2211yescharacteristic, complemented
40C4⋊(C67⋊C8)C2211yescharacteristic, complemented
41#14523967C2211yescharacteristic, complemented
42#14523967C2211yescharacteristic, complemented
43#14523966C2211yescharacteristic, complemented
44#14523964C2211yescharacteristic, complemented
45#14523963C2211yescharacteristic
46#14523963C2211yescharacteristic
47C22⋊(C67⋊C8)C2211yescharacteristic
48C22⋊(C67⋊C8)C2211yescharacteristic, complemented
49#14523954C2211yescharacteristic, complemented
50C2×#41×C67C2211yescharacteristic
51C2×#41×C67C2211yescharacteristic, complemented
52C4⋊Q8×C67C2211yescharacteristic, complemented
53#123×C67—21no—
54#123×C67—21no—
55C22⋊Q8×C67C2211yescharacteristic
56C22⋊Q8×C67C2211yescharacteristic, complemented
57#121×C67—21no—
58C4×D4×C67C2211yescharacteristic
59C4×D4×C67C2211yescharacteristic, complemented
60C2×C4⋊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#11761618—41no—
76#11761616—22no—
77#11761615—22no—
78#11761615—22no—
79#11761615—22no—
80#11761615—41no—
81#11761613—22no—
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—
96#41×C67—22no—
97#41×C67—22no—
98#41×C67—41no—
99#41×C67—41no—
100C2×Q8×C67C2311yescharacteristic
101C2×Q8×C67C2311yescharacteristic
102C2×D4×C67—21no—
103C2×D4×C67—21no—
104C2×D4×C67D411yescharacteristic
105C2×D4×C67C2311yescharacteristic
106C22×C4×C67—21no—
107C22×C4×C67—21no—
108C22×C4×C67—21no—
109C22×C4×C67C2311yescharacteristic
110C22×C4×C67C2311yescharacteristic
111C4⋊C4×C67—21no—
112C4⋊C4×C67—21no—
113C4⋊C4×C67—21no—
114C4⋊C4×C67—21no—
115C4⋊C4×C67D411yescharacteristic
116C4⋊C4×C67C2311yescharacteristic
117C22⋊C4×C67—21no—
118C22⋊C4×C67—21no—
119C22⋊C4×C67—21no—
120C22⋊C4×C67D411yescharacteristic
121C22⋊C4×C67D411yescharacteristic
122C42×C67C2311yescharacteristic
123C23×C67—21no—
124C23×C67C2×D411yescharacteristic
125C23×C67C2×D411yescharacteristic
126C22×D67—21no—
127C22×D67—21no—
128C22×D67—42no—
129C22×D67—42no—
130C22×D67—81no—
131Q8×C67—21no—
132Q8×C67—21no—
133Q8×C67—21no—
134Q8×C67—21no—
135D4×C67—21no—
136D4×C67—41no—
137D4×C67—42no—
138D4×C67—42no—
139D4×C67C2×D412yes—
140C2×C4×C67—21no—
141C2×C4×C67—21no—
142C2×C4×C67—21no—
143C2×C4×C67—21no—
144C2×C4×C67—21no—
145C2×C4×C67—21no—
146C2×C4×C67—21no—
147C2×C4×C67—42no—
148C2×C4×C67—42no—
149C2×C4×C67—42no—
150C2×C4×C67C2×D411yescharacteristic
151C2×C4×C67C2×D411yescharacteristic
152C2×C4×C67C2×D411yescharacteristic
153C2×C4×C67C2×D411yescharacteristic
154C2×C4×C67C2411yescharacteristic
155C22⋊D67—42no—
156C22⋊D67—81no—
157C2×Dic67—21no—
158C2×Dic67—21no—
159D268—21no—
160D268—21no—
161D268—21no—
162D268—42no—
163D268C2×D412yes—
164C4×D67—42no—
165C4×D67—81no—
166Dic134—21no—
167Dic134—21no—
168C67⋊C8—21no—
169C67⋊C8—21no—
170C67⋊C8—21no—
171C67⋊C8C2×D412yes—
172C22×C67—21no—
173C22×C67—21no—
174C22×C67—22no—
175C22×C67—21no—
176C22×C67—21no—
177C22×C67—21no—
178C22×C67—21no—
179C22×C67—41no—
180C22×C67—41no—
181C22×C67C22×D411yescharacteristic
182C2×D67—22no—
183C2×D67—41no—
184C2×D67—41no—
185C2×D67—41no—
186C2×D67—41no—
187C2×D67—41no—
188C2×D67—84no—
189C4×C67—21no—
190C4×C67—21no—
191C4×C67—21no—
192C4×C67—21no—
193C4×C67—41no—
194C4×C67—41no—
195C4×C67—41no—
196C4×C67C22×D411yescharacteristic
197C4×C67H(5,𝔽2)11yescharacteristic
198Dic67—41no—
199Dic67—41no—
200C2×C67—41no—
201C2×C67—41no—
202C2×C67—42no—
203C2×C67D4211yescharacteristic
204C2×C67#57511yescharacteristic
205C2×C67#57711yescharacteristic
206D67—42no—
207D67—81no—
208#3267—671nomaximal, Sylow
209C67#326711yescharacteristic, complemented, Sylow
210C2×AGL(1,ℤ/8)—671no—
211C2×#135—671no—
212#547D6711yescharacteristic, complemented
213#546—671no—
214#497—671no—
215#470—671no—
216#465—671no—
217#463—671no—
218#461—671no—
219#452—671no—
220#452—671no—
221#450—671no—
222#449—671no—
223#442—671no—
224#435—671no—
225C2×#41—671no—
226C2×#41C2×D6711yescharacteristic
227C2×#41C2×D6711yescharacteristic, complemented
228C22×D4—671no—
229AGL(1,ℤ/8)—1344no—
230#135—1344no—
231C2×Q16—671no—
232C2×Q16—671no—
233C2×SD16—671no—
234C2×SD16—671no—
235C2×SD16—671no—
236C2×SD16—671no—
237C2×SD16—671no—
238C2×D8—671no—
239C2×D8—671no—
240C2×M16—671no—
241C22×C8—671no—
242C4⋊Q8C2×D6711yescharacteristic, complemented
243#124—671no—
244#123—21no—
245#123—21no—
246#123—1341no—
247C22⋊Q8—671no—
248C22⋊Q8C2×D6711yescharacteristic
249C22⋊Q8C2×D6711yescharacteristic, complemented
250#121—21no—
251#121—671no—
252#121—1341no—
253#120—1341no—
254C4×D4—671no—
255C4×D4C2×D6711yescharacteristic
256C4×D4C2×D6711yescharacteristic, complemented
257C2×C4⋊C4—21no—
258C2×C22⋊C4—1341no—
259#106—671no—
260C4⋊C8—671no—
261#103—671no—
262#103—671no—
263#103—671no—
264#102—671no—
265#102—671no—
266#102—671no—
267C22⋊C8—671no—
268C22⋊C8—671no—
269#97—671no—
270C24—1341no—
271#41—22no—
272#41—22no—
273#41—41no—
274#41—41no—
275#41—2681no—
276#41—1342no—
277C2×Q8—671no—
278C2×Q8C22×D6711yescharacteristic
279C2×Q8C22×D6711yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—2681no—
283C2×D4—671no—
284C2×D4—671no—
285C2×D4—1342no—
286C2×D4—1341no—
287C2×D4—1341no—
288C2×D4—1341no—
289C2×D4C22⋊D6711yescharacteristic
290C2×D4C22×D6711yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—1341no—
295C22×C4—1341no—
296C22×C4C22×D6711yescharacteristic
297C22×C4C22×D6711yescharacteristic
298Q16—2681no—
299Q16—1342no—
300SD16—2681no—
301SD16—1342no—
302SD16—1342no—
303SD16—1342no—
304SD16—1342no—
305D8—674no—
306D8—1342no—
307M16—1342no—
308C2×C8—671no—
309C2×C8—671no—
310C2×C8—671no—
311C2×C8—671no—
312C2×C8—1342no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—671no—
318C4⋊C4—1341no—
319C4⋊C4C22⋊D6711yescharacteristic
320C4⋊C4C22×D6711yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—1341no—
325C22⋊C4—1341no—
326C22⋊C4—1341no—
327C22⋊C4—1341no—
328C22⋊C4—1341no—
329C22⋊C4C22⋊D6711yescharacteristic
330C22⋊C4C22⋊D6711yescharacteristic
331C42C22×D6711yescharacteristic
332C23—21no—
333C23—2682no—
334C23—2682no—
335C23—5361no—
336C23—1341no—
337C23—1341no—
338C23C2×C22⋊D6711yescharacteristic
339C23C2×C22⋊D6711yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—1341no—
345Q8—1341no—
346D4—21no—
347D4—41no—
348D4—42no—
349D4—42no—
350D4—2682no—
351D4—2682no—
352D4—5361no—
353D4—672no—
354D4—1341no—
355D4—1341no—
356D4—1341no—
357D4D4×D6712yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—2682no—
369C2×C4—5361no—
370C2×C4—1341no—
371C2×C4—1341no—
372C2×C4C2×C22⋊D6711yescharacteristic
373C2×C4C2×C22⋊D6711yescharacteristic
374C2×C4C2×C22⋊D6711yescharacteristic
375C2×C4C2×C22⋊D6711yescharacteristic
376C2×C4C23×D6711yescharacteristic
377C8—672no—
378C8—1341no—
379C8—1341no—
380C8—1341no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—41no—
389C22—41no—
390C22—2681no—
391C22—2681no—
392C22—2681no—
393C22—2681no—
394C22—2681no—
395C22—5364no—
396C22—1342no—
397C22C22×C22⋊D6711yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—2681no—
406C4—2681no—
407C4C2×D4×D6711yescharacteristic
408C4#1452408811yescharacteristic
409C2—41no—
410C2—41no—
411C2—42no—
412C2—2682no—
413C2—5361no—
414C2D4×C22⋊D6711yescharacteristic
415C2#1686419411yescharacteristic
416C2#1686420411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#547×C67C2
socleC22×C67C22×D4
hypercenter#547D67
hypocenterC67#3267
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#14524068C2semidirect
2. C2×#14524063C2semidirect
3. #16864042C2semidirect
4. #16863788C2semidirect
5. #16863652C2semidirect
6. #16863627C2semidirect
7. #16863611C2semidirect
8. #16863598C2semidirect
9. #16863558C2semidirect
10. #16863556C2semidirect
11. #16863546C2semidirect
12. #16863541C2semidirect
13. #16863506C2semidirect
14. #16863468C2semidirect
15. #547×C67C2semidirect
20. C2×#11761618C22semidirect
21. C2×#11761618C22semidirect
22. C2×#11761616C22semidirect
30. C22×C67⋊C8C22semidirect
39. #14523974C22semidirect
40. C4⋊(C67⋊C8)C22semidirect
41. #14523967C22semidirect
42. #14523967C22semidirect
43. #14523966C22semidirect
44. #14523964C22semidirect
48. C22⋊(C67⋊C8)C22semidirect
49. #14523954C22semidirect
51. C2×#41×C67C22semidirect
52. C4⋊Q8×C67C22semidirect
56. C22⋊Q8×C67C22semidirect
59. C4×D4×C67C22semidirect
82. #11761613D4semidirect
209. C67#3267semidirect
212. #547D67semidirect
227. C2×#41C2×D67semidirect
242. C4⋊Q8C2×D67semidirect
249. C22⋊Q8C2×D67semidirect
256. C4×D4C2×D67semidirect