Finite Group Databasebeta

← #22573862

Subgroups — #22573862

Order 8576.

1536 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14524090C211yesmaximal, characteristic, complemented
2C2×#41×D67C211yesmaximal, characteristic, complemented
3#16864066C213yesmaximal, complemented
4#16864065C211yesmaximal, characteristic, complemented
5#16864064C213yesmaximal, complemented
6#16864058C211yesmaximal, characteristic, complemented
7#16864055C213yesmaximal, complemented
8#16864053C211yesmaximal, characteristic, complemented
9#16864052C213yesmaximal, complemented
10#16863962C213yesmaximal, complemented
11#16863959C213yesmaximal, complemented
12#16863906C213yesmaximal, complemented
13#16863905C213yesmaximal, complemented
14#16863872C211yesmaximal, characteristic, complemented
15#550×C67C211yesmaximal, characteristic, complemented
16#14524090—28no—
17C2×#11761629C2211yescharacteristic
18C2×#11761628C2213yes—
19C2×#11761628C2213yes—
20C2×#11761627C2213yes—
21C2×#11761627C2213yes—
22C2×#11761627C2213yescomplemented
23#41×D67—28no—
24C22×Dic134C2213yescomplemented
25C2×Q8×D67C2211yescharacteristic
26C2×Q8×D67C2211yescharacteristic, complemented
27C2×D4×D67C2213yes—
28C22×C4×D67C2213yes—
29#14524039C2213yescomplemented
30#14524039C2216yescomplemented
31#14524031C2213yescomplemented
32#14524030C2213yescomplemented
33#14524029C2216yescomplemented
34#14524028C2211yescharacteristic
35#14524025C2213yes—
36#14524023C2213yescomplemented
37#14524022C2213yescomplemented
38#14524021C2213yescomplemented
39#14524020C2213yescomplemented
40#14524018C2213yescomplemented
41#14524017C2213yescomplemented
42#14524016C2213yescomplemented
43#14524015C2216yescomplemented
44#14524015C2216yescomplemented
45#14524015C2213yescomplemented
46#14524014C2213yescomplemented
47#14524014C2213yescomplemented
48#14524013C2213yes—
49#14524011C2213yes—
50#14524011C2216yes—
51Q8×Dic67C2211yescharacteristic, complemented
52C4×Dic134C2213yescomplemented
53#14524008C2213yescomplemented
54C4×D268C2213yescomplemented
55D4×Dic67C2216yescomplemented
56C4×C22⋊D67C2213yescomplemented
57C4×C22⋊D67C2213yescomplemented
58(C22⋊D67)⋊C4C2216yescomplemented
59#14524001C2213yes—
60#14524000C2213yes—
61C2×#41×C67C2211yescharacteristic
62C2×#41×C67C2211yescharacteristic, complemented
63#127×C67C2213yescomplemented
64#121×C67C2213yescomplemented
65#121×C67C2213yescomplemented
66C4×Q8×C67C2211yescharacteristic, complemented
67C4×D4×C67C2213yescomplemented
68C23×D67C2313yes—
69#11761629—24no—
70#11761628—212no—
71#11761628—212no—
72#11761627—212no—
73#11761627—212no—
74#11761627—212no—
75C2×Dic134—212no—
76C2×Dic134C2313yes—
77C2×Dic134C2313yes—
78C2×Dic134C2313yes—
79Q8×D67—24no—
80Q8×D67D418yescomplemented
81C2×D268C2311yescharacteristic
82C2×D268C2313yes—
83C2×D268C2313yes—
84C2×C22⋊D67C2313yes—
85C2×C22⋊D67C2316yes—
86C2×C22⋊D67C2313yes—
87C2×C22⋊D67C2316yes—
88C2×C22⋊D67C2313yes—
89C2×C22⋊D67C2313yes—
90D4×D67—212no—
91C22×Dic67C2316yes—
92C22×Dic67C2313yes—
93C2×C4×D67—212no—
94C2×C4×D67C2313yes—
95C2×C4×D67C2313yes—
96C2×C4×D67C2313yes—
97C2×C4×D67C2311yescharacteristic
98C2×C4×D67C2313yes—
99C4⋊Dic67C2313yes—
100C4⋊Dic67C2313yes—
101Dic67⋊C4C2313yes—
102Dic67⋊C4C2313yes—
103Dic67⋊C4C2313yes—
104Dic67⋊C4C2311yescharacteristic
105C22⋊Dic67C2313yes—
106C22⋊Dic67C2313yes—
107C22⋊Dic67C2316yes—
108(C2×D67)⋊C4C2311yescharacteristic
109(C2×D67)⋊C4C2313yes—
110(C2×D67)⋊C4C2316yes—
111(C2×D67)⋊C4C2311yescharacteristic
112(C2×D67)⋊C4C2313yes—
113(C2×D67)⋊C4C2311yescharacteristic
114(C2×D67)⋊C4C2313yes—
115C4×Dic67C2313yes—
116C4×Dic67C2316yes—
117#41×C67—24no—
118#41×C67—24no—
119C2×Q8×C67C2311yescharacteristic
120C2×D4×C67C2313yes—
121C2×D4×C67C2313yes—
122C2×D4×C67C2313yes—
123C2×D4×C67C2316yes—
124C22×C4×C67C2313yes—
125C22×C4×C67C2313yes—
126C4⋊C4×C67C2311yescharacteristic
127C4⋊C4×C67C2313yes—
128C22⋊C4×C67C2313yes—
129C22⋊C4×C67C2313yes—
130C42×C67C2313yes—
131C23×C67C2413yes—
132C23×C67C2413yes—
133C22×D67—26no—
134C22×D67—43no—
135C22×D67—43no—
136C22×D67C2413yes—
137C22×D67C2411yescharacteristic
138C22×D67C2411yescharacteristic
139Q8×C67C2×D414yes—
140D4×C67—26no—
141D4×C67—26no—
142D4×C67—46no—
143D4×C67—43no—
144C2×C4×C67—26no—
145C2×C4×C67—26no—
146C2×C4×C67C2411yescharacteristic
147C2×C4×C67C2413yes—
148C2×C4×C67C2411yescharacteristic
149C2×C4×C67C2411yescharacteristic
150C2×C4×C67C2413yes—
151C22⋊D67—26no—
152C22⋊D67—212no—
153C22⋊D67—26no—
154C22⋊D67—43no—
155C22⋊D67—43no—
156C22⋊D67—46no—
157C2×Dic67—26no—
158C2×Dic67—212no—
159C2×Dic67C2411yescharacteristic
160C2×Dic67C2413yes—
161C2×Dic67C2413yes—
162C2×Dic67C2411yescharacteristic
163C2×Dic67C2413yes—
164D268—22no—
165D268—26no—
166D268—43no—
167C4×D67—22no—
168C4×D67—26no—
169C4×D67—26no—
170C4×D67—26no—
171C4×D67C2×D4112yes—
172Dic134—26no—
173Dic134—26no—
174Dic134C2×D4112yes—
175C22×C67—23no—
176C22×C67—23no—
177C22×C67—23no—
178C22×C67—23no—
179C22×C67—23no—
180C22×C67—23no—
181C22×C67C2511yescharacteristic
182C2×D67—21no—
183C2×D67—21no—
184C2×D67—21no—
185C2×D67—23no—
186C2×D67—21no—
187C2×D67—23no—
188C2×D67—21no—
189C2×D67—23no—
190C2×D67—412no—
191C2×D67C22×D412yes—
192C4×C67—21no—
193C4×C67—21no—
194C4×C67—23no—
195C4×C67—21no—
196C4×C67C22×D416yes—
197Dic67—23no—
198Dic67—21no—
199Dic67—21no—
200Dic67—23no—
201Dic67C22×D416yes—
202C2×C67—43no—
203C2×C67—43no—
204C2×C67C23×D411yescharacteristic
205C2×C67#58511yescharacteristic
206C2×C67#58511yescharacteristic
207D67—22no—
208D67—41no—
209D67—43no—
210#3487—671nomaximal, Sylow
211C67#348711yescharacteristic, complemented, Sylow
212C2×2−1+4—671no—
213C22×#41—671no—
214#550—671no—
215#550—673no—
216#550—673no—
217#550D6711yescharacteristic, complemented
218#548—671no—
219#548—673no—
220#548—671no—
221#548—673no—
222#535—673no—
223#535—673no—
224#525—673no—
225#525—673no—
226#519—671no—
2272−1+4—1348no—
228C2×#41—671no—
229C2×#41—673no—
230C2×#41—673no—
231C2×#41—673no—
232C2×#41—673no—
233C2×#41—673no—
234C2×#41—1348no—
235C2×#41C2×D6711yescharacteristic
236C2×#41C2×D6711yescharacteristic, complemented
237C22×Q8—671no—
238C22×Q8—671no—
239C22×Q8—673no—
240C22×D4—673no—
241C23×C4—673no—
242#127—673no—
243#127—676no—
244#127C2×D6713yescomplemented
245#124—673no—
246#124—676no—
247#124—673no—
248#123—671no—
249#123—673no—
250C22⋊Q8—673no—
251C22⋊Q8—673no—
252C22⋊Q8—673no—
253C22⋊Q8—673no—
254#121—673no—
255#121—673no—
256#121—673no—
257#121—676no—
258#121—676no—
259#121—673no—
260#121—673no—
261#121—673no—
262#121C2×D6713yescomplemented
263#121C2×D6713yescomplemented
264#120—676no—
265#120—673no—
266#120—673no—
267C4×Q8—671no—
268C4×Q8—673no—
269C4×Q8—673no—
270C4×Q8C2×D6711yescharacteristic, complemented
271C4×D4—676no—
272C4×D4—673no—
273C4×D4—673no—
274C4×D4—676no—
275C4×D4—673no—
276C4×D4C2×D6713yescomplemented
277#117—673no—
278#117—673no—
279C24—673no—
280#41—24no—
281#41—24no—
282#41—13412no—
283#41—13412no—
284#41—13412no—
285#41—13412no—
286#41—13412no—
287#41—1344no—
288C2×Q8—678no—
289C2×Q8—673no—
290C2×Q8—673no—
291C2×Q8—673no—
292C2×Q8—13412no—
293C2×Q8—1344no—
294C2×Q8C22×D6711yescharacteristic
295C2×D4—673no—
296C2×D4—676no—
297C2×D4—673no—
298C2×D4—673no—
299C2×D4—676no—
300C2×D4—673no—
301C2×D4—673no—
302C2×D4—671no—
303C2×D4—673no—
304C2×D4—13412no—
305C2×D4C22×D6713yes—
306C2×D4C22×D6713yes—
307C2×D4C22×D6716yes—
308C2×D4C22×D6713yes—
309C22×C4—673no—
310C22×C4—676no—
311C22×C4—673no—
312C22×C4—673no—
313C22×C4—673no—
314C22×C4—673no—
315C22×C4—671no—
316C22×C4—13412no—
317C22×C4C22×D6713yes—
318C22×C4C22×D6713yes—
319C4⋊C4—673no—
320C4⋊C4—673no—
321C4⋊C4—673no—
322C4⋊C4—673no—
323C4⋊C4—671no—
324C4⋊C4—673no—
325C4⋊C4C22×D6713yes—
326C4⋊C4C22×D6711yescharacteristic
327C22⋊C4—673no—
328C22⋊C4—676no—
329C22⋊C4—673no—
330C22⋊C4—673no—
331C22⋊C4—671no—
332C22⋊C4—671no—
333C22⋊C4—671no—
334C22⋊C4—673no—
335C22⋊C4—676no—
336C22⋊C4—673no—
337C22⋊C4C22×D6713yes—
338C22⋊C4C22×D6713yes—
339C42—676no—
340C42—673no—
341C42C22×D6713yes—
342C23—2683no—
343C23—2683no—
344C23—671no—
345C23—673no—
346C23—671no—
347C23—1346no—
348C23C23×D6713yes—
349C23C23×D6713yes—
350Q8—6712no—
351Q8—1346no—
352Q8—1346no—
353Q8D4×D6714yes—
354D4—26no—
355D4—26no—
356D4—46no—
357D4—43no—
358D4—2683no—
359D4—2683no—
360D4—2686no—
361D4—2683no—
362D4—1342no—
363D4—13412no—
364D4—1346no—
365D4—1346no—
366D4—1346no—
367C2×C4—26no—
368C2×C4—26no—
369C2×C4—673no—
370C2×C4—6712no—
371C2×C4—671no—
372C2×C4—673no—
373C2×C4—673no—
374C2×C4—671no—
375C2×C4—1346no—
376C2×C4—1342no—
377C2×C4—1346no—
378C2×C4—13412no—
379C2×C4—1346no—
380C2×C4—1346no—
381C2×C4C23×D6713yes—
382C2×C4C23×D6711yescharacteristic
383C2×C4C23×D6711yescharacteristic
384C2×C4C23×D6711yescharacteristic
385C2×C4C23×D6713yes—
386C22—23no—
387C22—23no—
388C22—23no—
389C22—23no—
390C22—23no—
391C22—23no—
392C22—26812no—
393C22—672no—
394C22—1341no—
395C22—1343no—
396C22—1341no—
397C22—1343no—
398C22—1341no—
399C22—1343no—
400C22—1341no—
401C22—1341no—
402C22C24×D6711yescharacteristic
403C4—21no—
404C4—21no—
405C4—23no—
406C4—21no—
407C4—676no—
408C4—1341no—
409C4—1343no—
410C4—1341no—
411C4—1343no—
412C4C2×D4×D6716yes—
413C2—43no—
414C2—43no—
415C2—2683no—
416C2—2681no—
417C2—1342no—
418C2C22×D4×D6711yescharacteristic
419C2#1686423111yescharacteristic
420C2#1686423211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D67
commutator, socleC22×C67C25
Fitting#550×C67C2
hypercenter#550D67
hypocenterC67#3487
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#14524090C2semidirect
2. C2×#41×D67C2semidirect
3. #16864066C2semidirect
4. #16864065C2semidirect
5. #16864064C2semidirect
6. #16864058C2semidirect
7. #16864055C2semidirect
8. #16864053C2semidirect
9. #16864052C2semidirect
10. #16863962C2semidirect
11. #16863959C2semidirect
12. #16863906C2semidirect
13. #16863905C2semidirect
14. #16863872C2semidirect
15. #550×C67C2semidirect
22. C2×#11761627C22semidirect
24. C22×Dic134C22semidirect
26. C2×Q8×D67C22semidirect
29. #14524039C22semidirect
30. #14524039C22semidirect
31. #14524031C22semidirect
32. #14524030C22semidirect
33. #14524029C22semidirect
36. #14524023C22semidirect
37. #14524022C22semidirect
38. #14524021C22semidirect
39. #14524020C22semidirect
40. #14524018C22semidirect
41. #14524017C22semidirect
42. #14524016C22semidirect
43. #14524015C22semidirect
44. #14524015C22semidirect
45. #14524015C22semidirect
46. #14524014C22semidirect
47. #14524014C22semidirect
51. Q8×Dic67C22semidirect
52. C4×Dic134C22semidirect
53. #14524008C22semidirect
54. C4×D268C22semidirect
55. D4×Dic67C22semidirect
56. C4×C22⋊D67C22semidirect
57. C4×C22⋊D67C22semidirect
58. (C22⋊D67)⋊C4C22semidirect
62. C2×#41×C67C22semidirect
63. #127×C67C22semidirect
64. #121×C67C22semidirect
65. #121×C67C22semidirect
66. C4×Q8×C67C22semidirect
67. C4×D4×C67C22semidirect
80. Q8×D67D4semidirect
211. C67#3487semidirect
217. #550D67semidirect
236. C2×#41C2×D67semidirect
244. #127C2×D67semidirect
262. #121C2×D67semidirect
263. #121C2×D67semidirect
270. C4×Q8C2×D67semidirect
276. C4×D4C2×D67semidirect