Finite Group Databasebeta

← #22569065

Subgroups — #22569065

Order 8576.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D67C211yesmaximal, characteristic
2#16863872C211yesmaximal, characteristic, complemented
3#16863868C211yesmaximal, characteristic
4#16863498C211yesmaximal, characteristic, complemented
5#16863497C211yesmaximal, characteristic, complemented
6#16863495C211yesmaximal, characteristic, complemented
7#16863487C211yesmaximal, characteristic, complemented
8C4×#11761615C211yesmaximal, characteristic, complemented
9C4×#11761616C211yesmaximal, characteristic, complemented
10#16863481C211yesmaximal, characteristic, complemented
11#16863433C211yesmaximal, characteristic, complemented
12#16863403C211yesmaximal, characteristic, complemented
13#16863399C211yesmaximal, characteristic
14#16863353C211yesmaximal, characteristic, complemented
15#440×C67C211yesmaximal, characteristic, complemented
16C2×Q8×D67C2211yescharacteristic
17C2×D4×D67C2211yescharacteristic
18C2×#11761617C2211yescharacteristic
19C2×#11761616C2211yescharacteristic
20C2×#11761615C2211yescharacteristic
21SD16×D67C418yescomplemented
22C2×#11761611C2211yescharacteristic
23C2×C8×D67C2211yescharacteristic
24Q8×Dic67C2211yescharacteristic, complemented
25C4×Dic134—21no—
26C4×Dic134C2211yescharacteristic, complemented
27#14524008—21no—
28#14524008C2211yescharacteristic, complemented
29C4×D268C2211yescharacteristic
30D4×Dic67C2211yescharacteristic
31#14524005C2211yescharacteristic
32C4×C22⋊D67—21no—
33(C22⋊D67)⋊C4—21no—
34#14524001—21no—
35#14524001C2211yescharacteristic
36#14524001C2211yescharacteristic
37#14524000—21no—
38#14524000C2211yescharacteristic
39C2×(C2×D67)⋊C4—21no—
40C22⋊C4×D67—21no—
41#14523977C2211yescharacteristic, complemented
42#14523974C2211yescharacteristic, complemented
43#14523974C2211yescharacteristic, complemented
44#14523968C2211yescharacteristic, complemented
45#14523967C2211yescharacteristic, complemented
46#14523966C2211yescharacteristic, complemented
47#14523965C2211yescharacteristic
48#14523964C2211yescharacteristic
49#14523963C2211yescharacteristic
50#14523953C2211yescharacteristic, complemented
51C4×C67⋊C8C2211yescharacteristic, complemented
52C8×Dic67C2211yescharacteristic, complemented
53C2×SD16×C67C2211yescharacteristic
54C4×Q8×C67C2211yescharacteristic, complemented
55C4×D4×C67C2211yescharacteristic
56#107×C67C2211yescharacteristic, complemented
57#103×C67C2211yescharacteristic, complemented
58#102×C67C2211yescharacteristic
59#97×C67C2211yescharacteristic, complemented
60C23×D67—21no—
61C2×Dic134—21no—
62C2×Dic134C2311yescharacteristic
63Q8×D67—22no—
64Q8×D67C2×C414yescomplemented
65C2×D268C2311yescharacteristic
66C2×C22⋊D67—21no—
67D4×D67—22no—
68D4×D67C2×C414yes—
69C22×Dic67—21no—
70C2×C4×D67—21no—
71C2×C4×D67—21no—
72C2×C4×D67—21no—
73C2×C4×D67C2311yescharacteristic
74#11761617C2×C414yescomplemented
75#11761616C2×C414yescomplemented
76#11761615C2×C414yescomplemented
77#11761611—22no—
78C2×C67⋊C8C2311yescharacteristic
79C2×C67⋊C8C2311yescharacteristic
80C8×D67C2×C414yescomplemented
81C4⋊Dic67—21no—
82C4⋊Dic67C2311yescharacteristic
83C4⋊Dic67C2311yescharacteristic
84Dic67⋊C4—21no—
85Dic67⋊C4—21no—
86Dic67⋊C4D411yescharacteristic
87Dic67⋊C4D411yescharacteristic
88C22⋊Dic67—21no—
89(C2×D67)⋊C4—21no—
90(C2×D67)⋊C4—21no—
91(C2×D67)⋊C4—21no—
92(C2×D67)⋊C4—21no—
93(C2×D67)⋊C4D411yescharacteristic
94(C2×D67)⋊C4D411yescharacteristic
95C4×Dic67—21no—
96C4×Dic67—21no—
97C4×Dic67—21no—
98C4×Dic67—21no—
99C4×Dic67—21no—
100C4×Dic67C2311yescharacteristic
101C2×Q8×C67C2311yescharacteristic
102C2×D4×C67C2311yescharacteristic
103C22×C4×C67—21no—
104SD16×C67C2×C414yescomplemented
105C2×C8×C67C2311yescharacteristic
106C2×C8×C67C2311yescharacteristic
107C4⋊C4×C67—21no—
108C4⋊C4×C67C2311yescharacteristic
109C4⋊C4×C67C2311yescharacteristic
110C22⋊C4×C67—21no—
111C42×C67—21no—
112C42×C67C2311yescharacteristic
113C23×C67—21no—
114C22×D67—21no—
115C22×D67—24no—
116C22×D67—81no—
117C22×D67—81no—
118C22×D67C2×D411yescharacteristic
119Q8×C67—21no—
120Q8×C67C22×C412yes—
121D4×C67—21no—
122D4×C67C22×C412yes—
123C2×C4×C67—21no—
124C2×C4×C67—21no—
125C2×C4×C67—21no—
126C2×C4×C67—81no—
127C2×C4×C67C2×D411yescharacteristic
128C2×C4×C67C2×D411yescharacteristic
129C2×C4×C67C2411yescharacteristic
130C22⋊D67—24no—
131C2×Dic67—21no—
132C2×Dic67—21no—
133C2×Dic67—21no—
134C2×Dic67—81no—
135C2×Dic67C2×D411yescharacteristic
136C2×Dic67C2×D411yescharacteristic
137C2×Dic67C2×D411yescharacteristic
138D268—21no—
139D268C22×C412yes—
140C4×D67—24no—
141C4×D67—81no—
142C4×D67—81no—
143C4×D67C22×C412yes—
144C4×D67#4112yes—
145Dic134—21no—
146Dic134—24no—
147Dic134C22×C412yes—
148C8×C67—21no—
149C8×C67C22×C412yes—
150C67⋊C8—21no—
151C67⋊C8C22×C412yes—
152C22×C67—22no—
153C22×C67—41no—
154C22×C67—41no—
155C22×C67C22×D411yescharacteristic
156C2×D67—21no—
157C2×D67—21no—
158C2×D67—22no—
159C2×D67—41no—
160C2×D67—41no—
161C2×D67—84no—
162C2×D67C4×D412yes—
163C4×C67—22no—
164C4×C67—21no—
165C4×C67—21no—
166C4×C67—41no—
167C4×C67—41no—
168C4×C67C23×C411yescharacteristic
169C4×C67C2×#4111yescharacteristic
170Dic67—22no—
171Dic67—21no—
172Dic67—21no—
173Dic67—41no—
174Dic67—41no—
175Dic67C4×D412yes—
176C2×C67—42no—
177C2×C67C2×C4×D411yescharacteristic
178C2×C67#57711yescharacteristic
179C2×C67#57711yescharacteristic
180D67—22no—
181D67—42no—
182#2949—671nomaximal, Sylow
183C67#294911yescharacteristic, complemented, Sylow
184C22×SD16—671no—
185#519—671no—
186#518—671no—
187#440—671no—
188#440—671no—
189#440—671no—
190#440D6711yescharacteristic, complemented
191C4×SD16—671no—
192C4×SD16—671no—
193C4×SD16—671no—
194C4×SD16—671no—
195#427—671no—
196#419—671no—
197#418—671no—
198#405—671no—
199C22×Q8—671no—
200C22×D4—671no—
201C2×SD16—671no—
202C2×SD16—671no—
203C2×SD16—678no—
204C2×SD16—671no—
205C2×SD16C2×D6711yescharacteristic
206C2×M16—671no—
207C22×C8—671no—
208C4×Q8—671no—
209C4×Q8—671no—
210C4×Q8—671no—
211C4×Q8—1341no—
212C4×Q8—1341no—
213C4×Q8C2×D6711yescharacteristic, complemented
214C4×D4—671no—
215C4×D4—671no—
216C4×D4—671no—
217C4×D4—1341no—
218C4×D4—1341no—
219C4×D4C2×D6711yescharacteristic
220#117—671no—
221#117—671no—
222#117—671no—
223#117—1341no—
224#117—1341no—
225C2×C22⋊C4—1341no—
226C2×C22⋊C4—1341no—
227#107—671no—
228#107C2×D6711yescharacteristic, complemented
229#106—671no—
230#106—671no—
231#103—671no—
232#103—671no—
233#103—671no—
234#103C2×D6711yescharacteristic, complemented
235#102—671no—
236#102—671no—
237#102—671no—
238#102C2×D6711yescharacteristic
239#97—671no—
240#97C2×D6711yescharacteristic, complemented
241C4×C8—671no—
242C4×C8—671no—
243C24—1341no—
244C2×Q8—674no—
245C2×Q8—671no—
246C2×Q8—1341no—
247C2×Q8—1342no—
248C2×Q8C22×D6711yescharacteristic
249C2×D4—674no—
250C2×D4—671no—
251C2×D4—1342no—
252C2×D4—1341no—
253C2×D4C22×D6711yescharacteristic
254C22×C4—21no—
255C22×C4—671no—
256C22×C4—1341no—
257C22×C4—1341no—
258C22×C4—1341no—
259C22×C4—1341no—
260SD16—674no—
261SD16—674no—
262SD16—674no—
263SD16C4×D6714yescomplemented
264M16—1342no—
265C2×C8—671no—
266C2×C8—674no—
267C2×C8—671no—
268C2×C8C22×D6711yescharacteristic
269C2×C8C22×D6711yescharacteristic
270C4⋊C4—21no—
271C4⋊C4—671no—
272C4⋊C4—671no—
273C4⋊C4—671no—
274C4⋊C4—671no—
275C4⋊C4—1341no—
276C4⋊C4—1341no—
277C4⋊C4—1341no—
278C4⋊C4C22×D6711yescharacteristic
279C4⋊C4C22×D6711yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—671no—
282C22⋊C4—671no—
283C22⋊C4—1341no—
284C22⋊C4—1341no—
285C22⋊C4—1341no—
286C22⋊C4—1341no—
287C22⋊C4—1341no—
288C42—21no—
289C42—671no—
290C42—1341no—
291C42—1341no—
292C42—1341no—
293C42—1341no—
294C42—1341no—
295C42C22×D6711yescharacteristic
296C23—21no—
297C23—5361no—
298C23—5361no—
299C23—671no—
300C23—1344no—
301C23—1341no—
302Q8—21no—
303Q8—672no—
304Q8—1344no—
305Q8—1341no—
306Q8C2×C4×D6712yes—
307D4—21no—
308D4—672no—
309D4—1344no—
310D4—1341no—
311D4C2×C4×D6712yes—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—81no—
316C2×C4—5361no—
317C2×C4—5361no—
318C2×C4—5361no—
319C2×C4—671no—
320C2×C4—672no—
321C2×C4—672no—
322C2×C4—671no—
323C2×C4—671no—
324C2×C4—1344no—
325C2×C4—1341no—
326C2×C4—1341no—
327C2×C4—1341no—
328C2×C4D4×D6711yescharacteristic
329C2×C4D4×D6711yescharacteristic
330C2×C4C23×D6711yescharacteristic
331C8—21no—
332C8—672no—
333C8—1341no—
334C8C2×C4×D6712yes—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—2681no—
339C22—2681no—
340C22—5364no—
341C22—672no—
342C22—1342no—
343C22—1341no—
344C22—1341no—
345C22C2×D4×D6711yescharacteristic
346C4—22no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—2681no—
352C4—2681no—
353C4—672no—
354C4—1342no—
355C4—1341no—
356C4—1341no—
357C4C22×C4×D6711yescharacteristic
358C4#41×D6711yescharacteristic
359C2—42no—
360C2—2682no—
361C2—1342no—
362C2C4×D4×D6711yescharacteristic
363C2#1686420511yescharacteristic
364C2#1686420611yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC4×C67C23×C4
FrattiniC2×C4C23×D67
Fitting#440×C67C2
socleC22×C67C22×D4
hypercenter#440D67
hypocenterC67#2949
LCSG ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. #16863872C2semidirect
4. #16863498C2semidirect
5. #16863497C2semidirect
6. #16863495C2semidirect
7. #16863487C2semidirect
8. C4×#11761615C2semidirect
9. C4×#11761616C2semidirect
10. #16863481C2semidirect
11. #16863433C2semidirect
12. #16863403C2semidirect
14. #16863353C2semidirect
15. #440×C67C2semidirect
21. SD16×D67C4semidirect
24. Q8×Dic67C22semidirect
26. C4×Dic134C22semidirect
28. #14524008C22semidirect
41. #14523977C22semidirect
42. #14523974C22semidirect
43. #14523974C22semidirect
44. #14523968C22semidirect
45. #14523967C22semidirect
46. #14523966C22semidirect
50. #14523953C22semidirect
51. C4×C67⋊C8C22semidirect
52. C8×Dic67C22semidirect
54. C4×Q8×C67C22semidirect
56. #107×C67C22semidirect
57. #103×C67C22semidirect
59. #97×C67C22semidirect
64. Q8×D67C2×C4semidirect
74. #11761617C2×C4semidirect
75. #11761616C2×C4semidirect
76. #11761615C2×C4semidirect
80. C8×D67C2×C4semidirect
104. SD16×C67C2×C4semidirect
183. C67#2949semidirect
190. #440D67semidirect
213. C4×Q8C2×D67semidirect
228. #107C2×D67semidirect
234. #103C2×D67semidirect
240. #97C2×D67semidirect
263. SD16C4×D67semidirect