Finite Group Databasebeta

← #22570306

Subgroups — #22570306

Order 8576.

444 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16863950C211yesmaximal, characteristic, complemented
2C4×D4×D67C211yesmaximal, characteristic
3#16863733C211yesmaximal, characteristic, complemented
4#16863730C211yesmaximal, characteristic, complemented
5#16863675C211yesmaximal, characteristic, complemented
6#16863674C211yesmaximal, characteristic, complemented
7#16863600C211yesmaximal, characteristic, complemented
8#16863599C211yesmaximal, characteristic, complemented
9#16863548C211yesmaximal, characteristic
10#16863545C211yesmaximal, characteristic
11#16863498C211yesmaximal, characteristic, complemented
12#16863496C211yesmaximal, characteristic, complemented
13#16863494C211yesmaximal, characteristic, complemented
14#16863451C211yesmaximal, characteristic, complemented
15#440×C67C211yesmaximal, characteristic, complemented
16C2×D4×D67C2211yescharacteristic
17C22×C4×D67—21no—
18C2×#11761617C2211yescharacteristic
19C2×#11761616C2211yescharacteristic
20C2×#11761615C2211yescharacteristic
21#14524042C2211yescharacteristic, complemented
22#14524041C2211yescharacteristic, complemented
23#14524033—21no—
24#14524023C2211yescharacteristic
25#14524021—21no—
26#14524019C2211yescharacteristic
27C4×Dic134—21no—
28C4×Dic134C2211yescharacteristic, complemented
29C4×D268C2211yescharacteristic
30D4×Dic67C2211yescharacteristic
31#14524005C2211yescharacteristic
32C4×C22⋊D67—21no—
33(C22⋊D67)⋊C4—21no—
34#14524000—21no—
35C4⋊C4×D67C2211yescharacteristic
36C22⋊C4×D67—21no—
37C42×D67C2211yescharacteristic
38#14523977C2211yescharacteristic, complemented
39#14523976C2211yescharacteristic, complemented
40#14523976C2211yescharacteristic, complemented
41Dic67⋊C8C2211yescharacteristic, complemented
42Dic67⋊C8C2211yescharacteristic, complemented
43#14523968C2211yescharacteristic, complemented
44#14523967C2211yescharacteristic, complemented
45#14523966C2211yescharacteristic, complemented
46#14523965C2211yescharacteristic
47#14523964C2211yescharacteristic
48#14523963C2211yescharacteristic
49(C2×D67)⋊C8C2211yescharacteristic
50(C2×D67)⋊C8C2211yescharacteristic
51#14523954C2211yescharacteristic, complemented
52C2×SD16×C67C2211yescharacteristic
53C4×Q8×C67C2211yescharacteristic, complemented
54C4×D4×C67C2211yescharacteristic
55#107×C67C2211yescharacteristic, complemented
56#103×C67C2211yescharacteristic, complemented
57#102×C67C2211yescharacteristic
58#97×C67C2211yescharacteristic, complemented
59C23×D67—21no—
60C2×Dic134—21no—
61C2×Dic134C2311yescharacteristic
62C2×D268C2311yescharacteristic
63C2×C22⋊D67—21no—
64D4×D67—22no—
65D4×D67—22no—
66C22×Dic67—21no—
67C2×C4×D67—21no—
68C2×C4×D67—21no—
69C2×C4×D67—21no—
70C2×C4×D67—44no—
71C2×C4×D67D411yescharacteristic
72C2×C4×D67D411yescharacteristic
73C2×C4×D67C2311yescharacteristic
74#11761617—22no—
75#11761616—22no—
76#11761615—22no—
77C2×C67⋊C8C2311yescharacteristic
78C2×C67⋊C8C2311yescharacteristic
79C4⋊Dic67—21no—
80C4⋊Dic67C2311yescharacteristic
81C4⋊Dic67C2311yescharacteristic
82Dic67⋊C4—21no—
83Dic67⋊C4—21no—
84Dic67⋊C4—21no—
85Dic67⋊C4—21no—
86Dic67⋊C4—21no—
87Dic67⋊C4—21no—
88C22⋊Dic67—21no—
89(C2×D67)⋊C4—21no—
90(C2×D67)⋊C4—21no—
91(C2×D67)⋊C4—21no—
92(C2×D67)⋊C4—21no—
93C4×Dic67—21no—
94C4×Dic67D411yescharacteristic
95C4×Dic67D411yescharacteristic
96C4×Dic67C2311yescharacteristic
97C2×Q8×C67C2311yescharacteristic
98C2×D4×C67C2311yescharacteristic
99C22×C4×C67—21no—
100SD16×C67—22no—
101C2×C8×C67C2311yescharacteristic
102C2×C8×C67C2311yescharacteristic
103C4⋊C4×C67—21no—
104C4⋊C4×C67C2311yescharacteristic
105C4⋊C4×C67C2311yescharacteristic
106C22⋊C4×C67—21no—
107C42×C67—21no—
108C42×C67C2311yescharacteristic
109C23×C67—21no—
110C22×D67—24no—
111C22×D67—21no—
112C22×D67—42no—
113C22×D67—42no—
114C22×D67C2×D411yescharacteristic
115Q8×C67—21no—
116Q8×C67—21no—
117D4×C67—21no—
118D4×C67#4112yes—
119C2×C4×C67—21no—
120C2×C4×C67—21no—
121C2×C4×C67—21no—
122C2×C4×C67—42no—
123C2×C4×C67C2×D411yescharacteristic
124C2×C4×C67C2×D411yescharacteristic
125C2×C4×C67C2411yescharacteristic
126C22⋊D67—24no—
127C2×Dic67—21no—
128C2×Dic67—21no—
129C2×Dic67—21no—
130C2×Dic67—42no—
131C2×Dic67C2×D411yescharacteristic
132C2×Dic67C2×D411yescharacteristic
133C2×Dic67C2×D411yescharacteristic
134D268—21no—
135D268#4112yes—
136C4×D67—22no—
137C4×D67—21no—
138C4×D67—21no—
139C4×D67—22no—
140C4×D67—42no—
141C4×D67—42no—
142C4×D67—42no—
143Dic134—21no—
144Dic134—21no—
145Dic134—42no—
146C8×C67—21no—
147C8×C67—21no—
148C67⋊C8—21no—
149C67⋊C8—21no—
150C22×C67—22no—
151C22×C67—41no—
152C22×C67—41no—
153C22×C67C22×D411yescharacteristic
154C2×D67—21no—
155C2×D67—21no—
156C2×D67—22no—
157C2×D67—48no—
158C2×D67—41no—
159C2×D67—41no—
160C2×D67#13712yes—
161C4×C67—21no—
162C4×C67—21no—
163C4×C67—41no—
164C4×C67—41no—
165C4×C67—41no—
166C4×C67C2×#4111yescharacteristic
167C4×C67C2×#4111yescharacteristic
168Dic67—21no—
169Dic67—21no—
170Dic67—41no—
171Dic67—41no—
172Dic67—41no—
173Dic67AGL(1,ℤ/8)12yes—
174C2×C67—42no—
175C2×C67#52511yescharacteristic
176C2×C67C2×AGL(1,ℤ/8)11yescharacteristic
177C2×C67C2×#13711yescharacteristic
178D67—22no—
179D67—42no—
180#3110—671nomaximal, Sylow
181C67#311011yescharacteristic, complemented, Sylow
182#533—671no—
183C2×C4×D4—671no—
184#484—671no—
185#484—671no—
186#474—671no—
187#474—671no—
188#461—671no—
189#461—671no—
190#450—671no—
191#450—671no—
192#440—671no—
193#440—671no—
194#440—671no—
195#440D6711yescharacteristic, complemented
196#432—671no—
197C22×D4—671no—
198C23×C4—1341no—
199C2×SD16—671no—
200C2×SD16—671no—
201C2×SD16—671no—
202C2×SD16C2×D6711yescharacteristic
203C4⋊Q8—671no—
204C4⋊Q8—671no—
205#125—1341no—
206C22⋊Q8—671no—
207C22⋊Q8—671no—
208C22⋊Q8—1341no—
209C4×Q8—671no—
210C4×Q8—1341no—
211C4×Q8C2×D6711yescharacteristic, complemented
212C4×D4—671no—
213C4×D4—671no—
214C4×D4—671no—
215C4×D4—1341no—
216C4×D4—1341no—
217C4×D4C2×D6711yescharacteristic
218#117—1341no—
219C2×C4⋊C4—671no—
220C2×C22⋊C4—1341no—
221C2×C42—671no—
222#107—671no—
223#107—671no—
224#107—671no—
225#107C2×D6711yescharacteristic, complemented
226C4⋊C8—671no—
227C4⋊C8—671no—
228#103—671no—
229#103—671no—
230#103—671no—
231#103C2×D6711yescharacteristic, complemented
232#102—671no—
233#102—671no—
234#102—671no—
235#102C2×D6711yescharacteristic
236C22⋊C8—671no—
237C22⋊C8—671no—
238#97—671no—
239#97C2×D6711yescharacteristic, complemented
240C24—1341no—
241C2×Q8—671no—
242C2×Q8—1341no—
243C2×Q8C22×D6711yescharacteristic
244C2×D4—671no—
245C2×D4—1341no—
246C2×D4—1342no—
247C2×D4—1342no—
248C2×D4C22×D6711yescharacteristic
249C22×C4—21no—
250C22×C4—2684no—
251C22×C4—671no—
252C22×C4—671no—
253C22×C4—671no—
254C22×C4—1341no—
255C22×C4—1341no—
256C22×C4—1341no—
257C22×C4—1341no—
258SD16—22no—
259SD16—1342no—
260SD16—1342no—
261SD16—1342no—
262C2×C8—671no—
263C2×C8—671no—
264C2×C8C22×D6711yescharacteristic
265C2×C8C22×D6711yescharacteristic
266C4⋊C4—21no—
267C4⋊C4—671no—
268C4⋊C4—671no—
269C4⋊C4—1341no—
270C4⋊C4—1341no—
271C4⋊C4—1341no—
272C4⋊C4—1341no—
273C4⋊C4—1341no—
274C4⋊C4—1341no—
275C4⋊C4—1341no—
276C4⋊C4C22×D6711yescharacteristic
277C4⋊C4C22×D6711yescharacteristic
278C22⋊C4—21no—
279C22⋊C4—1341no—
280C22⋊C4—1341no—
281C22⋊C4—1341no—
282C22⋊C4—1341no—
283C22⋊C4—1341no—
284C42—21no—
285C42—671no—
286C42—671no—
287C42—671no—
288C42—1341no—
289C42C22×D6711yescharacteristic
290C23—21no—
291C23—2682no—
292C23—2682no—
293C23—671no—
294C23—1344no—
295C23—1341no—
296Q8—21no—
297Q8—21no—
298Q8—2682no—
299Q8—1341no—
300Q8—1341no—
301D4—21no—
302D4—672no—
303D4—1344no—
304D4—1341no—
305D4#1176162812yes—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—2682no—
311C2×C4—2682no—
312C2×C4—2682no—
313C2×C4—2682no—
314C2×C4—671no—
315C2×C4—671no—
316C2×C4—671no—
317C2×C4—1342no—
318C2×C4—1342no—
319C2×C4—1341no—
320C2×C4—1341no—
321C2×C4—1341no—
322C2×C4—1341no—
323C2×C4—1341no—
324C2×C4D4×D6711yescharacteristic
325C2×C4D4×D6711yescharacteristic
326C2×C4C23×D6711yescharacteristic
327C8—21no—
328C8—21no—
329C8—1341no—
330C8—1341no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—2688no—
335C22—2681no—
336C22—2681no—
337C22—672no—
338C22—1342no—
339C22—1341no—
340C22—1341no—
341C22C2×D4×D6711yescharacteristic
342C4—21no—
343C4—21no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—2681no—
348C4—2681no—
349C4—2681no—
350C4—672no—
351C4—1341no—
352C4—1341no—
353C4#41×D6711yescharacteristic
354C4C2×#1176162811yescharacteristic
355C2—42no—
356C2—2682no—
357C2—1342no—
358C2#1686390511yescharacteristic
359C2AGL(1,ℤ/8)×D6711yescharacteristic
360C2#137×D6711yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#440×C67C2
socleC22×C67C22×D4
hypercenter#440D67
hypocenterC67#3110
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #16863950C2semidirect
3. #16863733C2semidirect
4. #16863730C2semidirect
5. #16863675C2semidirect
6. #16863674C2semidirect
7. #16863600C2semidirect
8. #16863599C2semidirect
11. #16863498C2semidirect
12. #16863496C2semidirect
13. #16863494C2semidirect
14. #16863451C2semidirect
15. #440×C67C2semidirect
21. #14524042C22semidirect
22. #14524041C22semidirect
28. C4×Dic134C22semidirect
38. #14523977C22semidirect
39. #14523976C22semidirect
40. #14523976C22semidirect
41. Dic67⋊C8C22semidirect
42. Dic67⋊C8C22semidirect
43. #14523968C22semidirect
44. #14523967C22semidirect
45. #14523966C22semidirect
51. #14523954C22semidirect
53. C4×Q8×C67C22semidirect
55. #107×C67C22semidirect
56. #103×C67C22semidirect
58. #97×C67C22semidirect
181. C67#3110semidirect
195. #440D67semidirect
211. C4×Q8C2×D67semidirect
225. #107C2×D67semidirect
231. #103C2×D67semidirect
239. #97C2×D67semidirect