Finite Group Databasebeta

← #22569040

Subgroups — #22569040

Order 8576.

506 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14524070C211yesmaximal, characteristic
2#16863878C211yesmaximal, characteristic, complemented
3#16863868C211yesmaximal, characteristic, complemented
4#16863509C211yesmaximal, characteristic, complemented
5#16863505C211yesmaximal, characteristic, complemented
6#16863499C211yesmaximal, characteristic, complemented
7SD16×Dic67C211yesmaximal, characteristic, complemented
8C4×#11761615C211yesmaximal, characteristic, complemented
9#16863478C211yesmaximal, characteristic, complemented
10C4×#11761613C211yesmaximal, characteristic, complemented
11#16863437C211yesmaximal, characteristic, complemented
12#16863391C211yesmaximal, characteristic, complemented
13#102×D67C211yesmaximal, characteristic, complemented
14#16863353C211yesmaximal, characteristic, complemented
15#440×C67C211yesmaximal, characteristic, complemented
16C2×#11761629C2211yescharacteristic
17C2×D4×D67C2211yescharacteristic
18#14524070C418yescomplemented
19C2×#11761615C2211yescharacteristic
20C2×D536C2211yescharacteristic
21C2×#11761613C2211yescharacteristic
22C2×#11761611C2211yescharacteristic
23C2×C8×D67C2211yescharacteristic
24Q8×Dic67C2211yescharacteristic, complemented
25C4×D268—21no—
26C4×D268C2211yescharacteristic
27C4×D268C2211yescharacteristic, complemented
28D4×Dic67C2211yescharacteristic
29#14524005—21no—
30#14524005C2211yescharacteristic
31#14524005C2211yescharacteristic, complemented
32C4×C22⋊D67—21no—
33(C22⋊D67)⋊C4—21no—
34#14524001C2211yescharacteristic
35#14524000—21no—
36#14524000C2211yescharacteristic
37C4⋊C4×D67—21no—
38C4⋊C4×D67C2211yescharacteristic
39C2×(C2×D67)⋊C4—21no—
40C22⋊C4×D67—21no—
41#14523976C2211yescharacteristic, complemented
42#14523975C2211yescharacteristic, complemented
43#14523974C2211yescharacteristic, complemented
44#14523967C2211yescharacteristic, complemented
45#14523965C2211yescharacteristic
46#14523965C2211yescharacteristic, complemented
47#14523964C2211yescharacteristic
48#14523963C2211yescharacteristic
49#14523963C2211yescharacteristic, complemented
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—
61#11761629—22no—
62#11761629C2×C414yescomplemented
63C2×D268—21no—
64C2×D268C2311yescharacteristic
65C2×D268C2311yescharacteristic
66C2×C22⋊D67—21no—
67D4×D67—22no—
68D4×D67C2×C414yescomplemented
69C22×Dic67—21no—
70C2×C4×D67—21no—
71C2×C4×D67—21no—
72C2×C4×D67—21no—
73C2×C4×D67—21no—
74C2×C4×D67—21no—
75C2×C4×D67—21no—
76C2×C4×D67C2311yescharacteristic
77#11761615C2×C414yescomplemented
78D536C2×C414yescomplemented
79#11761613C2×C414yescomplemented
80#11761611C2×C414yescomplemented
81C2×C67⋊C8C2311yescharacteristic
82C2×C67⋊C8C2311yescharacteristic
83C8×D67—22no—
84C4⋊Dic67—21no—
85C4⋊Dic67C2311yescharacteristic
86C4⋊Dic67C2311yescharacteristic
87Dic67⋊C4—21no—
88Dic67⋊C4D411yescharacteristic
89Dic67⋊C4D411yescharacteristic
90C22⋊Dic67—21no—
91(C2×D67)⋊C4—21no—
92(C2×D67)⋊C4—21no—
93(C2×D67)⋊C4—21no—
94(C2×D67)⋊C4—21no—
95(C2×D67)⋊C4—21no—
96(C2×D67)⋊C4D411yescharacteristic
97(C2×D67)⋊C4D411yescharacteristic
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—21no—
116C22×D67—24no—
117C22×D67—81no—
118C22×D67—81no—
119C22×D67C2×D411yescharacteristic
120Q8×C67—21no—
121Q8×C67C22×C412yes—
122D4×C67—21no—
123D4×C67C22×C412yes—
124C2×C4×C67—21no—
125C2×C4×C67—21no—
126C2×C4×C67—21no—
127C2×C4×C67—81no—
128C2×C4×C67C2×D411yescharacteristic
129C2×C4×C67C2×D411yescharacteristic
130C2×C4×C67C2411yescharacteristic
131C22⋊D67—24no—
132C2×Dic67—21no—
133C2×Dic67—21no—
134C2×Dic67—81no—
135C2×Dic67C2×D411yescharacteristic
136C2×Dic67C2×D411yescharacteristic
137C2×Dic67C2×D411yescharacteristic
138D268—24no—
139D268—21no—
140D268—21no—
141D268C22×C412yes—
142D268C22×C412yes—
143C4×D67—24no—
144C4×D67—81no—
145C4×D67—81no—
146C4×D67—81no—
147C4×D67—81no—
148C4×D67—81no—
149C4×D67C22×C412yes—
150C4×D67#4112yes—
151C8×C67—21no—
152C8×C67C22×C412yes—
153C67⋊C8—21no—
154C67⋊C8C22×C412yes—
155C22×C67—22no—
156C22×C67—41no—
157C22×C67—41no—
158C22×C67C22×D411yescharacteristic
159C2×D67—21no—
160C2×D67—21no—
161C2×D67—22no—
162C2×D67—22no—
163C2×D67—41no—
164C2×D67—41no—
165C2×D67—41no—
166C2×D67—41no—
167C2×D67—84no—
168C2×D67C4×D412yes—
169C4×C67—22no—
170C4×C67—21no—
171C4×C67—21no—
172C4×C67—41no—
173C4×C67—41no—
174C4×C67C23×C411yescharacteristic
175C4×C67C2×#4111yescharacteristic
176Dic67—21no—
177Dic67—21no—
178Dic67—41no—
179Dic67—41no—
180Dic67C4×D412yes—
181C2×C67—42no—
182C2×C67C2×C4×D411yescharacteristic
183C2×C67#57611yescharacteristic
184C2×C67#57711yescharacteristic
185D67—22no—
186D67—42no—
187D67—42no—
188#2946—671nomaximal, Sylow
189C67#294611yescharacteristic, complemented, Sylow
190C2×AGL(1,ℤ/8)—671no—
191#520—671no—
192#518—671no—
193#442—671no—
194#442—671no—
195#440—671no—
196#440D6711yescharacteristic, complemented
197C4×SD16—671no—
198C4×SD16—671no—
199C4×D8—671no—
200C4×D8—671no—
201#428—671no—
202#416—671no—
203C2×#102—671no—
204#405—671no—
205C2×#41—671no—
206C22×D4—671no—
207AGL(1,ℤ/8)—678no—
208C2×SD16—671no—
209C2×SD16C2×D6711yescharacteristic
210C2×D8—671no—
211C2×D8—671no—
212C2×M16—671no—
213C22×C8—671no—
214C4×Q8—671no—
215C4×Q8C2×D6711yescharacteristic, complemented
216C4×D4—671no—
217C4×D4—671no—
218C4×D4—671no—
219C4×D4—671no—
220C4×D4—671no—
221C4×D4—1341no—
222C4×D4—1341no—
223C4×D4—1341no—
224C4×D4—1341no—
225C4×D4C2×D6711yescharacteristic
226#117—671no—
227#117—671no—
228#117—1341no—
229C2×C4⋊C4—671no—
230C2×C4⋊C4—1341no—
231C2×C22⋊C4—1341no—
232C2×C22⋊C4—1341no—
233#107—671no—
234#107C2×D6711yescharacteristic, complemented
235#106—671no—
236#106—671no—
237#103—671no—
238#103C2×D6711yescharacteristic, complemented
239#102—671no—
240#102—671no—
241#102—671no—
242#102—671no—
243#102—671no—
244#102C2×D6711yescharacteristic
245#97—671no—
246#97C2×D6711yescharacteristic, complemented
247C4×C8—671no—
248C4×C8—671no—
249C24—1341no—
250#41—674no—
251#41—1342no—
252C2×Q8C22×D6711yescharacteristic
253C2×D4—674no—
254C2×D4—671no—
255C2×D4—671no—
256C2×D4—1341no—
257C2×D4—1342no—
258C2×D4—1341no—
259C2×D4C22×D6711yescharacteristic
260C22×C4—21no—
261C22×C4—671no—
262C22×C4—1341no—
263C22×C4—1341no—
264C22×C4—1341no—
265C22×C4—1341no—
266C22×C4—1341no—
267C22×C4—1341no—
268C22×C4—1341no—
269SD16—674no—
270SD16C4×D6714yescomplemented
271D8—674no—
272D8—674no—
273M16—674no—
274C2×C8—671no—
275C2×C8—671no—
276C2×C8—1342no—
277C2×C8C22×D6711yescharacteristic
278C2×C8C22×D6711yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—671no—
281C4⋊C4—671no—
282C4⋊C4—671no—
283C4⋊C4—671no—
284C4⋊C4—1341no—
285C4⋊C4—1341no—
286C4⋊C4C22×D6711yescharacteristic
287C4⋊C4C22×D6711yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—671no—
290C22⋊C4—671no—
291C22⋊C4—1341no—
292C22⋊C4—1341no—
293C22⋊C4—1341no—
294C22⋊C4—1341no—
295C22⋊C4—1341no—
296C22⋊C4—1341no—
297C42—21no—
298C42—671no—
299C42—1341no—
300C42—1341no—
301C42C22×D6711yescharacteristic
302C23—21no—
303C23—5361no—
304C23—5361no—
305C23—671no—
306C23—1344no—
307C23—1341no—
308C23—1341no—
309Q8—21no—
310Q8C2×C4×D6712yes—
311D4—21no—
312D4—672no—
313D4—672no—
314D4—1344no—
315D4—1344no—
316D4—1341no—
317D4—1341no—
318D4C2×C4×D6712yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—81no—
323C2×C4—5361no—
324C2×C4—5361no—
325C2×C4—5361no—
326C2×C4—5361no—
327C2×C4—5361no—
328C2×C4—5361no—
329C2×C4—671no—
330C2×C4—672no—
331C2×C4—672no—
332C2×C4—671no—
333C2×C4—671no—
334C2×C4—1344no—
335C2×C4—1341no—
336C2×C4—1341no—
337C2×C4D4×D6711yescharacteristic
338C2×C4D4×D6711yescharacteristic
339C2×C4C23×D6711yescharacteristic
340C8—21no—
341C8—672no—
342C8—1341no—
343C8C2×C4×D6712yes—
344C22—22no—
345C22—41no—
346C22—41no—
347C22—2681no—
348C22—2681no—
349C22—2681no—
350C22—2681no—
351C22—5364no—
352C22—672no—
353C22—1342no—
354C22—1342no—
355C22—1341no—
356C22—1341no—
357C22C2×D4×D6711yescharacteristic
358C4—22no—
359C4—21no—
360C4—21no—
361C4—41no—
362C4—41no—
363C4—2681no—
364C4—2681no—
365C4—672no—
366C4—1341no—
367C4—1341no—
368C4C22×C4×D6711yescharacteristic
369C4#41×D6711yescharacteristic
370C2—42no—
371C2—2682no—
372C2—2682no—
373C2—1342no—
374C2C4×D4×D6711yescharacteristic
375C2#1686420011yescharacteristic
376C2#1686420511yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC4×C67C23×C4
FrattiniC2×C4C23×D67
Fitting#440×C67C2
socleC22×C67C22×D4
hypercenter#440D67
hypocenterC67#2946
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. #16863878C2semidirect
3. #16863868C2semidirect
4. #16863509C2semidirect
5. #16863505C2semidirect
6. #16863499C2semidirect
7. SD16×Dic67C2semidirect
8. C4×#11761615C2semidirect
9. #16863478C2semidirect
10. C4×#11761613C2semidirect
11. #16863437C2semidirect
12. #16863391C2semidirect
13. #102×D67C2semidirect
14. #16863353C2semidirect
15. #440×C67C2semidirect
18. #14524070C4semidirect
24. Q8×Dic67C22semidirect
27. C4×D268C22semidirect
31. #14524005C22semidirect
41. #14523976C22semidirect
42. #14523975C22semidirect
43. #14523974C22semidirect
44. #14523967C22semidirect
46. #14523965C22semidirect
49. #14523963C22semidirect
50. #14523953C22semidirect
51. C4×C67⋊C8C22semidirect
52. C8×Dic67C22semidirect
54. C4×Q8×C67C22semidirect
56. #107×C67C22semidirect
57. #103×C67C22semidirect
59. #97×C67C22semidirect
62. #11761629C2×C4semidirect
68. D4×D67C2×C4semidirect
77. #11761615C2×C4semidirect
78. D536C2×C4semidirect
79. #11761613C2×C4semidirect
80. #11761611C2×C4semidirect
104. SD16×C67C2×C4semidirect
189. C67#2946semidirect
196. #440D67semidirect
215. C4×Q8C2×D67semidirect
234. #107C2×D67semidirect
238. #103C2×D67semidirect
246. #97C2×D67semidirect
270. SD16C4×D67semidirect