Finite Group Databasebeta

← #22570562

Subgroups — #22570562

Order 8576.

404 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16864026C211yesmaximal, characteristic, complemented
2#16863964C211yesmaximal, characteristic
3#16863762C211yesmaximal, characteristic, complemented
4#16863759C211yesmaximal, characteristic, complemented
5#16863757C211yesmaximal, characteristic, complemented
6#16863755C211yesmaximal, characteristic, complemented
7#16863752C211yesmaximal, characteristic, complemented
8#16863743C211yesmaximal, characteristic, complemented
9#16863741C211yesmaximal, characteristic, complemented
10#16863399C211yesmaximal, characteristic, complemented
11#16863396C211yesmaximal, characteristic, complemented
12#16863391C211yesmaximal, characteristic, complemented
13#102×D67C211yesmaximal, characteristic, complemented
14#16863350C211yesmaximal, characteristic, complemented
15#488×C67C211yesmaximal, characteristic, complemented
16C2×#11761629C2211yescharacteristic
17C2×D4×D67C2211yescharacteristic
18C2×#11761611C2211yescharacteristic
19C2×C8×D67C2211yescharacteristic
20#14524043C2211yescharacteristic, complemented
21#14524042—21no—
22#14524042C2211yescharacteristic, complemented
23#14524040C2211yescharacteristic
24#14524039C2211yescharacteristic
25#14524034C2211yescharacteristic, complemented
26#14524033—21no—
27#14524032C2211yescharacteristic
28#14524023—21no—
29#14524021—21no—
30#14524016—21no—
31#14524016—21no—
32#14524013—21no—
33#14524013—21no—
34#14524001C2211yescharacteristic
35#14524000C2211yescharacteristic
36C4⋊C4×D67C2211yescharacteristic
37#14523967C2211yescharacteristic, complemented
38#14523967C2211yescharacteristic, complemented
39#14523965C2211yescharacteristic
40#14523965C2211yescharacteristic
41#14523965C2211yescharacteristic, complemented
42#14523965C2211yescharacteristic, complemented
43#14523964C2211yescharacteristic
44#14523964C2211yescharacteristic
45#14523963C2211yescharacteristic
46#14523963C2211yescharacteristic
47#14523963C2211yescharacteristic, complemented
48#14523963C2211yescharacteristic, complemented
49#14523954C2211yescharacteristic, complemented
50#14523953C2211yescharacteristic, complemented
51C8×Dic67C2211yescharacteristic, complemented
52#125×C67C2211yescharacteristic, complemented
53#124×C67C2211yescharacteristic
54#103×C67C2211yescharacteristic, complemented
55#103×C67C2211yescharacteristic, complemented
56#102×C67C2211yescharacteristic
57#102×C67C2211yescharacteristic
58C4×C8×C67C2211yescharacteristic, complemented
59C23×D67—21no—
60#11761629—22no—
61#11761629—22no—
62C2×Dic134—21no—
63C2×Dic134—21no—
64C2×D268—21no—
65C2×D268—21no—
66C2×D268—21no—
67C2×D268C2311yescharacteristic
68C2×D268C2311yescharacteristic
69C2×C22⋊D67—21no—
70C2×C22⋊D67—21no—
71C2×C22⋊D67—21no—
72D4×D67—22no—
73D4×D67—22no—
74C2×C4×D67—21no—
75C2×C4×D67—21no—
76C2×C4×D67C2311yescharacteristic
77#11761611—22no—
78C2×C67⋊C8C2311yescharacteristic
79C2×C67⋊C8C2311yescharacteristic
80C8×D67—22no—
81C4⋊Dic67—21no—
82C4⋊Dic67—21no—
83C4⋊Dic67C2311yescharacteristic
84C4⋊Dic67C2311yescharacteristic
85Dic67⋊C4—21no—
86Dic67⋊C4—21no—
87Dic67⋊C4—21no—
88Dic67⋊C4D411yescharacteristic
89Dic67⋊C4D411yescharacteristic
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×Dic67C2311yescharacteristic
97C2×Q8×C67C2311yescharacteristic
98C2×D4×C67C2311yescharacteristic
99C2×C8×C67C2311yescharacteristic
100C2×C8×C67C2311yescharacteristic
101C4⋊C4×C67—21no—
102C4⋊C4×C67—21no—
103C4⋊C4×C67C2311yescharacteristic
104C4⋊C4×C67C2311yescharacteristic
105C22⋊C4×C67—21no—
106C22⋊C4×C67—21no—
107C42×C67C2311yescharacteristic
108C23×C67—21no—
109C22×D67—21no—
110C22×D67—21no—
111C22×D67—42no—
112C22×D67—81no—
113C22×D67—81no—
114C22×D67C2×D411yescharacteristic
115Q8×C67—21no—
116Q8×C67—21no—
117D4×C67—21no—
118D4×C67—21no—
119C2×C4×C67—21no—
120C2×C4×C67—21no—
121C2×C4×C67—21no—
122C2×C4×C67C2×D411yescharacteristic
123C2×C4×C67C2×D411yescharacteristic
124C2×C4×C67C2411yescharacteristic
125C22⋊D67—42no—
126C22⋊D67—81no—
127C22⋊D67—81no—
128C2×Dic67—21no—
129C2×Dic67—21no—
130C2×Dic67C2×D411yescharacteristic
131C2×Dic67C2×D411yescharacteristic
132C2×Dic67C2×D411yescharacteristic
133D268—21no—
134D268—21no—
135D268—21no—
136D268—21no—
137D268—42no—
138D268—81no—
139D268—81no—
140C4×D67—42no—
141C4×D67—81no—
142C4×D67#4112yes—
143C4×D67#4112yes—
144Dic134—81no—
145Dic134—81no—
146C8×C67—21no—
147C8×C67—21no—
148C67⋊C8—21no—
149C67⋊C8—21no—
150C22×C67—41no—
151C22×C67—41no—
152C22×C67—41no—
153C22×C67C22×D411yescharacteristic
154C2×D67—21no—
155C2×D67—21no—
156C2×D67—41no—
157C2×D67—41no—
158C2×D67—41no—
159C2×D67—41no—
160C2×D67—41no—
161C2×D67—41no—
162C2×D67—84no—
163C2×D67#12412yes—
164C4×C67—21no—
165C4×C67—21no—
166C4×C67—41no—
167C4×C67—41no—
168C4×C67—41no—
169C4×C67C2×#4111yescharacteristic
170C4×C67C2×#4111yescharacteristic
171Dic67—21no—
172Dic67—21no—
173Dic67—41no—
174Dic67—41no—
175Dic67#12412yes—
176C2×C67—81no—
177C2×C67C2×#12411yescharacteristic
178C2×C67#57611yescharacteristic
179C2×C67#57711yescharacteristic
180D67—22no—
181D67—81no—
182D67—81no—
183#3136—671nomaximal, Sylow
184C67#313611yescharacteristic, complemented, Sylow
185#544—671no—
186#535—671no—
187#490—671no—
188#490—671no—
189#489—671no—
190#489—671no—
191#488—671no—
192#488D6711yescharacteristic, complemented
193#486—671no—
194#486—671no—
195#418—671no—
196#417—671no—
197#416—671no—
198C2×#102—671no—
199#405—671no—
200C2×#41—671no—
201C22×D4—671no—
202C2×M16—671no—
203C22×C8—671no—
204C4⋊Q8—671no—
205C4⋊Q8—671no—
206C4⋊Q8—1341no—
207#127—671no—
208#127—671no—
209#125—671no—
210#125—1341no—
211#125C2×D6711yescharacteristic, complemented
212#124—671no—
213#124C2×D6711yescharacteristic
214C22⋊Q8—1341no—
215C22⋊Q8—1341no—
216#121—1341no—
217#121—1341no—
218#120—1341no—
219#120—1341no—
220#117—671no—
221#117—671no—
222C2×C4⋊C4—671no—
223#103—671no—
224#103—671no—
225#103C2×D6711yescharacteristic, complemented
226#103C2×D6711yescharacteristic, complemented
227#102—671no—
228#102—671no—
229#102—671no—
230#102—671no—
231#102—671no—
232#102—671no—
233#102—671no—
234#102—671no—
235#102—671no—
236#102—671no—
237#102C2×D6711yescharacteristic
238#102C2×D6711yescharacteristic
239#97—671no—
240#97—671no—
241C4×C8—671no—
242C4×C8C2×D6711yescharacteristic, complemented
243C24—1341no—
244#41—1342no—
245#41—1342no—
246C2×Q8—1341no—
247C2×Q8—1341no—
248C2×Q8C22×D6711yescharacteristic
249C2×D4—671no—
250C2×D4—671no—
251C2×D4—1341no—
252C2×D4—1342no—
253C2×D4—1342no—
254C2×D4—1341no—
255C2×D4—1341no—
256C2×D4—1341no—
257C2×D4—1341no—
258C2×D4—1341no—
259C2×D4C22×D6711yescharacteristic
260C22×C4—671no—
261C22×C4—1341no—
262C22×C4—1341no—
263M16—1342no—
264C2×C8—671no—
265C2×C8—671no—
266C2×C8—1342no—
267C2×C8C22×D6711yescharacteristic
268C2×C8C22×D6711yescharacteristic
269C4⋊C4—21no—
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⋊C4—1341no—
279C4⋊C4—1341no—
280C4⋊C4C22×D6711yescharacteristic
281C4⋊C4C22×D6711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—671no—
285C22⋊C4—671no—
286C22⋊C4—1341no—
287C22⋊C4—1341no—
288C22⋊C4—1341no—
289C42—671no—
290C42—1341no—
291C42C22×D6711yescharacteristic
292C23—21no—
293C23—2682no—
294C23—5361no—
295C23—5361no—
296C23—671no—
297C23—1341no—
298C23—1341no—
299Q8—21no—
300Q8—21no—
301Q8—5361no—
302Q8—5361no—
303D4—21no—
304D4—21no—
305D4—2682no—
306D4—2682no—
307D4—5361no—
308D4—5361no—
309D4—5361no—
310D4—5361no—
311D4—1341no—
312D4—1341no—
313D4—1341no—
314D4—1341no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—2682no—
319C2×C4—5361no—
320C2×C4—671no—
321C2×C4—672no—
322C2×C4—672no—
323C2×C4—671no—
324C2×C4—671no—
325C2×C4—1341no—
326C2×C4—1341no—
327C2×C4D4×D6711yescharacteristic
328C2×C4D4×D6711yescharacteristic
329C2×C4C23×D6711yescharacteristic
330C8—21no—
331C8—21no—
332C8—1341no—
333C8—1341no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—2681no—
338C22—2681no—
339C22—2681no—
340C22—2681no—
341C22—2681no—
342C22—2681no—
343C22—5364no—
344C22—672no—
345C22—1341no—
346C22—1341no—
347C22C2×D4×D6711yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—2681no—
354C4—2681no—
355C4—672no—
356C4—1341no—
357C4—1341no—
358C4#41×D6711yescharacteristic
359C4#41×D6711yescharacteristic
360C2—81no—
361C2—5361no—
362C2—5361no—
363C2—1342no—
364C2#124×D6711yescharacteristic
365C2#1686419811yescharacteristic
366C2#1686420311yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#488×C67C2
socleC22×C67C22×D4
hypercenter#488D67
hypocenterC67#3136
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #16864026C2semidirect
3. #16863762C2semidirect
4. #16863759C2semidirect
5. #16863757C2semidirect
6. #16863755C2semidirect
7. #16863752C2semidirect
8. #16863743C2semidirect
9. #16863741C2semidirect
10. #16863399C2semidirect
11. #16863396C2semidirect
12. #16863391C2semidirect
13. #102×D67C2semidirect
14. #16863350C2semidirect
15. #488×C67C2semidirect
20. #14524043C22semidirect
22. #14524042C22semidirect
25. #14524034C22semidirect
37. #14523967C22semidirect
38. #14523967C22semidirect
41. #14523965C22semidirect
42. #14523965C22semidirect
47. #14523963C22semidirect
48. #14523963C22semidirect
49. #14523954C22semidirect
50. #14523953C22semidirect
51. C8×Dic67C22semidirect
52. #125×C67C22semidirect
54. #103×C67C22semidirect
55. #103×C67C22semidirect
58. C4×C8×C67C22semidirect
184. C67#3136semidirect
192. #488D67semidirect
211. #125C2×D67semidirect
225. #103C2×D67semidirect
226. #103C2×D67semidirect
242. C4×C8C2×D67semidirect