Finite Group Databasebeta

← #22570166

Subgroups — #22570166

Order 8576.

418 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
1#16863962C211yesmaximal, characteristic
2#16863877C211yesmaximal, characteristic, complemented
3#16863722C211yesmaximal, characteristic, complemented
4#16863721C211yesmaximal, characteristic, complemented
5#16863719C211yesmaximal, characteristic, complemented
6#16863717C211yesmaximal, characteristic, complemented
7#16863716C211yesmaximal, characteristic, complemented
8#16863709C211yesmaximal, characteristic, complemented
9#16863708C211yesmaximal, characteristic, complemented
10#16863435C211yesmaximal, characteristic, complemented
11#16863431C211yesmaximal, characteristic, complemented
12#16863399C211yesmaximal, characteristic, complemented
13#102×D67C211yesmaximal, characteristic, complemented
14#16863361C211yesmaximal, characteristic, complemented
15#481×C67C211yesmaximal, characteristic, complemented
16C2×#11761628C2211yescharacteristic
17C2×D4×D67C2211yescharacteristic
18C2×#11761611C2211yescharacteristic
19C2×C8×D67C2211yescharacteristic
20#14524039—21no—
21#14524029—21no—
22#14524018C2211yescharacteristic
23#14524017C2211yescharacteristic
24#14524014C2211yescharacteristic
25#14524013—21no—
26#14524011—21no—
27#14524008—21no—
28#14524005—21no—
29C4×C22⋊D67—21no—
30C4×C22⋊D67—21no—
31#14524002C2211yescharacteristic, complemented
32#14524001C2211yescharacteristic
33#14524001C2211yescharacteristic
34#14524001C2211yescharacteristic, complemented
35C4⋊C4×D67C2211yescharacteristic
36C4⋊C4×D67C2211yescharacteristic, complemented
37#14523977C2211yescharacteristic, complemented
38#14523976C2211yescharacteristic, complemented
39#14523976C2211yescharacteristic, complemented
40#14523976C2211yescharacteristic, complemented
41#14523975C2211yescharacteristic, complemented
42#14523974C2211yescharacteristic, complemented
43#14523965C2211yescharacteristic
44#14523965C2211yescharacteristic
45#14523964C2211yescharacteristic
46#14523964C2211yescharacteristic
47#14523963C2211yescharacteristic
48#14523963C2211yescharacteristic
49C22⋊(C67⋊C8)C2211yescharacteristic, complemented
50(C2×D67)⋊C8C2211yescharacteristic, complemented
51(C2×D67)⋊C8C2211yescharacteristic, complemented
52#121×C67C2211yescharacteristic
53C2×C4⋊C4×C67C2211yescharacteristic, complemented
54#106×C67C2211yescharacteristic, complemented
55#106×C67C2211yescharacteristic, complemented
56#102×C67C2211yescharacteristic
57#102×C67C2211yescharacteristic
58C22⋊C8×C67C2211yescharacteristic, complemented
59C23×D67—21no—
60#11761628—22no—
61#11761628—22no—
62C2×Dic134D411yescharacteristic
63C2×D268—21no—
64C2×D268D411yescharacteristic
65C2×D268C2311yescharacteristic
66C2×C22⋊D67—21no—
67C2×C22⋊D67—21no—
68C2×C22⋊D67—21no—
69C2×C22⋊D67D411yescharacteristic
70C2×C22⋊D67D411yescharacteristic
71D4×D67—22no—
72D4×D67—22no—
73C2×C4×D67—21no—
74C2×C4×D67—21no—
75C2×C4×D67C2311yescharacteristic
76C2×C4×D67C2311yescharacteristic
77#11761611—22no—
78C2×C67⋊C8C2311yescharacteristic
79C2×C67⋊C8C2311yescharacteristic
80C8×D67—22no—
81C4⋊Dic67C2311yescharacteristic
82C4⋊Dic67C2311yescharacteristic
83C4⋊Dic67C2311yescharacteristic
84Dic67⋊C4—21no—
85Dic67⋊C4—21no—
86C22⋊Dic67—21no—
87C22⋊Dic67—21no—
88(C2×D67)⋊C4—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×Dic67—21no—
95C4×Dic67—21no—
96C4×Dic67—21no—
97C2×D4×C67—21no—
98C2×D4×C67C2311yescharacteristic
99C22×C4×C67—21no—
100C22×C4×C67C2311yescharacteristic
101C2×C8×C67C2311yescharacteristic
102C2×C8×C67C2311yescharacteristic
103C4⋊C4×C67—21no—
104C4⋊C4×C67C2311yescharacteristic
105C4⋊C4×C67C2311yescharacteristic
106C4⋊C4×C67C2311yescharacteristic
107C22⋊C4×C67—21no—
108C23×C67—21no—
109C23×C67C2×D411yescharacteristic
110C22×D67—21no—
111C22×D67—42no—
112C22×D67—81no—
113C22×D67—81no—
114C22×D67C2×D411yescharacteristic
115C22×D67C2×D411yescharacteristic
116D4×C67—21no—
117D4×C67—21no—
118D4×C67—81no—
119C2×C4×C67—21no—
120C2×C4×C67—21no—
121C2×C4×C67—21no—
122C2×C4×C67—21no—
123C2×C4×C67—21no—
124C2×C4×C67—81no—
125C2×C4×C67C2×D411yescharacteristic
126C2×C4×C67C2411yescharacteristic
127C22⋊D67—22no—
128C22⋊D67—22no—
129C22⋊D67—42no—
130C22⋊D67—81no—
131C22⋊D67—81no—
132C2×Dic67—21no—
133C2×Dic67—21no—
134C2×Dic67—21no—
135C2×Dic67C2×D411yescharacteristic
136C2×Dic67C2×D411yescharacteristic
137D268—22no—
138D268—21no—
139D268—21no—
140D268—81no—
141C4×D67—21no—
142C4×D67—21no—
143C4×D67—81no—
144C4×D67—81no—
145C4×D67#4112yes—
146C4×D67#4112yes—
147Dic134—22no—
148C8×C67—21no—
149C8×C67—21no—
150C67⋊C8—21no—
151C67⋊C8—21no—
152C22×C67—21no—
153C22×C67—21no—
154C22×C67—21no—
155C22×C67—41no—
156C22×C67—41no—
157C22×C67—41no—
158C22×C67C22×D411yescharacteristic
159C2×D67—21no—
160C2×D67—21no—
161C2×D67—21no—
162C2×D67—21no—
163C2×D67—21no—
164C2×D67—41no—
165C2×D67—41no—
166C2×D67—41no—
167C2×D67—84no—
168C2×D67#12312yes—
169C4×C67—21no—
170C4×C67—41no—
171C4×C67—41no—
172C4×C67—41no—
173C4×C67C2×#4111yescharacteristic
174C4×C67C2×#4111yescharacteristic
175Dic67—21no—
176Dic67—41no—
177Dic67—41no—
178Dic67—41no—
179Dic67#12312yes—
180C2×C67—41no—
181C2×C67—81no—
182C2×C67C2×#12311yescharacteristic
183C2×C67#57611yescharacteristic
184C2×C67#57711yescharacteristic
185D67—22no—
186D67—41no—
187D67—81no—
188#3096—671nomaximal, Sylow
189C67#309611yescharacteristic, complemented, Sylow
190#535—671no—
191#520—671no—
192#482—671no—
193#482—671no—
194#482—671no—
195#482—671no—
196#481—671no—
197#481D6711yescharacteristic, complemented
198#480—671no—
199#480—671no—
200#428—671no—
201#427—671no—
202#418—671no—
203C2×#102—671no—
204#408—671no—
205C2×#41—671no—
206C22×D4—671no—
207C2×M16—671no—
208C22×C8—671no—
209#127—1341no—
210#124—1341no—
211#121—671no—
212#121—671no—
213#121—671no—
214#121C2×D6711yescharacteristic
215#120—1341no—
216#120—1341no—
217C4×Q8—1341no—
218C4×D4—1341no—
219C4×D4—1341no—
220C4×D4—1341no—
221#117—671no—
222#117—671no—
223#117—671no—
224#117—671no—
225C2×C4⋊C4—671no—
226C2×C4⋊C4—671no—
227C2×C4⋊C4C2×D6711yescharacteristic, complemented
228#107—671no—
229#107—671no—
230#107—671no—
231#107—671no—
232#106—671no—
233#106—671no—
234#106C2×D6711yescharacteristic, complemented
235#106C2×D6711yescharacteristic, complemented
236#102—671no—
237#102—671no—
238#102—671no—
239#102—671no—
240#102—671no—
241#102—671no—
242#102C2×D6711yescharacteristic
243#102C2×D6711yescharacteristic
244C22⋊C8—671no—
245C22⋊C8—671no—
246C22⋊C8—671no—
247C22⋊C8C2×D6711yescharacteristic, complemented
248C24—1341no—
249#41—1342no—
250#41—1342no—
251C2×Q8—671no—
252C2×D4—21no—
253C2×D4—671no—
254C2×D4—671no—
255C2×D4—671no—
256C2×D4—671no—
257C2×D4—1341no—
258C2×D4—1342no—
259C2×D4—1342no—
260C2×D4—1341no—
261C2×D4—1341no—
262C2×D4—1341no—
263C2×D4C22×D6711yescharacteristic
264C22×C4—21no—
265C22×C4—671no—
266C22×C4—671no—
267C22×C4—1341no—
268C22×C4—1341no—
269C22×C4C22×D6711yescharacteristic
270M16—1342no—
271C2×C8—671no—
272C2×C8—671no—
273C2×C8—1342no—
274C2×C8C22×D6711yescharacteristic
275C2×C8C22×D6711yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—671no—
278C4⋊C4—671no—
279C4⋊C4—671no—
280C4⋊C4—1341no—
281C4⋊C4—1341no—
282C4⋊C4C22×D6711yescharacteristic
283C4⋊C4C22×D6711yescharacteristic
284C4⋊C4C22×D6711yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—1341no—
287C22⋊C4—1341no—
288C22⋊C4—1341no—
289C22⋊C4—1341no—
290C22⋊C4—1341no—
291C22⋊C4—1341no—
292C22⋊C4—1341no—
293C42—1341no—
294C42—1341no—
295C42—1341no—
296C42—1341no—
297C23—21no—
298C23—2682no—
299C23—5361no—
300C23—5361no—
301C23—671no—
302C23—671no—
303C23—1341no—
304C23D4×D6711yescharacteristic
305Q8—1342no—
306D4—21no—
307D4—21no—
308D4—81no—
309D4—2682no—
310D4—5361no—
311D4—5361no—
312D4—5361no—
313D4—1342no—
314D4—1342no—
315D4—1342no—
316D4—1341no—
317D4—1341no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—81no—
324C2×C4—5361no—
325C2×C4—5361no—
326C2×C4—671no—
327C2×C4—672no—
328C2×C4—672no—
329C2×C4—671no—
330C2×C4—1341no—
331C2×C4—1341no—
332C2×C4—1341no—
333C2×C4—1341no—
334C2×C4—1341no—
335C2×C4D4×D6711yescharacteristic
336C2×C4C23×D6711yescharacteristic
337C8—21no—
338C8—21no—
339C8—1341no—
340C8—1341no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—2681no—
348C22—2681no—
349C22—2681no—
350C22—5364no—
351C22—672no—
352C22—1341no—
353C22—1341no—
354C22—1341no—
355C22—1341no—
356C22—1341no—
357C22C2×D4×D6711yescharacteristic
358C4—21no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—2681no—
363C4—2681no—
364C4—2681no—
365C4—672no—
366C4—1341no—
367C4#41×D6711yescharacteristic
368C4#41×D6711yescharacteristic
369C2—41no—
370C2—81no—
371C2—2681no—
372C2—5361no—
373C2—1342no—
374C2#123×D6711yescharacteristic
375C2#1686420011yescharacteristic
376C2#1686420211yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#481×C67C2
socleC22×C67C22×D4
hypercenter#481D67
hypocenterC67#3096
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. #16863877C2semidirect
3. #16863722C2semidirect
4. #16863721C2semidirect
5. #16863719C2semidirect
6. #16863717C2semidirect
7. #16863716C2semidirect
8. #16863709C2semidirect
9. #16863708C2semidirect
10. #16863435C2semidirect
11. #16863431C2semidirect
12. #16863399C2semidirect
13. #102×D67C2semidirect
14. #16863361C2semidirect
15. #481×C67C2semidirect
31. #14524002C22semidirect
34. #14524001C22semidirect
36. C4⋊C4×D67C22semidirect
37. #14523977C22semidirect
38. #14523976C22semidirect
39. #14523976C22semidirect
40. #14523976C22semidirect
41. #14523975C22semidirect
42. #14523974C22semidirect
49. C22⋊(C67⋊C8)C22semidirect
50. (C2×D67)⋊C8C22semidirect
51. (C2×D67)⋊C8C22semidirect
53. C2×C4⋊C4×C67C22semidirect
54. #106×C67C22semidirect
55. #106×C67C22semidirect
58. C22⋊C8×C67C22semidirect
189. C67#3096semidirect
197. #481D67semidirect
227. C2×C4⋊C4C2×D67semidirect
234. #106C2×D67semidirect
235. #106C2×D67semidirect
247. C22⋊C8C2×D67semidirect