Finite Group Databasebeta

← #22570393

Subgroups — #22570393

Order 8576.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16864017C211yesmaximal, characteristic, complemented
2#16863737C211yesmaximal, characteristic, complemented
3#16863720C211yesmaximal, characteristic, complemented
4#16863686C211yesmaximal, characteristic, complemented
5#16863676C211yesmaximal, characteristic, complemented
6#16863611C211yesmaximal, characteristic, complemented
7#16863611C211yesmaximal, characteristic, complemented
8#16863547C211yesmaximal, characteristic, complemented
9#16863530C211yesmaximal, characteristic
10#16863506C211yesmaximal, characteristic, complemented
11#16863496C211yesmaximal, characteristic, complemented
12C4×#11761616C211yesmaximal, characteristic, complemented
13C4×#11761613C211yesmaximal, characteristic, complemented
14#16863457C211yesmaximal, characteristic, complemented
15#518×C67C211yesmaximal, characteristic, complemented
16C2×#11761616C2211yescharacteristic
17C2×#11761616C2211yescharacteristic
18C2×#11761613C2211yescharacteristic
19C2×#11761613C2211yescharacteristic
20#14524043C2211yescharacteristic, complemented
21#14524038—21no—
22#14524031C2211yescharacteristic, complemented
23#14524029—21no—
24#14524025—21no—
25#14524022C2211yescharacteristic
26#14524021—21no—
27#14524017C2211yescharacteristic
28C4×Dic134C2211yescharacteristic, complemented
29C4×D268C2211yescharacteristic, complemented
30#14523976C2211yescharacteristic, complemented
31#14523976C2211yescharacteristic, complemented
32#14523974C2211yescharacteristic, complemented
33#14523974C2211yescharacteristic, complemented
34C4⋊(C67⋊C8)C2211yescharacteristic, complemented
35C4⋊(C67⋊C8)C2211yescharacteristic, complemented
36#14523966C2211yescharacteristic, complemented
37#14523966C2211yescharacteristic, complemented
38#14523964C2211yescharacteristic
39#14523964C2211yescharacteristic
40#14523964C2211yescharacteristic
41#14523964C2211yescharacteristic
42#14523963C2211yescharacteristic, complemented
43#14523963C2211yescharacteristic, complemented
44C22⋊(C67⋊C8)C2211yescharacteristic
45C22⋊(C67⋊C8)C2211yescharacteristic
46#14523954C2211yescharacteristic, complemented
47C4×C67⋊C8C2211yescharacteristic, complemented
48C22×D4×C67C2211yescharacteristic
49C4×D4×C67—21no—
50C4×D4×C67—21no—
51C4×D4×C67C2211yescharacteristic
52C4×D4×C67C2211yescharacteristic
53C4×D4×C67C2211yescharacteristic
54C4×D4×C67C2211yescharacteristic
55#117×C67C2211yescharacteristic
56#117×C67C2211yescharacteristic
57C2×C22⋊C4×C67—21no—
58C2×C22⋊C4×C67—21no—
59C2×Dic134—21no—
60C2×Dic134C2311yescharacteristic
61C2×D268C2311yescharacteristic
62C2×C22⋊D67—21no—
63C2×C4×D67—21no—
64#11761616—22no—
65#11761616—22no—
66#11761613—22no—
67#11761613—22no—
68C2×C67⋊C8C2311yescharacteristic
69C2×C67⋊C8C2311yescharacteristic
70C2×C67⋊C8C2311yescharacteristic
71C2×C67⋊C8C2311yescharacteristic
72C4⋊Dic67—21no—
73C4⋊Dic67C2311yescharacteristic
74C4⋊Dic67C2311yescharacteristic
75Dic67⋊C4—21no—
76Dic67⋊C4—21no—
77C22⋊Dic67—21no—
78(C2×D67)⋊C4—21no—
79(C2×D67)⋊C4—21no—
80(C2×D67)⋊C4—21no—
81(C2×D67)⋊C4—21no—
82C4×Dic67—21no—
83C24×C67—21no—
84C2×D4×C67—22no—
85C2×D4×C67—22no—
86C2×D4×C67—21no—
87C2×D4×C67C2311yescharacteristic
88C2×D4×C67C2311yescharacteristic
89C22×C4×C67—21no—
90C22×C4×C67—21no—
91C22×C4×C67—21no—
92C22×C4×C67—21no—
93C22×C4×C67C2311yescharacteristic
94C4⋊C4×C67D411yescharacteristic
95C4⋊C4×C67D411yescharacteristic
96C4⋊C4×C67C2311yescharacteristic
97C4⋊C4×C67C2311yescharacteristic
98C22⋊C4×C67—21no—
99C22⋊C4×C67—21no—
100C22⋊C4×C67—21no—
101C22⋊C4×C67—21no—
102C22⋊C4×C67—21no—
103C22⋊C4×C67D411yescharacteristic
104C22⋊C4×C67D411yescharacteristic
105C42×C67—21no—
106C42×C67C2311yescharacteristic
107C42×C67C2311yescharacteristic
108C23×C67—24no—
109C23×C67—21no—
110C23×C67—21no—
111C23×C67—81no—
112C23×C67—81no—
113C23×C67C2×D411yescharacteristic
114C22×D67—21no—
115D4×C67—24no—
116D4×C67—21no—
117D4×C67—21no—
118D4×C67#4112yes—
119D4×C67#4112yes—
120C2×C4×C67—21no—
121C2×C4×C67—21no—
122C2×C4×C67—21no—
123C2×C4×C67—21no—
124C2×C4×C67—81no—
125C2×C4×C67—81no—
126C2×C4×C67—81no—
127C2×C4×C67—81no—
128C2×C4×C67C2×D411yescharacteristic
129C2×C4×C67C2×D411yescharacteristic
130C2×C4×C67C2×D411yescharacteristic
131C2×C4×C67C2×D411yescharacteristic
132C2×C4×C67C2×D411yescharacteristic
133C2×C4×C67C2411yescharacteristic
134C22⋊D67—81no—
135C2×Dic67—21no—
136C2×Dic67—21no—
137C2×Dic67—21no—
138D268—21no—
139D268—21no—
140C4×D67—81no—
141Dic134—21no—
142Dic134—21no—
143Dic134—81no—
144C67⋊C8—21no—
145C67⋊C8—21no—
146C67⋊C8—21no—
147C67⋊C8—21no—
148C22×C67—21no—
149C22×C67—22no—
150C22×C67—22no—
151C22×C67—21no—
152C22×C67—21no—
153C22×C67—41no—
154C22×C67—41no—
155C22×C67—41no—
156C22×C67—41no—
157C22×C67—84no—
158C22×C67C22×D411yescharacteristic
159C2×D67—41no—
160C2×D67—41no—
161C2×D67—41no—
162C4×C67—21no—
163C4×C67—21no—
164C4×C67—21no—
165C4×C67—21no—
166C4×C67—21no—
167C4×C67—41no—
168C4×C67—41no—
169C4×C67C2×#4111yescharacteristic
170C4×C67C2×#4111yescharacteristic
171Dic67—41no—
172Dic67—41no—
173Dic67—41no—
174C2×C67—41no—
175C2×C67—42no—
176C2×C67—42no—
177C2×C67#52511yescharacteristic
178C2×C67#57611yescharacteristic
179C2×C67#57711yescharacteristic
180D67—81no—
181#3118—671nomaximal, Sylow
182C67#311811yescharacteristic, complemented, Sylow
183#542—671no—
184#518D6711yescharacteristic, complemented
185#485—671no—
186#482—671no—
187#476—671no—
188#474—671no—
189#463—671no—
190#463—671no—
191#450—671no—
192#447—671no—
193#442—671no—
194#440—671no—
195C4×SD16—671no—
196C4×D8—671no—
197#433—671no—
198C22×D4C2×D6711yescharacteristic
199C2×SD16—671no—
200C2×SD16—671no—
201C2×D8—671no—
202C2×D8—671no—
203C4⋊Q8—671no—
204#126—1341no—
205#124—671no—
206#124—1341no—
207#123—1341no—
208C22⋊Q8—671no—
209C22⋊Q8—1341no—
210#121—671no—
211C4×Q8—671no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—671no—
215C4×D4C2×D6711yescharacteristic
216C4×D4C2×D6711yescharacteristic
217C4×D4C2×D6711yescharacteristic
218C4×D4C2×D6711yescharacteristic
219#117C2×D6711yescharacteristic
220#117C2×D6711yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223#107—671no—
224#107—671no—
225#106—671no—
226#106—671no—
227C4⋊C8—671no—
228C4⋊C8—671no—
229#103—671no—
230#103—671no—
231#102—671no—
232#102—671no—
233#102—671no—
234#102—671no—
235#102—671no—
236#102—671no—
237C22⋊C8—671no—
238C22⋊C8—671no—
239#97—671no—
240C4×C8—671no—
241C24—21no—
242C2×Q8—671no—
243C2×Q8—1341no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—671no—
248C2×D4—1341no—
249C2×D4C22×D6711yescharacteristic
250C2×D4C22×D6711yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—1341no—
256C22×C4C22×D6711yescharacteristic
257SD16—1342no—
258SD16—1342no—
259D8—1342no—
260D8—1342no—
261C2×C8—671no—
262C2×C8—671no—
263C2×C8—671no—
264C2×C8—671no—
265C4⋊C4—671no—
266C4⋊C4—671no—
267C4⋊C4—1341no—
268C4⋊C4—1341no—
269C4⋊C4—1341no—
270C4⋊C4C22⋊D6711yescharacteristic
271C4⋊C4C22⋊D6711yescharacteristic
272C4⋊C4C22×D6711yescharacteristic
273C4⋊C4C22×D6711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—1341no—
280C22⋊C4—1341no—
281C22⋊C4—1341no—
282C22⋊C4—1341no—
283C22⋊C4—1341no—
284C22⋊C4C22⋊D6711yescharacteristic
285C22⋊C4C22⋊D6711yescharacteristic
286C42—21no—
287C42—1341no—
288C42C22×D6711yescharacteristic
289C42C22×D6711yescharacteristic
290C23—24no—
291C23—21no—
292C23—21no—
293C23—81no—
294C23—81no—
295C23—1341no—
296C23C2×C22⋊D6711yescharacteristic
297Q8—5361no—
298Q8—1341no—
299Q8—1341no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—5361no—
304D4—1341no—
305D4—1341no—
306D4#1176162812yes—
307D4#1176162812yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—5361no—
317C2×C4—1341no—
318C2×C4—1341no—
319C2×C4—1341no—
320C2×C4C2×C22⋊D6711yescharacteristic
321C2×C4C2×C22⋊D6711yescharacteristic
322C2×C4C2×C22⋊D6711yescharacteristic
323C2×C4C2×C22⋊D6711yescharacteristic
324C2×C4C2×C22⋊D6711yescharacteristic
325C2×C4C23×D6711yescharacteristic
326C8—1341no—
327C8—1341no—
328C8—1341no—
329C8—1341no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—84no—
340C22—2681no—
341C22—2681no—
342C22—2681no—
343C22C22×C22⋊D6711yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—2681no—
352C4—2681no—
353C4—2681no—
354C4C2×#1176162811yescharacteristic
355C4C2×#1176162811yescharacteristic
356C2—41no—
357C2—42no—
358C2—42no—
359C2—5361no—
360C2#1686390711yescharacteristic
361C2#1686419911yescharacteristic
362C2#1686420711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#518×C67C2
socleC22×C67C22×D4
hypercenter#518D67
hypocenterC67#3118
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #16864017C2semidirect
2. #16863737C2semidirect
3. #16863720C2semidirect
4. #16863686C2semidirect
5. #16863676C2semidirect
6. #16863611C2semidirect
7. #16863611C2semidirect
8. #16863547C2semidirect
10. #16863506C2semidirect
11. #16863496C2semidirect
12. C4×#11761616C2semidirect
13. C4×#11761613C2semidirect
14. #16863457C2semidirect
15. #518×C67C2semidirect
20. #14524043C22semidirect
22. #14524031C22semidirect
28. C4×Dic134C22semidirect
29. C4×D268C22semidirect
30. #14523976C22semidirect
31. #14523976C22semidirect
32. #14523974C22semidirect
33. #14523974C22semidirect
34. C4⋊(C67⋊C8)C22semidirect
35. C4⋊(C67⋊C8)C22semidirect
36. #14523966C22semidirect
37. #14523966C22semidirect
42. #14523963C22semidirect
43. #14523963C22semidirect
46. #14523954C22semidirect
47. C4×C67⋊C8C22semidirect
182. C67#3118semidirect
184. #518D67semidirect