Finite Group Databasebeta

← #22571037

Subgroups — #22571037

Order 8576.

414 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16863903C211yesmaximal, characteristic
2C22⋊Q8×D67C211yesmaximal, characteristic, complemented
3#16863738C211yesmaximal, characteristic, complemented
4#16863734C211yesmaximal, characteristic, complemented
5#16863730C211yesmaximal, characteristic, complemented
6#16863713C211yesmaximal, characteristic, complemented
7#16863653C211yesmaximal, characteristic, complemented
8#16863651C211yesmaximal, characteristic, complemented
9#16863641C211yesmaximal, characteristic, complemented
10#16863639C211yesmaximal, characteristic, complemented
11#16863558C211yesmaximal, characteristic, complemented
12#16863550C211yesmaximal, characteristic, complemented
13#16863533C211yesmaximal, characteristic, complemented
14#16863358C211yesmaximal, characteristic, complemented
15#452×C67C211yesmaximal, characteristic, complemented
16C2×Q8×D67C2211yescharacteristic
17C22×C4×D67C2211yescharacteristic
18C2×#11761618C2211yescharacteristic, complemented
19C2×#11761616C2211yescharacteristic, complemented
20C2×#11761611C2211yescharacteristic
21C2×#11761611C2211yescharacteristic
22#14524024—21no—
23#14524023C2211yescharacteristic
24#14524022C2211yescharacteristic
25#14524021—21no—
26#14524020—21no—
27#14524019C2211yescharacteristic
28#14524019C2211yescharacteristic
29#14524019C2211yescharacteristic
30#14524014C2211yescharacteristic
31#14524014C2211yescharacteristic
32#14524011—21no—
33C4×C22⋊D67—21no—
34C4×C22⋊D67—21no—
35#14524002C2211yescharacteristic
36C4⋊C4×D67—21no—
37C4⋊C4×D67C2211yescharacteristic
38C22⋊C4×D67—21no—
39#14523977C2211yescharacteristic, complemented
40#14523976C2211yescharacteristic, complemented
41#14523975C2211yescharacteristic, complemented
42#14523974C2211yescharacteristic, complemented
43#14523968C2211yescharacteristic, complemented
44#14523968C2211yescharacteristic, complemented
45#14523967C2211yescharacteristic, complemented
46#14523967C2211yescharacteristic, complemented
47#14523967C2211yescharacteristic, complemented
48#14523964C2211yescharacteristic, complemented
49C22⋊(C67⋊C8)C2211yescharacteristic
50(C2×D67)⋊C8C2211yescharacteristic
51(C2×D67)⋊C8C2211yescharacteristic
52C2×#41×C67C2211yescharacteristic
53C2×Q16×C67C2211yescharacteristic, complemented
54C2×SD16×C67C2211yescharacteristic, complemented
55C22⋊Q8×C67C2211yescharacteristic
56#103×C67C2211yescharacteristic, complemented
57#102×C67C2211yescharacteristic, complemented
58C22⋊C8×C67C2211yescharacteristic
59C23×D67D411yescharacteristic
60C2×Dic134—21no—
61C2×Dic134C2311yescharacteristic
62Q8×D67—41no—
63Q8×D67—41no—
64C2×C22⋊D67—21no—
65C2×C22⋊D67—21no—
66C22×Dic67D411yescharacteristic
67C2×C4×D67—21no—
68C2×C4×D67—21no—
69C2×C4×D67—41no—
70C2×C4×D67—41no—
71C2×C4×D67D411yescharacteristic
72C2×C4×D67D411yescharacteristic
73C2×C4×D67C2311yescharacteristic
74C2×C4×D67C2311yescharacteristic
75#11761618—41no—
76#11761616—41no—
77#11761611—41no—
78#11761611—41no—
79C2×C67⋊C8C2311yescharacteristic
80C2×C67⋊C8C2311yescharacteristic
81C4⋊Dic67—21no—
82C4⋊Dic67C2311yescharacteristic
83C4⋊Dic67C2311yescharacteristic
84C4⋊Dic67C2311yescharacteristic
85Dic67⋊C4—21no—
86Dic67⋊C4—21no—
87Dic67⋊C4—21no—
88Dic67⋊C4—21no—
89Dic67⋊C4—21no—
90C22⋊Dic67—21no—
91C22⋊Dic67—21no—
92C22⋊Dic67—21no—
93C22⋊Dic67—21no—
94(C2×D67)⋊C4—21no—
95(C2×D67)⋊C4—21no—
96(C2×D67)⋊C4—21no—
97(C2×D67)⋊C4—21no—
98C4×Dic67—21no—
99#41×C67—41no—
100#41×C67—41no—
101C2×Q8×C67C2311yescharacteristic
102C2×Q8×C67C2311yescharacteristic
103C2×D4×C67—21no—
104C2×D4×C67C2311yescharacteristic
105C22×C4×C67—21no—
106C22×C4×C67C2311yescharacteristic
107Q16×C67—41no—
108SD16×C67—41no—
109C2×C8×C67C2311yescharacteristic
110C2×C8×C67C2311yescharacteristic
111C4⋊C4×C67—21no—
112C4⋊C4×C67C2311yescharacteristic
113C22⋊C4×C67—21no—
114C23×C67—21no—
115C23×C67C2×D411yescharacteristic
116C22×D67—22no—
117C22×D67—41no—
118C22×D67—41no—
119C22×D67C2×D411yescharacteristic
120C22×D67C2×D411yescharacteristic
121Q8×C67—21no—
122Q8×C67—21no—
123Q8×C67—21no—
124Q8×C67—21no—
125D4×C67—21no—
126D4×C67—21no—
127D4×C67—42no—
128C2×C4×C67—21no—
129C2×C4×C67—21no—
130C2×C4×C67—21no—
131C2×C4×C67—21no—
132C2×C4×C67—21no—
133C2×C4×C67—42no—
134C2×C4×C67C2×D411yescharacteristic
135C2×C4×C67C2411yescharacteristic
136C22⋊D67—42no—
137C22⋊D67—42no—
138C2×Dic67—21no—
139C2×Dic67—21no—
140C2×Dic67—21no—
141C2×Dic67—21no—
142C2×Dic67—41no—
143C2×Dic67C2×D411yescharacteristic
144C2×Dic67C2×D411yescharacteristic
145C4×D67—21no—
146C4×D67—21no—
147C4×D67—21no—
148C4×D67—21no—
149C4×D67—41no—
150C4×D67—41no—
151C4×D67—42no—
152C4×D67—42no—
153Dic134—21no—
154Dic134—21no—
155Dic134—42no—
156C8×C67—21no—
157C8×C67—21no—
158C67⋊C8—21no—
159C67⋊C8—21no—
160C22×C67—21no—
161C22×C67—21no—
162C22×C67—21no—
163C22×C67—41no—
164C22×C67—41no—
165C22×C67—41no—
166C22×C67C22×D411yescharacteristic
167C2×D67—21no—
168C2×D67—21no—
169C2×D67—21no—
170C2×D67—21no—
171C2×D67—21no—
172C2×D67—44no—
173C2×D67#13712yes—
174C4×C67—21no—
175C4×C67—41no—
176C4×C67—41no—
177C4×C67—41no—
178C4×C67H(5,𝔽2)11yescharacteristic
179C4×C67H(5,𝔽2)11yescharacteristic
180Dic67—21no—
181Dic67—21no—
182Dic67—41no—
183Dic67—41no—
184Dic67—41no—
185Dic67—41no—
186C2×C67—41no—
187C2×C67—81no—
188C2×C67#53411yescharacteristic
189C2×C67C2×#13711yescharacteristic
190C2×C67#57511yescharacteristic
191D67—22no—
192D67—41no—
193#3196—671nomaximal, Sylow
194C67#319611yescharacteristic, complemented, Sylow
195#525—671no—
196C2×C22⋊Q8—671no—
197#485—671no—
198#485—671no—
199#484—671no—
200#481—671no—
201#470—671no—
202#470—671no—
203#468—671no—
204#468—671no—
205#452—671no—
206#452D6711yescharacteristic, complemented
207#451—671no—
208#448—671no—
209#407—671no—
210C2×#41C2×D6711yescharacteristic
211C22×Q8—671no—
212C23×C4—671no—
213C2×Q16—671no—
214C2×Q16C2×D6711yescharacteristic, complemented
215C2×SD16—671no—
216C2×SD16C2×D6711yescharacteristic, complemented
217C2×M16—671no—
218C2×M16—671no—
219#123—1341no—
220C22⋊Q8—671no—
221C22⋊Q8—671no—
222C22⋊Q8—671no—
223C22⋊Q8—671no—
224C22⋊Q8—671no—
225C22⋊Q8—1341no—
226C22⋊Q8—1341no—
227C22⋊Q8C2×D6711yescharacteristic
228#121—671no—
229#121—671no—
230#120—1341no—
231C4×D4—1341no—
232C4×D4—1341no—
233#117—671no—
234C2×C4⋊C4—671no—
235C2×C4⋊C4—1341no—
236C2×C22⋊C4—1341no—
237#107—671no—
238#107—671no—
239#106—671no—
240#106—671no—
241#103—671no—
242#103—671no—
243#103—671no—
244#103—671no—
245#103—671no—
246#103C2×D6711yescharacteristic, complemented
247#102—671no—
248#102C2×D6711yescharacteristic, complemented
249C22⋊C8—671no—
250C22⋊C8—671no—
251C22⋊C8—671no—
252C22⋊C8C2×D6711yescharacteristic
253C24—671no—
254#41—41no—
255#41—41no—
256C2×Q8—2681no—
257C2×Q8—2681no—
258C2×Q8—671no—
259C2×Q8—1341no—
260C2×Q8C22×D6711yescharacteristic
261C2×Q8C22×D6711yescharacteristic
262C2×D4—21no—
263C2×D4—1341no—
264C2×D4—1341no—
265C2×D4C22×D6711yescharacteristic
266C22×C4—21no—
267C22×C4—2681no—
268C22×C4—2681no—
269C22×C4—671no—
270C22×C4—671no—
271C22×C4—671no—
272C22×C4—671no—
273C22×C4—671no—
274C22×C4—1341no—
275C22×C4—1341no—
276C22×C4C22×D6711yescharacteristic
277Q16—41no—
278Q16—2681no—
279SD16—41no—
280SD16—2681no—
281M16—2681no—
282M16—2681no—
283C2×C8—671no—
284C2×C8—671no—
285C2×C8C22×D6711yescharacteristic
286C2×C8C22×D6711yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—671no—
289C4⋊C4—671no—
290C4⋊C4—671no—
291C4⋊C4—1341no—
292C4⋊C4—1341no—
293C4⋊C4—1341no—
294C4⋊C4—1341no—
295C4⋊C4—1341no—
296C4⋊C4—1341no—
297C4⋊C4C22×D6711yescharacteristic
298C22⋊C4—21no—
299C22⋊C4—1341no—
300C22⋊C4—1341no—
301C22⋊C4—1341no—
302C22⋊C4—1341no—
303C22⋊C4—1341no—
304C22⋊C4—1341no—
305C22⋊C4—1341no—
306C22⋊C4—1341no—
307C42—1341no—
308C23—21no—
309C23—2681no—
310C23—2681no—
311C23—671no—
312C23—671no—
313C23—1342no—
314C23D4×D6711yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—21no—
318Q8—21no—
319Q8—2682no—
320Q8—1341no—
321Q8—1341no—
322D4—21no—
323D4—21no—
324D4—42no—
325D4—2682no—
326D4—2682no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—42no—
333C2×C4—2681no—
334C2×C4—2681no—
335C2×C4—2682no—
336C2×C4—2682no—
337C2×C4—2681no—
338C2×C4—671no—
339C2×C4—671no—
340C2×C4—1341no—
341C2×C4—1341no—
342C2×C4—1341no—
343C2×C4—1341no—
344C2×C4—1341no—
345C2×C4—1341no—
346C2×C4—1341no—
347C2×C4—1341no—
348C2×C4D4×D6711yescharacteristic
349C2×C4C23×D6711yescharacteristic
350C8—21no—
351C8—21no—
352C8—1341no—
353C8—1341no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—2684no—
361C22—672no—
362C22—1341no—
363C22—1341no—
364C22—1341no—
365C22—1341no—
366C22—1341no—
367C22C2×D4×D6711yescharacteristic
368C4—21no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—2681no—
373C4—2681no—
374C4—2681no—
375C4—2681no—
376C4—1341no—
377C4—1341no—
378C4#1452408811yescharacteristic
379C4#1452408811yescharacteristic
380C2—41no—
381C2—81no—
382C2—2681no—
383C2—1342no—
384C2#1686395611yescharacteristic
385C2#137×D6711yescharacteristic
386C2#1686419311yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#452×C67C2
socleC22×C67C22×D4
hypercenter#452D67
hypocenterC67#3196
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. C22⋊Q8×D67C2semidirect
3. #16863738C2semidirect
4. #16863734C2semidirect
5. #16863730C2semidirect
6. #16863713C2semidirect
7. #16863653C2semidirect
8. #16863651C2semidirect
9. #16863641C2semidirect
10. #16863639C2semidirect
11. #16863558C2semidirect
12. #16863550C2semidirect
13. #16863533C2semidirect
14. #16863358C2semidirect
15. #452×C67C2semidirect
18. C2×#11761618C22semidirect
19. C2×#11761616C22semidirect
39. #14523977C22semidirect
40. #14523976C22semidirect
41. #14523975C22semidirect
42. #14523974C22semidirect
43. #14523968C22semidirect
44. #14523968C22semidirect
45. #14523967C22semidirect
46. #14523967C22semidirect
47. #14523967C22semidirect
48. #14523964C22semidirect
53. C2×Q16×C67C22semidirect
54. C2×SD16×C67C22semidirect
56. #103×C67C22semidirect
57. #102×C67C22semidirect
194. C67#3196semidirect
206. #452D67semidirect
214. C2×Q16C2×D67semidirect
216. C2×SD16C2×D67semidirect
246. #103C2×D67semidirect
248. #102C2×D67semidirect