Finite Group Databasebeta

← #22572435

Subgroups — #22572435

Order 8576.

448 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16864117C211yesmaximal, characteristic, complemented
2#16864030C211yesmaximal, characteristic
3#16863741C211yesmaximal, characteristic, complemented
4#16863723C211yesmaximal, characteristic, complemented
5#16863717C211yesmaximal, characteristic, complemented
6#16863675C211yesmaximal, characteristic, complemented
7#16863645C211yesmaximal, characteristic, complemented
8#16863594C211yesmaximal, characteristic, complemented
9#16863546C211yesmaximal, characteristic, complemented
10#16863545C211yesmaximal, characteristic, complemented
11#16863481C211yesmaximal, characteristic, complemented
12#16863472C211yesmaximal, characteristic, complemented
13#16863402C211yesmaximal, characteristic, complemented
14#16863399C211yesmaximal, characteristic, complemented
15#468×C67C211yesmaximal, characteristic, complemented
16C22×D268C2211yescharacteristic
17C2×D4×D67—21no—
18C2×D4×D67C2211yescharacteristic
19C2×#11761615C2211yescharacteristic
20C2×#11761615C2211yescharacteristic, complemented
21C2×#11761611C2211yescharacteristic, complemented
22#14524042C2211yescharacteristic, complemented
23#14524039C2211yescharacteristic
24#14524038—21no—
25#14524036—21no—
26#14524029—21no—
27#14524019C2211yescharacteristic, complemented
28#14524018C2211yescharacteristic
29#14524016—21no—
30#14524013—21no—
31#14524011—21no—
32Q8×Dic67C2211yescharacteristic, complemented
33#14524008—21no—
34#14524005C2211yescharacteristic
35C4×C22⋊D67C2211yescharacteristic
36#14524002C2211yescharacteristic, complemented
37#14524001C2211yescharacteristic, complemented
38#14523976C2211yescharacteristic, complemented
39#14523974C2211yescharacteristic, complemented
40Dic67⋊C8C2211yescharacteristic, complemented
41#14523965C2211yescharacteristic
42#14523965C2211yescharacteristic
43#14523965C2211yescharacteristic, complemented
44#14523965C2211yescharacteristic, complemented
45#14523964C2211yescharacteristic, complemented
46#14523964C2211yescharacteristic, complemented
47#14523963C2211yescharacteristic
48#14523963C2211yescharacteristic, complemented
49C22⋊(C67⋊C8)C2211yescharacteristic, complemented
50(C2×D67)⋊C8C2211yescharacteristic, complemented
51C8×Dic67C2211yescharacteristic, complemented
52C2×SD16×C67C2211yescharacteristic, complemented
53C2×M16×C67C2211yescharacteristic, complemented
54C22⋊Q8×C67C2211yescharacteristic, complemented
55#121×C67C2211yescharacteristic
56#107×C67C2211yescharacteristic, complemented
57#103×C67C2211yescharacteristic, complemented
58#102×C67C2211yescharacteristic, complemented
59C23×D67—21no—
60C23×D67—21no—
61C2×Dic134—21no—
62C2×D268—22no—
63C2×D268—21no—
64C2×D268—21no—
65C2×D268—41no—
66C2×D268C2311yescharacteristic
67C2×D268C2311yescharacteristic
68C2×C22⋊D67—21no—
69C2×C22⋊D67—21no—
70C2×C22⋊D67—21no—
71C2×C22⋊D67D411yescharacteristic
72D4×D67—22no—
73D4×D67—41no—
74D4×D67—44no—
75C2×C4×D67—21no—
76C2×C4×D67C2311yescharacteristic
77#11761615—22no—
78#11761615—41no—
79#11761611—41no—
80C2×C67⋊C8C2311yescharacteristic
81C2×C67⋊C8C2311yescharacteristic
82C4⋊Dic67—21no—
83C4⋊Dic67C2311yescharacteristic
84C4⋊Dic67C2311yescharacteristic
85Dic67⋊C4—21no—
86Dic67⋊C4—21no—
87Dic67⋊C4D411yescharacteristic
88C22⋊Dic67—21no—
89C22⋊Dic67D411yescharacteristic
90(C2×D67)⋊C4—21no—
91(C2×D67)⋊C4—21no—
92(C2×D67)⋊C4—21no—
93(C2×D67)⋊C4—21no—
94(C2×D67)⋊C4D411yescharacteristic
95C4×Dic67—21no—
96C4×Dic67—21no—
97C4×Dic67—21no—
98C4×Dic67C2311yescharacteristic
99C2×Q8×C67C2311yescharacteristic
100C2×D4×C67—21no—
101C2×D4×C67C2311yescharacteristic
102C22×C4×C67C2311yescharacteristic
103SD16×C67—41no—
104M16×C67—41no—
105C2×C8×C67C2311yescharacteristic
106C2×C8×C67C2311yescharacteristic
107C4⋊C4×C67—21no—
108C4⋊C4×C67C2311yescharacteristic
109C4⋊C4×C67C2311yescharacteristic
110C22⋊C4×C67—21no—
111C22⋊C4×C67—21no—
112C23×C67—21no—
113C23×C67C2×D411yescharacteristic
114C22×D67—21no—
115C22×D67—21no—
116C22×D67—42no—
117C22×D67—42no—
118C22×D67—42no—
119C22×D67—42no—
120C22×D67—81no—
121C22×D67—81no—
122C22×D67C2×D411yescharacteristic
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×C67C2×D411yescharacteristic
134C2×C4×C67C2411yescharacteristic
135C22⋊D67—22no—
136C22⋊D67—42no—
137C22⋊D67—42no—
138C22⋊D67—81no—
139C2×Dic67—21no—
140C2×Dic67—21no—
141C2×Dic67C2×D411yescharacteristic
142C2×Dic67C2×D411yescharacteristic
143C2×Dic67C2×D411yescharacteristic
144D268—21no—
145D268—21no—
146D268—21no—
147D268—42no—
148D268—42no—
149D268#4112yes—
150C4×D67—21no—
151C4×D67—21no—
152C4×D67—42no—
153Dic134—42no—
154C8×C67—21no—
155C8×C67—21no—
156C67⋊C8—21no—
157C67⋊C8—21no—
158C22×C67—21no—
159C22×C67—21no—
160C22×C67—21no—
161C22×C67—41no—
162C22×C67—41no—
163C22×C67—41no—
164C22×C67C22×D411yescharacteristic
165C2×D67—21no—
166C2×D67—22no—
167C2×D67—21no—
168C2×D67—21no—
169C2×D67—41no—
170C2×D67—41no—
171C2×D67—41no—
172C2×D67—41no—
173C2×D67—41no—
174C2×D67—84no—
175C2×D67—84no—
176C4×C67—21no—
177C4×C67—41no—
178C4×C67—41no—
179C4×C67—41no—
180C4×C67C2×#4111yescharacteristic
181C4×C67H(5,𝔽2)11yescharacteristic
182Dic67—21no—
183Dic67—21no—
184Dic67—41no—
185Dic67—41no—
186Dic67AGL(1,ℤ/8)12yes—
187C2×C67—41no—
188C2×C67—81no—
189C2×C67#54611yescharacteristic
190C2×C67C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C67#57711yescharacteristic
192D67—41no—
193D67—42no—
194D67—81no—
195#3324—671nomaximal, Sylow
196C67#332411yescharacteristic, complemented, Sylow
197#556—671no—
198D42—671no—
199#486—671no—
200#482—671no—
201#482—671no—
202#474—671no—
203#468—671no—
204#468D6711yescharacteristic, complemented
205#460—671no—
206#450—671no—
207#450—671no—
208C4×SD16—671no—
209#436—671no—
210#418—671no—
211#418—671no—
212C22×D4—671no—
213C22×D4—671no—
214C22×D4—1341no—
215C2×SD16—671no—
216C2×SD16—671no—
217C2×SD16C2×D6711yescharacteristic, complemented
218C2×M16—671no—
219C2×M16C2×D6711yescharacteristic, complemented
220C4⋊Q8—671no—
221#127—671no—
222#126—1341no—
223#126—1341no—
224#124—1341no—
225C22⋊Q8—671no—
226C22⋊Q8C2×D6711yescharacteristic, complemented
227#121—671no—
228#121—1341no—
229#121C2×D6711yescharacteristic
230#120—1341no—
231#120—1341no—
232C4×Q8—671no—
233C4×Q8—1341no—
234C4×D4—671no—
235C4×D4—671no—
236#117—671no—
237#117—671no—
238#107—671no—
239#107C2×D6711yescharacteristic, complemented
240#106—671no—
241C4⋊C8—671no—
242#103C2×D6711yescharacteristic, complemented
243#102—671no—
244#102—671no—
245#102—671no—
246#102—671no—
247#102—671no—
248#102—671no—
249#102—671no—
250#102—671no—
251#102C2×D6711yescharacteristic, complemented
252C22⋊C8—671no—
253C22⋊C8—671no—
254C4×C8—671no—
255C24—1341no—
256C24—1341no—
257C2×Q8—1341no—
258C2×Q8C22×D6711yescharacteristic
259C2×D4—21no—
260C2×D4—2681no—
261C2×D4—2684no—
262C2×D4—2681no—
263C2×D4—671no—
264C2×D4—671no—
265C2×D4—671no—
266C2×D4—1341no—
267C2×D4—1342no—
268C2×D4—1341no—
269C2×D4—1341no—
270C2×D4—1341no—
271C2×D4—1342no—
272C2×D4—1341no—
273C2×D4C22×D6711yescharacteristic
274C22×C4—671no—
275C22×C4—1341no—
276C22×C4C22×D6711yescharacteristic
277SD16—41no—
278SD16—2681no—
279SD16—1342no—
280M16—41no—
281M16—2681no—
282C2×C8—671no—
283C2×C8—671no—
284C2×C8C22×D6711yescharacteristic
285C2×C8C22×D6711yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—671no—
288C4⋊C4—671no—
289C4⋊C4—671no—
290C4⋊C4—1341no—
291C4⋊C4—1341no—
292C4⋊C4—1341no—
293C4⋊C4C22×D6711yescharacteristic
294C4⋊C4C22×D6711yescharacteristic
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—671no—
298C22⋊C4—671no—
299C22⋊C4—1341no—
300C22⋊C4—1341no—
301C22⋊C4—1341no—
302C22⋊C4—1341no—
303C22⋊C4—1341no—
304C42—671no—
305C42—1341no—
306C42—1341no—
307C42—1341no—
308C23—21no—
309C23—2682no—
310C23—2682no—
311C23—2682no—
312C23—2682no—
313C23—5361no—
314C23—5361no—
315C23—671no—
316C23—1341no—
317C23—1341no—
318C23D4×D6711yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—2682no—
322D4—21no—
323D4—21no—
324D4—42no—
325D4—2682no—
326D4—2682no—
327D4—2682no—
328D4—2682no—
329D4—5361no—
330D4—672no—
331D4—1342no—
332D4—1341no—
333D4—1341no—
334D4—1341no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—2682no—
341C2×C4—671no—
342C2×C4—671no—
343C2×C4—671no—
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—2681no—
361C22—2681no—
362C22—2681no—
363C22—2681no—
364C22—2681no—
365C22—5364no—
366C22—5364no—
367C22—1341no—
368C22—1342no—
369C22—1341no—
370C22—1341no—
371C22C2×D4×D6711yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—2681no—
377C4—2681no—
378C4—672no—
379C4—1341no—
380C4—1341no—
381C4#41×D6711yescharacteristic
382C4#1452408811yescharacteristic
383C2—41no—
384C2—81no—
385C2—2682no—
386C2—2681no—
387C2—5361no—
388C2#1686403511yescharacteristic
389C2AGL(1,ℤ/8)×D6711yescharacteristic
390C2#1686420511yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#468×C67C2
socleC22×C67C22×D4
hypercenter#468D67
hypocenterC67#3324
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #16864117C2semidirect
3. #16863741C2semidirect
4. #16863723C2semidirect
5. #16863717C2semidirect
6. #16863675C2semidirect
7. #16863645C2semidirect
8. #16863594C2semidirect
9. #16863546C2semidirect
10. #16863545C2semidirect
11. #16863481C2semidirect
12. #16863472C2semidirect
13. #16863402C2semidirect
14. #16863399C2semidirect
15. #468×C67C2semidirect
20. C2×#11761615C22semidirect
21. C2×#11761611C22semidirect
22. #14524042C22semidirect
27. #14524019C22semidirect
32. Q8×Dic67C22semidirect
36. #14524002C22semidirect
37. #14524001C22semidirect
38. #14523976C22semidirect
39. #14523974C22semidirect
40. Dic67⋊C8C22semidirect
43. #14523965C22semidirect
44. #14523965C22semidirect
45. #14523964C22semidirect
46. #14523964C22semidirect
48. #14523963C22semidirect
49. C22⋊(C67⋊C8)C22semidirect
50. (C2×D67)⋊C8C22semidirect
51. C8×Dic67C22semidirect
52. C2×SD16×C67C22semidirect
53. C2×M16×C67C22semidirect
54. C22⋊Q8×C67C22semidirect
56. #107×C67C22semidirect
57. #103×C67C22semidirect
58. #102×C67C22semidirect
196. C67#3324semidirect
204. #468D67semidirect
217. C2×SD16C2×D67semidirect
219. C2×M16C2×D67semidirect
226. C22⋊Q8C2×D67semidirect
239. #107C2×D67semidirect
242. #103C2×D67semidirect
251. #102C2×D67semidirect