Finite Group Databasebeta

← #22570120

Subgroups — #22570120

Order 8576.

420 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16863903C211yesmaximal, characteristic
2#117×D67C211yesmaximal, characteristic, complemented
3#16863736C211yesmaximal, characteristic, complemented
4#16863735C211yesmaximal, characteristic, complemented
5#16863730C211yesmaximal, characteristic, complemented
6#16863725C211yesmaximal, characteristic, complemented
7#16863719C211yesmaximal, characteristic, complemented
8#16863718C211yesmaximal, characteristic, complemented
9#16863713C211yesmaximal, characteristic, complemented
10#16863708C211yesmaximal, characteristic, complemented
11#16863434C211yesmaximal, characteristic, complemented
12#16863432C211yesmaximal, characteristic, complemented
13#16863392C211yesmaximal, characteristic, complemented
14C2×(C2×D67)⋊C8C211yesmaximal, characteristic, complemented
15#416×C67C211yesmaximal, characteristic, complemented
16C22×C4×D67C2211yescharacteristic
17C22×C67⋊C8C2211yescharacteristic
18#14524024—21no—
19#14524019C2211yescharacteristic
20#14524019C2211yescharacteristic
21#14524014C2211yescharacteristic
22#14524014C2211yescharacteristic
23#14524011—21no—
24C4×C22⋊D67—21no—
25C4×C22⋊D67—21no—
26#14524002C2211yescharacteristic
27#14524002C2211yescharacteristic
28#14524001C2211yescharacteristic
29#14524001C2211yescharacteristic
30#14524000—21no—
31#14523999—21no—
32C4⋊C4×D67C2211yescharacteristic
33C4⋊C4×D67C2211yescharacteristic
34C22⋊C4×D67—21no—
35C42×D67—21no—
36#14523977C2211yescharacteristic, complemented
37#14523977C2211yescharacteristic, complemented
38#14523976C2211yescharacteristic, complemented
39#14523976C2211yescharacteristic, complemented
40#14523975C2211yescharacteristic, complemented
41#14523975C2211yescharacteristic, complemented
42#14523974C2211yescharacteristic, complemented
43#14523974C2211yescharacteristic, complemented
44#14523967C2211yescharacteristic, complemented
45#14523967C2211yescharacteristic, complemented
46#14523964C2211yescharacteristic, complemented
47#14523964C2211yescharacteristic, complemented
48(C2×D67)⋊C8C2211yescharacteristic
49(C2×D67)⋊C8C2211yescharacteristic
50(C2×D67)⋊C8C2211yescharacteristic
51(C2×D67)⋊C8C2211yescharacteristic
52C2×#41×C67C2211yescharacteristic
53C22×C8×C67C2211yescharacteristic
54#117×C67C2211yescharacteristic
55#103×C67C2211yescharacteristic, complemented
56#103×C67C2211yescharacteristic, complemented
57#102×C67C2211yescharacteristic, complemented
58#102×C67C2211yescharacteristic, complemented
59C23×D67D411yescharacteristic
60C2×C22⋊D67—21no—
61C2×C22⋊D67—21no—
62C22×Dic67D411yescharacteristic
63C2×C4×D67—22no—
64C2×C4×D67—22no—
65C2×C4×D67—21no—
66C2×C4×D67—21no—
67C2×C4×D67D411yescharacteristic
68C2×C4×D67D411yescharacteristic
69C2×C4×D67C2311yescharacteristic
70C2×C4×D67C2311yescharacteristic
71C2×C67⋊C8—22no—
72C2×C67⋊C8C2311yescharacteristic
73C2×C67⋊C8C2311yescharacteristic
74C4⋊Dic67C2311yescharacteristic
75C4⋊Dic67C2311yescharacteristic
76C4⋊Dic67C2311yescharacteristic
77C4⋊Dic67C2311yescharacteristic
78Dic67⋊C4—21no—
79Dic67⋊C4—21no—
80Dic67⋊C4—21no—
81Dic67⋊C4—21no—
82C22⋊Dic67—21no—
83C22⋊Dic67—21no—
84C22⋊Dic67—21no—
85C22⋊Dic67—21no—
86(C2×D67)⋊C4—21no—
87(C2×D67)⋊C4—21no—
88(C2×D67)⋊C4—21no—
89(C2×D67)⋊C4—21no—
90C4×Dic67—21no—
91C4×Dic67—21no—
92C4×Dic67—21no—
93C4×Dic67—21no—
94#41×C67—22no—
95#41×C67—22no—
96C2×Q8×C67C2311yescharacteristic
97C2×D4×C67—21no—
98C2×D4×C67C2311yescharacteristic
99C22×C4×C67—21no—
100C22×C4×C67C2311yescharacteristic
101C2×C8×C67—22no—
102C2×C8×C67C2311yescharacteristic
103C2×C8×C67C2311yescharacteristic
104C4⋊C4×C67C2311yescharacteristic
105C4⋊C4×C67C2311yescharacteristic
106C22⋊C4×C67—21no—
107C42×C67—21no—
108C23×C67—21no—
109C23×C67C2×D411yescharacteristic
110C22×D67—22no—
111C22×D67—22no—
112C22×D67—41no—
113C22×D67C2×D411yescharacteristic
114C22×D67C2×D411yescharacteristic
115Q8×C67—21no—
116Q8×C67—21no—
117D4×C67—21no—
118D4×C67—21no—
119D4×C67—42no—
120C2×C4×C67—21no—
121C2×C4×C67—21no—
122C2×C4×C67—21no—
123C2×C4×C67—42no—
124C2×C4×C67C2×D411yescharacteristic
125C2×C4×C67#4112yes—
126C2×C4×C67#4112yes—
127C2×C4×C67C2411yescharacteristic
128C22⋊D67—42no—
129C22⋊D67—42no—
130C2×Dic67—22no—
131C2×Dic67—21no—
132C2×Dic67—21no—
133C2×Dic67—21no—
134C2×Dic67—21no—
135C2×Dic67C2×D411yescharacteristic
136C2×Dic67C2×D411yescharacteristic
137C4×D67—22no—
138C4×D67—22no—
139C4×D67—21no—
140C4×D67—21no—
141C4×D67—21no—
142C4×D67—21no—
143C4×D67—42no—
144C4×D67—42no—
145C8×C67—21no—
146C8×C67—21no—
147C67⋊C8—21no—
148C67⋊C8—21no—
149C22×C67—21no—
150C22×C67—21no—
151C22×C67—41no—
152C22×C67—41no—
153C22×C67—41no—
154C22×C67#12312yes—
155C22×C67C22×D411yescharacteristic
156C2×D67—21no—
157C2×D67—21no—
158C2×D67—21no—
159C2×D67—21no—
160C2×D67—21no—
161C2×D67—44no—
162C2×D67#13512yes—
163C4×C67—41no—
164C4×C67—41no—
165C4×C67—41no—
166C4×C67#12312yes—
167C4×C67C2×#4111yescharacteristic
168C4×C67C2×#4111yescharacteristic
169Dic67—21no—
170Dic67—21no—
171Dic67—41no—
172Dic67—41no—
173Dic67—41no—
174Dic67—41no—
175C2×C67—22no—
176C2×C67—81no—
177C2×C67C2×#12311yescharacteristic
178C2×C67C2×#13511yescharacteristic
179C2×C67#57511yescharacteristic
180D67—22no—
181D67—41no—
182#3091—671nomaximal, Sylow
183C67#309111yescharacteristic, complemented, Sylow
184#525—671no—
185C2×#117—671no—
186#485—671no—
187#485—671no—
188#484—671no—
189#483—671no—
190#482—671no—
191#482—671no—
192#481—671no—
193#480—671no—
194#427—671no—
195#427—671no—
196#416—671no—
197#416D6711yescharacteristic, complemented
198C2×C22⋊C8—671no—
199C2×#41C2×D6711yescharacteristic
200C23×C4—671no—
201C22×C8—671no—
202C22×C8C2×D6711yescharacteristic
203#123—1341no—
204C22⋊Q8—671no—
205C22⋊Q8—671no—
206#121—671no—
207#121—671no—
208#120—1341no—
209C4×D4—1341no—
210C4×D4—1341no—
211#117—671no—
212#117—671no—
213#117—671no—
214#117—671no—
215#117—1341no—
216#117—1341no—
217#117C2×D6711yescharacteristic
218C2×C4⋊C4—671no—
219C2×C4⋊C4—671no—
220C2×C22⋊C4—1341no—
221C2×C42—1341no—
222#107—671no—
223#107—671no—
224#107—671no—
225#107—671no—
226#106—671no—
227#106—671no—
228#106—671no—
229#106—671no—
230#103—671no—
231#103—671no—
232#103C2×D6711yescharacteristic, complemented
233#103C2×D6711yescharacteristic, complemented
234#102—671no—
235#102—671no—
236#102C2×D6711yescharacteristic, complemented
237#102C2×D6711yescharacteristic, complemented
238C22⋊C8—671no—
239C22⋊C8—671no—
240C22⋊C8—671no—
241C22⋊C8—671no—
242C24—671no—
243#41—22no—
244#41—22no—
245C2×Q8C22×D6711yescharacteristic
246C2×D4—21no—
247C2×D4—1341no—
248C2×D4—1341no—
249C2×D4C22×D6711yescharacteristic
250C22×C4—21no—
251C22×C4—671no—
252C22×C4—671no—
253C22×C4—671no—
254C22×C4—671no—
255C22×C4—671no—
256C22×C4—1341no—
257C22×C4—1341no—
258C22×C4—1342no—
259C22×C4—1342no—
260C22×C4C22×D6711yescharacteristic
261C2×C8—22no—
262C2×C8—671no—
263C2×C8—671no—
264C2×C8—1342no—
265C2×C8C22×D6711yescharacteristic
266C2×C8C22×D6711yescharacteristic
267C4⋊C4—671no—
268C4⋊C4—671no—
269C4⋊C4—671no—
270C4⋊C4—671no—
271C4⋊C4—1341no—
272C4⋊C4—1341no—
273C4⋊C4—1341no—
274C4⋊C4—1341no—
275C4⋊C4C22×D6711yescharacteristic
276C4⋊C4C22×D6711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—1341no—
279C22⋊C4—1341no—
280C22⋊C4—1341no—
281C22⋊C4—1341no—
282C22⋊C4—1341no—
283C22⋊C4—1341no—
284C22⋊C4—1341no—
285C22⋊C4—1341no—
286C42—21no—
287C42—1341no—
288C42—1341no—
289C42—1341no—
290C42—1341no—
291C23—21no—
292C23—2681no—
293C23—671no—
294C23—671no—
295C23—1342no—
296C23—1342no—
297C23D4×D6711yescharacteristic
298Q8—21no—
299Q8—21no—
300D4—21no—
301D4—21no—
302D4—42no—
303D4—2682no—
304D4—2682no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—2682no—
310C2×C4—2682no—
311C2×C4—671no—
312C2×C4—671no—
313C2×C4—1341no—
314C2×C4—1341no—
315C2×C4—1341no—
316C2×C4—1341no—
317C2×C4—1342no—
318C2×C4—1342no—
319C2×C4—1342no—
320C2×C4—1341no—
321C2×C4—1341no—
322C2×C4—1341no—
323C2×C4—1341no—
324C2×C4D4×D6711yescharacteristic
325C2×C4#1176162712yes—
326C2×C4#1176162812yes—
327C2×C4C23×D6711yescharacteristic
328C8—21no—
329C8—21no—
330C8—1341no—
331C8—1341no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—2684no—
338C22—672no—
339C22—1341no—
340C22—1341no—
341C22—1341no—
342C22—1341no—
343C22—1341no—
344C22#1452402512yes—
345C22C2×D4×D6711yescharacteristic
346C4—41no—
347C4—41no—
348C4—41no—
349C4—2681no—
350C4—2681no—
351C4—2681no—
352C4—2681no—
353C4—1341no—
354C4—1341no—
355C4#1452402512yes—
356C4C2×#1176162711yescharacteristic
357C4C2×#1176162811yescharacteristic
358C2—22no—
359C2—81no—
360C2—2681no—
361C2—1342no—
362C2C2×#1452402511yescharacteristic
363C2#135×D6711yescharacteristic
364C2#1686419311yescharacteristic
subgroupG/H
centerC2×C4D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#416×C67C2
socleC22×C67C22×D4
hypercenter#416D67
hypocenterC67#3091
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. #117×D67C2semidirect
3. #16863736C2semidirect
4. #16863735C2semidirect
5. #16863730C2semidirect
6. #16863725C2semidirect
7. #16863719C2semidirect
8. #16863718C2semidirect
9. #16863713C2semidirect
10. #16863708C2semidirect
11. #16863434C2semidirect
12. #16863432C2semidirect
13. #16863392C2semidirect
14. C2×(C2×D67)⋊C8C2semidirect
15. #416×C67C2semidirect
36. #14523977C22semidirect
37. #14523977C22semidirect
38. #14523976C22semidirect
39. #14523976C22semidirect
40. #14523975C22semidirect
41. #14523975C22semidirect
42. #14523974C22semidirect
43. #14523974C22semidirect
44. #14523967C22semidirect
45. #14523967C22semidirect
46. #14523964C22semidirect
47. #14523964C22semidirect
55. #103×C67C22semidirect
56. #103×C67C22semidirect
57. #102×C67C22semidirect
58. #102×C67C22semidirect
183. C67#3091semidirect
197. #416D67semidirect
232. #103C2×D67semidirect
233. #103C2×D67semidirect
236. #102C2×D67semidirect
237. #102C2×D67semidirect