Finite Group Databasebeta

← #22570486

Subgroups — #22570486

Order 8576.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16864017C211yesmaximal, characteristic, complemented
2#16863726C211yesmaximal, characteristic, complemented
3#16863714C211yesmaximal, characteristic, complemented
4#16863686C211yesmaximal, characteristic, complemented
5#16863681C211yesmaximal, characteristic, complemented
6#16863618C211yesmaximal, characteristic, complemented
7#16863611C211yesmaximal, characteristic, complemented
8#16863558C211yesmaximal, characteristic, complemented
9#16863541C211yesmaximal, characteristic
10#16863506C211yesmaximal, characteristic, complemented
11#16863501C211yesmaximal, characteristic, complemented
12C4×#11761615C211yesmaximal, characteristic, complemented
13C4×#11761616C211yesmaximal, characteristic, complemented
14#16863457C211yesmaximal, characteristic, complemented
15#520×C67C211yesmaximal, characteristic, complemented
16C2×#11761618C2211yescharacteristic
17C2×#11761616C2211yescharacteristic
18C2×#11761615C2211yescharacteristic
19C2×#11761613C2211yescharacteristic
20#14524043C2211yescharacteristic
21#14524038—21no—
22#14524031C2211yescharacteristic
23#14524029—21no—
24#14524025—21no—
25#14524022C2211yescharacteristic
26#14524021—21no—
27#14524017C2211yescharacteristic
28C4×Dic134C2211yescharacteristic
29C4×D268C2211yescharacteristic
30#14523974C2211yescharacteristic
31#14523974C2211yescharacteristic
32#14523974C2211yescharacteristic
33#14523974C2211yescharacteristic
34C4⋊(C67⋊C8)C2211yescharacteristic
35C4⋊(C67⋊C8)C2211yescharacteristic
36#14523967C2211yescharacteristic
37#14523967C2211yescharacteristic
38#14523966C2211yescharacteristic
39#14523966C2211yescharacteristic
40#14523964C2211yescharacteristic
41#14523964C2211yescharacteristic
42#14523963C2211yescharacteristic
43#14523963C2211yescharacteristic
44C22⋊(C67⋊C8)C2211yescharacteristic
45C22⋊(C67⋊C8)C2211yescharacteristic
46#14523954C2211yescharacteristic
47C4×C67⋊C8C2211yescharacteristic
48C2×#41×C67C2211yescharacteristic
49C4×Q8×C67C2211yescharacteristic
50C4×Q8×C67C2211yescharacteristic
51C4×D4×C67—21no—
52C4×D4×C67—21no—
53C4×D4×C67C2211yescharacteristic
54C4×D4×C67C2211yescharacteristic
55#117×C67—21no—
56#117×C67C2211yescharacteristic
57C2×C4⋊C4×C67—21no—
58C2×C4⋊C4×C67C2211yescharacteristic
59C2×Dic134—21no—
60C2×Dic134C2311yescharacteristic
61C2×D268C2311yescharacteristic
62C2×C22⋊D67—21no—
63C2×C4×D67—21no—
64#11761618—22no—
65#11761616—22no—
66#11761615—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—
83#41×C67—22no—
84#41×C67—22no—
85C2×Q8×C67C2311yescharacteristic
86C2×D4×C67—21no—
87C2×D4×C67C2311yescharacteristic
88C22×C4×C67—21no—
89C22×C4×C67—21no—
90C22×C4×C67—21no—
91C22×C4×C67—21no—
92C22×C4×C67C2311yescharacteristic
93C4⋊C4×C67—21no—
94C4⋊C4×C67—21no—
95C4⋊C4×C67—21no—
96C4⋊C4×C67D411yescharacteristic
97C4⋊C4×C67D411yescharacteristic
98C4⋊C4×C67C2311yescharacteristic
99C4⋊C4×C67C2311yescharacteristic
100C22⋊C4×C67—21no—
101C22⋊C4×C67—21no—
102C22⋊C4×C67D411yescharacteristic
103C22⋊C4×C67D411yescharacteristic
104C42×C67—21no—
105C42×C67—21no—
106C42×C67C2311yescharacteristic
107C42×C67C2311yescharacteristic
108C23×C67—21no—
109C23×C67C2×D411yescharacteristic
110C22×D67—21no—
111Q8×C67—21no—
112Q8×C67#4112yes—
113D4×C67—24no—
114D4×C67—21no—
115D4×C67#4112yes—
116C2×C4×C67—24no—
117C2×C4×C67—21no—
118C2×C4×C67—21no—
119C2×C4×C67—21no—
120C2×C4×C67—21no—
121C2×C4×C67—21no—
122C2×C4×C67—81no—
123C2×C4×C67—81no—
124C2×C4×C67—81no—
125C2×C4×C67C2×D411yescharacteristic
126C2×C4×C67C2×D411yescharacteristic
127C2×C4×C67C2×D411yescharacteristic
128C2×C4×C67C2×D411yescharacteristic
129C2×C4×C67C2×D411yescharacteristic
130C2×C4×C67C2411yescharacteristic
131C22⋊D67—81no—
132C2×Dic67—21no—
133C2×Dic67—21no—
134C2×Dic67—21no—
135D268—21no—
136D268—21no—
137C4×D67—81no—
138Dic134—21no—
139Dic134—21no—
140Dic134—81no—
141C67⋊C8—21no—
142C67⋊C8—21no—
143C67⋊C8—21no—
144C67⋊C8—21no—
145C22×C67—21no—
146C22×C67—22no—
147C22×C67—21no—
148C22×C67—21no—
149C22×C67—41no—
150C22×C67—41no—
151C22×C67C22×D411yescharacteristic
152C2×D67—41no—
153C2×D67—41no—
154C2×D67—41no—
155C4×C67—21no—
156C4×C67—22no—
157C4×C67—21no—
158C4×C67—21no—
159C4×C67—21no—
160C4×C67—21no—
161C4×C67—41no—
162C4×C67—41no—
163C4×C67C2×#4111yescharacteristic
164C4×C67C2×#4111yescharacteristic
165Dic67—41no—
166Dic67—41no—
167Dic67—41no—
168C2×C67—41no—
169C2×C67—42no—
170C2×C67#52511yescharacteristic
171C2×C67#57711yescharacteristic
172C2×C67#57711yescharacteristic
173D67—81no—
174#3125—671nomaximal, Sylow
175C67#312511yescharacteristic, complemented, Sylow
176#542—671no—
177#520D6711yescharacteristic, complemented
178#483—671no—
179#481—671no—
180#476—671no—
181#475—671no—
182#464—671no—
183#463—671no—
184#452—671no—
185#449—671no—
186#442—671no—
187#441—671no—
188C4×SD16—671no—
189C4×SD16—671no—
190#433—671no—
191C2×#41C2×D6711yescharacteristic
192C2×Q16—671no—
193C2×SD16—671no—
194C2×SD16—671no—
195C2×D8—671no—
196C4⋊Q8—671no—
197#126—1341no—
198#124—671no—
199#124—1341no—
200#123—1341no—
201C22⋊Q8—671no—
202C22⋊Q8—1341no—
203#121—671no—
204C4×Q8—671no—
205C4×Q8C2×D6711yescharacteristic
206C4×Q8C2×D6711yescharacteristic
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—671no—
210C4×D4C2×D6711yescharacteristic
211C4×D4C2×D6711yescharacteristic
212#117—21no—
213#117C2×D6711yescharacteristic
214C2×C4⋊C4—21no—
215C2×C4⋊C4C2×D6711yescharacteristic
216#106—671no—
217#106—671no—
218#106—671no—
219#106—671no—
220C4⋊C8—671no—
221C4⋊C8—671no—
222#103—671no—
223#103—671no—
224#103—671no—
225#103—671no—
226#102—671no—
227#102—671no—
228#102—671no—
229#102—671no—
230C22⋊C8—671no—
231C22⋊C8—671no—
232#97—671no—
233C4×C8—671no—
234#41—22no—
235#41—22no—
236C2×Q8—671no—
237C2×Q8—1341no—
238C2×Q8C22×D6711yescharacteristic
239C2×D4—21no—
240C2×D4—671no—
241C2×D4—1341no—
242C2×D4C22×D6711yescharacteristic
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—1341no—
248C22×C4C22×D6711yescharacteristic
249Q16—1342no—
250SD16—1342no—
251SD16—1342no—
252D8—1342no—
253C2×C8—671no—
254C2×C8—671no—
255C2×C8—671no—
256C2×C8—671no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—671no—
261C4⋊C4—671no—
262C4⋊C4—1341no—
263C4⋊C4—1341no—
264C4⋊C4—1341no—
265C4⋊C4C22⋊D6711yescharacteristic
266C4⋊C4C22⋊D6711yescharacteristic
267C4⋊C4C22×D6711yescharacteristic
268C4⋊C4C22×D6711yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—1341no—
272C22⋊C4—1341no—
273C22⋊C4—1341no—
274C22⋊C4—1341no—
275C22⋊C4—1341no—
276C22⋊C4C22⋊D6711yescharacteristic
277C22⋊C4C22⋊D6711yescharacteristic
278C42—21no—
279C42—21no—
280C42—1341no—
281C42C22×D6711yescharacteristic
282C42C22×D6711yescharacteristic
283C23—21no—
284C23—1341no—
285C23C2×C22⋊D6711yescharacteristic
286Q8—21no—
287Q8—5361no—
288Q8—1341no—
289Q8—1341no—
290Q8#1176162812yes—
291D4—24no—
292D4—21no—
293D4—5361no—
294D4—1341no—
295D4—1341no—
296D4#1176162812yes—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—5361no—
307C2×C4—1341no—
308C2×C4—1341no—
309C2×C4—1341no—
310C2×C4C2×C22⋊D6711yescharacteristic
311C2×C4C2×C22⋊D6711yescharacteristic
312C2×C4C2×C22⋊D6711yescharacteristic
313C2×C4C2×C22⋊D6711yescharacteristic
314C2×C4C2×C22⋊D6711yescharacteristic
315C2×C4C23×D6711yescharacteristic
316C8—1341no—
317C8—1341no—
318C8—1341no—
319C8—1341no—
320C22—21no—
321C22—22no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—2681no—
327C22—2681no—
328C22—2681no—
329C22C22×C22⋊D6711yescharacteristic
330C4—21no—
331C4—22no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—41no—
337C4—41no—
338C4—2681no—
339C4—2681no—
340C4—2681no—
341C4C2×#1176162811yescharacteristic
342C4C2×#1176162811yescharacteristic
343C2—41no—
344C2—42no—
345C2—5361no—
346C2#1686390711yescharacteristic
347C2#1686420411yescharacteristic
348C2#1686420711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#520×C67C2
socleC22×C67C22×D4
hypercenter#520D67
hypocenterC67#3125
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ #117 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #16864017C2semidirect
2. #16863726C2semidirect
3. #16863714C2semidirect
4. #16863686C2semidirect
5. #16863681C2semidirect
6. #16863618C2semidirect
7. #16863611C2semidirect
8. #16863558C2semidirect
10. #16863506C2semidirect
11. #16863501C2semidirect
12. C4×#11761615C2semidirect
13. C4×#11761616C2semidirect
14. #16863457C2semidirect
15. #520×C67C2semidirect
175. C67#3125semidirect
177. #520D67semidirect