Finite Group Databasebeta

← #22571720

Subgroups — #22571720

Order 8576.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14524074C211yesmaximal, characteristic, complemented
2C22×#11761615C211yesmaximal, characteristic, complemented
3#16864042C211yesmaximal, characteristic
4#16863788C211yesmaximal, characteristic, complemented
5#16863644C211yesmaximal, characteristic, complemented
6#16863635C211yesmaximal, characteristic, complemented
7#16863618C211yesmaximal, characteristic, complemented
8#16863598C211yesmaximal, characteristic, complemented
9#16863556C211yesmaximal, characteristic, complemented
10#16863552C211yesmaximal, characteristic, complemented
11#16863546C211yesmaximal, characteristic, complemented
12#16863535C211yesmaximal, characteristic, complemented
13#16863494C211yesmaximal, characteristic, complemented
14#16863468C211yesmaximal, characteristic, complemented
15D4×Q8×C67C211yesmaximal, characteristic, complemented
16C2×#11761628C2211yescharacteristic
17C22×D268C2211yescharacteristic
18#14524074—24no—
19C2×#11761618C2211yescharacteristic, complemented
20C2×#11761618C2211yescharacteristic, complemented
21C2×#11761618C2211yescharacteristic, complemented
22C2×#11761615—24no—
23C2×#11761615C2211yescharacteristic
24C2×#11761615C2211yescharacteristic
25C2×#11761615C2211yescharacteristic
26C2×#11761615C2211yescharacteristic
27C2×#11761615C2211yescharacteristic
28C2×#11761615C2211yescharacteristic
29C2×#11761612C2211yescharacteristic, complemented
30C22×C67⋊C8C2211yescharacteristic
31#14524031C2211yescharacteristic
32#14524028—21no—
33#14524022C2211yescharacteristic
34#14524017C2211yescharacteristic
35#14524016—21no—
36#14524011—21no—
37C4×D268C2211yescharacteristic
38C22⋊C4×D67—21no—
39#14523976C2211yescharacteristic, complemented
40C4⋊(C67⋊C8)C2211yescharacteristic, complemented
41#14523967C2211yescharacteristic, complemented
42#14523967C2211yescharacteristic, complemented
43#14523967C2211yescharacteristic, complemented
44#14523966C2211yescharacteristic, complemented
45#14523963C2211yescharacteristic
46#14523963C2211yescharacteristic
47C22⋊(C67⋊C8)C2211yescharacteristic
48C22⋊(C67⋊C8)C2211yescharacteristic, complemented
49#14523954C2211yescharacteristic, complemented
50C22×Q8×C67C2211yescharacteristic
51C22×Q8×C67C2211yescharacteristic, complemented
52C4⋊Q8×C67—21no—
53C4⋊Q8×C67C2211yescharacteristic, complemented
54C22⋊Q8×C67—21no—
55C22⋊Q8×C67—21no—
56C22⋊Q8×C67C2211yescharacteristic
57C22⋊Q8×C67C2211yescharacteristic, complemented
58C4×Q8×C67C2211yescharacteristic, complemented
59C4×D4×C67—21no—
60C4×D4×C67C2211yescharacteristic
61C23×D67—21no—
62#11761628—22no—
63#11761628—41no—
64C2×Dic134C2311yescharacteristic
65C2×D268—22no—
66C2×D268—21no—
67C2×D268—41no—
68C2×D268C2311yescharacteristic
69C2×D268C2311yescharacteristic
70C2×C22⋊D67—21no—
71C2×C22⋊D67—21no—
72C2×C4×D67—21no—
73C2×C4×D67—21no—
74#11761618—22no—
75#11761618—22no—
76#11761618—41no—
77#11761615—22no—
78#11761615—22no—
79#11761615—22no—
80#11761615—22no—
81#11761615—41no—
82#11761615D414yescomplemented
83#11761612—22no—
84C2×C67⋊C8—22no—
85C2×C67⋊C8C2311yescharacteristic
86C2×C67⋊C8C2311yescharacteristic
87C2×C67⋊C8C2311yescharacteristic
88C2×C67⋊C8C2311yescharacteristic
89C4⋊Dic67C2311yescharacteristic
90Dic67⋊C4—21no—
91C22⋊Dic67—21no—
92(C2×D67)⋊C4—21no—
93(C2×D67)⋊C4—21no—
94(C2×D67)⋊C4—21no—
95(C2×D67)⋊C4—21no—
96C2×Q8×C67—21no—
97C2×Q8×C67—21no—
98C2×Q8×C67—22no—
99C2×Q8×C67—22no—
100C2×Q8×C67—41no—
101C2×Q8×C67—41no—
102C2×Q8×C67C2311yescharacteristic
103C2×Q8×C67C2311yescharacteristic
104C2×Q8×C67C2311yescharacteristic
105C2×D4×C67D411yescharacteristic
106C22×C4×C67—21no—
107C22×C4×C67—21no—
108C22×C4×C67C2311yescharacteristic
109C22×C4×C67C2311yescharacteristic
110C4⋊C4×C67—21no—
111C4⋊C4×C67—21no—
112C4⋊C4×C67—21no—
113C4⋊C4×C67—21no—
114C4⋊C4×C67—21no—
115C4⋊C4×C67D411yescharacteristic
116C4⋊C4×C67C2311yescharacteristic
117C22⋊C4×C67—21no—
118C22⋊C4×C67—21no—
119C22⋊C4×C67D411yescharacteristic
120C22⋊C4×C67D411yescharacteristic
121C42×C67—21no—
122C42×C67C2311yescharacteristic
123C23×C67C2×D411yescharacteristic
124C23×C67C2×D411yescharacteristic
125C22×D67—21no—
126C22×D67—21no—
127C22×D67—42no—
128C22×D67—42no—
129C22×D67—81no—
130Q8×C67—21no—
131Q8×C67—21no—
132Q8×C67—21no—
133Q8×C67—21no—
134Q8×C67—21no—
135Q8×C67—42no—
136Q8×C67—42no—
137Q8×C67C2×D412yes—
138D4×C67—22no—
139C2×C4×C67—21no—
140C2×C4×C67—21no—
141C2×C4×C67—21no—
142C2×C4×C67—21no—
143C2×C4×C67—21no—
144C2×C4×C67—21no—
145C2×C4×C67—21no—
146C2×C4×C67—21no—
147C2×C4×C67—42no—
148C2×C4×C67—42no—
149C2×C4×C67C2×D411yescharacteristic
150C2×C4×C67C2×D411yescharacteristic
151C2×C4×C67C2×D411yescharacteristic
152C2×C4×C67C2×D411yescharacteristic
153C2×C4×C67C2411yescharacteristic
154C22⋊D67—42no—
155C22⋊D67—81no—
156C2×Dic67—21no—
157C2×Dic67—21no—
158D268—21no—
159D268—21no—
160D268—21no—
161D268—42no—
162D268C2×D412yes—
163C4×D67—42no—
164C4×D67—81no—
165Dic134—21no—
166Dic134—21no—
167C67⋊C8—21no—
168C67⋊C8—21no—
169C67⋊C8—21no—
170C67⋊C8C2×D412yes—
171C22×C67—21no—
172C22×C67—21no—
173C22×C67—21no—
174C22×C67—21no—
175C22×C67—21no—
176C22×C67#13712yescomplemented
177C22×C67C22×D411yescharacteristic
178C2×D67—22no—
179C2×D67—41no—
180C2×D67—41no—
181C2×D67—41no—
182C2×D67—41no—
183C2×D67—41no—
184C2×D67—84no—
185C4×C67—21no—
186C4×C67—21no—
187C4×C67—22no—
188C4×C67—21no—
189C4×C67—21no—
190C4×C67—41no—
191C4×C67—41no—
192C4×C67—41no—
193C4×C67C22×D411yescharacteristic
194C4×C67H(5,𝔽2)11yescharacteristic
195Dic67—41no—
196Dic67—41no—
197C2×C67—22no—
198C2×C67—41no—
199C2×C67D4211yescharacteristic
200C2×C67C2×#13711yescharacteristic
201C2×C67#57711yescharacteristic
202D67—42no—
203D67—81no—
204#3266—671nomaximal, Sylow
205C67#326611yescharacteristic, complemented, Sylow
206C2×#137—671no—
207C22×SD16—671no—
208D4×Q8D6711yescharacteristic, complemented
209#546—671no—
210#497—671no—
211#468—671no—
212#467—671no—
213#464—671no—
214#461—671no—
215#452—671no—
216#451—671no—
217#450—671no—
218#448—671no—
219#440—671no—
220#435—671no—
221C2×#41—671no—
222C22×Q8C2×D6711yescharacteristic
223C22×Q8C2×D6711yescharacteristic, complemented
224C22×D4—671no—
225#137—1344no—
226C2×Q16—671no—
227C2×Q16—671no—
228C2×Q16—671no—
229C2×SD16—671no—
230C2×SD16—671no—
231C2×SD16—671no—
232C2×SD16—671no—
233C2×SD16—671no—
234C2×SD16—671no—
235C2×SD16—1344no—
236C2×M16—671no—
237C22×C8—671no—
238C4⋊Q8—21no—
239C4⋊Q8C2×D6711yescharacteristic, complemented
240#124—671no—
241#123—1341no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—671no—
245C22⋊Q8C2×D6711yescharacteristic
246C22⋊Q8C2×D6711yescharacteristic, complemented
247#121—671no—
248#121—1341no—
249#120—1341no—
250C4×Q8C2×D6711yescharacteristic, complemented
251C4×D4—21no—
252C4×D4—671no—
253C4×D4C2×D6711yescharacteristic
254C2×C22⋊C4—1341no—
255#107—671no—
256C4⋊C8—671no—
257#103—671no—
258#103—671no—
259#103—671no—
260#103—671no—
261#102—671no—
262#102—671no—
263C22⋊C8—671no—
264C22⋊C8—671no—
265#97—671no—
266C24—1341no—
267#41—2681no—
268#41—1342no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—671no—
276C2×Q8C22×D6711yescharacteristic
277C2×Q8C22×D6711yescharacteristic
278C2×Q8C22×D6711yescharacteristic
279C2×D4—2681no—
280C2×D4—671no—
281C2×D4—671no—
282C2×D4—1342no—
283C2×D4—1341no—
284C2×D4—1341no—
285C2×D4—1341no—
286C2×D4C22⋊D6711yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—1341no—
290C22×C4—1341no—
291C22×C4C22×D6711yescharacteristic
292C22×C4C22×D6711yescharacteristic
293Q16—2681no—
294Q16—1342no—
295Q16—1342no—
296SD16—2681no—
297SD16—674no—
298SD16—1342no—
299SD16—1342no—
300SD16—1342no—
301SD16—1342no—
302M16—1342no—
303C2×C8—671no—
304C2×C8—671no—
305C2×C8—671no—
306C2×C8—671no—
307C2×C8—1342no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—671no—
314C4⋊C4—1341no—
315C4⋊C4C22⋊D6711yescharacteristic
316C4⋊C4C22×D6711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—1341no—
320C22⋊C4—1341no—
321C22⋊C4—1341no—
322C22⋊C4—1341no—
323C22⋊C4—1341no—
324C22⋊C4C22⋊D6711yescharacteristic
325C22⋊C4C22⋊D6711yescharacteristic
326C42—21no—
327C42C22×D6711yescharacteristic
328C23—2682no—
329C23—2682no—
330C23—5361no—
331C23—1341no—
332C23—1341no—
333C23C2×C22⋊D6711yescharacteristic
334C23C2×C22⋊D6711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—42no—
342Q8—1341no—
343Q8—1341no—
344Q8D4×D6712yes—
345D4—22no—
346D4—2682no—
347D4—2682no—
348D4—5361no—
349D4—672no—
350D4—1341no—
351D4—1341no—
352D4—1341no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—2682no—
364C2×C4—5361no—
365C2×C4—1341no—
366C2×C4—1341no—
367C2×C4C2×C22⋊D6711yescharacteristic
368C2×C4C2×C22⋊D6711yescharacteristic
369C2×C4C2×C22⋊D6711yescharacteristic
370C2×C4C2×C22⋊D6711yescharacteristic
371C2×C4C23×D6711yescharacteristic
372C8—672no—
373C8—1341no—
374C8—1341no—
375C8—1341no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—2681no—
382C22—2681no—
383C22—2681no—
384C22—2681no—
385C22—2681no—
386C22—5364no—
387C22—1342no—
388C22#1452407412yescomplemented
389C22C22×C22⋊D6711yescharacteristic
390C4—21no—
391C4—21no—
392C4—22no—
393C4—21no—
394C4—21no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—2681no—
399C4—2681no—
400C4C2×D4×D6711yescharacteristic
401C4#1452408811yescharacteristic
402C2—22no—
403C2—41no—
404C2—2682no—
405C2—5361no—
406C2D4×C22⋊D6711yescharacteristic
407C2C2×#1452407411yescharacteristic
408C2#1686420411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
FittingD4×Q8×C67C2
socleC22×C67C22×D4
hypercenterD4×Q8D67
hypocenterC67#3266
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#14524074C2semidirect
2. C22×#11761615C2semidirect
4. #16863788C2semidirect
5. #16863644C2semidirect
6. #16863635C2semidirect
7. #16863618C2semidirect
8. #16863598C2semidirect
9. #16863556C2semidirect
10. #16863552C2semidirect
11. #16863546C2semidirect
12. #16863535C2semidirect
13. #16863494C2semidirect
14. #16863468C2semidirect
15. D4×Q8×C67C2semidirect
19. C2×#11761618C22semidirect
20. C2×#11761618C22semidirect
21. C2×#11761618C22semidirect
29. C2×#11761612C22semidirect
39. #14523976C22semidirect
40. C4⋊(C67⋊C8)C22semidirect
41. #14523967C22semidirect
42. #14523967C22semidirect
43. #14523967C22semidirect
44. #14523966C22semidirect
48. C22⋊(C67⋊C8)C22semidirect
49. #14523954C22semidirect
51. C22×Q8×C67C22semidirect
53. C4⋊Q8×C67C22semidirect
57. C22⋊Q8×C67C22semidirect
58. C4×Q8×C67C22semidirect
82. #11761615D4semidirect
176. C22×C67#137semidirect
205. C67#3266semidirect
208. D4×Q8D67semidirect
223. C22×Q8C2×D67semidirect
239. C4⋊Q8C2×D67semidirect
246. C22⋊Q8C2×D67semidirect
250. C4×Q8C2×D67semidirect
388. C22#14524074semidirect