Finite Group Databasebeta

← #29667922

Subgroups — #29667922

Order 15616.

784 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456769C211yesmaximal, characteristic, complemented
2#21456739C211yesmaximal, characteristic
3#21456378C211yesmaximal, characteristic, complemented
4#21456213C211yesmaximal, characteristic, complemented
5#21456194C211yesmaximal, characteristic, complemented
6C2×#16840109C211yesmaximal, characteristic, complemented
7#21445479C212yesmaximal, complemented
8#21445458C212yesmaximal, complemented
9#21445431C212yesmaximal, complemented
10#21445425C212yesmaximal, complemented
11C8×M16×C61C211yesmaximal, characteristic, complemented
12#16840319C2211yescharacteristic
13#16840312C2211yescharacteristic
14C2×#14519777C2211yescharacteristic
15C2×#14519774C2211yescharacteristic
16#16840160C2211yescharacteristic, complemented
17#16840142C2212yescomplemented
18#16840139C2212yes—
19#16840135C2211yescharacteristic, complemented
20#16840127—22no—
21#16840122—22no—
22#16840113C2211yescharacteristic, complemented
23#16840113C2212yescomplemented
24#16840113C2212yescomplemented
25#16840109—22no—
26#16840109C2212yes—
27#16840109C2212yes—
28#16840109C2212yes—
29#16840109C2212yes—
30#16840109C2212yes—
31#16840109C2211yescharacteristic, complemented
32#16840109C2212yescomplemented
33#16839991—22no—
34#16839973—22no—
35#16839958—22no—
36#16839920—22no—
37#16839898—22no—
38C4×#11758190—22no—
39#16839768—22no—
40#16839764—22no—
41C2×#14519699—22no—
42C61×#431C2211yescharacteristic
43C4×M16×C61C2211yescharacteristic
44C2×C4×C8×C61C2211yescharacteristic
45C61×#322C2212yescomplemented
46C82×C61C2212yescomplemented
47C2×#11758222—21no—
48C22×Dic122—21no—
49C22×D244—21no—
50C22×D244—21no—
51C22×D244—21no—
52C2×Dic244—24no—
53C2×#11758190—22no—
54C2×D488—24no—
55#14519777—21no—
56#14519777—21no—
57#14519777—21no—
58#14519777C2312yes—
59#14519777C2311yescharacteristic
60#14519777C2311yescharacteristic
61#14519777C2311yescharacteristic
62#14519774—21no—
63#14519774—21no—
64#14519774—21no—
65#14519774C2312yes—
66#14519774C2311yescharacteristic
67#14519769—21no—
68#14519765—21no—
69#14519756—21no—
70#14519756—21no—
71#14519751—21no—
72#14519751—21no—
73C4×Dic122—21no—
74C4×Dic122—21no—
75C4×D244—21no—
76C4×D244—21no—
77#14519736—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—21no—
80#14519709—22no—
81#14519702—24no—
82#14519702—22no—
83#14519702—24no—
84#14519699—24no—
85#14519699—22no—
86#14519699—24no—
87#14519699—24no—
88#14519699—24no—
89#14519699—24no—
90#14519699—24no—
91C2×M16×C61D412yes—
92C22×C8×C61D412yes—
93C2×C42×C61C2311yescharacteristic
94C4⋊C8×C61D412yes—
95C22⋊C8×C61D412yes—
96#97×C61D412yes—
97C4×C8×C61D412yes—
98C4×C8×C61C2311yescharacteristic
99C4×C8×C61C2312yes—
100C4×C8×C61C2312yes—
101C4×C8×C61C2311yescharacteristic
102C23×D61—41no—
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C2×C22⋊D61—41no—
107C22×Dic61—41no—
108#11758222—44no—
109C2×D244—22no—
110C2×D244—22no—
111C2×D244—22no—
112C2×D244—21no—
113C2×D244—21no—
114C2×D244—21no—
115C2×D244—44no—
116C2×D244—44no—
117C2×D244—41no—
118C2×D244—41no—
119C2×D244—41no—
120C2×D244—82no—
121C2×C4×D61—41no—
122C2×Dic122—22no—
123C2×Dic122—21no—
124C2×Dic122—21no—
125C2×Dic122—21no—
126C2×Dic122—41no—
127C2×Dic122—82no—
128M16×C61—24no—
129C2×C8×C61—42no—
130C2×C8×C61C2×D412yes—
131C2×C8×C61C2×D412yes—
132C2×C8×C61C2×D414yes—
133C2×C8×C61C2×D414yes—
134C42×C61C2×D411yescharacteristic
135C42×C61C2×D411yescharacteristic
136C42×C61C2×D411yescharacteristic
137C42×C61C2411yescharacteristic
138C22⋊Dic61—41no—
139#11758198—41no—
140#11758198—41no—
141C4⋊Dic61—22no—
142C4⋊Dic61—21no—
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145C4⋊Dic61—22no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149C4⋊Dic61—41no—
150C4⋊Dic61—41no—
151#11758196—41no—
152#11758196—41no—
153C4×Dic61—41no—
154Dic244—84no—
155D488—84no—
156#11758190—44no—
157C23×C61#12711yescharacteristic
158C22×D61—41no—
159C22×D61—42no—
160C22×D61—82no—
161C22×D61—82no—
162C22×D61—82no—
163C2×C4×C61—22no—
164C2×C4×C61—22no—
165C2×C4×C61#12711yescharacteristic
166C2×C4×C61#12711yescharacteristic
167C2×C4×C61#12711yescharacteristic
168C2×C4×C61#13514yes—
169C2×C4×C61C22×D411yescharacteristic
170C2×C4×C61C22×D411yescharacteristic
171C2×C4×C61C22×D411yescharacteristic
172C22⋊D61—82no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175C2×Dic61—42no—
176C2×Dic61—41no—
177C2×Dic61—82no—
178D244—44no—
179D244—42no—
180D244—42no—
181D244—44no—
182D244—42no—
183D244—44no—
184D244—82no—
185D244—82no—
186D244—82no—
187C4×D61—82no—
188Dic122—42no—
189Dic122—42no—
190Dic122—42no—
191Dic122—44no—
192Dic122—82no—
193C8×C61—24no—
194C8×C61—22no—
195C8×C61—22no—
196C8×C61—24no—
197#19627—611nomaximal, Sylow
198C22×C61—21no—
199C22×C61—21no—
200C22×C61#49512yes—
201C22×C61C2×#12711yescharacteristic
202C2×D61—81no—
203C2×D61—82no—
204C2×D61—82no—
205C2×D61—81no—
206C2×D61—82no—
207C2×D61—81no—
208C2×D61—164no—
209C4×C61—21no—
210C4×C61—21no—
211C4×C61#49512yes—
212C4×C61C2×#13512yes—
213C4×C61#57512yes—
214C4×C61#57612yes—
215Dic61—81no—
216Dic61—81no—
217Dic61—82no—
218Dic61—81no—
219#3209—611no—
220#3205—611no—
221#3167—611no—
222#3141—611no—
223#3138—611no—
224C2×#486—611no—
225#1720—612no—
226#1717—612no—
227#1711—612no—
228#1709—612no—
229C8×M16D6111yescharacteristic, complemented
230C2×C61—22no—
231C2×C61C2×#49511yescharacteristic
232C2×C61#315311yescharacteristic
233C2×C61#315511yescharacteristic
234D61—161no—
235D61—162no—
236#533—611no—
237#532—611no—
238C2×C4⋊Q8—611no—
239C2×#127—611no—
240#498—611no—
241#494—612no—
242#493—612no—
243#492—611no—
244#490—1222no—
245#489—1222no—
246#487—612no—
247#487—611no—
248#487—612no—
249#486—612no—
250#486—612no—
251#486—612no—
252#486—612no—
253#486—612no—
254#486—611no—
255#486—612no—
256#486—1222no—
257#465—1222no—
258#462—1222no—
259#459—1222no—
260#451—1222no—
261#447—1222no—
262C4×SD16—1222no—
263#431C2×D6111yescharacteristic
264#418—1222no—
265#417—1222no—
266C2×#102—1222no—
267C4×M16C2×D6111yescharacteristic
268C2×C4×C8C2×D6111yescharacteristic
269#322C2×D6112yescomplemented
270C82C2×D6112yescomplemented
271C61#1962711yescharacteristic, complemented, Sylow
272C2×#41—1221no—
273C22×Q8—1221no—
274C22×D4—1221no—
275C22×D4—1221no—
276C22×D4—1221no—
277C2×Q16—1224no—
278C2×SD16—1222no—
279C2×D8—1224no—
280C2×M16D24412yes—
281C22×C8D24412yes—
282C4⋊Q8—612no—
283C4⋊Q8—611no—
284C4⋊Q8—611no—
285C4⋊Q8—611no—
286C4⋊Q8—1221no—
287C4⋊Q8—1221no—
288C4⋊Q8—1221no—
289#127—612no—
290#127—611no—
291#127—1221no—
292#127—1221no—
293#127—1221no—
294#125—1221no—
295#124—1221no—
296C22⋊Q8—1221no—
297C22⋊Q8—1221no—
298#121—1221no—
299#121—1221no—
300C4×Q8—1221no—
301C4×Q8—1221no—
302C4×D4—1221no—
303C4×D4—1221no—
304#117—1221no—
305C2×C4⋊C4—1221no—
306C2×C4⋊C4—1221no—
307C2×C42C22×D6111yescharacteristic
308#106—1222no—
309C4⋊C8D24412yes—
310#103—1224no—
311#103—1222no—
312#103—1224no—
313#102—1224no—
314#102—1224no—
315#102—1224no—
316#102—1224no—
317#102—1224no—
318#102—1222no—
319#102—1224no—
320C22⋊C8D24412yes—
321#97D24412yes—
322C4×C8D24412yes—
323C4×C8C22×D6111yescharacteristic
324C4×C8C22×D6112yes—
325C4×C8C22×D6112yes—
326C4×C8C22×D6111yescharacteristic
327C24—2441no—
328#41—2444no—
329C2×Q8—1221no—
330C2×Q8—1221no—
331C2×Q8—1222no—
332C2×Q8—1221no—
333C2×Q8—4882no—
334C2×Q8—2441no—
335C2×D4—1221no—
336C2×D4—1222no—
337C2×D4—1222no—
338C2×D4—1221no—
339C2×D4—1222no—
340C2×D4—1221no—
341C2×D4—4882no—
342C2×D4—2444no—
343C2×D4—2441no—
344C2×D4—2441no—
345C2×D4—2441no—
346C2×D4—2441no—
347C2×D4—2444no—
348C22×C4—2441no—
349C22×C4—2441no—
350C22×C4C2×D24411yescharacteristic
351C22×C4C2×D24411yescharacteristic
352C22×C4C2×D24411yescharacteristic
353Q16—4884no—
354SD16—2444no—
355D8—4884no—
356M16—24no—
357C2×C8—42no—
358C2×C8C2×D24412yes—
359C2×C8C2×D24412yes—
360C2×C8C2×D24414yes—
361C2×C8C2×D24414yes—
362C4⋊C4—1222no—
363C4⋊C4—1221no—
364C4⋊C4—1221no—
365C4⋊C4—1221no—
366C4⋊C4—1222no—
367C4⋊C4—1221no—
368C4⋊C4—1221no—
369C4⋊C4—1221no—
370C4⋊C4—2441no—
371C4⋊C4—2441no—
372C4⋊C4—2441no—
373C4⋊C4—2441no—
374C22⋊C4—2441no—
375C22⋊C4—2441no—
376C22⋊C4—2441no—
377C42—2441no—
378C42C2×D24411yescharacteristic
379C42C2×D24411yescharacteristic
380C42C2×D24411yescharacteristic
381C42C23×D6111yescharacteristic
382C23—4882no—
383C23—4882no—
384C23—4882no—
385C23—2441no—
386C23—2442no—
387C23#1451977411yescharacteristic
388Q8—4882no—
389Q8—2442no—
390Q8—2442no—
391Q8—2442no—
392Q8—2444no—
393D4—4882no—
394D4—4882no—
395D4—4882no—
396D4—4882no—
397D4—2444no—
398D4—2442no—
399D4—2442no—
400D4—2444no—
401D4—2442no—
402D4—2444no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—4882no—
406C2×C4—4882no—
407C2×C4—2441no—
408C2×C4—2441no—
409C2×C4—2441no—
410C2×C4—2442no—
411C2×C4#1451977411yescharacteristic
412C2×C4#1451977411yescharacteristic
413C2×C4#1451977411yescharacteristic
414C2×C4#1451979914yes—
415C2×C4C22×D24411yescharacteristic
416C2×C4C22×D24411yescharacteristic
417C2×C4C22×D24411yescharacteristic
418C8—24no—
419C8—22no—
420C8—22no—
421C8—24no—
422C22—21no—
423C22—21no—
424C22—9764no—
425C22—4881no—
426C22—4882no—
427C22—4882no—
428C22—4881no—
429C22—4882no—
430C22—4881no—
431C22#1684014612yes—
432C22C2×#1451977411yescharacteristic
433C4—21no—
434C4—21no—
435C4—4881no—
436C4—4881no—
437C4—4882no—
438C4—4881no—
439C4#1684014612yes—
440C4C2×#1451979912yes—
441C4#1684056212yes—
442C4#1684056712yes—
443C2—22no—
444C2—9762no—
445C2—9761no—
446C2C2×#1684014611yescharacteristic
447C2#2145628511yescharacteristic
448C2#2145630511yescharacteristic
subgroupG/H
centerC22C2×#14519774
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC8×M16×C61C2
socleC22×C61C2×#127
prime radical#21456739C2
hypercenterC8×M16D61
hypocenterC61#19627
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×M16

Complements

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

NH ≅ G/Ntype
1. #21456769C2semidirect
3. #21456378C2semidirect
4. #21456213C2semidirect
5. #21456194C2semidirect
6. C2×#16840109C2semidirect
7. #21445479C2semidirect
8. #21445458C2semidirect
9. #21445431C2semidirect
10. #21445425C2semidirect
11. C8×M16×C61C2semidirect
16. #16840160C22semidirect
17. #16840142C22semidirect
19. #16840135C22semidirect
22. #16840113C22semidirect
23. #16840113C22semidirect
24. #16840113C22semidirect
31. #16840109C22semidirect
32. #16840109C22semidirect
45. C61×#322C22semidirect
46. C82×C61C22semidirect
229. C8×M16D61semidirect
269. #322C2×D61semidirect
270. C82C2×D61semidirect
271. C61#19627semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214567692
#214567392
#214563782
#214562132
#214561942
C2×#168401092
#214454792
#214454582
#214454312
#214454252
C8×M16×C612
#1962761