Finite Group Databasebeta

← #29667878

Subgroups — #29667878

Order 15616.

848 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456762C211yesmaximal, characteristic, complemented
2#21456746C211yesmaximal, characteristic, complemented
3#21456378C211yesmaximal, characteristic, complemented
4#21456365C211yesmaximal, characteristic, complemented
5#21456271C211yesmaximal, characteristic
6C2×#16840109C211yesmaximal, characteristic, complemented
7#21445447C212yesmaximal, complemented
8#21445441C212yesmaximal, complemented
9#21445420C212yesmaximal, complemented
10#21445411C212yesmaximal, complemented
11C8×M16×C61C211yesmaximal, characteristic, complemented
12C2×#14519805—22no—
13#16840319C2211yescharacteristic
14#16840312C2211yescharacteristic
15C2×#14519777C2211yescharacteristic
16C2×#14519774C2211yescharacteristic
17#16840171—22no—
18#16840167C2212yescomplemented
19#16840160C2212yescomplemented
20#16840160C2211yescharacteristic, complemented
21#16840151—22no—
22#16840139C2212yes—
23#16840135C2212yes—
24#16840135C2212yes—
25#16840135C2212yescomplemented
26#16840135C2211yescharacteristic, complemented
27#16840113C2211yescharacteristic, complemented
28#16840113C2212yescomplemented
29#16840109—22no—
30#16840109C2212yes—
31#16840109C2212yes—
32#16840109C2212yes—
33#16840109C2211yescharacteristic, complemented
34#16839991—22no—
35#16839968—22no—
36#16839963—22no—
37#16839915—22no—
38#16839903—22no—
39C4×#11758190—22no—
40#16839804—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—
52#14519805—48no—
53C2×#11758190—22no—
54C2×#11758190—24no—
55C2×#11758190—24no—
56C2×#11758190—24no—
57C2×D488—24no—
58#14519777—21no—
59#14519777—21no—
60#14519777—21no—
61#14519777C2312yes—
62#14519777C2311yescharacteristic
63#14519777C2311yescharacteristic
64#14519777C2311yescharacteristic
65#14519774—21no—
66#14519774—21no—
67#14519774—21no—
68#14519774C2312yes—
69#14519774C2311yescharacteristic
70#14519769—21no—
71#14519765—21no—
72#14519756—21no—
73#14519756—21no—
74#14519751—21no—
75#14519751—21no—
76C4×Dic122—21no—
77C4×Dic122—21no—
78C4×D244—21no—
79C4×D244—21no—
80#14519736—21no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61—21no—
83#14519711—24no—
84#14519709—22no—
85#14519709—24no—
86#14519702—24no—
87#14519702—22no—
88#14519699—24no—
89#14519699—22no—
90#14519699—24no—
91#14519699—24no—
92C2×M16×C61D412yes—
93C22×C8×C61D412yes—
94C2×C42×C61C2311yescharacteristic
95C4⋊C8×C61D412yes—
96C22⋊C8×C61D412yes—
97#97×C61D412yes—
98C4×C8×C61D412yes—
99C4×C8×C61C2311yescharacteristic
100C4×C8×C61C2312yes—
101C4×C8×C61C2312yes—
102C4×C8×C61C2311yescharacteristic
103C23×D61—41no—
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C2×C22⋊D61—41no—
108C22×Dic61—41no—
109#11758222—44no—
110C2×D244—22no—
111C2×D244—22no—
112C2×D244—22no—
113C2×D244—21no—
114C2×D244—21no—
115C2×D244—21no—
116C2×D244—44no—
117C2×D244—44no—
118C2×D244—41no—
119C2×D244—41no—
120C2×D244—41no—
121C2×D244—82no—
122C2×C4×D61—41no—
123C2×Dic122—22no—
124C2×Dic122—21no—
125C2×Dic122—21no—
126C2×Dic122—21no—
127C2×Dic122—41no—
128C2×Dic122—82no—
129M16×C61SD1618yescomplemented
130C2×C8×C61—42no—
131C2×C8×C61C2×D412yes—
132C2×C8×C61C2×D412yes—
133C2×C8×C61C2×D414yes—
134C2×C8×C61C2×D414yes—
135C42×C61C2×D411yescharacteristic
136C42×C61C2×D411yescharacteristic
137C42×C61C2×D411yescharacteristic
138C42×C61C2411yescharacteristic
139C22⋊Dic61—41no—
140#11758198—41no—
141#11758198—41no—
142C4⋊Dic61—22no—
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145C4⋊Dic61—21no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61—22no—
149C4⋊Dic61—21no—
150C4⋊Dic61—41no—
151C4⋊Dic61—41no—
152#11758196—41no—
153#11758196—41no—
154C4×Dic61—41no—
155D488—48no—
156#11758190—44no—
157#11758190—48no—
158#11758190—84no—
159#11758190—84no—
160C23×C61#12711yescharacteristic
161C22×D61—41no—
162C22×D61—42no—
163C22×D61—82no—
164C22×D61—82no—
165C22×D61—82no—
166C2×C4×C61—22no—
167C2×C4×C61—22no—
168C2×C4×C61#12711yescharacteristic
169C2×C4×C61#12711yescharacteristic
170C2×C4×C61#12711yescharacteristic
171C2×C4×C61C2×SD1614yes—
172C2×C4×C61C22×D411yescharacteristic
173C2×C4×C61C22×D411yescharacteristic
174C2×C4×C61C22×D411yescharacteristic
175C22⋊D61—82no—
176C2×Dic61—41no—
177C2×Dic61—41no—
178C2×Dic61—42no—
179C2×Dic61—41no—
180C2×Dic61—82no—
181D244—44no—
182D244—42no—
183D244—42no—
184D244—44no—
185D244—42no—
186D244—44no—
187D244—82no—
188D244—82no—
189D244—82no—
190C4×D61—82no—
191Dic122—42no—
192Dic122—42no—
193Dic122—42no—
194Dic122—44no—
195Dic122—82no—
196C8×C61—22no—
197C8×C61—22no—
198C8×C61—24no—
199C8×C61C2×SD1618yes—
200#19623—611nomaximal, Sylow
201C22×C61—21no—
202C22×C61—21no—
203C22×C61#49212yes—
204C22×C61C2×#12711yescharacteristic
205C2×D61—81no—
206C2×D61—82no—
207C2×D61—82no—
208C2×D61—81no—
209C2×D61—82no—
210C2×D61—81no—
211C2×D61—164no—
212C4×C61—21no—
213C4×C61—21no—
214C4×C61#49212yes—
215C4×C61C22×SD1612yes—
216C4×C61#57512yes—
217C4×C61#57612yes—
218Dic61—81no—
219Dic61—81no—
220Dic61—82no—
221Dic61—81no—
222#3208—611no—
223#3206—611no—
224#3167—611no—
225#3165—611no—
226#3151—611no—
227C2×#486—611no—
228#1715—612no—
229#1714—612no—
230#1708—612no—
231#1706—612no—
232C8×M16D6111yescharacteristic, complemented
233C2×C61—22no—
234C2×C61C2×#49211yescharacteristic
235C2×C61#315311yescharacteristic
236C2×C61#315511yescharacteristic
237D61—161no—
238D61—162no—
239C2×AGL(1,ℤ/8)—1222no—
240#533—611no—
241#532—611no—
242C2×C4⋊Q8—611no—
243C2×#127—611no—
244#501—1222no—
245#500—612no—
246#498—612no—
247#498—611no—
248#496—1222no—
249#493—612no—
250#492—612no—
251#492—611no—
252#492—612no—
253#492—612no—
254#487—611no—
255#487—612no—
256#486—612no—
257#486—612no—
258#486—611no—
259#486—612no—
260#486—1222no—
261#465—1222no—
262#461—1222no—
263#460—1222no—
264#450—1222no—
265#448—1222no—
266C4×SD16—1222no—
267#431C2×D6111yescharacteristic
268#428—1222no—
269C2×#102—1222no—
270C4×M16C2×D6111yescharacteristic
271C2×C4×C8C2×D6111yescharacteristic
272#322C2×D6112yescomplemented
273C82C2×D6112yescomplemented
274C61#1962311yescharacteristic, complemented, Sylow
275C2×#41—1221no—
276C22×Q8—1221no—
277C22×D4—1221no—
278C22×D4—1221no—
279C22×D4—1221no—
280AGL(1,ℤ/8)—2448no—
281C2×SD16—1224no—
282C2×SD16—1224no—
283C2×SD16—1224no—
284C2×SD16—1222no—
285C2×D8—1224no—
286C2×M16D24412yes—
287C22×C8D24412yes—
288C4⋊Q8—612no—
289C4⋊Q8—611no—
290C4⋊Q8—611no—
291C4⋊Q8—611no—
292C4⋊Q8—1221no—
293C4⋊Q8—1221no—
294C4⋊Q8—1221no—
295#127—612no—
296#127—611no—
297#127—1221no—
298#127—1221no—
299#127—1221no—
300#125—1221no—
301#124—1221no—
302C22⋊Q8—1221no—
303C22⋊Q8—1221no—
304#121—1221no—
305#121—1221no—
306C4×Q8—1221no—
307C4×Q8—1221no—
308C4×D4—1221no—
309C4×D4—1221no—
310#117—1221no—
311C2×C4⋊C4—1221no—
312C2×C4⋊C4—1221no—
313C2×C42C22×D6111yescharacteristic
314#107—1224no—
315#106—1224no—
316#106—1222no—
317C4⋊C8D24412yes—
318#103—1222no—
319#103—1224no—
320#102—1224no—
321#102—1224no—
322#102—1222no—
323#102—1224no—
324C22⋊C8D24412yes—
325#97D24412yes—
326C4×C8D24412yes—
327C4×C8C22×D6111yescharacteristic
328C4×C8C22×D6112yes—
329C4×C8C22×D6112yes—
330C4×C8C22×D6111yescharacteristic
331C24—2441no—
332#41—2444no—
333C2×Q8—1221no—
334C2×Q8—1222no—
335C2×Q8—1221no—
336C2×Q8—1221no—
337C2×Q8—4882no—
338C2×Q8—2441no—
339C2×D4—1222no—
340C2×D4—1221no—
341C2×D4—1222no—
342C2×D4—1221no—
343C2×D4—1221no—
344C2×D4—1222no—
345C2×D4—4882no—
346C2×D4—2444no—
347C2×D4—2441no—
348C2×D4—2441no—
349C2×D4—2441no—
350C2×D4—2441no—
351C2×D4—2444no—
352C22×C4—2441no—
353C22×C4—2441no—
354C22×C4C2×D24411yescharacteristic
355C22×C4C2×D24411yescharacteristic
356C22×C4C2×D24411yescharacteristic
357SD16—4884no—
358SD16—4884no—
359SD16—2444no—
360SD16—2448no—
361D8—2448no—
362M16#1175819018yescomplemented
363C2×C8—42no—
364C2×C8C2×D24412yes—
365C2×C8C2×D24412yes—
366C2×C8C2×D24414yes—
367C2×C8C2×D24414yes—
368C4⋊C4—1222no—
369C4⋊C4—1221no—
370C4⋊C4—1221no—
371C4⋊C4—1221no—
372C4⋊C4—1221no—
373C4⋊C4—1221no—
374C4⋊C4—1222no—
375C4⋊C4—1221no—
376C4⋊C4—2441no—
377C4⋊C4—2441no—
378C4⋊C4—2441no—
379C4⋊C4—2441no—
380C22⋊C4—2441no—
381C22⋊C4—2441no—
382C22⋊C4—2441no—
383C42—2441no—
384C42C2×D24411yescharacteristic
385C42C2×D24411yescharacteristic
386C42C2×D24411yescharacteristic
387C42C23×D6111yescharacteristic
388C23—4882no—
389C23—4882no—
390C23—4882no—
391C23—2441no—
392C23—2442no—
393C23#1451977411yescharacteristic
394Q8—4882no—
395Q8—2442no—
396Q8—2442no—
397Q8—2442no—
398Q8—2444no—
399D4—4882no—
400D4—4882no—
401D4—4882no—
402D4—4882no—
403D4—2442no—
404D4—2444no—
405D4—2442no—
406D4—2444no—
407D4—2442no—
408D4—2444no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—4882no—
412C2×C4—4882no—
413C2×C4—2441no—
414C2×C4—2441no—
415C2×C4—2441no—
416C2×C4—2442no—
417C2×C4#1451977411yescharacteristic
418C2×C4#1451977411yescharacteristic
419C2×C4#1451977411yescharacteristic
420C2×C4C2×#1175819014yes—
421C2×C4C22×D24411yescharacteristic
422C2×C4C22×D24411yescharacteristic
423C2×C4C22×D24411yescharacteristic
424C8—22no—
425C8—22no—
426C8—24no—
427C8C2×#1175819018yes—
428C22—21no—
429C22—21no—
430C22—9764no—
431C22—4881no—
432C22—4882no—
433C22—4882no—
434C22—4881no—
435C22—4882no—
436C22—4881no—
437C22#1684013512yes—
438C22C2×#1451977411yescharacteristic
439C4—21no—
440C4—21no—
441C4—4881no—
442C4—4881no—
443C4—4882no—
444C4—4881no—
445C4#1684013512yes—
446C4C22×#1175819012yes—
447C4#1684056212yes—
448C4#1684056712yes—
449C2—22no—
450C2—9762no—
451C2—9761no—
452C2C2×#1684013511yescharacteristic
453C2#2145628511yescharacteristic
454C2#2145630511yescharacteristic
subgroupG/H
centerC22C2×#14519774
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC8×M16×C61C2
socleC22×C61C2×#127
prime radical#21456271C2
hypercenterC8×M16D61
hypocenterC61#19623
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. #21456762C2semidirect
2. #21456746C2semidirect
3. #21456378C2semidirect
4. #21456365C2semidirect
6. C2×#16840109C2semidirect
7. #21445447C2semidirect
8. #21445441C2semidirect
9. #21445420C2semidirect
10. #21445411C2semidirect
11. C8×M16×C61C2semidirect
18. #16840167C22semidirect
19. #16840160C22semidirect
20. #16840160C22semidirect
25. #16840135C22semidirect
26. #16840135C22semidirect
27. #16840113C22semidirect
28. #16840113C22semidirect
33. #16840109C22semidirect
45. C61×#322C22semidirect
46. C82×C61C22semidirect
129. M16×C61SD16semidirect
232. C8×M16D61semidirect
272. #322C2×D61semidirect
273. C82C2×D61semidirect
274. C61#19623semidirect
362. M16#11758190semidirect

Maximal subgroups

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

subgroupindex
#214567622
#214567462
#214563782
#214563652
#214562712
C2×#168401092
#214454472
#214454412
#214454202
#214454112
C8×M16×C612
#1962361