Finite Group Databasebeta

← #29667900

Subgroups — #29667900

Order 15616.

836 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456972C211yesmaximal, characteristic, complemented
2#21456739C211yesmaximal, characteristic
3#21456378C211yesmaximal, characteristic, complemented
4C2×#16840135C211yesmaximal, characteristic, complemented
5#21456213C211yesmaximal, characteristic, complemented
6#21456194C211yesmaximal, characteristic, complemented
7#21445472C212yesmaximal, complemented
8#21445458C212yesmaximal, complemented
9#21445420C212yesmaximal, complemented
10#21445411C212yesmaximal, complemented
11C8×M16×C61C211yesmaximal, characteristic, complemented
12C22×#11758190—22no—
13#16840319C2211yescharacteristic
14#16840312C2211yescharacteristic
15C2×#14519777C2211yescharacteristic
16C2×#14519774C2211yescharacteristic
17#16840167C2212yescomplemented
18#16840160C2211yescharacteristic, complemented
19#16840139C2212yes—
20#16840135—22no—
21#16840135C2212yes—
22#16840135C2212yes—
23#16840135C2211yescharacteristic, complemented
24#16840127—22no—
25#16840122—22no—
26#16840113C2212yescomplemented
27#16840113C2212yescomplemented
28#16840113C2211yescharacteristic, complemented
29#16840109C2212yes—
30#16840109C2212yes—
31#16840109C2212yes—
32#16840109C2212yescomplemented
33#16840109C2211yescharacteristic, complemented
34#16840098—22no—
35#16840058—22no—
36#16839991—22no—
37#16839958—22no—
38#16839898—22no—
39C4×#11758190—22no—
40#16839768—22no—
41#16839764—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×#11758190—22no—
53C2×#11758190—24no—
54C2×#11758190—24no—
55C2×#11758190—48no—
56C2×D488—24no—
57#14519777—21no—
58#14519777—21no—
59#14519777—21no—
60#14519777C2312yes—
61#14519777C2311yescharacteristic
62#14519777C2311yescharacteristic
63#14519777C2311yescharacteristic
64#14519774—21no—
65#14519774—21no—
66#14519774—21no—
67#14519774C2312yes—
68#14519774C2311yescharacteristic
69#14519769—21no—
70#14519765—21no—
71#14519756—21no—
72#14519756—21no—
73#14519751—21no—
74#14519751—21no—
75C4×Dic122—21no—
76C4×Dic122—21no—
77C4×D244—21no—
78C4×D244—21no—
79#14519736—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61—21no—
82#14519711—24no—
83#14519709—22no—
84#14519702—22no—
85#14519702—24no—
86#14519702—24no—
87#14519699—24no—
88#14519699—22no—
89#14519699—24no—
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—44no—
128C2×Dic122—41no—
129M16×C61—24no—
130C2×C8×C61—24no—
131C2×C8×C61C2×D412yes—
132C2×C8×C61C2×D414yes—
133C2×C8×C61C2×D414yes—
134C2×C8×C61C2×D412yes—
135C42×C61C2×D411yescharacteristic
136C42×C61C2×D411yescharacteristic
137C42×C61C2×D411yescharacteristic
138C42×C61C2411yescharacteristic
139C22⋊Dic61—41no—
140#11758198—41no—
141#11758198—41no—
142C4⋊Dic61—21no—
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145C4⋊Dic61—22no—
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—84no—
156#11758190—48no—
157#11758190—44no—
158#11758190—48no—
159C23×C61#12711yescharacteristic
160C22×D61—41no—
161C22×D61—42no—
162C22×D61—82no—
163C22×D61—82no—
164C22×D61—82no—
165C2×C4×C61—22no—
166C2×C4×C61—22no—
167C2×C4×C61#12711yescharacteristic
168C2×C4×C61#12711yescharacteristic
169C2×C4×C61#12711yescharacteristic
170C2×C4×C61#13514yes—
171C2×C4×C61C22×D411yescharacteristic
172C2×C4×C61C22×D411yescharacteristic
173C2×C4×C61C22×D411yescharacteristic
174C22⋊D61—82no—
175C2×Dic61—41no—
176C2×Dic61—41no—
177C2×Dic61—42no—
178C2×Dic61—41no—
179C2×Dic61—82no—
180D244—44no—
181D244—42no—
182D244—44no—
183D244—42no—
184D244—42no—
185D244—44no—
186D244—82no—
187D244—82no—
188D244—82no—
189C4×D61—82no—
190Dic122—42no—
191Dic122—44no—
192Dic122—42no—
193Dic122—42no—
194Dic122—82no—
195C8×C61—24no—
196C8×C61—22no—
197C8×C61—24no—
198C8×C61AGL(1,ℤ/8)14yes—
199#19625—611nomaximal, Sylow
200C22×C61—21no—
201C22×C61—21no—
202C22×C61#49512yes—
203C22×C61C2×#12711yescharacteristic
204C2×D61—81no—
205C2×D61—82no—
206C2×D61—82no—
207C2×D61—81no—
208C2×D61—82no—
209C2×D61—81no—
210C2×D61—164no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61#49512yes—
214C4×C61C2×#13512yes—
215C4×C61C2×AGL(1,ℤ/8)12yes—
216C4×C61#57712yes—
217Dic61—81no—
218Dic61—82no—
219Dic61—81no—
220Dic61—81no—
221#3230—611no—
222#3205—611no—
223#3167—611no—
224C2×#492—611no—
225#3141—611no—
226#3138—611no—
227#1719—612no—
228#1717—612no—
229#1708—612no—
230#1706—612no—
231C8×M16D6111yescharacteristic, complemented
232C2×C61—22no—
233C2×C61C2×#49511yescharacteristic
234C2×C61#315111yescharacteristic
235C2×C61#315411yescharacteristic
236D61—161no—
237D61—162no—
238C22×SD16—1222no—
239#533—611no—
240#532—611no—
241C2×C4⋊Q8—611no—
242C2×#127—611no—
243#500—612no—
244#498—611no—
245#493—612no—
246#492—612no—
247#492—612no—
248#492—611no—
249#492—1222no—
250#490—1222no—
251#489—1222no—
252#487—612no—
253#487—611no—
254#487—612no—
255#486—612no—
256#486—612no—
257#486—612no—
258#486—611no—
259#486—612no—
260#484—1222no—
261#477—1222no—
262#465—1222no—
263#459—1222no—
264#447—1222no—
265C4×SD16—1222no—
266#431C2×D6111yescharacteristic
267#418—1222no—
268#417—1222no—
269C4×M16C2×D6111yescharacteristic
270C2×C4×C8C2×D6111yescharacteristic
271#322C2×D6112yescomplemented
272C82C2×D6112yescomplemented
273C61#1962511yescharacteristic, complemented, Sylow
274C2×#41—1221no—
275C22×Q8—1221no—
276C22×D4—1221no—
277C22×D4—1221no—
278C22×D4—1221no—
279C2×SD16—1224no—
280C2×SD16—1224no—
281C2×SD16—1222no—
282C2×SD16—2448no—
283C2×D8—1224no—
284C2×M16D24412yes—
285C22×C8D24412yes—
286C4⋊Q8—612no—
287C4⋊Q8—611no—
288C4⋊Q8—611no—
289C4⋊Q8—611no—
290C4⋊Q8—1221no—
291C4⋊Q8—1221no—
292C4⋊Q8—1221no—
293#127—612no—
294#127—611no—
295#127—1221no—
296#127—1221no—
297#127—1221no—
298#125—1221no—
299#124—1221no—
300C22⋊Q8—1221no—
301C22⋊Q8—1221no—
302#121—1221no—
303#121—1221no—
304C4×Q8—1221no—
305C4×Q8—1221no—
306C4×D4—1221no—
307C4×D4—1221no—
308#117—1221no—
309C2×C4⋊C4—1221no—
310C2×C4⋊C4—1221no—
311C2×C42C22×D6111yescharacteristic
312#107—1224no—
313#106—1222no—
314C4⋊C8D24412yes—
315#103—1224no—
316#103—1222no—
317#103—1224no—
318#102—1224no—
319#102—1224no—
320#102—1224no—
321#102—1222no—
322#102—1224no—
323C22⋊C8D24412yes—
324#97D24412yes—
325C4×C8D24412yes—
326C4×C8C22×D6111yescharacteristic
327C4×C8C22×D6112yes—
328C4×C8C22×D6112yes—
329C4×C8C22×D6111yescharacteristic
330C24—2441no—
331#41—2444no—
332C2×Q8—1221no—
333C2×Q8—1221no—
334C2×Q8—1221no—
335C2×Q8—1222no—
336C2×Q8—2444no—
337C2×Q8—2441no—
338C2×D4—1221no—
339C2×D4—1222no—
340C2×D4—1221no—
341C2×D4—1222no—
342C2×D4—1221no—
343C2×D4—1222no—
344C2×D4—4882no—
345C2×D4—2444no—
346C2×D4—2441no—
347C2×D4—2441no—
348C2×D4—2441no—
349C2×D4—2441no—
350C2×D4—2444no—
351C22×C4—2441no—
352C22×C4—2441no—
353C22×C4C2×D24411yescharacteristic
354C22×C4C2×D24411yescharacteristic
355C22×C4C2×D24411yescharacteristic
356SD16—2448no—
357SD16—2444no—
358SD16—2448no—
359D8—4884no—
360M16—24no—
361C2×C8—24no—
362C2×C8C2×D24412yes—
363C2×C8C2×D24414yes—
364C2×C8C2×D24414yes—
365C2×C8C2×D24412yes—
366C4⋊C4—1221no—
367C4⋊C4—1221no—
368C4⋊C4—1221no—
369C4⋊C4—1222no—
370C4⋊C4—1221no—
371C4⋊C4—1221no—
372C4⋊C4—1222no—
373C4⋊C4—1221no—
374C4⋊C4—2441no—
375C4⋊C4—2441no—
376C4⋊C4—2441no—
377C4⋊C4—2441no—
378C22⋊C4—2441no—
379C22⋊C4—2441no—
380C22⋊C4—2441no—
381C42—2441no—
382C42C2×D24411yescharacteristic
383C42C2×D24411yescharacteristic
384C42C2×D24411yescharacteristic
385C42C23×D6111yescharacteristic
386C23—4882no—
387C23—4882no—
388C23—4882no—
389C23—2441no—
390C23—2442no—
391C23#1451977411yescharacteristic
392Q8—4882no—
393Q8—2442no—
394Q8—2444no—
395Q8—2442no—
396Q8—2442no—
397D4—4882no—
398D4—4882no—
399D4—4882no—
400D4—4882no—
401D4—2444no—
402D4—2442no—
403D4—2444no—
404D4—2442no—
405D4—2442no—
406D4—2444no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—4882no—
410C2×C4—4882no—
411C2×C4—2441no—
412C2×C4—2441no—
413C2×C4—2441no—
414C2×C4—2442no—
415C2×C4#1451977411yescharacteristic
416C2×C4#1451977411yescharacteristic
417C2×C4#1451977411yescharacteristic
418C2×C4#1451979914yes—
419C2×C4C22×D24411yescharacteristic
420C2×C4C22×D24411yescharacteristic
421C2×C4C22×D24411yescharacteristic
422C8—24no—
423C8—22no—
424C8—24no—
425C8#1451980514yes—
426C22—21no—
427C22—21no—
428C22—9764no—
429C22—4881no—
430C22—4882no—
431C22—4882no—
432C22—4881no—
433C22—4882no—
434C22—4881no—
435C22#1684014612yes—
436C22C2×#1451977411yescharacteristic
437C4—21no—
438C4—21no—
439C4—4881no—
440C4—4882no—
441C4—4881no—
442C4—4881no—
443C4#1684014612yes—
444C4C2×#1451979912yes—
445C4C2×#1451980512yes—
446C4#1684057412yes—
447C2—22no—
448C2—9762no—
449C2—9761no—
450C2C2×#1684014611yescharacteristic
451C2#2145627111yescharacteristic
452C2#2145629511yescharacteristic
subgroupG/H
centerC22C2×#14519774
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC8×M16×C61C2
socleC22×C61C2×#127
prime radical#21456739C2
hypercenterC8×M16D61
hypocenterC61#19625
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. #21456972C2semidirect
3. #21456378C2semidirect
4. C2×#16840135C2semidirect
5. #21456213C2semidirect
6. #21456194C2semidirect
7. #21445472C2semidirect
8. #21445458C2semidirect
9. #21445420C2semidirect
10. #21445411C2semidirect
11. C8×M16×C61C2semidirect
17. #16840167C22semidirect
18. #16840160C22semidirect
23. #16840135C22semidirect
26. #16840113C22semidirect
27. #16840113C22semidirect
28. #16840113C22semidirect
32. #16840109C22semidirect
33. #16840109C22semidirect
45. C61×#322C22semidirect
46. C82×C61C22semidirect
231. C8×M16D61semidirect
271. #322C2×D61semidirect
272. C82C2×D61semidirect
273. C61#19625semidirect

Maximal subgroups

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

subgroupindex
#214569722
#214567392
#214563782
C2×#168401352
#214562132
#214561942
#214454722
#214454582
#214454202
#214454112
C8×M16×C612
#1962561