Finite Group Databasebeta

← #29657597

Subgroups — #29657597

Order 15616.

980 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457131C211yesmaximal, characteristic, complemented
2#21456379C211yesmaximal, characteristic, complemented
3#21455283C211yesmaximal, characteristic
4#21445082C212yesmaximal, complemented
5#21445026C212yesmaximal, complemented
6#21444922C212yesmaximal, complemented
7#21444891C212yesmaximal, complemented
8#21444509C211yesmaximal, characteristic, complemented
9#21444274C211yesmaximal, characteristic, complemented
10C2×D4⋊#92418C211yesmaximal, characteristic, complemented
11C61×#3501C211yesmaximal, characteristic, complemented
12C2×#14519803—21no—
13#16840312C2211yescharacteristic
14#16840158C2211yescharacteristic, complemented
15#16840136C2211yescharacteristic, complemented
16#16840112C2211yescharacteristic, complemented
17#16840110C2211yescharacteristic, complemented
18#16840049—21no—
19#16840047C2212yescomplemented
20#16839993—22no—
21#16839993—22no—
22#16839977—22no—
23#16839967C2212yes—
24#16839964—21no—
25#16839962C2212yescomplemented
26#16839957C2212yes—
27C4×#11758201—22no—
28#16839786C2211yescharacteristic
29C2×C4⋊#92418C2211yescharacteristic
30#16839761—21no—
31C2×C4×#92418C2211yescharacteristic
32#16839511C2212yescomplemented
33#16839508C2212yescomplemented
34Q8⋊#92418C2212yescomplemented
35Q8⋊#92418C2212yescomplemented
36D4⋊#92418C2212yes—
37D4⋊#92418C2212yes—
38C61×#554—21no—
39C61×#554—22no—
40C61×#554—22no—
41C61×#554C2212yes—
42D4×Q8×C61—21no—
43D4×Q8×C61C2212yes—
44C61×#544—21no—
45C61×#543—22no—
46C61×#533C2211yescharacteristic
47C61×#533C2211yescharacteristic
48C2×C22⋊Q8×C61—22no—
49C61×#518—21no—
50C2×C4×D4×C61C2211yescharacteristic
51C2×#11758222—21no—
52#14519803—44no—
53C2×#11758200—22no—
54C2×#11758201—22no—
55C2×#11758201—22no—
56C2×#11758199—22no—
57C2×#11758194—21no—
58C22×#92418—21no—
59C22×#92418—21no—
60#14519777C2311yescharacteristic
61#14519774C2311yescharacteristic
62#14519765—21no—
63#14519751—21no—
64#14519751—21no—
65C4×D244—21no—
66C4×D244—21no—
67#14519708—22no—
68#14519708—22no—
69C4⋊#92418—21no—
70C4⋊#92418—21no—
71C4⋊#92418C2312yes—
72C4⋊#92418C2312yes—
73#14519701—22no—
74#14519701—22no—
75#14519700—22no—
76#14519697—22no—
77#14519697—22no—
78#14519697—22no—
79C4×#92418—21no—
80C4×#92418C2311yescharacteristic
81C4×#92418C2311yescharacteristic
82C22×Q8×C61—22no—
83C22×D4×C61D411yescharacteristic
84C23×C4×C61—21no—
85C4⋊Q8×C61—22no—
86C4⋊Q8×C61—22no—
87C4⋊Q8×C61D411yescharacteristic
88C4⋊Q8×C61D411yescharacteristic
89C4⋊Q8×C61C2311yescharacteristic
90C4⋊Q8×C61C2311yescharacteristic
91#125×C61—22no—
92#125×C61—22no—
93#125×C61—21no—
94#125×C61—21no—
95C22⋊Q8×C61—24no—
96C22⋊Q8×C61—24no—
97C22⋊Q8×C61—24no—
98C22⋊Q8×C61—24no—
99C22⋊Q8×C61—24no—
100C22⋊Q8×C61—22no—
101C22⋊Q8×C61—21no—
102C22⋊Q8×C61—21no—
103C22⋊Q8×C61—24no—
104C22⋊Q8×C61—22no—
105C22⋊Q8×C61D412yes—
106C22⋊Q8×C61D412yes—
107C4×Q8×C61—21no—
108C4×Q8×C61—21no—
109C4×Q8×C61D412yes—
110C4×Q8×C61C2312yes—
111C4×D4×C61—22no—
112C4×D4×C61—22no—
113C4×D4×C61—22no—
114C4×D4×C61—22no—
115C4×D4×C61—21no—
116C4×D4×C61—21no—
117C4×D4×C61D412yes—
118C4×D4×C61C2312yes—
119#117×C61—21no—
120#117×C61—21no—
121C2×C4⋊C4×C61—24no—
122C2×C4⋊C4×C61—22no—
123C2×C4⋊C4×C61D411yescharacteristic
124C2×C22⋊C4×C61—24no—
125C2×C22⋊C4×C61—21no—
126C2×C42×C61C2311yescharacteristic
127C24×C61—21no—
128C2×Q8×C61—24no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—82no—
132C2×Q8×C61—82no—
133C2×Q8×C61C2×D412yes—
134C2×Q8×C61C2×D412yes—
135C2×D4×C61—21no—
136C2×D4×C61—22no—
137C2×D4×C61SD1614yescomplemented
138C2×D4×C61C2×D412yes—
139C22×C4×C61—24no—
140C22×C4×C61—24no—
141C22×C4×C61—21no—
142C22×C4×C61—22no—
143C22×C4×C61—22no—
144C22×C4×C61—81no—
145C22×C4×C61—81no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C2×C22⋊D61—41no—
150#11758222—42no—
151#11758222—42no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×C4×D61—41no—
156C2×Dic122—21no—
157C4⋊C4×C61—24no—
158C4⋊C4×C61—24no—
159C4⋊C4×C61—24no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61C2×D412yes—
172C4⋊C4×C61C2×D412yes—
173C4⋊C4×C61C2×D412yes—
174C22⋊C4×C61—24no—
175C22⋊C4×C61—24no—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C42×C61—22no—
181C42×C61—22no—
182C42×C61C2×D411yescharacteristic
183C42×C61C2×D411yescharacteristic
184C42×C61C2×D411yescharacteristic
185C42×C61C2411yescharacteristic
186#11758201—44no—
187#11758201—82no—
188#11758200—44no—
189#11758199—44no—
190#11758198—41no—
191#11758198—41no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194#11758194—42no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—42no—
200C2×#92418—81no—
201C23×C61—22no—
202C23×C61—24no—
203C23×C61—81no—
204C23×C61—81no—
205C23×C61#12011yescharacteristic
206C22×D61—41no—
207Q8×C61—22no—
208Q8×C61—22no—
209Q8×C61—42no—
210Q8×C61—41no—
211Q8×C61—41no—
212Q8×C61—81no—
213Q8×C61—84no—
214D4×C61—22no—
215D4×C61—24no—
216D4×C61C2×SD1614yes—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—41no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—82no—
228C2×C4×C61—81no—
229C2×C4×C61—84no—
230C2×C4×C61—84no—
231C2×C4×C61#12011yescharacteristic
232C2×C4×C61#12011yescharacteristic
233C2×C4×C61#12011yescharacteristic
234C2×C4×C61C2×SD1612yes—
235C2×C4×C61#13712yes—
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C22⋊D61—82no—
240C2×Dic61—41no—
241D244—42no—
242D244—41no—
243D244—41no—
244D244—41no—
245D244—41no—
246C4×D61—82no—
247Dic122—41no—
248Dic122—41no—
249#92418—22no—
250#92418—41no—
251#92418—42no—
252#92418—42no—
253#18664—611nomaximal, Sylow
254C22×C61—21no—
255C22×C61—21no—
256C22×C61—24no—
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—84no—
260C22×C61#44812yes—
261C22×C61C2×#12011yescharacteristic
262C2×D61—81no—
263C2×D61—81no—
264C2×D61—81no—
265C4×C61—21no—
266C4×C61—21no—
267C4×C61—21no—
268C4×C61—42no—
269C4×C61—42no—
270C4×C61—42no—
271C4×C61—42no—
272C4×C61—42no—
273C4×C61#44812yes—
274C4×C61C22×SD1611yescharacteristic
275C4×C61C2×#13711yescharacteristic
276C4×C61#57611yescharacteristic
277C4×C61#57711yescharacteristic
278Dic61—81no—
279#3501D6111yescharacteristic, complemented
280#3246—611no—
281#3167—611no—
282#3036—611no—
283#1651—612no—
284#1643—612no—
285#1627—612no—
286#1622—612no—
287#1550—611no—
288#1493—611no—
289C2×D4⋊C8—611no—
290C2×C61—22no—
291C2×C61—44no—
292C2×C61C2×#44811yescharacteristic
293C2×C61#301211yescharacteristic
294C2×C61#301811yescharacteristic
295D61—161no—
296C2×AGL(1,ℤ/8)—1221no—
297#554—21no—
298#554—22no—
299#554—22no—
300#554C2×D6112yes—
301D4×Q8—21no—
302D4×Q8C2×D6112yes—
303#544—21no—
304#543—22no—
305#533C2×D6111yescharacteristic
306#533C2×D6111yescharacteristic
307#532—611no—
308C2×C22⋊Q8—22no—
309#518—21no—
310C2×C4×D4C2×D6111yescharacteristic
311#498—611no—
312#492—611no—
313#487—611no—
314#486—611no—
315#476—1221no—
316#475—612no—
317#465—1222no—
318#465—1222no—
319#463—1222no—
320#461—612no—
321#461—1221no—
322#460—612no—
323#459—612no—
324C4×SD16—1222no—
325#423—611no—
326C2×C4⋊C8—611no—
327#417—1221no—
328C2×C4×C8—611no—
329#333—612no—
330#332—612no—
331Q8⋊C8—612no—
332Q8⋊C8—612no—
333D4⋊C8—612no—
334D4⋊C8—612no—
335C61#1866411yescharacteristic, complemented, Sylow
336C2×#41—1221no—
337C22×Q8—22no—
338C22×D4C22⋊D6111yescharacteristic
339C23×C4—21no—
340AGL(1,ℤ/8)—2444no—
341C2×SD16—1222no—
342C2×SD16—1222no—
343C2×SD16—1222no—
344C2×D8—1222no—
345C2×M16—1221no—
346C22×C8—1221no—
347C22×C8—1221no—
348C4⋊Q8—22no—
349C4⋊Q8—22no—
350C4⋊Q8—611no—
351C4⋊Q8C22⋊D6111yescharacteristic
352C4⋊Q8C22⋊D6111yescharacteristic
353C4⋊Q8C22×D6111yescharacteristic
354C4⋊Q8C22×D6111yescharacteristic
355#127—611no—
356#125—22no—
357#125—22no—
358#125—21no—
359#125—21no—
360#124—1221no—
361C22⋊Q8—24no—
362C22⋊Q8—24no—
363C22⋊Q8—24no—
364C22⋊Q8—24no—
365C22⋊Q8—22no—
366C22⋊Q8—24no—
367C22⋊Q8—21no—
368C22⋊Q8—21no—
369C22⋊Q8—22no—
370C22⋊Q8—24no—
371C22⋊Q8C22⋊D6112yes—
372C22⋊Q8C22⋊D6112yes—
373#121—1221no—
374#121—1221no—
375C4×Q8—21no—
376C4×Q8—21no—
377C4×Q8C22⋊D6112yes—
378C4×Q8C22×D6112yes—
379C4×D4—22no—
380C4×D4—22no—
381C4×D4—22no—
382C4×D4—22no—
383C4×D4—21no—
384C4×D4—21no—
385C4×D4—1221no—
386C4×D4—1221no—
387C4×D4C22⋊D6112yes—
388C4×D4C22×D6112yes—
389#117—21no—
390#117—21no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4C22⋊D6111yescharacteristic
394C2×C22⋊C4—24no—
395C2×C22⋊C4—21no—
396C2×C42C22×D6111yescharacteristic
397#106—1222no—
398#106—1222no—
399C4⋊C8—612no—
400C4⋊C8—612no—
401C4⋊C8—1221no—
402C4⋊C8—1221no—
403#103—1222no—
404#103—1222no—
405#103—1222no—
406#102—1222no—
407#102—1222no—
408#102—1222no—
409C4×C8—611no—
410C4×C8—611no—
411C4×C8—1221no—
412C24—21no—
413#41—2442no—
414#41—2442no—
415C2×Q8—24no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—82no—
419C2×Q8—82no—
420C2×Q8—1221no—
421C2×Q8C2×C22⋊D6112yes—
422C2×Q8C2×C22⋊D6112yes—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—1221no—
426C2×D4—1221no—
427C2×D4—1221no—
428C2×D4—2441no—
429C2×D4#1175820114yescomplemented
430C2×D4C2×C22⋊D6112yes—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—2441no—
439C22×C4C2×C22⋊D6111yescharacteristic
440C22×C4C2×C22⋊D6111yescharacteristic
441C22×C4C2×C22⋊D6111yescharacteristic
442SD16—4882no—
443SD16—2444no—
444SD16—2444no—
445D8—2444no—
446M16—2442no—
447C2×C8—1221no—
448C2×C8—1221no—
449C2×C8—1222no—
450C2×C8—1222no—
451C2×C8—4881no—
452C2×C8—2442no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4C2×C22⋊D6112yes—
470C4⋊C4C2×C22⋊D6112yes—
471C4⋊C4C2×C22⋊D6112yes—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—2441no—
479C22⋊C4—2441no—
480C42—22no—
481C42—22no—
482C42C2×C22⋊D6111yescharacteristic
483C42C2×C22⋊D6111yescharacteristic
484C42C2×C22⋊D6111yescharacteristic
485C42C23×D6111yescharacteristic
486C23—22no—
487C23—24no—
488C23—81no—
489C23—81no—
490C23—2441no—
491C23#1451974611yescharacteristic
492Q8—22no—
493Q8—22no—
494Q8—42no—
495Q8—41no—
496Q8—41no—
497Q8—81no—
498Q8—84no—
499Q8—2441no—
500Q8—2441no—
501D4—22no—
502D4—24no—
503D4—4882no—
504D4—2442no—
505D4—2441no—
506D4—2441no—
507D4—2441no—
508D4—2441no—
509D4C2×#1175820114yes—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—82no—
521C2×C4—81no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—4882no—
525C2×C4—2441no—
526C2×C4#1451974611yescharacteristic
527C2×C4#1451974611yescharacteristic
528C2×C4#1451974611yescharacteristic
529C2×C4C2×#1175820112yes—
530C2×C4#1451980812yes—
531C2×C4C22×C22⋊D6111yescharacteristic
532C2×C4C22×C22⋊D6111yescharacteristic
533C2×C4C22×C22⋊D6111yescharacteristic
534C8—1222no—
535C8—2441no—
536C8—2442no—
537C8—2442no—
538C22—21no—
539C22—21no—
540C22—24no—
541C22—42no—
542C22—42no—
543C22—84no—
544C22—4881no—
545C22—4881no—
546C22—4881no—
547C22#1683990112yes—
548C22C2×#1451974611yescharacteristic
549C4—21no—
550C4—21no—
551C4—21no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—42no—
556C4—42no—
557C4—4881no—
558C4#1683990112yes—
559C4C22×#1175820111yescharacteristic
560C4C2×#1451980811yescharacteristic
561C4#1684056511yescharacteristic
562C4#1684057311yescharacteristic
563C2—22no—
564C2—44no—
565C2—9761no—
566C2C2×#1683990111yescharacteristic
567C2#2145512311yescharacteristic
568C2#2145516811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3501C2
socleC22×C61C2×#120
prime radical#21455283C2
hypercenter#3501D61
hypocenterC61#18664
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #21457131C2semidirect
2. #21456379C2semidirect
4. #21445082C2semidirect
5. #21445026C2semidirect
6. #21444922C2semidirect
7. #21444891C2semidirect
8. #21444509C2semidirect
9. #21444274C2semidirect
10. C2×D4⋊#92418C2semidirect
11. C61×#3501C2semidirect
14. #16840158C22semidirect
15. #16840136C22semidirect
16. #16840112C22semidirect
17. #16840110C22semidirect
19. #16840047C22semidirect
25. #16839962C22semidirect
32. #16839511C22semidirect
33. #16839508C22semidirect
34. Q8⋊#92418C22semidirect
35. Q8⋊#92418C22semidirect
137. C2×D4×C61SD16semidirect
279. #3501D61semidirect
335. C61#18664semidirect
429. C2×D4#11758201semidirect

Maximal subgroups

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

subgroupindex
#214571312
#214563792
#214552832
#214450822
#214450262
#214449222
#214448912
#214445092
#214442742
C2×D4⋊#924182
C61×#35012
#1866461