Finite Group Databasebeta

← #29671432

Subgroups — #29671432

Order 15616.

624 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458727C211yesmaximal, characteristic, complemented
2Q8×#11758200C211yesmaximal, characteristic, complemented
3#21458176C211yesmaximal, characteristic
4#21445413C211yesmaximal, characteristic, complemented
5#21445342C211yesmaximal, characteristic, complemented
6#21445310C211yesmaximal, characteristic, complemented
7#21445199C211yesmaximal, characteristic, complemented
8#21445192C211yesmaximal, characteristic, complemented
9#21445038C211yesmaximal, characteristic, complemented
10#21445024C211yesmaximal, characteristic, complemented
11#21444984C211yesmaximal, characteristic, complemented
12#21444970C211yesmaximal, characteristic
13#21444707C211yesmaximal, characteristic, complemented
14SD16×#92418C211yesmaximal, characteristic, complemented
15C61×#3276C211yesmaximal, characteristic
16Q8×Dic122C2211yescharacteristic
17#16840469C2211yescharacteristic
18#16840169—21no—
19#16840167C2211yescharacteristic
20#16840158—21no—
21#16840158C2211yescharacteristic
22#16840158C2211yescharacteristic
23#16840114C2211yescharacteristic
24#16840114C2211yescharacteristic
25#16840092—21no—
26#16840078—21no—
27#16840071—21no—
28#16840057C2211yescharacteristic
29#16840052—21no—
30#16840052C2211yescharacteristic
31#16840052C2211yescharacteristic
32#16840052C2211yescharacteristic
33#16840044—21no—
34#16840044C2211yescharacteristic
35#16840044C2211yescharacteristic
36#16840040—21no—
37#16839989—21no—
38#16839892C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40#16839863—21no—
41C4×#11758200—21no—
42C4×#11758200C2211yescharacteristic
43D4×#92418C2211yescharacteristic
44C2×#14519708—21no—
45#16839513C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839510C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839508C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52#16839492C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55C8×#92418C2211yescharacteristic
56C2×#137×C61—21no—
57D4×Q8×C61C2211yescharacteristic
58C61×#547C2211yescharacteristic
59C61×#494C2211yescharacteristic
60C61×#470—21no—
61C61×#462C2211yescharacteristic
62C61×#461C2211yescharacteristic
63C61×#452—21no—
64C61×#451—21no—
65C4×SD16×C61C2211yescharacteristic
66C61×#436C2211yescharacteristic
67C2×#11758200D411yescharacteristic
68C22×#92418—21no—
69#14519777—21no—
70#14519777C2311yescharacteristic
71#14519777C2311yescharacteristic
72#14519776—21no—
73#14519776—41no—
74#14519768—21no—
75#14519756—21no—
76#14519754—41no—
77Q8×Dic61—21no—
78C4×Dic122—21no—
79C4×Dic122C2311yescharacteristic
80D4×Dic61—21no—
81C2×C4⋊Dic61—21no—
82#14519711—21no—
83#14519711—21no—
84#14519710—21no—
85#14519708—21no—
86#14519708—21no—
87#14519708—21no—
88#14519708—21no—
89#14519708—21no—
90#14519708—21no—
91#14519708—21no—
92#14519708—21no—
93#14519708—21no—
94#14519708D411yescharacteristic
95C4⋊#92418—21no—
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99#14519706—21no—
100#14519701—21no—
101#14519700—21no—
102#14519700—21no—
103#14519700—41no—
104#14519700—41no—
105#14519700D411yescharacteristic
106#14519698—21no—
107#14519698—21no—
108#14519698—21no—
109#14519698D411yescharacteristic
110C22⋊#92418—21no—
111#14519687—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×#41×C61—21no—
116C22×Q8×C61—21no—
117#137×C61—44no—
118C2×Q16×C61—21no—
119C2×Q16×C61—21no—
120C2×Q16×C61—21no—
121C2×SD16×C61—21no—
122C2×SD16×C61D411yescharacteristic
123C2×M16×C61—21no—
124C4⋊Q8×C61—21no—
125C4⋊Q8×C61C2311yescharacteristic
126C4⋊Q8×C61C2311yescharacteristic
127#123×C61—41no—
128C22⋊Q8×C61—21no—
129C22⋊Q8×C61—21no—
130C22⋊Q8×C61—41no—
131#121×C61—21no—
132C4×Q8×C61C2311yescharacteristic
133C4×D4×C61—21no—
134C4×D4×C61C2311yescharacteristic
135C4×D4×C61C2311yescharacteristic
136C2×C4⋊C4×C61—21no—
137#106×C61D411yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61—21no—
141#103×C61D411yescharacteristic
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145#41×C61—42no—
146#41×C61—81no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61—41no—
150C2×Q8×C61—42no—
151C2×Q8×C61—81no—
152C2×Q8×C61C2×D411yescharacteristic
153C2×D4×C61—41no—
154C2×D4×C61C2×D411yescharacteristic
155C2×D4×C61C2×D411yescharacteristic
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—41no—
159C22×C4×C61—41no—
160C22×Dic61—41no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—41no—
164C2×Dic122C2×D411yescharacteristic
165Q16×C61—42no—
166Q16×C61—42no—
167Q16×C61—81no—
168SD16×C61—42no—
169SD16×C61SD1614yescomplemented
170M16×C61—42no—
171C2×C8×C61—21no—
172C2×C8×C61C2×D411yescharacteristic
173C2×C8×C61C2×D411yescharacteristic
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—41no—
180C4⋊C4×C61—41no—
181C4⋊C4×C61C2×D411yescharacteristic
182C4⋊C4×C61C2×D411yescharacteristic
183C4⋊C4×C61C2×D411yescharacteristic
184C22⋊C4×C61—21no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—41no—
187C22⋊C4×C61—41no—
188C42×C61—21no—
189C42×C61C2411yescharacteristic
190C22⋊Dic61—41no—
191#11758200—22no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—41no—
197C4⋊Dic61—41no—
198C4⋊Dic61C2×D411yescharacteristic
199#11758196—21no—
200#11758196—41no—
201#11758196—41no—
202C4×Dic61—21no—
203C4×Dic61—41no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—42no—
209C2×#92418C2×D411yescharacteristic
210C2×#92418C2×D411yescharacteristic
211C23×C61—21no—
212C23×C61—21no—
213Q8×C61—21no—
214Q8×C61—41no—
215Q8×C61—41no—
216Q8×C61—41no—
217Q8×C61—41no—
218Q8×C61—82no—
219Q8×C61C2×SD1612yes—
220D4×C61—21no—
221D4×C61—21no—
222D4×C61—21no—
223D4×C61—82no—
224D4×C61C2×SD1612yes—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—82no—
236C2×C4×C61—82no—
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61H(5,𝔽2)11yescharacteristic
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—161no—
245Dic122—21no—
246Dic122—21no—
247Dic122—81no—
248Dic122—81no—
249Dic122—161no—
250C8×C61—21no—
251C8×C61—41no—
252C8×C61C2×SD1612yes—
253#92418—21no—
254#92418—41no—
255#92418—41no—
256#92418—41no—
257#92418—41no—
258#92418#13712yescomplemented
259#19895—611nomaximal, Sylow
260C22×C61—22no—
261C22×C61—41no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61—41no—
265C22×C61—41no—
266C22×C61D4211yescharacteristic
267C4×C61—22no—
268C4×C61—21no—
269C4×C61—41no—
270C4×C61—41no—
271C4×C61—41no—
272C4×C61—81no—
273C4×C61—81no—
274C4×C61C22×SD1611yescharacteristic
275C4×C61C2×#13711yescharacteristic
276C4×C61#57711yescharacteristic
277C4×C61#57811yescharacteristic
278Dic61—41no—
279Dic61—41no—
280Dic61—81no—
281Dic61—81no—
282#3385—611no—
283Q8×SD16—611no—
284#3335—611no—
285#3276D6111yescharacteristic
286#1707—611no—
287#1695—611no—
288#1690—611no—
289#1672—611no—
290#1671—611no—
291#1645—611no—
292#1643—611no—
293#1637—611no—
294#1635—611no—
295#1590—611no—
296C8×SD16—611no—
297C2×C61—42no—
298C2×C61—81no—
299C2×C61#327611yescharacteristic
300C2×C61D4×SD1611yescharacteristic
301C2×C61#329011yescharacteristic
302C2×#137—21no—
303Q82—611no—
304#554—611no—
305D4×Q8C2×D6111yescharacteristic
306#547C2×D6111yescharacteristic
307#501—1221no—
308#500—611no—
309#498—611no—
310#498—611no—
311#498—1221no—
312#494C2×D6111yescharacteristic
313#487—611no—
314#487—611no—
315#483—1221no—
316#481—1221no—
317#479—1221no—
318#477—611no—
319#476—611no—
320#476—611no—
321#476—611no—
322#476—1221no—
323#475—611no—
324#475—611no—
325#475—1221no—
326#474—1221no—
327#470—21no—
328#465—1221no—
329#462C2×D6111yescharacteristic
330#461C2×D6111yescharacteristic
331#452—21no—
332#451—21no—
333#446—611no—
334C8×Q8—611no—
335#440—1221no—
336C4×SD16—611no—
337C4×SD16—1221no—
338C4×SD16C2×D6111yescharacteristic
339#436C2×D6111yescharacteristic
340C8×D4—611no—
341C2×#106—1221no—
342#334—611no—
343#334—611no—
344#333—611no—
345#333—611no—
346#332—611no—
347#332—611no—
348Q8⋊C8—611no—
349Q8⋊C8—611no—
350D4⋊C8—611no—
351D4⋊C8—611no—
352C82—611no—
353C61#1989511yescharacteristic, complemented, Sylow
354C2×#41—21no—
355C22×Q8—21no—
356#137—44no—
357C2×Q16—21no—
358C2×Q16—21no—
359C2×Q16—21no—
360C2×SD16—21no—
361C2×SD16—611no—
362C2×SD16C22⋊D6111yescharacteristic
363C2×M16—21no—
364C22×C8—1221no—
365C4⋊Q8—21no—
366C4⋊Q8—611no—
367C4⋊Q8—611no—
368C4⋊Q8—1221no—
369C4⋊Q8—1221no—
370C4⋊Q8—2441no—
371C4⋊Q8C22×D6111yescharacteristic
372C4⋊Q8C22×D6111yescharacteristic
373#125—1221no—
374#123—41no—
375C22⋊Q8—21no—
376C22⋊Q8—21no—
377C22⋊Q8—41no—
378C22⋊Q8—1221no—
379C22⋊Q8—2441no—
380#121—21no—
381C4×Q8—611no—
382C4×Q8—1221no—
383C4×Q8—1221no—
384C4×Q8C22×D6111yescharacteristic
385C4×D4—21no—
386C4×D4—1221no—
387C4×D4C22×D6111yescharacteristic
388C4×D4C22×D6111yescharacteristic
389C2×C4⋊C4—21no—
390C2×C4⋊C4—1221no—
391#107—1221no—
392#107—1221no—
393#107—1221no—
394#106—611no—
395#106—1221no—
396#106—1221no—
397#106—1221no—
398#106—1221no—
399#106—1221no—
400#106—1221no—
401#106—1221no—
402#106—1221no—
403#106—1221no—
404#106C22⋊D6111yescharacteristic
405C4⋊C8—611no—
406C4⋊C8—611no—
407C4⋊C8—611no—
408C4⋊C8—1221no—
409C4⋊C8—1221no—
410C4⋊C8C22×D6111yescharacteristic
411#103—21no—
412#103—21no—
413#103—611no—
414#103—1221no—
415#103—1221no—
416#103—1221no—
417#103—2441no—
418#103—2441no—
419#103C22⋊D6111yescharacteristic
420#102—611no—
421#102—1221no—
422#102—1221no—
423#102—1221no—
424#102C22⋊D6111yescharacteristic
425C22⋊C8—21no—
426C22⋊C8—1221no—
427#97—1221no—
428C4×C8—611no—
429C4×C8—611no—
430C4×C8—1221no—
431C4×C8C22×D6111yescharacteristic
432#41—42no—
433#41—81no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—41no—
437C2×Q8—42no—
438C2×Q8—81no—
439C2×Q8—611no—
440C2×Q8—1221no—
441C2×Q8—1221no—
442C2×Q8—2441no—
443C2×Q8C2×C22⋊D6111yescharacteristic
444C2×D4—41no—
445C2×D4D4×D6111yescharacteristic
446C2×D4C2×C22⋊D6111yescharacteristic
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—2441no—
452Q16—42no—
453Q16—42no—
454Q16—81no—
455SD16—42no—
456SD16—1222no—
457SD16#1175820014yescomplemented
458M16—42no—
459C2×C8—21no—
460C2×C8—611no—
461C2×C8—611no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—2442no—
467C2×C8C2×C22⋊D6111yescharacteristic
468C2×C8C2×C22⋊D6111yescharacteristic
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—611no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—2441no—
483C4⋊C4—2441no—
484C4⋊C4—2441no—
485C4⋊C4—2441no—
486C4⋊C4D4×D6111yescharacteristic
487C4⋊C4C2×C22⋊D6111yescharacteristic
488C4⋊C4C2×C22⋊D6111yescharacteristic
489C22⋊C4—21no—
490C22⋊C4—21no—
491C22⋊C4—41no—
492C22⋊C4—41no—
493C22⋊C4—2441no—
494C42—21no—
495C42—1221no—
496C42—2441no—
497C42C23×D6111yescharacteristic
498C23—21no—
499C23—21no—
500Q8—21no—
501Q8—41no—
502Q8—41no—
503Q8—41no—
504Q8—41no—
505Q8—82no—
506Q8—1221no—
507Q8—1221no—
508Q8—9761no—
509Q8—4881no—
510Q8—4881no—
511Q8C2×#1175820012yes—
512D4—21no—
513D4—21no—
514D4—21no—
515D4—82no—
516D4C2×#1175820012yes—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—9761no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4C2×D4×D6111yescharacteristic
535C2×C4C22×C22⋊D6111yescharacteristic
536C2×C4#1451982211yescharacteristic
537C8—21no—
538C8—41no—
539C8—612no—
540C8—1221no—
541C8—2441no—
542C8—2441no—
543C8—2441no—
544C8—2441no—
545C8C2×#1175820012yes—
546C22—22no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22D4×C22⋊D6111yescharacteristic
553C4—22no—
554C4—21no—
555C4—41no—
556C4—41no—
557C4—41no—
558C4—81no—
559C4—81no—
560C4—4881no—
561C4—4881no—
562C4—2441no—
563C4—2441no—
564C4C22×#1175820011yescharacteristic
565C4#137×D6111yescharacteristic
566C4#1684057311yescharacteristic
567C4#1684057811yescharacteristic
568C2—42no—
569C2—81no—
570C2#2145745911yescharacteristic
571C2D4×#1175820011yescharacteristic
572C2#2145760511yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3276C2
socleC22×C61D42
prime radicalC61×#547C22
hypercenter#3276D61
hypocenterC61#19895
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3276

Complements

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

NH ≅ G/Ntype
1. #21458727C2semidirect
2. Q8×#11758200C2semidirect
4. #21445413C2semidirect
5. #21445342C2semidirect
6. #21445310C2semidirect
7. #21445199C2semidirect
8. #21445192C2semidirect
9. #21445038C2semidirect
10. #21445024C2semidirect
11. #21444984C2semidirect
13. #21444707C2semidirect
14. SD16×#92418C2semidirect
169. SD16×C61SD16semidirect
258. #92418#137semidirect
353. C61#19895semidirect
457. SD16#11758200semidirect

Maximal subgroups

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

subgroupindex
#214587272
Q8×#117582002
#214581762
#214454132
#214453422
#214453102
#214451992
#214451922
#214450382
#214450242
#214449842
#214449702
#214447072
SD16×#924182
C61×#32762
#1989561