Finite Group Databasebeta

← #29655193

Subgroups — #29655193

Order 15616.

972 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457037C211yesmaximal, characteristic, complemented
2#21456937C211yesmaximal, characteristic, complemented
3#21451759C211yesmaximal, characteristic, complemented
4#21444833C214yesmaximal, complemented
5#21444805C214yesmaximal, complemented
6#21444327C211yesmaximal, characteristic, complemented
7#21444305C211yesmaximal, characteristic, complemented
8#21444171C211yesmaximal, characteristic, complemented
9C61×#3558C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic, complemented
11#16840172—21no—
12#16840165—21no—
13#16840121—21no—
14#16840108—21no—
15#16840090C2212yescomplemented
16#16840073C2212yes—
17#16840073C2214yescomplemented
18#16840057—21no—
19#16840052—21no—
20#16840042—22no—
21#16839819C2211yescharacteristic, complemented
22#16839815C2211yescharacteristic, complemented
23#16839687C2214yescomplemented
24#16839674—22no—
25#16839669—21no—
26#16839665—21no—
27#16839608C2211yescharacteristic, complemented
28#16839499C2212yescomplemented
29#16839496C2212yes—
30#16839496C2214yescomplemented
31#16839486C2214yescomplemented
32C61×#562—21no—
33C61×H(3,𝔽4)—21no—
34C61×#559—21no—
35C61×#548—21no—
36C61×#546—21no—
37C61×#546—22no—
38D42×C61—21no—
39C61×#542—21no—
40C61×#538—22no—
41C61×#538C2212yes—
42C61×#535—21no—
43C61×#534C2212yes—
44C61×#532C2211yescharacteristic
45C61×#529C2211yescharacteristic
46C2×#124×C61C2211yescharacteristic
47C22×Dic122—21no—
48#14519777—21no—
49#14519777—21no—
50#14519777—21no—
51#14519777—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61C2312yes—
54C2×C22⋊Dic61—24no—
55C2×C4×Dic61—21no—
56#14519710—24no—
57#14519710—22no—
58#14519708—22no—
59C4⋊#92418—21no—
60C4⋊#92418—21no—
61#14519701—22no—
62#14519698—24no—
63#14519698—22no—
64C22⋊#92418C2312yes—
65C22⋊#92418C2312yes—
66#14519688—21no—
67C4×#92418—21no—
68#14519684—24no—
69#14519684—21no—
70#14519684C2312yes—
71C2×#41×C61—21no—
72C2×#41×C61—21no—
73C22×Q8×C61D411yescharacteristic
74C22×D4×C61—21no—
75C22×D4×C61—22no—
76C22×D4×C61D411yescharacteristic
77C4⋊Q8×C61D411yescharacteristic
78#127×C61—21no—
79#127×C61D411yescharacteristic
80#126×C61—22no—
81#126×C61—21no—
82#125×C61D411yescharacteristic
83#124×C61—21no—
84#124×C61—22no—
85#124×C61—22no—
86#124×C61—22no—
87#124×C61—21no—
88#124×C61—21no—
89#124×C61—21no—
90#124×C61—21no—
91#124×C61—21no—
92#124×C61D411yescharacteristic
93#123×C61—22no—
94#123×C61—21no—
95#123×C61—22no—
96#123×C61—22no—
97#123×C61—22no—
98C22⋊Q8×C61—22no—
99C22⋊Q8×C61—21no—
100C22⋊Q8×C61—22no—
101C22⋊Q8×C61C2311yescharacteristic
102#121×C61—22no—
103#121×C61—22no—
104#121×C61—22no—
105#121×C61—21no—
106#121×C61—22no—
107#121×C61—21no—
108#121×C61—22no—
109#121×C61C2312yes—
110#121×C61C2311yescharacteristic
111#120×C61—22no—
112#120×C61—22no—
113#120×C61—22no—
114#120×C61—22no—
115#120×C61—22no—
116C4×Q8×C61—21no—
117C4×Q8×C61D411yescharacteristic
118C4×D4×C61—22no—
119C4×D4×C61—21no—
120C4×D4×C61—22no—
121C4×D4×C61—21no—
122C4×D4×C61—21no—
123C4×D4×C61D412yes—
124C4×D4×C61D411yescharacteristic
125#117×C61—21no—
126C2×C22⋊C4×C61—22no—
127C2×C22⋊C4×C61D412yes—
128C2×C22⋊C4×C61C2312yes—
129C2×C42×C61C2311yescharacteristic
130C24×C61—22no—
131C24×C61C2×D412yes—
132#41×C61—82no—
133#41×C61—82no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—42no—
138C2×Q8×C61C2×D411yescharacteristic
139C2×D4×C61—22no—
140C2×D4×C61—21no—
141C2×D4×C61—22no—
142C2×D4×C61—22no—
143C2×D4×C61—22no—
144C2×D4×C61—22no—
145C2×D4×C61—22no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—42no—
151C2×D4×C61—82no—
152C2×D4×C61—84no—
153C2×D4×C61C2×D411yescharacteristic
154C2×D4×C61C2×D412yes—
155C22×C4×C61—22no—
156C22×C4×C61—22no—
157C22×C4×C61—21no—
158C22×C4×C61—22no—
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2×D412yes—
162C22×C4×C61C2411yescharacteristic
163C22×Dic61—22no—
164C22×Dic61—22no—
165C2×Dic122—41no—
166C2×Dic122—41no—
167C2×Dic122—82no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61C2×D412yes—
176C4⋊C4×C61C2×D411yescharacteristic
177C4⋊C4×C61C2×D411yescharacteristic
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—21no—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—21no—
188C22⋊C4×C61—21no—
189C22⋊C4×C61—22no—
190C22⋊C4×C61—22no—
191C22⋊C4×C61—22no—
192C22⋊C4×C61—21no—
193C22⋊C4×C61—22no—
194C42×C61—22no—
195C42×C61—21no—
196C42×C61C2×D411yescharacteristic
197C42×C61C2×D411yescharacteristic
198C42×C61C2×D411yescharacteristic
199C42×C61C2×D411yescharacteristic
200C22⋊Dic61—84no—
201C4⋊Dic61—22no—
202C4⋊Dic61—22no—
203C4⋊Dic61—41no—
204C4⋊Dic61—41no—
205C4×Dic61—42no—
206C2×#92418—22no—
207C2×#92418—22no—
208C23×C61—22no—
209C23×C61—21no—
210C23×C61—22no—
211C23×C61—24no—
212C23×C61—42no—
213C23×C61—42no—
214C23×C61—82no—
215C23×C61—82no—
216C23×C61—82no—
217C23×C61C22×D411yescharacteristic
218Q8×C61—41no—
219Q8×C61—42no—
220Q8×C61—81no—
221Q8×C61—81no—
222D4×C61—42no—
223D4×C61—41no—
224D4×C61—42no—
225D4×C61—44no—
226D4×C61—42no—
227D4×C61—42no—
228D4×C61—81no—
229D4×C61—81no—
230D4×C61—82no—
231D4×C61—82no—
232D4×C61—82no—
233D4×C61—82no—
234D4×C61—82no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—21no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—41no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61—82no—
246C2×C4×C61—82no—
247C2×C4×C61—82no—
248C2×C4×C61#12011yescharacteristic
249C2×C4×C61#12011yescharacteristic
250C2×C4×C61#12011yescharacteristic
251C2×C4×C61#12011yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61—42no—
258C2×Dic61—44no—
259Dic122—82no—
260Dic122—82no—
261#92418—42no—
262#92418—42no—
263#18436—611nomaximal, Sylow
264C22×C61—21no—
265C22×C61—21no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—42no—
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—41no—
274C22×C61—42no—
275C22×C61—84no—
276C22×C61—164no—
277C22×C61C2×#12011yescharacteristic
278C22×C61#57612yes—
279C4×C61—21no—
280C4×C61—21no—
281C4×C61—21no—
282C4×C61—21no—
283C4×C61—21no—
284C4×C61—41no—
285C4×C61—42no—
286C4×C61—41no—
287C4×C61—42no—
288C4×C61—41no—
289C4×C61#45512yes—
290Dic61—82no—
291Dic61—82no—
292#3558D6111yescharacteristic
293#3238—611no—
294#3226—611no—
295#2640—611no—
296#1612—614no—
297#1607—614no—
298#1505—611no—
299#1500—611no—
300#1469—611no—
301C2×C61—22no—
302C2×C61—82no—
303C2×C61—82no—
304C2×C61—81no—
305C2×C61#300811yescharacteristic
306C2×C61C2×#45511yescharacteristic
307C2×C61#302811yescharacteristic
308#562—21no—
309H(3,𝔽4)—21no—
310#559—21no—
311#548—21no—
312#546—21no—
313#546—22no—
314D42—21no—
315#542—21no—
316#538—22no—
317#538C2×D6112yes—
318#535—21no—
319#534C2×D6112yes—
320#532C2×D6111yescharacteristic
321C2×C4⋊Q8—611no—
322#529C2×D6111yescharacteristic
323C2×#124C2×D6111yescharacteristic
324#501—1221no—
325#500—1221no—
326#489—1221no—
327#486—1221no—
328#483—612no—
329#480—612no—
330#480—614no—
331#477—1221no—
332#476—1221no—
333#474—1222no—
334#432—611no—
335#431—611no—
336#397—614no—
337#393—1222no—
338#391—1221no—
339#390—1221no—
340#376—611no—
341#328—612no—
342#327—612no—
343#327—614no—
344#323—614no—
345C61#1843611yescharacteristic, complemented, Sylow
346C2×#41—21no—
347C2×#41—21no—
348C22×Q8—1221no—
349C22×Q8C22⋊D6111yescharacteristic
350C22×D4—21no—
351C22×D4—22no—
352C22×D4C22⋊D6111yescharacteristic
353C4⋊Q8—1221no—
354C4⋊Q8—1221no—
355C4⋊Q8—1221no—
356C4⋊Q8—1221no—
357C4⋊Q8C22⋊D6111yescharacteristic
358#127—21no—
359#127C22⋊D6111yescharacteristic
360#126—22no—
361#126—21no—
362#125C22⋊D6111yescharacteristic
363#124—21no—
364#124—22no—
365#124—22no—
366#124—22no—
367#124—21no—
368#124—21no—
369#124—21no—
370#124—21no—
371#124—21no—
372#124C22⋊D6111yescharacteristic
373#123—22no—
374#123—21no—
375#123—22no—
376#123—22no—
377#123—22no—
378C22⋊Q8—22no—
379C22⋊Q8—21no—
380C22⋊Q8—22no—
381C22⋊Q8C22×D6111yescharacteristic
382#121—22no—
383#121—22no—
384#121—22no—
385#121—21no—
386#121—22no—
387#121—21no—
388#121—22no—
389#121C22×D6111yescharacteristic
390#121C22×D6112yes—
391#120—22no—
392#120—22no—
393#120—22no—
394#120—22no—
395#120—22no—
396C4×Q8—21no—
397C4×Q8C22⋊D6111yescharacteristic
398C4×D4—22no—
399C4×D4—21no—
400C4×D4—21no—
401C4×D4—22no—
402C4×D4—21no—
403C4×D4C22⋊D6111yescharacteristic
404C4×D4C22⋊D6112yes—
405#117—21no—
406C2×C4⋊C4—612no—
407C2×C4⋊C4—1221no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—1224no—
410C2×C22⋊C4C22⋊D6112yes—
411C2×C22⋊C4C22×D6112yes—
412C2×C42—1221no—
413C2×C42C22×D6111yescharacteristic
414#107—1224no—
415#107—1222no—
416#106—1222no—
417C4⋊C8—1221no—
418C4⋊C8—1221no—
419#103—1222no—
420#102—1224no—
421#102—1222no—
422C22⋊C8—612no—
423C22⋊C8—612no—
424#97—1221no—
425C4×C8—1221no—
426#95—612no—
427#95—1221no—
428#95—1224no—
429C24—22no—
430C24C2×C22⋊D6112yes—
431#41—82no—
432#41—82no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—42no—
437C2×Q8—4882no—
438C2×Q8—2441no—
439C2×Q8—2441no—
440C2×Q8C2×C22⋊D6111yescharacteristic
441C2×D4—22no—
442C2×D4—21no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—42no—
453C2×D4—82no—
454C2×D4—84no—
455C2×D4C2×C22⋊D6111yescharacteristic
456C2×D4C2×C22⋊D6112yes—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4C2×C22⋊D6111yescharacteristic
464C22×C4C2×C22⋊D6111yescharacteristic
465C22×C4C2×C22⋊D6112yes—
466C22×C4C23×D6111yescharacteristic
467C2×C8—1222no—
468C2×C8—1222no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—1222no—
477C4⋊C4—1222no—
478C4⋊C4—2441no—
479C4⋊C4—2441no—
480C4⋊C4C2×C22⋊D6111yescharacteristic
481C4⋊C4C2×C22⋊D6112yes—
482C4⋊C4C2×C22⋊D6111yescharacteristic
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—21no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—21no—
492C22⋊C4—22no—
493C22⋊C4—21no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—21no—
497C22⋊C4—21no—
498C22⋊C4—22no—
499C22⋊C4—4884no—
500C42—22no—
501C42—21no—
502C42—2442no—
503C42C2×C22⋊D6111yescharacteristic
504C42C2×C22⋊D6111yescharacteristic
505C42C2×C22⋊D6111yescharacteristic
506C42C2×C22⋊D6111yescharacteristic
507C23—22no—
508C23—21no—
509C23—22no—
510C23—24no—
511C23—42no—
512C23—42no—
513C23—82no—
514C23—82no—
515C23—82no—
516C23C22×C22⋊D6111yescharacteristic
517Q8—41no—
518Q8—42no—
519Q8—81no—
520Q8—81no—
521Q8—4882no—
522Q8—4882no—
523D4—42no—
524D4—41no—
525D4—42no—
526D4—44no—
527D4—42no—
528D4—42no—
529D4—81no—
530D4—81no—
531D4—82no—
532D4—82no—
533D4—82no—
534D4—82no—
535D4—82no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4—2444no—
554C2×C4#1451974611yescharacteristic
555C2×C4#1451974611yescharacteristic
556C2×C4#1451974611yescharacteristic
557C2×C4#1451974611yescharacteristic
558C2×C4C22×C22⋊D6111yescharacteristic
559C2×C4C22×C22⋊D6111yescharacteristic
560C8—2442no—
561C8—2442no—
562C22—21no—
563C22—21no—
564C22—41no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—41no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—41no—
573C22—84no—
574C22—164no—
575C22C2×#1451974611yescharacteristic
576C22#1684056512yes—
577C4—21no—
578C4—21no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—41no—
583C4—42no—
584C4—41no—
585C4—41no—
586C4—42no—
587C4—4882no—
588C4—4882no—
589C4#1683994012yes—
590C2—22no—
591C2—82no—
592C2—81no—
593C2—82no—
594C2#2145508711yescharacteristic
595C2C2×#1683994011yescharacteristic
596C2#2145522511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3558C2
socleC22×C61C2×#120
hypercenter#3558D61
hypocenterC61#18436
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3558

Complements

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

NH ≅ G/Ntype
1. #21457037C2semidirect
2. #21456937C2semidirect
3. #21451759C2semidirect
4. #21444833C2semidirect
5. #21444805C2semidirect
6. #21444327C2semidirect
7. #21444305C2semidirect
8. #21444171C2semidirect
10. C2×#14519777C22semidirect
15. #16840090C22semidirect
17. #16840073C22semidirect
21. #16839819C22semidirect
22. #16839815C22semidirect
23. #16839687C22semidirect
27. #16839608C22semidirect
28. #16839499C22semidirect
30. #16839496C22semidirect
31. #16839486C22semidirect
345. C61#18436semidirect

Maximal subgroups

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

subgroupindex
#214570372
#214569372
#214517592
#214448332
#214448052
#214443272
#214443052
#214441712
C61×#35582
#1843661