Finite Group Databasebeta

← #29655100

Subgroups — #29655100

Order 15616.

996 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456843C211yesmaximal, characteristic, complemented
2#21456741C211yesmaximal, characteristic, complemented
3#21451759C211yesmaximal, characteristic, complemented
4#21444799C214yesmaximal, complemented
5#21444771C214yesmaximal, complemented
6#21444327C211yesmaximal, characteristic, complemented
7#21444305C211yesmaximal, characteristic, complemented
8#21444171C211yesmaximal, characteristic, complemented
9C61×#3558C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic, complemented
11#16840157—21no—
12#16840134—21no—
13#16840132—21no—
14#16840114—21no—
15#16839972—21no—
16#16839967—22no—
17#16839967—21no—
18#16839917C2212yescomplemented
19#16839900C2212yes—
20#16839900C2214yescomplemented
21#16839819C2211yescharacteristic, complemented
22#16839815C2211yescharacteristic, complemented
23#16839687—22no—
24#16839674C2214yescomplemented
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×Dic122C2312yes—
48C2×#11758202—22no—
49C2×#11758200—24no—
50C2×#11758200—22no—
51#14519777—21no—
52#14519777—21no—
53#14519777—21no—
54#14519777—21no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57C2×C22⋊Dic61—24no—
58C2×C4×Dic61—21no—
59C4⋊#92418—21no—
60C4⋊#92418—21no—
61#14519700—24no—
62#14519700—22no—
63#14519700—22no—
64C22⋊#92418C2312yes—
65C22⋊#92418C2312yes—
66#14519688—21no—
67C4×#92418—21no—
68#14519684—21no—
69#14519684—24no—
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—22no—
166C2×Dic122—22no—
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—
201#11758202—82no—
202#11758200—84no—
203#11758200—82no—
204C4⋊Dic61—41no—
205C4⋊Dic61—41no—
206C4⋊Dic61—41no—
207C4⋊Dic61—41no—
208C4×Dic61—42no—
209C2×#92418—22no—
210C2×#92418—22no—
211C23×C61—22no—
212C23×C61—21no—
213C23×C61—22no—
214C23×C61—24no—
215C23×C61—42no—
216C23×C61—42no—
217C23×C61—82no—
218C23×C61—82no—
219C23×C61—82no—
220C23×C61C22×D411yescharacteristic
221Q8×C61—41no—
222Q8×C61—42no—
223Q8×C61—81no—
224Q8×C61—81no—
225D4×C61—42no—
226D4×C61—41no—
227D4×C61—42no—
228D4×C61—44no—
229D4×C61—42no—
230D4×C61—42no—
231D4×C61—81no—
232D4×C61—81no—
233D4×C61—82no—
234D4×C61—82no—
235D4×C61—82no—
236D4×C61—82no—
237D4×C61—82no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—41no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61—82no—
249C2×C4×C61—82no—
250C2×C4×C61—82no—
251C2×C4×C61#12011yescharacteristic
252C2×C4×C61#12011yescharacteristic
253C2×C4×C61#12011yescharacteristic
254C2×C4×C61#12011yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×Dic61—42no—
258C2×Dic61—42no—
259C2×Dic61—44no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262Dic122—44no—
263Dic122—42no—
264Dic122—42no—
265#92418—42no—
266#92418—42no—
267#18427—611nomaximal, Sylow
268C22×C61—21no—
269C22×C61—21no—
270C22×C61—42no—
271C22×C61—42no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—42no—
275C22×C61—42no—
276C22×C61—42no—
277C22×C61—41no—
278C22×C61—42no—
279C22×C61—84no—
280C22×C61—164no—
281C22×C61C2×#12011yescharacteristic
282C22×C61#57712yes—
283C4×C61—21no—
284C4×C61—21no—
285C4×C61—21no—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—41no—
289C4×C61—42no—
290C4×C61—41no—
291C4×C61—42no—
292C4×C61—41no—
293C4×C61#45512yes—
294Dic61—82no—
295Dic61—82no—
296#3558D6111yescharacteristic
297#3218—611no—
298#3206—611no—
299#2640—611no—
300#1606—614no—
301#1601—614no—
302#1505—611no—
303#1500—611no—
304#1469—611no—
305C2×C61—22no—
306C2×C61—82no—
307C2×C61—82no—
308C2×C61—81no—
309C2×C61#300911yescharacteristic
310C2×C61C2×#45511yescharacteristic
311C2×C61#302711yescharacteristic
312#562—21no—
313H(3,𝔽4)—21no—
314#559—21no—
315#548—21no—
316#546—21no—
317#546—22no—
318D42—21no—
319#542—21no—
320#538—22no—
321#538C2×D6112yes—
322#535—21no—
323#534C2×D6112yes—
324#532C2×D6111yescharacteristic
325C2×C4⋊Q8—611no—
326#529C2×D6111yescharacteristic
327C2×#124C2×D6111yescharacteristic
328#497—1221no—
329#492—1221no—
330#491—1221no—
331#487—1221no—
332#462—1221no—
333#461—1222no—
334#461—1221no—
335#451—612no—
336#448—612no—
337#448—614no—
338#432—611no—
339#431—611no—
340#397—1222no—
341#393—614no—
342#391—1221no—
343#390—1221no—
344#376—611no—
345#328—612no—
346#327—612no—
347#327—614no—
348#323—614no—
349C61#1842711yescharacteristic, complemented, Sylow
350C2×#41—21no—
351C2×#41—21no—
352C22×Q8—612no—
353C22×Q8C22⋊D6111yescharacteristic
354C22×D4—21no—
355C22×D4—22no—
356C22×D4C22⋊D6111yescharacteristic
357C2×Q16—1222no—
358C2×SD16—1224no—
359C2×SD16—1222no—
360C4⋊Q8—1221no—
361C4⋊Q8—1221no—
362C4⋊Q8—1221no—
363C4⋊Q8—1221no—
364C4⋊Q8C22⋊D6111yescharacteristic
365#127—21no—
366#127C22⋊D6111yescharacteristic
367#126—22no—
368#126—21no—
369#125C22⋊D6111yescharacteristic
370#124—21no—
371#124—22no—
372#124—22no—
373#124—22no—
374#124—21no—
375#124—21no—
376#124—21no—
377#124—21no—
378#124—21no—
379#124C22⋊D6111yescharacteristic
380#123—22no—
381#123—21no—
382#123—22no—
383#123—22no—
384#123—22no—
385C22⋊Q8—22no—
386C22⋊Q8—21no—
387C22⋊Q8—22no—
388C22⋊Q8C22×D6111yescharacteristic
389#121—22no—
390#121—22no—
391#121—22no—
392#121—21no—
393#121—22no—
394#121—21no—
395#121—22no—
396#121C22×D6111yescharacteristic
397#121C22×D6112yes—
398#120—22no—
399#120—22no—
400#120—22no—
401#120—22no—
402#120—22no—
403C4×Q8—21no—
404C4×Q8C22⋊D6111yescharacteristic
405C4×D4—22no—
406C4×D4—21no—
407C4×D4—21no—
408C4×D4—22no—
409C4×D4—21no—
410C4×D4C22⋊D6111yescharacteristic
411C4×D4C22⋊D6112yes—
412#117—21no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C22⋊C4—22no—
416C2×C22⋊C4—1224no—
417C2×C22⋊C4C22⋊D6112yes—
418C2×C22⋊C4C22×D6112yes—
419C2×C42—1221no—
420C2×C42C22×D6111yescharacteristic
421C4⋊C8—1221no—
422C4⋊C8—1221no—
423#103—1224no—
424#103—1222no—
425#103—1222no—
426C22⋊C8—612no—
427C22⋊C8—612no—
428#97—1221no—
429C4×C8—1221no—
430#95—612no—
431#95—1221no—
432#95—1224no—
433C24—22no—
434C24C2×C22⋊D6112yes—
435#41—82no—
436#41—82no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—42no—
441C2×Q8—1222no—
442C2×Q8—1222no—
443C2×Q8—4882no—
444C2×Q8C2×C22⋊D6111yescharacteristic
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—42no—
457C2×D4—82no—
458C2×D4—84no—
459C2×D4C2×C22⋊D6111yescharacteristic
460C2×D4C2×C22⋊D6112yes—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4C2×C22⋊D6111yescharacteristic
468C22×C4C2×C22⋊D6111yescharacteristic
469C22×C4C2×C22⋊D6112yes—
470C22×C4C23×D6111yescharacteristic
471Q16—4882no—
472SD16—4884no—
473SD16—4882no—
474C2×C8—1222no—
475C2×C8—1222no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—2441no—
484C4⋊C4—2441no—
485C4⋊C4—2441no—
486C4⋊C4—2441no—
487C4⋊C4C2×C22⋊D6112yes—
488C4⋊C4C2×C22⋊D6111yescharacteristic
489C4⋊C4C2×C22⋊D6111yescharacteristic
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—21no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—21no—
499C22⋊C4—22no—
500C22⋊C4—21no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—22no—
506C22⋊C4—4884no—
507C42—22no—
508C42—21no—
509C42—2442no—
510C42C2×C22⋊D6111yescharacteristic
511C42C2×C22⋊D6111yescharacteristic
512C42C2×C22⋊D6111yescharacteristic
513C42C2×C22⋊D6111yescharacteristic
514C23—22no—
515C23—21no—
516C23—22no—
517C23—24no—
518C23—42no—
519C23—42no—
520C23—82no—
521C23—82no—
522C23—82no—
523C23C22×C22⋊D6111yescharacteristic
524Q8—41no—
525Q8—42no—
526Q8—81no—
527Q8—81no—
528Q8—2444no—
529Q8—2442no—
530Q8—2442no—
531D4—42no—
532D4—41no—
533D4—42no—
534D4—44no—
535D4—42no—
536D4—42no—
537D4—81no—
538D4—81no—
539D4—82no—
540D4—82no—
541D4—82no—
542D4—82no—
543D4—82no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—2442no—
558C2×C4—2442no—
559C2×C4—2444no—
560C2×C4—2442no—
561C2×C4—2442no—
562C2×C4#1451974611yescharacteristic
563C2×C4#1451974611yescharacteristic
564C2×C4#1451974611yescharacteristic
565C2×C4#1451974611yescharacteristic
566C2×C4C22×C22⋊D6111yescharacteristic
567C2×C4C22×C22⋊D6111yescharacteristic
568C8—2442no—
569C8—2442no—
570C22—21no—
571C22—21no—
572C22—41no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—41no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—41no—
581C22—84no—
582C22—164no—
583C22C2×#1451974611yescharacteristic
584C22#1684057312yes—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—21no—
590C4—41no—
591C4—42no—
592C4—41no—
593C4—41no—
594C4—42no—
595C4—4882no—
596C4—4882no—
597C4#1683994012yes—
598C2—22no—
599C2—82no—
600C2—81no—
601C2—82no—
602C2#2145509611yescharacteristic
603C2C2×#1683994011yescharacteristic
604C2#2145522111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3558C2
socleC22×C61C2×#120
hypercenter#3558D61
hypocenterC61#18427
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. #21456843C2semidirect
2. #21456741C2semidirect
3. #21451759C2semidirect
4. #21444799C2semidirect
5. #21444771C2semidirect
6. #21444327C2semidirect
7. #21444305C2semidirect
8. #21444171C2semidirect
10. C2×#14519777C22semidirect
18. #16839917C22semidirect
20. #16839900C22semidirect
21. #16839819C22semidirect
22. #16839815C22semidirect
24. #16839674C22semidirect
27. #16839608C22semidirect
28. #16839499C22semidirect
30. #16839496C22semidirect
31. #16839486C22semidirect
349. C61#18427semidirect

Maximal subgroups

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

subgroupindex
#214568432
#214567412
#214517592
#214447992
#214447712
#214443272
#214443052
#214441712
C61×#35582
#1842761