Finite Group Databasebeta

← #29661736

Subgroups — #29661736

Order 15616.

950 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456822C211yesmaximal, characteristic, complemented
2#21456529C211yesmaximal, characteristic, complemented
3#21456164C211yesmaximal, characteristic, complemented
4#21445142C212yesmaximal, complemented
5#21445136C212yesmaximal, complemented
6#21445125C212yesmaximal, complemented
7#21445125C212yesmaximal, complemented
8#21444439C211yesmaximal, characteristic, complemented
9#21444430C211yesmaximal, characteristic, complemented
10C2×#16839502C211yesmaximal, characteristic, complemented
11C61×#3558C211yesmaximal, characteristic
12#16840319C2211yescharacteristic, complemented
13#16840172C2211yescharacteristic, complemented
14#16840157C2211yescharacteristic, complemented
15#16840157C2212yescomplemented
16#16840132C2211yescharacteristic, complemented
17#16840130—21no—
18#16840126—21no—
19#16840121C2211yescharacteristic, complemented
20#16840121C2212yescomplemented
21#16840121C2212yescomplemented
22#16840121C2212yescomplemented
23#16840001—21no—
24#16839989—21no—
25#16839981—22no—
26#16839918—21no—
27#16839913—21no—
28#16839867—21no—
29#16839862—21no—
30#16839819C2211yescharacteristic, complemented
31#16839758—22no—
32C2×#14519688C2211yescharacteristic, complemented
33C2×#14519688C2211yescharacteristic, complemented
34#16839504C2212yescomplemented
35#16839502C2212yescomplemented
36#16839502C2212yescomplemented
37#16839502C2212yescomplemented
38#16839502C2212yescomplemented
39#16839502C2212yescomplemented
40C61×#562C2212yes—
41C61×H(3,𝔽4)C2212yes—
42C61×#559—21no—
43C61×#548—21no—
44C61×#546—22no—
45C61×#546—21no—
46D42×C61—21no—
47C61×#542—21no—
48C61×#538—22no—
49C61×#538—21no—
50C61×#535—21no—
51C61×#534—21no—
52C61×#532C2211yescharacteristic
53C61×#529C2211yescharacteristic
54C2×#124×C61C2211yescharacteristic
55C2×#11758202—21no—
56C2×#11758202—22no—
57C2×#11758200—21no—
58C2×#11758200—22no—
59C22×#92418—21no—
60C22×#92418—21no—
61#14519777C2311yescharacteristic
62#14519777C2311yescharacteristic
63#14519769—21no—
64#14519756—21no—
65#14519756—21no—
66C4×Dic122—21no—
67C4×Dic122—21no—
68#14519736—21no—
69#14519710—21no—
70#14519708—21no—
71C4⋊#92418—21no—
72#14519701—22no—
73#14519701—21no—
74#14519701—22no—
75#14519701—22no—
76#14519700—21no—
77#14519700—21no—
78#14519698—22no—
79#14519698—21no—
80#14519698—22no—
81#14519698—22no—
82C22⋊#92418—21no—
83#14519688—21no—
84#14519688—21no—
85#14519688C2311yescharacteristic
86#14519688C2311yescharacteristic
87#14519688C2312yes—
88#14519688C2312yes—
89C2×#41×C61—21no—
90C2×#41×C61D412yes—
91C22×Q8×C61D411yescharacteristic
92C22×D4×C61—22no—
93C22×D4×C61—21no—
94C22×D4×C61D411yescharacteristic
95C4⋊Q8×C61D411yescharacteristic
96#127×C61—21no—
97#127×C61D411yescharacteristic
98#126×C61—22no—
99#126×C61—21no—
100#125×C61C2311yescharacteristic
101#124×C61—21no—
102#124×C61—22no—
103#124×C61—22no—
104#124×C61—22no—
105#124×C61—21no—
106#124×C61—21no—
107#124×C61D412yes—
108#124×C61C2312yes—
109#124×C61C2311yescharacteristic
110#124×C61C2312yes—
111#123×C61—22no—
112#123×C61—22no—
113#123×C61—21no—
114#123×C61—22no—
115#123×C61—22no—
116C22⋊Q8×C61—21no—
117C22⋊Q8×C61—22no—
118C22⋊Q8×C61—22no—
119C22⋊Q8×C61D411yescharacteristic
120#121×C61—22no—
121#121×C61—22no—
122#121×C61—22no—
123#121×C61—21no—
124#121×C61—22no—
125#121×C61—22no—
126#121×C61—21no—
127#121×C61—21no—
128#121×C61D411yescharacteristic
129#120×C61—22no—
130#120×C61—22no—
131#120×C61—22no—
132#120×C61—22no—
133#120×C61—22no—
134C4×Q8×C61—21no—
135C4×Q8×C61D411yescharacteristic
136C4×D4×C61—22no—
137C4×D4×C61—22no—
138C4×D4×C61—21no—
139C4×D4×C61—21no—
140C4×D4×C61—21no—
141C4×D4×C61—21no—
142C4×D4×C61D411yescharacteristic
143#117×C61—21no—
144C2×C22⋊C4×C61—22no—
145C2×C22⋊C4×C61—21no—
146C2×C22⋊C4×C61—21no—
147C2×C42×C61C2311yescharacteristic
148C24×C61—22no—
149C24×C61—21no—
150#41×C61—44no—
151#41×C61—82no—
152C2×Q8×C61—21no—
153C2×Q8×C61—81no—
154C2×Q8×C61C2×D411yescharacteristic
155C2×Q8×C61C2×D412yes—
156C2×Q8×C61C2×D412yes—
157C2×D4×C61—22no—
158C2×D4×C61—21no—
159C2×D4×C61—22no—
160C2×D4×C61—22no—
161C2×D4×C61—22no—
162C2×D4×C61—22no—
163C2×D4×C61—22no—
164C2×D4×C61—21no—
165C2×D4×C61—21no—
166C2×D4×C61—21no—
167C2×D4×C61—81no—
168C2×D4×C61—82no—
169C2×D4×C61—84no—
170C2×D4×C61C2×D411yescharacteristic
171C2×D4×C61C2×D412yes—
172C2×D4×C61C2×D412yes—
173C22×C4×C61—22no—
174C22×C4×C61—22no—
175C22×C4×C61—21no—
176C22×C4×C61—21no—
177C22×C4×C61—22no—
178C22×C4×C61C2×D411yescharacteristic
179C22×C4×C61C2×D411yescharacteristic
180C22×C4×C61C2×D411yescharacteristic
181C2×Dic122—21no—
182C2×Dic122—21no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61C2×D411yescharacteristic
192C4⋊C4×C61C2×D411yescharacteristic
193C22⋊C4×C61—22no—
194C22⋊C4×C61—22no—
195C22⋊C4×C61—22no—
196C22⋊C4×C61—22no—
197C22⋊C4×C61—22no—
198C22⋊C4×C61—21no—
199C22⋊C4×C61—22no—
200C22⋊C4×C61—21no—
201C22⋊C4×C61—22no—
202C22⋊C4×C61—21no—
203C22⋊C4×C61—22no—
204C22⋊C4×C61—22no—
205C22⋊C4×C61—21no—
206C22⋊C4×C61—21no—
207C22⋊C4×C61—22no—
208C22⋊C4×C61—22no—
209C42×C61—22no—
210C42×C61—21no—
211C42×C61C2×D411yescharacteristic
212C42×C61C2×D411yescharacteristic
213C42×C61C2×D411yescharacteristic
214C42×C61C2411yescharacteristic
215C22⋊Dic61—41no—
216#11758202—42no—
217#11758202—82no—
218#11758200—42no—
219#11758200—82no—
220C4⋊Dic61—21no—
221C4⋊Dic61—21no—
222C4⋊Dic61—21no—
223C4⋊Dic61—21no—
224#11758196—41no—
225#11758196—41no—
226C4×Dic61—41no—
227C2×#92418—21no—
228C2×#92418—22no—
229C2×#92418—21no—
230C2×#92418—22no—
231C2×#92418—42no—
232C2×#92418—42no—
233C23×C61—22no—
234C23×C61—21no—
235C23×C61—22no—
236C23×C61—42no—
237C23×C61—81no—
238C23×C61—81no—
239C23×C61—82no—
240C23×C61—82no—
241C23×C61—82no—
242C23×C61#12011yescharacteristic
243Q8×C61—42no—
244Q8×C61—41no—
245Q8×C61—42no—
246Q8×C61—42no—
247D4×C61—41no—
248D4×C61—42no—
249D4×C61—42no—
250D4×C61—42no—
251D4×C61—42no—
252D4×C61—42no—
253D4×C61—44no—
254D4×C61—81no—
255D4×C61—82no—
256D4×C61—82no—
257D4×C61—82no—
258D4×C61—82no—
259D4×C61—82no—
260C2×C4×C61—22no—
261C2×C4×C61—21no—
262C2×C4×C61—22no—
263C2×C4×C61—22no—
264C2×C4×C61—21no—
265C2×C4×C61—21no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—42no—
269C2×C4×C61—82no—
270C2×C4×C61—81no—
271C2×C4×C61—82no—
272C2×C4×C61—82no—
273C2×C4×C61#12011yescharacteristic
274C2×C4×C61#12011yescharacteristic
275C2×C4×C61#12011yescharacteristic
276C2×C4×C61C22×D411yescharacteristic
277C2×C4×C61C22×D411yescharacteristic
278C2×C4×C61C22×D411yescharacteristic
279C2×Dic61—41no—
280C2×Dic61—41no—
281Dic122—41no—
282Dic122—42no—
283Dic122—41no—
284#92418—22no—
285#92418—42no—
286#92418—41no—
287#92418—42no—
288#19043—611nomaximal, Sylow
289C22×C61—21no—
290C22×C61—21no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61—41no—
294C22×C61—41no—
295C22×C61—42no—
296C22×C61—42no—
297C22×C61—41no—
298C22×C61—42no—
299C22×C61—42no—
300C22×C61—162no—
301C22×C61—164no—
302C22×C61#45512yes—
303C22×C61C2×#12011yescharacteristic
304C4×C61—21no—
305C4×C61—21no—
306C4×C61—21no—
307C4×C61—21no—
308C4×C61—21no—
309C4×C61—21no—
310C4×C61—41no—
311C4×C61—42no—
312C4×C61—41no—
313C4×C61—41no—
314C4×C61—42no—
315Dic61—81no—
316Dic61—81no—
317#3558D6111yescharacteristic
318#3215—611no—
319#3181—611no—
320#3135—611no—
321#1662—612no—
322#1661—612no—
323#1659—612no—
324#1659—612no—
325#1532—611no—
326#1530—611no—
327C2×#329—611no—
328C2×C61—22no—
329C2×C61—82no—
330C2×C61—82no—
331C2×C61—81no—
332C2×C61C2×#45511yescharacteristic
333C2×C61#301811yescharacteristic
334C2×C61#301911yescharacteristic
335#562C2×D6112yes—
336H(3,𝔽4)C2×D6112yes—
337#559—21no—
338#548—21no—
339#546—22no—
340#546—21no—
341D42—21no—
342#542—21no—
343#538—22no—
344#538—21no—
345#535—21no—
346#534—21no—
347#533—611no—
348#532C2×D6111yescharacteristic
349#529C2×D6111yescharacteristic
350C2×#124C2×D6111yescharacteristic
351#501—611no—
352#497—611no—
353#497—612no—
354#491—611no—
355#491—1221no—
356#490—1221no—
357#489—612no—
358#489—612no—
359#489—611no—
360#489—612no—
361#467—1221no—
362#465—1221no—
363#464—1222no—
364#451—1221no—
365#450—1221no—
366#441—1221no—
367#440—1221no—
368#432—611no—
369#416—1222no—
370C2×#97—611no—
371C2×#97—611no—
372#330—612no—
373#329—612no—
374#329—612no—
375#329—612no—
376#329—612no—
377#329—612no—
378C61#1904311yescharacteristic, complemented, Sylow
379C2×#41—21no—
380C2×#41C22⋊D6112yes—
381C22×Q8C22⋊D6111yescharacteristic
382C22×D4—22no—
383C22×D4—21no—
384C22×D4C22⋊D6111yescharacteristic
385C2×Q16—1222no—
386C2×Q16—1221no—
387C2×SD16—1222no—
388C2×SD16—1221no—
389C22×C8—1221no—
390C22×C8—1221no—
391C4⋊Q8—611no—
392C4⋊Q8—611no—
393C4⋊Q8C22⋊D6111yescharacteristic
394#127—21no—
395#127C22⋊D6111yescharacteristic
396#126—22no—
397#126—21no—
398#125—1221no—
399#125C22×D6111yescharacteristic
400#124—21no—
401#124—22no—
402#124—22no—
403#124—22no—
404#124—21no—
405#124—21no—
406#124C22⋊D6112yes—
407#124C22×D6112yes—
408#124C22×D6111yescharacteristic
409#124C22×D6112yes—
410#123—22no—
411#123—22no—
412#123—21no—
413#123—22no—
414#123—22no—
415C22⋊Q8—21no—
416C22⋊Q8—22no—
417C22⋊Q8—22no—
418C22⋊Q8—1221no—
419C22⋊Q8—1221no—
420C22⋊Q8C22⋊D6111yescharacteristic
421#121—22no—
422#121—22no—
423#121—22no—
424#121—21no—
425#121—22no—
426#121—22no—
427#121—21no—
428#121—21no—
429#121C22⋊D6111yescharacteristic
430#120—22no—
431#120—22no—
432#120—22no—
433#120—22no—
434#120—22no—
435C4×Q8—21no—
436C4×Q8—1221no—
437C4×Q8—1221no—
438C4×Q8C22⋊D6111yescharacteristic
439C4×D4—22no—
440C4×D4—22no—
441C4×D4—21no—
442C4×D4—21no—
443C4×D4—21no—
444C4×D4—21no—
445C4×D4C22⋊D6111yescharacteristic
446#117—21no—
447#117—1221no—
448C2×C22⋊C4—22no—
449C2×C22⋊C4—21no—
450C2×C22⋊C4—21no—
451C2×C42C22×D6111yescharacteristic
452#107—1221no—
453#106—1221no—
454C4⋊C8—1221no—
455#103—1222no—
456#103—1222no—
457#103—1221no—
458#103—1222no—
459#103—1221no—
460#103—1221no—
461#102—1222no—
462#102—1222no—
463#102—1221no—
464#102—1222no—
465C22⋊C8—1221no—
466#97—611no—
467#97—611no—
468#97—612no—
469#97—612no—
470#97—1221no—
471#97—1221no—
472C24—22no—
473C24—21no—
474#41—44no—
475#41—82no—
476C2×Q8—21no—
477C2×Q8—81no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×Q8C2×C22⋊D6112yes—
481C2×Q8C2×C22⋊D6111yescharacteristic
482C2×Q8C2×C22⋊D6112yes—
483C2×D4—22no—
484C2×D4—21no—
485C2×D4—22no—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—81no—
494C2×D4—82no—
495C2×D4—84no—
496C2×D4C2×C22⋊D6112yes—
497C2×D4C2×C22⋊D6111yescharacteristic
498C2×D4C2×C22⋊D6112yes—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—22no—
504C22×C4C2×C22⋊D6111yescharacteristic
505C22×C4C2×C22⋊D6111yescharacteristic
506C22×C4C2×C22⋊D6111yescharacteristic
507Q16—4882no—
508Q16—2442no—
509SD16—4882no—
510SD16—2442no—
511C2×C8—1221no—
512C2×C8—1222no—
513C2×C8—1221no—
514C2×C8—1222no—
515C2×C8—2442no—
516C2×C8—2442no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—2441no—
530C4⋊C4—2441no—
531C4⋊C4C2×C22⋊D6111yescharacteristic
532C4⋊C4C2×C22⋊D6111yescharacteristic
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—21no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—21no—
542C22⋊C4—21no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—21no—
547C22⋊C4—22no—
548C22⋊C4—21no—
549C22⋊C4—2441no—
550C42—22no—
551C42—21no—
552C42—2441no—
553C42C2×C22⋊D6111yescharacteristic
554C42C2×C22⋊D6111yescharacteristic
555C42C2×C22⋊D6111yescharacteristic
556C42C23×D6111yescharacteristic
557C23—22no—
558C23—21no—
559C23—22no—
560C23—42no—
561C23—81no—
562C23—81no—
563C23—82no—
564C23—82no—
565C23—82no—
566C23#1451974611yescharacteristic
567Q8—42no—
568Q8—41no—
569Q8—42no—
570Q8—42no—
571Q8—2442no—
572Q8—2441no—
573Q8—2441no—
574D4—42no—
575D4—41no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—42no—
580D4—44no—
581D4—81no—
582D4—82no—
583D4—82no—
584D4—82no—
585D4—82no—
586D4—82no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—82no—
597C2×C4—81no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4#1451974611yescharacteristic
603C2×C4#1451974611yescharacteristic
604C2×C4#1451974611yescharacteristic
605C2×C4C22×C22⋊D6111yescharacteristic
606C2×C4C22×C22⋊D6111yescharacteristic
607C2×C4C22×C22⋊D6111yescharacteristic
608C8—1222no—
609C8—2442no—
610C8—2441no—
611C8—2442no—
612C22—21no—
613C22—21no—
614C22—42no—
615C22—42no—
616C22—41no—
617C22—41no—
618C22—42no—
619C22—42no—
620C22—41no—
621C22—42no—
622C22—42no—
623C22—162no—
624C22—164no—
625C22#1683994012yes—
626C22C2×#1451974611yescharacteristic
627C4—21no—
628C4—21no—
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—41no—
634C4—42no—
635C4—41no—
636C4—41no—
637C4—42no—
638C4—4881no—
639C4—4881no—
640C2—22no—
641C2—82no—
642C2—81no—
643C2—82no—
644C2C2×#1683994011yescharacteristic
645C2#2145516811yescharacteristic
646C2#2145517611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3558C2
socleC22×C61C2×#120
hypercenter#3558D61
hypocenterC61#19043
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3558

Complements

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

NH ≅ G/Ntype
1. #21456822C2semidirect
2. #21456529C2semidirect
3. #21456164C2semidirect
4. #21445142C2semidirect
5. #21445136C2semidirect
6. #21445125C2semidirect
7. #21445125C2semidirect
8. #21444439C2semidirect
9. #21444430C2semidirect
10. C2×#16839502C2semidirect
12. #16840319C22semidirect
13. #16840172C22semidirect
14. #16840157C22semidirect
15. #16840157C22semidirect
16. #16840132C22semidirect
19. #16840121C22semidirect
20. #16840121C22semidirect
21. #16840121C22semidirect
22. #16840121C22semidirect
30. #16839819C22semidirect
32. C2×#14519688C22semidirect
33. C2×#14519688C22semidirect
34. #16839504C22semidirect
35. #16839502C22semidirect
36. #16839502C22semidirect
37. #16839502C22semidirect
38. #16839502C22semidirect
39. #16839502C22semidirect
378. C61#19043semidirect

Maximal subgroups

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

subgroupindex
#214568222
#214565292
#214561642
#214451422
#214451362
#214451252
#214444392
#214444302
C2×#168395022
C61×#35582
#1904361