Finite Group Databasebeta

← #29656758

Subgroups — #29656758

Order 15616.

878 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456843C211yesmaximal, characteristic
2#21456366C211yesmaximal, characteristic, complemented
3C2×#16840047C211yesmaximal, characteristic, complemented
4#21445096C212yesmaximal, complemented
5#21445082C212yesmaximal, complemented
6#21445075C212yesmaximal, complemented
7#21445061C212yesmaximal, complemented
8#21444519C211yesmaximal, characteristic, complemented
9#21444266C211yesmaximal, characteristic, complemented
10#21444232C211yesmaximal, characteristic
11C61×#3523C211yesmaximal, characteristic
12C2×#14519777C2211yescharacteristic
13#16840172—22no—
14#16840165C2212yes—
15#16840158C2212yes—
16#16840157—21no—
17#16840132—21no—
18#16840047—21no—
19#16840047—21no—
20#16840047C2212yes—
21#16840047C2212yes—
22#16839972C2212yes—
23#16839967C2212yes—
24#16839917—21no—
25#16839900—21no—
26#16839819C2211yescharacteristic
27#16839805—22no—
28C2×#14519708—21no—
29C2×C4⋊#92418C2211yescharacteristic
30C2×#14519701—21no—
31C4×#11758194C2211yescharacteristic
32#16839508C2212yes—
33#16839508C2212yes—
34Q8⋊#92418C2212yes—
35Q8⋊#92418C2212yes—
36Q8⋊#92418C2212yes—
37D4⋊#92418C2212yes—
38C61×#557—21no—
39C61×#557C2211yescharacteristic
40C61×#557C2211yescharacteristic
41C61×#556—21no—
42C61×#556C2211yescharacteristic
43C61×#552—22no—
44C61×#552—21no—
45D4×Q8×C61—21no—
46D4×Q8×C61C2211yescharacteristic
47C61×#541—22no—
48C61×#541—21no—
49C61×#533—21no—
50C61×#533C2211yescharacteristic
51C61×#529—21no—
52C61×#529C2211yescharacteristic
53C61×#519—21no—
54C2×C4×Q8×C61C2211yescharacteristic
55C22×Dic122—21no—
56C2×#11758202—22no—
57C2×#11758200—22no—
58C2×#11758194—21no—
59C22×#92418—21no—
60#14519777—21no—
61#14519777—21no—
62#14519777—21no—
63#14519777C2312yes—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—21no—
66#14519710—24no—
67#14519708—22no—
68#14519708—22no—
69#14519708—24no—
70C4⋊#92418—21no—
71C4⋊#92418C2312yes—
72C4⋊#92418C2312yes—
73#14519701—22no—
74#14519701—22no—
75#14519700—22no—
76#14519700—22no—
77C22⋊#92418—21no—
78#14519688—21no—
79#14519688—21no—
80C4×#92418C2312yes—
81C22×Q8×C61—21no—
82C22×Q8×C61D411yescharacteristic
83C4⋊Q8×C61—22no—
84C4⋊Q8×C61—21no—
85C4⋊Q8×C61—21no—
86C4⋊Q8×C61—21no—
87C4⋊Q8×C61C2311yescharacteristic
88C4⋊Q8×C61C2311yescharacteristic
89#126×C61—22no—
90#126×C61—22no—
91#126×C61—22no—
92#126×C61—22no—
93#126×C61—21no—
94#125×C61—22no—
95#125×C61—22no—
96#125×C61—22no—
97#125×C61—22no—
98#125×C61—22no—
99#125×C61—21no—
100#125×C61—21no—
101#125×C61D412yes—
102#125×C61D411yescharacteristic
103#124×C61—21no—
104#124×C61D411yescharacteristic
105#123×C61—22no—
106#123×C61—21no—
107C22⋊Q8×C61—22no—
108C22⋊Q8×C61—22no—
109C22⋊Q8×C61—21no—
110C22⋊Q8×C61—21no—
111C22⋊Q8×C61—21no—
112C22⋊Q8×C61—21no—
113C22⋊Q8×C61—22no—
114C22⋊Q8×C61D411yescharacteristic
115#121×C61D411yescharacteristic
116C4×Q8×C61—22no—
117C4×Q8×C61—22no—
118C4×Q8×C61—21no—
119C4×Q8×C61—21no—
120C4×Q8×C61—21no—
121C4×Q8×C61—21no—
122C4×Q8×C61—21no—
123C4×Q8×C61—21no—
124C4×Q8×C61—21no—
125C4×Q8×C61—21no—
126C4×Q8×C61—21no—
127C4×Q8×C61D411yescharacteristic
128C4×Q8×C61D411yescharacteristic
129C4×Q8×C61C2311yescharacteristic
130C4×Q8×C61C2311yescharacteristic
131C4×Q8×C61C2311yescharacteristic
132C4×D4×C61—22no—
133C4×D4×C61—21no—
134C4×D4×C61—21no—
135C4×D4×C61C2311yescharacteristic
136#117×C61—22no—
137#117×C61—22no—
138#117×C61—22no—
139#117×C61—21no—
140#117×C61—21no—
141#117×C61D412yes—
142C2×C4⋊C4×C61—21no—
143C2×C4⋊C4×C61D411yescharacteristic
144C2×C42×C61—21no—
145C2×C42×C61C2311yescharacteristic
146C2×Q8×C61—22no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61—21no—
150C2×Q8×C61—22no—
151C2×Q8×C61—41no—
152C2×Q8×C61—42no—
153C2×Q8×C61—42no—
154C2×Q8×C61C2×D411yescharacteristic
155C2×Q8×C61C2×D411yescharacteristic
156C2×Q8×C61C2×D411yescharacteristic
157C2×D4×C61—21no—
158C2×D4×C61C2×D411yescharacteristic
159C22×C4×C61—22no—
160C22×C4×C61—21no—
161C22×C4×C61—21no—
162C22×C4×C61—21no—
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61C2×D411yescharacteristic
165C22×C4×C61C2×D411yescharacteristic
166C22×Dic61—41no—
167C2×Dic122—22no—
168C2×Dic122—41no—
169C2×Dic122—82no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—22no—
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
193C4⋊C4×C61C2×D412yes—
194C4⋊C4×C61C2×D412yes—
195C4⋊C4×C61C2×D411yescharacteristic
196C4⋊C4×C61C2×D411yescharacteristic
197C22⋊C4×C61—22no—
198C22⋊C4×C61—22no—
199C22⋊C4×C61—22no—
200C22⋊C4×C61—21no—
201C22⋊C4×C61—22no—
202C22⋊C4×C61—21no—
203C22⋊C4×C61—21no—
204C22⋊C4×C61—21no—
205C22⋊C4×C61—21no—
206C42×C61—22no—
207C42×C61—22no—
208C42×C61—22no—
209C42×C61—21no—
210C42×C61—21no—
211C42×C61—22no—
212C42×C61—21no—
213C42×C61—21no—
214C42×C61—21no—
215C42×C61C2×D411yescharacteristic
216C42×C61C2×D411yescharacteristic
217C42×C61C2×D411yescharacteristic
218C42×C61C2411yescharacteristic
219#11758202—82no—
220#11758200—82no—
221C4⋊Dic61—22no—
222C4⋊Dic61—22no—
223C4⋊Dic61—41no—
224C4⋊Dic61—41no—
225#11758194—24no—
226C2×#92418—22no—
227C2×#92418—22no—
228C2×#92418—22no—
229C2×#92418—42no—
230C23×C61—21no—
231C23×C61#12011yescharacteristic
232Q8×C61—21no—
233Q8×C61—21no—
234Q8×C61—21no—
235Q8×C61—22no—
236Q8×C61—42no—
237Q8×C61—44no—
238Q8×C61—41no—
239Q8×C61—41no—
240Q8×C61—42no—
241Q8×C61#13712yes—
242D4×C61—22no—
243D4×C61—42no—
244C2×C4×C61—21no—
245C2×C4×C61—22no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—22no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—22no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—22no—
257C2×C4×C61—42no—
258C2×C4×C61—44no—
259C2×C4×C61—42no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61#12011yescharacteristic
263C2×C4×C61#12011yescharacteristic
264C2×C4×C61#12011yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C22×D411yescharacteristic
268C2×Dic61—42no—
269C2×Dic61—82no—
270Dic122—42no—
271Dic122—42no—
272Dic122—82no—
273#92418—24no—
274#92418—42no—
275#92418—42no—
276#18589—611nomaximal, Sylow
277C22×C61—21no—
278C22×C61—21no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61#45612yes—
283C22×C61C2×#12011yescharacteristic
284C4×C61—21no—
285C4×C61—22no—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—21no—
289C4×C61—41no—
290C4×C61—42no—
291C4×C61—41no—
292C4×C61—41no—
293C4×C61—42no—
294C4×C61—41no—
295C4×C61—41no—
296C4×C61—41no—
297C4×C61C2×#13711yescharacteristic
298C4×C61#57511yescharacteristic
299C4×C61#57712yes—
300Dic61—82no—
301#3523D6111yescharacteristic
302#3218—611no—
303#3165—611no—
304C2×#475—611no—
305#1653—612no—
306#1651—612no—
307#1650—612no—
308#1648—612no—
309#1552—611no—
310#1491—611no—
311#1483—611no—
312C2×C61—22no—
313C2×C61—81no—
314C2×C61#300711yescharacteristic
315C2×C61#301011yescharacteristic
316C2×C61C2×#45611yescharacteristic
317#557—21no—
318#557C2×D6111yescharacteristic
319#557C2×D6111yescharacteristic
320#556—21no—
321#556C2×D6111yescharacteristic
322#552—22no—
323#552—21no—
324D4×Q8—21no—
325D4×Q8C2×D6111yescharacteristic
326#541—22no—
327#541—21no—
328#533—21no—
329#533C2×D6111yescharacteristic
330C2×C4⋊Q8—611no—
331#529—21no—
332#529C2×D6111yescharacteristic
333#519—21no—
334C2×C4×Q8C2×D6111yescharacteristic
335#501—1222no—
336#500—612no—
337#498—612no—
338#497—1221no—
339#491—1221no—
340#475—612no—
341#475—612no—
342#475—1221no—
343#475—1221no—
344#462—612no—
345#461—612no—
346#451—1221no—
347#448—1221no—
348#432—611no—
349#428—1222no—
350C2×#106—1221no—
351C2×C4⋊C8—611no—
352C2×#103—1221no—
353C4×M16—611no—
354#332—612no—
355#332—612no—
356Q8⋊C8—612no—
357Q8⋊C8—612no—
358Q8⋊C8—612no—
359D4⋊C8—612no—
360C61#1858911yescharacteristic, complemented, Sylow
361C22×Q8—21no—
362C22×Q8—1221no—
363C22×Q8C22⋊D6111yescharacteristic
364C2×Q16—1222no—
365C2×SD16—1222no—
366C2×M16—1221no—
367C22×C8—1221no—
368C4⋊Q8—22no—
369C4⋊Q8—21no—
370C4⋊Q8—21no—
371C4⋊Q8—21no—
372C4⋊Q8—612no—
373C4⋊Q8—1221no—
374C4⋊Q8—1221no—
375C4⋊Q8—1221no—
376C4⋊Q8C22×D6111yescharacteristic
377C4⋊Q8C22×D6111yescharacteristic
378#126—22no—
379#126—22no—
380#126—22no—
381#126—22no—
382#126—21no—
383#125—22no—
384#125—22no—
385#125—22no—
386#125—21no—
387#125—22no—
388#125—22no—
389#125—21no—
390#125C22⋊D6111yescharacteristic
391#125C22⋊D6112yes—
392#124—21no—
393#124C22⋊D6111yescharacteristic
394#123—22no—
395#123—21no—
396C22⋊Q8—22no—
397C22⋊Q8—22no—
398C22⋊Q8—21no—
399C22⋊Q8—21no—
400C22⋊Q8—21no—
401C22⋊Q8—21no—
402C22⋊Q8—22no—
403C22⋊Q8C22⋊D6111yescharacteristic
404#121C22⋊D6111yescharacteristic
405C4×Q8—22no—
406C4×Q8—22no—
407C4×Q8—21no—
408C4×Q8—21no—
409C4×Q8—21no—
410C4×Q8—21no—
411C4×Q8—21no—
412C4×Q8—21no—
413C4×Q8—21no—
414C4×Q8—21no—
415C4×Q8—21no—
416C4×Q8C22⋊D6111yescharacteristic
417C4×Q8C22⋊D6111yescharacteristic
418C4×Q8C22×D6111yescharacteristic
419C4×Q8C22×D6111yescharacteristic
420C4×Q8C22×D6111yescharacteristic
421C4×D4—22no—
422C4×D4—21no—
423C4×D4—21no—
424C4×D4C22×D6111yescharacteristic
425#117—22no—
426#117—22no—
427#117—22no—
428#117—21no—
429#117—21no—
430#117C22⋊D6112yes—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4C22⋊D6111yescharacteristic
435C2×C42—21no—
436C2×C42C22×D6111yescharacteristic
437#107—1224no—
438#106—1222no—
439#106—1222no—
440#106—1224no—
441C4⋊C8—612no—
442C4⋊C8—612no—
443C4⋊C8—1221no—
444#103—1222no—
445#103—1222no—
446#103—1222no—
447#103—1222no—
448C22⋊C8—1221no—
449#97—1221no—
450#97—1221no—
451C4×C8—612no—
452C2×Q8—22no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—22no—
457C2×Q8—41no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8—1222no—
461C2×Q8—4882no—
462C2×Q8—2441no—
463C2×Q8C2×C22⋊D6111yescharacteristic
464C2×Q8C2×C22⋊D6111yescharacteristic
465C2×Q8C2×C22⋊D6111yescharacteristic
466C2×D4—21no—
467C2×D4C2×C22⋊D6111yescharacteristic
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—2441no—
473C22×C4C2×C22⋊D6111yescharacteristic
474C22×C4C2×C22⋊D6111yescharacteristic
475C22×C4C2×C22⋊D6111yescharacteristic
476Q16—4882no—
477SD16—4882no—
478M16—1224no—
479C2×C8—1222no—
480C2×C8—1222no—
481C2×C8—1222no—
482C2×C8—2442no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—1222no—
505C4⋊C4—1222no—
506C4⋊C4—2441no—
507C4⋊C4—2441no—
508C4⋊C4C2×C22⋊D6111yescharacteristic
509C4⋊C4C2×C22⋊D6111yescharacteristic
510C4⋊C4C2×C22⋊D6111yescharacteristic
511C4⋊C4C2×C22⋊D6111yescharacteristic
512C4⋊C4C2×C22⋊D6112yes—
513C4⋊C4C2×C22⋊D6112yes—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—21no—
518C22⋊C4—22no—
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—21no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—22no—
532C42C2×C22⋊D6111yescharacteristic
533C42C2×C22⋊D6111yescharacteristic
534C42C2×C22⋊D6111yescharacteristic
535C42C23×D6111yescharacteristic
536C23—21no—
537C23#1451974611yescharacteristic
538Q8—21no—
539Q8—21no—
540Q8—21no—
541Q8—22no—
542Q8—42no—
543Q8—44no—
544Q8—41no—
545Q8—41no—
546Q8—42no—
547Q8—4882no—
548Q8—2442no—
549Q8—2442no—
550Q8#1451980712yes—
551D4—22no—
552D4—42no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—4882no—
572C2×C4—2442no—
573C2×C4#1451974611yescharacteristic
574C2×C4#1451974611yescharacteristic
575C2×C4#1451974611yescharacteristic
576C2×C4C22×C22⋊D6111yescharacteristic
577C2×C4C22×C22⋊D6111yescharacteristic
578C2×C4C22×C22⋊D6111yescharacteristic
579C8—1224no—
580C8—2442no—
581C8—2442no—
582C22—21no—
583C22—21no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22#1683994512yes—
588C22C2×#1451974611yescharacteristic
589C4—21no—
590C4—21no—
591C4—21no—
592C4—21no—
593C4—22no—
594C4—41no—
595C4—42no—
596C4—41no—
597C4—41no—
598C4—41no—
599C4—41no—
600C4—42no—
601C4—41no—
602C4—4882no—
603C4C2×#1451980711yescharacteristic
604C4#1684056011yescharacteristic
605C4#1684057012yes—
606C2—22no—
607C2—81no—
608C2#2145507911yescharacteristic
609C2#2145510511yescharacteristic
610C2C2×#1683994511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3523C2
socleC22×C61C2×#120
prime radical#121×C61D4
hypercenter#3523D61
hypocenterC61#18589
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3523

Complements

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

NH ≅ G/Ntype
2. #21456366C2semidirect
3. C2×#16840047C2semidirect
4. #21445096C2semidirect
5. #21445082C2semidirect
6. #21445075C2semidirect
7. #21445061C2semidirect
8. #21444519C2semidirect
9. #21444266C2semidirect
360. C61#18589semidirect

Maximal subgroups

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

subgroupindex
#214568432
#214563662
C2×#168400472
#214450962
#214450822
#214450752
#214450612
#214445192
#214442662
#214442322
C61×#35232
#1858961