Finite Group Databasebeta

← #29656670

Subgroups — #29656670

Order 15616.

866 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#21457037C211yesmaximal, characteristic
2#21456201C211yesmaximal, characteristic, complemented
3C2×#16840057C211yesmaximal, characteristic, complemented
4#21444991C212yesmaximal, complemented
5#21444984C212yesmaximal, complemented
6#21444977C212yesmaximal, complemented
7#21444977C212yesmaximal, complemented
8#21444519C211yesmaximal, characteristic, complemented
9#21444266C211yesmaximal, characteristic, complemented
10#21444232C211yesmaximal, characteristic
11C61×#3523C211yesmaximal, characteristic
12C2×#14519777C2211yescharacteristic
13#16840172—21no—
14#16840132—22no—
15#16840121—21no—
16#16840114C2212yes—
17#16840114C2212yes—
18#16840090—21no—
19#16840073—21no—
20#16840057—21no—
21#16840057—21no—
22#16840057C2212yes—
23#16840057C2212yes—
24#16840057C2212yes—
25#16840052C2212yes—
26#16839819C2211yescharacteristic
27C2×#14519710—21no—
28C2×C4⋊#92418C2211yescharacteristic
29#16839770—22no—
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×#11758194—21no—
57C22×#92418—21no—
58#14519777—21no—
59#14519777—21no—
60#14519777—21no—
61#14519777C2312yes—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61—21no—
64#14519710—22no—
65#14519710—22no—
66#14519710—22no—
67#14519708—22no—
68C4⋊#92418—21no—
69C4⋊#92418C2312yes—
70C4⋊#92418C2312yes—
71#14519701—22no—
72#14519701—22no—
73#14519701—22no—
74#14519700—24no—
75#14519700—24no—
76#14519698—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—44no—
169C2×Dic122—41no—
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
219C4⋊Dic61—22no—
220C4⋊Dic61—22no—
221C4⋊Dic61—41no—
222C4⋊Dic61—41no—
223#11758194—42no—
224C2×#92418—22no—
225C2×#92418—22no—
226C2×#92418—22no—
227C2×#92418—42no—
228C23×C61—21no—
229C23×C61#12011yescharacteristic
230Q8×C61—21no—
231Q8×C61—21no—
232Q8×C61—21no—
233Q8×C61—22no—
234Q8×C61—42no—
235Q8×C61—44no—
236Q8×C61—41no—
237Q8×C61—41no—
238Q8×C61—42no—
239Q8×C61#13712yes—
240D4×C61—22no—
241D4×C61—42no—
242C2×C4×C61—21no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—22no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—22no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—22no—
255C2×C4×C61—42no—
256C2×C4×C61—44no—
257C2×C4×C61—42no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61#12011yescharacteristic
261C2×C4×C61#12011yescharacteristic
262C2×C4×C61#12011yescharacteristic
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×Dic61—42no—
267C2×Dic61—82no—
268Dic122—44no—
269Dic122—82no—
270#92418—42no—
271#92418—42no—
272#92418—42no—
273#18581—611nomaximal, Sylow
274C22×C61—21no—
275C22×C61—21no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61#45412yes—
280C22×C61C2×#12011yescharacteristic
281C4×C61—21no—
282C4×C61—22no—
283C4×C61—21no—
284C4×C61—21no—
285C4×C61—21no—
286C4×C61—41no—
287C4×C61—42no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61—42no—
291C4×C61—41no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61C2×#13711yescharacteristic
295C4×C61#57511yescharacteristic
296C4×C61#57812yes—
297Dic61—82no—
298#3523D6111yescharacteristic
299#3238—611no—
300#3139—611no—
301C2×#477—611no—
302#1638—612no—
303#1637—612no—
304#1636—612no—
305#1636—612no—
306#1552—611no—
307#1491—611no—
308#1483—611no—
309C2×C61—22no—
310C2×C61—81no—
311C2×C61#300711yescharacteristic
312C2×C61#301111yescharacteristic
313C2×C61C2×#45411yescharacteristic
314#557—21no—
315#557C2×D6111yescharacteristic
316#557C2×D6111yescharacteristic
317#556—21no—
318#556C2×D6111yescharacteristic
319#552—22no—
320#552—21no—
321D4×Q8—21no—
322D4×Q8C2×D6111yescharacteristic
323#541—22no—
324#541—21no—
325#533—21no—
326#533C2×D6111yescharacteristic
327C2×C4⋊Q8—611no—
328#529—21no—
329#529C2×D6111yescharacteristic
330#519—21no—
331C2×C4×Q8C2×D6111yescharacteristic
332#501—1221no—
333#491—1222no—
334#489—1221no—
335#487—612no—
336#487—612no—
337#483—1221no—
338#480—1221no—
339#477—612no—
340#477—612no—
341#477—612no—
342#477—1221no—
343#477—1221no—
344#476—612no—
345#432—611no—
346C2×#107—1221no—
347C2×C4⋊C8—611no—
348#419—1222no—
349C2×#103—1221no—
350C4×M16—611no—
351#332—612no—
352#332—612no—
353Q8⋊C8—612no—
354Q8⋊C8—612no—
355Q8⋊C8—612no—
356D4⋊C8—612no—
357C61#1858111yescharacteristic, complemented, Sylow
358C22×Q8—21no—
359C22×Q8—1221no—
360C22×Q8C22⋊D6111yescharacteristic
361C2×M16—1221no—
362C22×C8—1221no—
363C4⋊Q8—22no—
364C4⋊Q8—21no—
365C4⋊Q8—21no—
366C4⋊Q8—21no—
367C4⋊Q8—612no—
368C4⋊Q8—1221no—
369C4⋊Q8—1221no—
370C4⋊Q8—1221no—
371C4⋊Q8C22×D6111yescharacteristic
372C4⋊Q8C22×D6111yescharacteristic
373#126—22no—
374#126—22no—
375#126—22no—
376#126—22no—
377#126—21no—
378#125—22no—
379#125—22no—
380#125—22no—
381#125—21no—
382#125—22no—
383#125—22no—
384#125—21no—
385#125C22⋊D6111yescharacteristic
386#125C22⋊D6112yes—
387#124—21no—
388#124C22⋊D6111yescharacteristic
389#123—22no—
390#123—21no—
391C22⋊Q8—22no—
392C22⋊Q8—22no—
393C22⋊Q8—21no—
394C22⋊Q8—21no—
395C22⋊Q8—21no—
396C22⋊Q8—21no—
397C22⋊Q8—22no—
398C22⋊Q8C22⋊D6111yescharacteristic
399#121C22⋊D6111yescharacteristic
400C4×Q8—22no—
401C4×Q8—22no—
402C4×Q8—21no—
403C4×Q8—21no—
404C4×Q8—21no—
405C4×Q8—21no—
406C4×Q8—21no—
407C4×Q8—21no—
408C4×Q8—21no—
409C4×Q8—21no—
410C4×Q8—21no—
411C4×Q8C22⋊D6111yescharacteristic
412C4×Q8C22⋊D6111yescharacteristic
413C4×Q8C22×D6111yescharacteristic
414C4×Q8C22×D6111yescharacteristic
415C4×Q8C22×D6111yescharacteristic
416C4×D4—22no—
417C4×D4—21no—
418C4×D4—21no—
419C4×D4C22×D6111yescharacteristic
420#117—22no—
421#117—22no—
422#117—22no—
423#117—21no—
424#117—21no—
425#117C22⋊D6112yes—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4C22⋊D6111yescharacteristic
430C2×C42—21no—
431C2×C42C22×D6111yescharacteristic
432#107—1222no—
433#107—1222no—
434#107—1222no—
435#106—1222no—
436C4⋊C8—612no—
437C4⋊C8—612no—
438C4⋊C8—1221no—
439#103—1222no—
440#103—1222no—
441#103—1222no—
442#103—1224no—
443#103—1224no—
444#102—1222no—
445C22⋊C8—1221no—
446#97—1221no—
447#97—1221no—
448C4×C8—612no—
449C2×Q8—22no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—22no—
454C2×Q8—41no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—1222no—
458C2×Q8—2444no—
459C2×Q8—2441no—
460C2×Q8C2×C22⋊D6111yescharacteristic
461C2×Q8C2×C22⋊D6111yescharacteristic
462C2×Q8C2×C22⋊D6111yescharacteristic
463C2×D4—21no—
464C2×D4C2×C22⋊D6111yescharacteristic
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—2441no—
470C22×C4C2×C22⋊D6111yescharacteristic
471C22×C4C2×C22⋊D6111yescharacteristic
472C22×C4C2×C22⋊D6111yescharacteristic
473M16—2442no—
474C2×C8—1222no—
475C2×C8—1222no—
476C2×C8—1222no—
477C2×C8—2442no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—1222no—
500C4⋊C4—1222no—
501C4⋊C4—2441no—
502C4⋊C4—2441no—
503C4⋊C4C2×C22⋊D6112yes—
504C4⋊C4C2×C22⋊D6112yes—
505C4⋊C4C2×C22⋊D6111yescharacteristic
506C4⋊C4C2×C22⋊D6111yescharacteristic
507C4⋊C4C2×C22⋊D6111yescharacteristic
508C4⋊C4C2×C22⋊D6111yescharacteristic
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—21no—
513C22⋊C4—22no—
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—21no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—22no—
527C42C2×C22⋊D6111yescharacteristic
528C42C2×C22⋊D6111yescharacteristic
529C42C2×C22⋊D6111yescharacteristic
530C42C23×D6111yescharacteristic
531C23—21no—
532C23#1451974611yescharacteristic
533Q8—21no—
534Q8—21no—
535Q8—21no—
536Q8—22no—
537Q8—42no—
538Q8—44no—
539Q8—41no—
540Q8—41no—
541Q8—42no—
542Q8—4882no—
543Q8—2444no—
544Q8#1451980712yes—
545D4—22no—
546D4—42no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—4882no—
566C2×C4—2442no—
567C2×C4#1451974611yescharacteristic
568C2×C4#1451974611yescharacteristic
569C2×C4#1451974611yescharacteristic
570C2×C4C22×C22⋊D6111yescharacteristic
571C2×C4C22×C22⋊D6111yescharacteristic
572C2×C4C22×C22⋊D6111yescharacteristic
573C8—2442no—
574C8—2442no—
575C8—2442no—
576C22—21no—
577C22—21no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22#1683993512yes—
582C22C2×#1451974611yescharacteristic
583C4—21no—
584C4—21no—
585C4—21no—
586C4—21no—
587C4—22no—
588C4—41no—
589C4—42no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—42no—
595C4—41no—
596C4—4882no—
597C4C2×#1451980711yescharacteristic
598C4#1684056011yescharacteristic
599C4#1684057712yes—
600C2—22no—
601C2—81no—
602C2#2145507911yescharacteristic
603C2#2145511411yescharacteristic
604C2C2×#1683993511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3523C2
socleC22×C61C2×#120
prime radical#121×C61D4
hypercenter#3523D61
hypocenterC61#18581
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. #21456201C2semidirect
3. C2×#16840057C2semidirect
4. #21444991C2semidirect
5. #21444984C2semidirect
6. #21444977C2semidirect
7. #21444977C2semidirect
8. #21444519C2semidirect
9. #21444266C2semidirect
357. C61#18581semidirect

Maximal subgroups

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

subgroupindex
#214570372
#214562012
C2×#168400572
#214449912
#214449842
#214449772
#214445192
#214442662
#214442322
C61×#35232
#1858161