Finite Group Databasebeta

← #29657932

Subgroups — #29657932

Order 15616.

866 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457037C211yesmaximal, characteristic, complemented
2#21457037C211yesmaximal, characteristic, complemented
3C2×#16840114C211yesmaximal, characteristic, complemented
4#21444991C212yesmaximal, complemented
5#21444984C212yesmaximal, complemented
6#21444977C212yesmaximal, complemented
7#21444970C212yesmaximal, complemented
8#21444536C211yesmaximal, characteristic, complemented
9#21444279C211yesmaximal, characteristic, complemented
10#21444279C211yesmaximal, characteristic, complemented
11C61×#3552C211yesmaximal, characteristic
12C2×#14519777C2211yescharacteristic
13#16840172—21no—
14#16840172—21no—
15#16840121—21no—
16#16840121—21no—
17#16840114—22no—
18#16840114C2212yes—
19#16840114C2212yes—
20#16840090—21no—
21#16840090—21no—
22#16840073—21no—
23#16840073—21no—
24#16840057C2212yes—
25#16840057C2212yes—
26#16840052C2212yes—
27#16840052C2212yes—
28#16839819C2211yescharacteristic
29#16839819C2211yescharacteristic
30C2×#14519700—22no—
31C2×C4×#92418C2211yescharacteristic
32#16839508C2212yes—
33#16839508C2212yes—
34Q8⋊#92418C2212yes—
35Q8⋊#92418C2212yes—
36D4⋊#92418C2212yes—
37D4⋊#92418C2212yes—
38C61×#563—22no—
39C61×#557C2211yescharacteristic
40C61×#556C2211yescharacteristic
41C61×#554—21no—
42C61×#553—21no—
43D4×Q8×C61—21no—
44D4×Q8×C61C2211yescharacteristic
45C61×#548—21no—
46C61×#547C2211yescharacteristic
47C61×#544—21no—
48C61×#542—21no—
49C61×#541—21no—
50C61×#539—22no—
51C61×#536—21no—
52C2×C4⋊Q8×C61C2211yescharacteristic
53C61×#529C2211yescharacteristic
54C61×#529C2211yescharacteristic
55C22×Dic122—21no—
56C22×#92418—21no—
57#14519777—21no—
58#14519777—21no—
59#14519777—21no—
60#14519777C2312yes—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—21no—
63#14519710—22no—
64#14519710—22no—
65#14519708—22no—
66#14519708—22no—
67C4⋊#92418C2312yes—
68C4⋊#92418C2312yes—
69#14519701—22no—
70#14519701—22no—
71#14519700—24no—
72#14519700—24no—
73#14519698—22no—
74#14519698—22no—
75C22⋊#92418—21no—
76C22⋊#92418—21no—
77#14519688—21no—
78#14519688—21no—
79C4×#92418—21no—
80C4×#92418C2312yes—
81C2×#41×C61—21no—
82C22×Q8×C61—21no—
83C22×Q8×C61—21no—
84C4⋊Q8×C61—22no—
85C4⋊Q8×C61—21no—
86C4⋊Q8×C61—21no—
87C4⋊Q8×C61—21no—
88C4⋊Q8×C61—21no—
89C4⋊Q8×C61D412yes—
90C4⋊Q8×C61C2311yescharacteristic
91C4⋊Q8×C61C2311yescharacteristic
92#126×C61—22no—
93#126×C61—22no—
94#126×C61—22no—
95#126×C61—21no—
96#126×C61—21no—
97#125×C61—22no—
98#125×C61—22no—
99#125×C61—22no—
100#125×C61—21no—
101#125×C61D411yescharacteristic
102#125×C61D411yescharacteristic
103#124×C61—22no—
104#124×C61—21no—
105#124×C61D411yescharacteristic
106#124×C61D411yescharacteristic
107#123×C61—22no—
108#123×C61—22no—
109#123×C61—21no—
110#123×C61—21no—
111#123×C61—22no—
112C22⋊Q8×C61—22no—
113C22⋊Q8×C61—22no—
114C22⋊Q8×C61—22no—
115C22⋊Q8×C61—22no—
116C22⋊Q8×C61—22no—
117C22⋊Q8×C61—21no—
118C22⋊Q8×C61—22no—
119C22⋊Q8×C61—21no—
120C22⋊Q8×C61—21no—
121C22⋊Q8×C61D411yescharacteristic
122C22⋊Q8×C61D411yescharacteristic
123#121×C61—22no—
124#121×C61—21no—
125#121×C61D411yescharacteristic
126#121×C61D411yescharacteristic
127C4×Q8×C61—21no—
128C4×Q8×C61—21no—
129C4×Q8×C61—21no—
130C4×Q8×C61C2311yescharacteristic
131C4×Q8×C61C2311yescharacteristic
132C4×D4×C61—22no—
133C4×D4×C61—22no—
134C4×D4×C61—21no—
135C4×D4×C61—21no—
136C4×D4×C61—21no—
137C4×D4×C61C2311yescharacteristic
138C4×D4×C61C2311yescharacteristic
139#117×C61—21no—
140#117×C61—21no—
141#117×C61—21no—
142C2×C4⋊C4×C61—21no—
143C2×C4⋊C4×C61—21no—
144C2×C4⋊C4×C61D412yes—
145C2×C42×C61C2311yescharacteristic
146#41×C61—82no—
147C2×Q8×C61—22no—
148C2×Q8×C61—22no—
149C2×Q8×C61—21no—
150C2×Q8×C61—21no—
151C2×Q8×C61—82no—
152C2×Q8×C61—82no—
153C2×Q8×C61C2×D411yescharacteristic
154C2×Q8×C61C2×D411yescharacteristic
155C2×D4×C61—21no—
156C2×D4×C61—21no—
157C2×D4×C61—22no—
158C2×D4×C61C2×D411yescharacteristic
159C2×D4×C61C2×D411yescharacteristic
160C22×C4×C61—22no—
161C22×C4×C61—22no—
162C22×C4×C61—22no—
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61C2×D411yescharacteristic
166C22×C4×C61C2×D411yescharacteristic
167C22×C4×C61C2×D411yescharacteristic
168C22×Dic61—41no—
169C2×Dic122—22no—
170C2×Dic122—44no—
171C2×Dic122—41no—
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—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61C2×D411yescharacteristic
191C4⋊C4×C61C2×D412yes—
192C4⋊C4×C61C2×D412yes—
193C4⋊C4×C61C2×D411yescharacteristic
194C4⋊C4×C61C2×D411yescharacteristic
195C4⋊C4×C61C2×D411yescharacteristic
196C22⋊C4×C61—22no—
197C22⋊C4×C61—22no—
198C22⋊C4×C61—22no—
199C22⋊C4×C61—22no—
200C22⋊C4×C61—21no—
201C22⋊C4×C61—21no—
202C22⋊C4×C61—21no—
203C22⋊C4×C61—21no—
204C22⋊C4×C61—21no—
205C22⋊C4×C61—21no—
206C22⋊C4×C61—21no—
207C22⋊C4×C61—21no—
208C22⋊C4×C61—21no—
209C22⋊C4×C61—21no—
210C42×C61—22no—
211C42×C61—22no—
212C42×C61—21no—
213C42×C61—21no—
214C42×C61C2×D411yescharacteristic
215C42×C61C2×D411yescharacteristic
216C42×C61C2×D411yescharacteristic
217C42×C61C2411yescharacteristic
218C4⋊Dic61—22no—
219C4⋊Dic61—22no—
220C4⋊Dic61—41no—
221C4⋊Dic61—41no—
222C2×#92418—22no—
223C2×#92418—22no—
224C2×#92418—22no—
225C2×#92418—42no—
226C23×C61—21no—
227C23×C61—21no—
228C23×C61#12011yescharacteristic
229Q8×C61—22no—
230Q8×C61—22no—
231Q8×C61—42no—
232Q8×C61—42no—
233Q8×C61—82no—
234Q8×C61—82no—
235D4×C61—22no—
236D4×C61—22no—
237D4×C61—42no—
238D4×C61—42no—
239D4×C61—82no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—21no—
251C2×C4×C61—42no—
252C2×C4×C61—42no—
253C2×C4×C61—82no—
254C2×C4×C61—82no—
255C2×C4×C61—82no—
256C2×C4×C61#12011yescharacteristic
257C2×C4×C61#12011yescharacteristic
258C2×C4×C61#12011yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C22×D411yescharacteristic
262C2×Dic61—42no—
263C2×Dic61—82no—
264Dic122—44no—
265Dic122—82no—
266#92418—42no—
267#92418—42no—
268#92418—42no—
269#18695—611nomaximal, Sylow
270C22×C61—21no—
271C22×C61—21no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61#45412yes—
279C22×C61C2×#12011yescharacteristic
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—41no—
290C4×C61—41no—
291C4×C61—41no—
292C4×C61#57712yes—
293C4×C61#57812yes—
294Dic61—82no—
295#3552D6111yescharacteristic
296#3238—611no—
297#3238—611no—
298C2×#487—611no—
299#1638—612no—
300#1637—612no—
301#1636—612no—
302#1635—612no—
303#1556—611no—
304#1494—611no—
305#1494—611no—
306C2×C61—22no—
307C2×C61—81no—
308C2×C61—81no—
309C2×C61#300911yescharacteristic
310C2×C61#301111yescharacteristic
311C2×C61C2×#45411yescharacteristic
312#563—22no—
313#557C2×D6111yescharacteristic
314#556C2×D6111yescharacteristic
315#554—21no—
316#553—21no—
317D4×Q8—21no—
318D4×Q8C2×D6111yescharacteristic
319#548—21no—
320#547C2×D6111yescharacteristic
321#544—21no—
322#542—21no—
323#541—21no—
324#539—22no—
325#536—21no—
326C2×C4⋊Q8—611no—
327C2×C4⋊Q8C2×D6111yescharacteristic
328#529C2×D6111yescharacteristic
329#529C2×D6111yescharacteristic
330#501—1221no—
331#501—1221no—
332#489—1221no—
333#489—1221no—
334#487—612no—
335#487—612no—
336#487—1222no—
337#483—1221no—
338#483—1221no—
339#480—1221no—
340#480—1221no—
341#477—612no—
342#477—612no—
343#476—612no—
344#476—612no—
345#432—611no—
346#432—611no—
347C2×#103—1222no—
348C2×C4×C8—611no—
349#332—612no—
350#332—612no—
351Q8⋊C8—612no—
352Q8⋊C8—612no—
353D4⋊C8—612no—
354D4⋊C8—612no—
355C61#1869511yescharacteristic, complemented, Sylow
356C2×#41—21no—
357C22×Q8—21no—
358C22×Q8—21no—
359C22×Q8—1221no—
360C22×C8—1221no—
361C4⋊Q8—22no—
362C4⋊Q8—21no—
363C4⋊Q8—21no—
364C4⋊Q8—21no—
365C4⋊Q8—21no—
366C4⋊Q8—612no—
367C4⋊Q8—1221no—
368C4⋊Q8—1221no—
369C4⋊Q8—1221no—
370C4⋊Q8C22⋊D6112yes—
371C4⋊Q8C22×D6111yescharacteristic
372C4⋊Q8C22×D6111yescharacteristic
373#126—22no—
374#126—22no—
375#126—22no—
376#126—21no—
377#126—21no—
378#125—22no—
379#125—22no—
380#125—22no—
381#125—21no—
382#125C22⋊D6111yescharacteristic
383#125C22⋊D6111yescharacteristic
384#124—22no—
385#124—21no—
386#124C22⋊D6111yescharacteristic
387#124C22⋊D6111yescharacteristic
388#123—22no—
389#123—22no—
390#123—21no—
391#123—21no—
392#123—22no—
393C22⋊Q8—22no—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8—22no—
397C22⋊Q8—22no—
398C22⋊Q8—21no—
399C22⋊Q8—22no—
400C22⋊Q8—21no—
401C22⋊Q8—21no—
402C22⋊Q8C22⋊D6111yescharacteristic
403C22⋊Q8C22⋊D6111yescharacteristic
404#121—22no—
405#121—21no—
406#121C22⋊D6111yescharacteristic
407#121C22⋊D6111yescharacteristic
408C4×Q8—21no—
409C4×Q8—21no—
410C4×Q8—21no—
411C4×Q8C22×D6111yescharacteristic
412C4×Q8C22×D6111yescharacteristic
413C4×D4—22no—
414C4×D4—22no—
415C4×D4—21no—
416C4×D4—21no—
417C4×D4—21no—
418C4×D4C22×D6111yescharacteristic
419C4×D4C22×D6111yescharacteristic
420#117—21no—
421#117—21no—
422#117—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4C22⋊D6112yes—
428C2×C42C22×D6111yescharacteristic
429#107—1222no—
430#107—1222no—
431#106—1222no—
432#106—1222no—
433C4⋊C8—612no—
434C4⋊C8—612no—
435#103—1222no—
436#103—1222no—
437#103—1224no—
438#103—1224no—
439#102—1222no—
440#102—1222no—
441C22⋊C8—1221no—
442C22⋊C8—1221no—
443#97—1221no—
444#97—1221no—
445C4×C8—612no—
446C4×C8—1221no—
447#41—82no—
448C2×Q8—22no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—22no—
452C2×Q8—82no—
453C2×Q8—82no—
454C2×Q8—1222no—
455C2×Q8—2444no—
456C2×Q8—2441no—
457C2×Q8C2×C22⋊D6111yescharacteristic
458C2×Q8C2×C22⋊D6111yescharacteristic
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—22no—
462C2×D4C2×C22⋊D6111yescharacteristic
463C2×D4C2×C22⋊D6111yescharacteristic
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—2441no—
470C22×C4C2×C22⋊D6111yescharacteristic
471C22×C4C2×C22⋊D6111yescharacteristic
472C22×C4C2×C22⋊D6111yescharacteristic
473C2×C8—1222no—
474C2×C8—1222no—
475C2×C8—1222no—
476C2×C8—2442no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—2441no—
498C4⋊C4—2441no—
499C4⋊C4C2×C22⋊D6112yes—
500C4⋊C4C2×C22⋊D6112yes—
501C4⋊C4C2×C22⋊D6111yescharacteristic
502C4⋊C4C2×C22⋊D6111yescharacteristic
503C4⋊C4C2×C22⋊D6111yescharacteristic
504C4⋊C4C2×C22⋊D6111yescharacteristic
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—21no—
513C22⋊C4—21no—
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—21no—
518C22⋊C4—21no—
519C42—22no—
520C42—22no—
521C42—21no—
522C42—21no—
523C42C2×C22⋊D6111yescharacteristic
524C42C2×C22⋊D6111yescharacteristic
525C42C2×C22⋊D6111yescharacteristic
526C42C23×D6111yescharacteristic
527C23—21no—
528C23—21no—
529C23#1451974611yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—42no—
533Q8—42no—
534Q8—82no—
535Q8—82no—
536Q8—4882no—
537Q8—2444no—
538D4—22no—
539D4—22no—
540D4—42no—
541D4—42no—
542D4—82no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—4882no—
560C2×C4—2442no—
561C2×C4#1451974611yescharacteristic
562C2×C4#1451974611yescharacteristic
563C2×C4#1451974611yescharacteristic
564C2×C4C22×C22⋊D6111yescharacteristic
565C2×C4C22×C22⋊D6111yescharacteristic
566C2×C4C22×C22⋊D6111yescharacteristic
567C8—2442no—
568C8—2442no—
569C8—2442no—
570C22—21no—
571C22—21no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22#1683993512yes—
579C22C2×#1451974611yescharacteristic
580C4—21no—
581C4—21no—
582C4—21no—
583C4—21no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—42no—
590C4—41no—
591C4—42no—
592C4—4882no—
593C4#1684057312yes—
594C4#1684057712yes—
595C2—22no—
596C2—81no—
597C2—81no—
598C2#2145509611yescharacteristic
599C2#2145511411yescharacteristic
600C2C2×#1683993511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3552C2
socleC22×C61C2×#120
hypercenter#3552D61
hypocenterC61#18695
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3552

Complements

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

NH ≅ G/Ntype
1. #21457037C2semidirect
2. #21457037C2semidirect
3. C2×#16840114C2semidirect
4. #21444991C2semidirect
5. #21444984C2semidirect
6. #21444977C2semidirect
7. #21444970C2semidirect
8. #21444536C2semidirect
9. #21444279C2semidirect
10. #21444279C2semidirect
355. C61#18695semidirect

Maximal subgroups

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

subgroupindex
#214570372
C2×#168401142
#214449912
#214449842
#214449772
#214449702
#214445362
#214442792
C61×#35522
#1869561