Finite Group Databasebeta

← #29661500

Subgroups — #29661500

Order 15616.

1012 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457046C211yesmaximal, characteristic, complemented
2C2×#16840157C211yesmaximal, characteristic, complemented
3#21456212C211yesmaximal, characteristic, complemented
4#21445136C214yesmaximal, complemented
5#21445125C214yesmaximal, complemented
6#21444430C211yesmaximal, characteristic, complemented
7#21444417C211yesmaximal, characteristic, complemented
8#21444407C211yesmaximal, characteristic, complemented
9C61×#3558C211yesmaximal, characteristic
10C22×#11758202—21no—
11C22×#11758200—21no—
12C2×#14519777C2211yescharacteristic, complemented
13#16840172C2212yescomplemented
14#16840157—22no—
15#16840157C2214yescomplemented
16#16840121C2212yescomplemented
17#16840121C2214yescomplemented
18#16840112—21no—
19#16840108—21no—
20#16840095—21no—
21#16840078—21no—
22#16840047—21no—
23#16840042—21no—
24#16839819C2211yescharacteristic, complemented
25#16839763—22no—
26C4×#11758194C2211yescharacteristic, complemented
27C2×#14519688C2211yescharacteristic, complemented
28#16839504C2212yescomplemented
29#16839502C2212yescomplemented
30#16839502C2214yescomplemented
31#16839502C2214yescomplemented
32C61×#562C2212yes—
33C61×H(3,𝔽4)C2212yes—
34C61×#559—21no—
35C61×#548—21no—
36C61×#546—22no—
37C61×#546—21no—
38D42×C61—21no—
39C61×#542—21no—
40C61×#538—22no—
41C61×#538—21no—
42C61×#535—21no—
43C61×#534—21no—
44C61×#532C2211yescharacteristic
45C61×#529C2211yescharacteristic
46C2×#124×C61C2211yescharacteristic
47C22×Dic122—21no—
48C2×#11758202—24no—
49C2×#11758202—44no—
50C2×#11758200—24no—
51C2×#11758200—44no—
52C2×#11758194—21no—
53C22×#92418—21no—
54#14519777—21no—
55#14519777—21no—
56#14519777—21no—
57#14519777C2312yes—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—21no—
60#14519710—22no—
61#14519708—22no—
62C4⋊#92418—21no—
63#14519701—24no—
64#14519701—22no—
65#14519698—24no—
66#14519698—22no—
67C22⋊#92418—21no—
68#14519688—21no—
69#14519688C2312yes—
70#14519688C2312yes—
71#14519688C2312yes—
72C4×#92418—21no—
73C2×#41×C61—21no—
74C2×#41×C61D412yes—
75C22×Q8×C61D411yescharacteristic
76C22×D4×C61—22no—
77C22×D4×C61—21no—
78C22×D4×C61D411yescharacteristic
79C4⋊Q8×C61D411yescharacteristic
80#127×C61—21no—
81#127×C61D411yescharacteristic
82#126×C61—22no—
83#126×C61—21no—
84#125×C61C2311yescharacteristic
85#124×C61—21no—
86#124×C61—22no—
87#124×C61—22no—
88#124×C61—22no—
89#124×C61—21no—
90#124×C61—21no—
91#124×C61D412yes—
92#124×C61C2312yes—
93#124×C61C2311yescharacteristic
94#124×C61C2312yes—
95#123×C61—22no—
96#123×C61—22no—
97#123×C61—21no—
98#123×C61—22no—
99#123×C61—22no—
100C22⋊Q8×C61—21no—
101C22⋊Q8×C61—22no—
102C22⋊Q8×C61—22no—
103C22⋊Q8×C61D411yescharacteristic
104#121×C61—22no—
105#121×C61—22no—
106#121×C61—22no—
107#121×C61—21no—
108#121×C61—22no—
109#121×C61—22no—
110#121×C61—21no—
111#121×C61—21no—
112#121×C61D411yescharacteristic
113#120×C61—22no—
114#120×C61—22no—
115#120×C61—22no—
116#120×C61—22no—
117#120×C61—22no—
118C4×Q8×C61—21no—
119C4×Q8×C61D411yescharacteristic
120C4×D4×C61—22no—
121C4×D4×C61—22no—
122C4×D4×C61—21no—
123C4×D4×C61—21no—
124C4×D4×C61—21no—
125C4×D4×C61—21no—
126C4×D4×C61D411yescharacteristic
127#117×C61—21no—
128C2×C22⋊C4×C61—22no—
129C2×C22⋊C4×C61—21no—
130C2×C22⋊C4×C61—21no—
131C2×C42×C61C2311yescharacteristic
132C24×C61—22no—
133C24×C61—21no—
134#41×C61—44no—
135#41×C61—82no—
136C2×Q8×C61—21no—
137C2×Q8×C61—42no—
138C2×Q8×C61C2×D411yescharacteristic
139C2×Q8×C61C2×D412yes—
140C2×Q8×C61C2×D412yes—
141C2×D4×C61—22no—
142C2×D4×C61—21no—
143C2×D4×C61—22no—
144C2×D4×C61—22no—
145C2×D4×C61—22no—
146C2×D4×C61—22no—
147C2×D4×C61—22no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—42no—
152C2×D4×C61—82no—
153C2×D4×C61—84no—
154C2×D4×C61C2×D411yescharacteristic
155C2×D4×C61C2×D412yes—
156C2×D4×C61C2×D412yes—
157C22×C4×C61—22no—
158C22×C4×C61—22no—
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61—22no—
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61C2×D411yescharacteristic
165C22×Dic61—41no—
166C2×Dic122—22no—
167C2×Dic122—42no—
168C2×Dic122—42no—
169C2×Dic122—41no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61C2×D411yescharacteristic
179C4⋊C4×C61C2×D411yescharacteristic
180C22⋊C4×C61—22no—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—22no—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—21no—
188C22⋊C4×C61—22no—
189C22⋊C4×C61—21no—
190C22⋊C4×C61—22no—
191C22⋊C4×C61—22no—
192C22⋊C4×C61—21no—
193C22⋊C4×C61—21no—
194C22⋊C4×C61—22no—
195C22⋊C4×C61—22no—
196C42×C61—22no—
197C42×C61—21no—
198C42×C61C2×D411yescharacteristic
199C42×C61C2×D411yescharacteristic
200C42×C61C2×D411yescharacteristic
201C42×C61C2411yescharacteristic
202#11758202—48no—
203#11758200—48no—
204C4⋊Dic61—22no—
205C4⋊Dic61—22no—
206C4⋊Dic61—41no—
207C4⋊Dic61—41no—
208#11758194—42no—
209C2×#92418—24no—
210C2×#92418—22no—
211C2×#92418—22no—
212C2×#92418—22no—
213C23×C61—21no—
214C23×C61—22no—
215C23×C61—22no—
216C23×C61—42no—
217C23×C61—81no—
218C23×C61—81no—
219C23×C61—82no—
220C23×C61—82no—
221C23×C61—82no—
222C23×C61#12011yescharacteristic
223Q8×C61—42no—
224Q8×C61—41no—
225Q8×C61—42no—
226Q8×C61—42no—
227D4×C61—42no—
228D4×C61—41no—
229D4×C61—42no—
230D4×C61—42no—
231D4×C61—44no—
232D4×C61—42no—
233D4×C61—42no—
234D4×C61—81no—
235D4×C61—82no—
236D4×C61—82no—
237D4×C61—82no—
238D4×C61—82no—
239D4×C61—82no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—21no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—41no—
248C2×C4×C61—42no—
249C2×C4×C61—82no—
250C2×C4×C61—81no—
251C2×C4×C61—82no—
252C2×C4×C61—82no—
253C2×C4×C61#12011yescharacteristic
254C2×C4×C61#12011yescharacteristic
255C2×C4×C61#12011yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×Dic61—42no—
260C2×Dic61—82no—
261Dic122—42no—
262Dic122—42no—
263Dic122—82no—
264#92418—24no—
265#92418—42no—
266#92418—42no—
267#19021—611nomaximal, Sylow
268C22×C61—21no—
269C22×C61—21no—
270C22×C61—41no—
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—42no—
274C22×C61—42no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—42no—
278C22×C61—42no—
279C22×C61—162no—
280C22×C61—164no—
281C22×C61#45512yes—
282C22×C61C2×#12011yescharacteristic
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#45312yes—
294Dic61—82no—
295#3558D6111yescharacteristic
296#3239—611no—
297C2×#497—611no—
298#3141—611no—
299#1661—614no—
300#1659—614no—
301#1530—611no—
302#1527—611no—
303#1524—611no—
304C2×C61—22no—
305C2×C61—82no—
306C2×C61—81no—
307C2×C61—82no—
308C2×C61C2×#45311yescharacteristic
309C2×C61C2×#45511yescharacteristic
310C2×C61#302211yescharacteristic
311C22×Q16—1221no—
312C22×SD16—1221no—
313#562C2×D6112yes—
314H(3,𝔽4)C2×D6112yes—
315#559—21no—
316#548—21no—
317#546—22no—
318#546—21no—
319D42—21no—
320#542—21no—
321#538—22no—
322#538—21no—
323#535—21no—
324#534—21no—
325#532C2×D6111yescharacteristic
326C2×C4⋊Q8—611no—
327#529C2×D6111yescharacteristic
328C2×#124C2×D6111yescharacteristic
329#501—612no—
330#497—614no—
331#497—1222no—
332#489—614no—
333#489—612no—
334#487—1221no—
335#486—1221no—
336#484—1221no—
337#481—1221no—
338#475—1221no—
339#474—1221no—
340#432—611no—
341#417—1222no—
342C4×M16—611no—
343C2×#97—611no—
344#330—612no—
345#329—612no—
346#329—614no—
347#329—614no—
348C61#1902111yescharacteristic, complemented, Sylow
349C2×#41—21no—
350C2×#41C22⋊D6112yes—
351C22×Q8—1221no—
352C22×Q8C22⋊D6111yescharacteristic
353C22×D4—22no—
354C22×D4—21no—
355C22×D4C22⋊D6111yescharacteristic
356C2×Q16—1224no—
357C2×Q16—2444no—
358C2×SD16—1224no—
359C2×SD16—2444no—
360C2×M16—1221no—
361C22×C8—1221no—
362C4⋊Q8—612no—
363C4⋊Q8—1221no—
364C4⋊Q8—1221no—
365C4⋊Q8—1221no—
366C4⋊Q8C22⋊D6111yescharacteristic
367#127—21no—
368#127C22⋊D6111yescharacteristic
369#126—22no—
370#126—21no—
371#125C22×D6111yescharacteristic
372#124—21no—
373#124—22no—
374#124—22no—
375#124—22no—
376#124—21no—
377#124—21no—
378#124C22⋊D6112yes—
379#124C22×D6112yes—
380#124C22×D6111yescharacteristic
381#124C22×D6112yes—
382#123—22no—
383#123—22no—
384#123—21no—
385#123—22no—
386#123—22no—
387C22⋊Q8—21no—
388C22⋊Q8—22no—
389C22⋊Q8—22no—
390C22⋊Q8C22⋊D6111yescharacteristic
391#121—22no—
392#121—22no—
393#121—22no—
394#121—21no—
395#121—22no—
396#121—22no—
397#121—21no—
398#121—21no—
399#121C22⋊D6111yescharacteristic
400#120—22no—
401#120—22no—
402#120—22no—
403#120—22no—
404#120—22no—
405C4×Q8—21no—
406C4×Q8C22⋊D6111yescharacteristic
407C4×D4—22no—
408C4×D4—22no—
409C4×D4—21no—
410C4×D4—21no—
411C4×D4—21no—
412C4×D4—21no—
413C4×D4C22⋊D6111yescharacteristic
414#117—21no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C22⋊C4—22no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C42C22×D6111yescharacteristic
421#107—1222no—
422#106—1222no—
423C4⋊C8—1221no—
424#103—1224no—
425#103—1222no—
426#102—1224no—
427#102—1222no—
428C22⋊C8—1221no—
429#97—612no—
430#97—612no—
431#97—612no—
432#97—1221no—
433C4×C8—1221no—
434C24—22no—
435C24—21no—
436#41—44no—
437#41—82no—
438C2×Q8—21no—
439C2×Q8—42no—
440C2×Q8—1222no—
441C2×Q8—2442no—
442C2×Q8—2442no—
443C2×Q8—2441no—
444C2×Q8C2×C22⋊D6111yescharacteristic
445C2×Q8C2×C22⋊D6112yes—
446C2×Q8C2×C22⋊D6112yes—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—42no—
458C2×D4—82no—
459C2×D4—84no—
460C2×D4C2×C22⋊D6111yescharacteristic
461C2×D4C2×C22⋊D6112yes—
462C2×D4C2×C22⋊D6112yes—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—2441no—
469C22×C4C2×C22⋊D6111yescharacteristic
470C22×C4C2×C22⋊D6111yescharacteristic
471C22×C4C2×C22⋊D6111yescharacteristic
472Q16—2448no—
473SD16—2448no—
474M16—2442no—
475C2×C8—1224no—
476C2×C8—1222no—
477C2×C8—1222no—
478C2×C8—1222no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C4⋊C4—2441no—
490C4⋊C4—2441no—
491C4⋊C4C2×C22⋊D6111yescharacteristic
492C4⋊C4C2×C22⋊D6111yescharacteristic
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—21no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—21no—
507C22⋊C4—22no—
508C22⋊C4—21no—
509C42—22no—
510C42—21no—
511C42C2×C22⋊D6111yescharacteristic
512C42C2×C22⋊D6111yescharacteristic
513C42C2×C22⋊D6111yescharacteristic
514C42C23×D6111yescharacteristic
515C23—21no—
516C23—22no—
517C23—22no—
518C23—42no—
519C23—81no—
520C23—81no—
521C23—82no—
522C23—82no—
523C23—82no—
524C23#1451974611yescharacteristic
525Q8—42no—
526Q8—41no—
527Q8—42no—
528Q8—42no—
529Q8—4882no—
530Q8—2442no—
531Q8—2442no—
532D4—42no—
533D4—41no—
534D4—42no—
535D4—42no—
536D4—44no—
537D4—42no—
538D4—42no—
539D4—81no—
540D4—82no—
541D4—82no—
542D4—82no—
543D4—82no—
544D4—82no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—82no—
555C2×C4—81no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—4882no—
559C2×C4—2442no—
560C2×C4#1451974611yescharacteristic
561C2×C4#1451974611yescharacteristic
562C2×C4#1451974611yescharacteristic
563C2×C4C22×C22⋊D6111yescharacteristic
564C2×C4C22×C22⋊D6111yescharacteristic
565C2×C4C22×C22⋊D6111yescharacteristic
566C8—1224no—
567C8—2442no—
568C8—2442no—
569C22—21no—
570C22—21no—
571C22—41no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—41no—
577C22—41no—
578C22—42no—
579C22—42no—
580C22—162no—
581C22—164no—
582C22#1683994012yes—
583C22C2×#1451974611yescharacteristic
584C4—21no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—41no—
590C4—42no—
591C4—41no—
592C4—42no—
593C4—41no—
594C4—4882no—
595C4#1683993012yes—
596C2—22no—
597C2—81no—
598C2—82no—
599C2—82no—
600C2C2×#1683993011yescharacteristic
601C2C2×#1683994011yescharacteristic
602C2#2145519811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3558C2
socleC22×C61C2×#120
hypercenter#3558D61
hypocenterC61#19021
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. #21457046C2semidirect
2. C2×#16840157C2semidirect
3. #21456212C2semidirect
4. #21445136C2semidirect
5. #21445125C2semidirect
6. #21444430C2semidirect
7. #21444417C2semidirect
8. #21444407C2semidirect
12. C2×#14519777C22semidirect
13. #16840172C22semidirect
15. #16840157C22semidirect
16. #16840121C22semidirect
17. #16840121C22semidirect
24. #16839819C22semidirect
26. C4×#11758194C22semidirect
27. C2×#14519688C22semidirect
28. #16839504C22semidirect
29. #16839502C22semidirect
30. #16839502C22semidirect
31. #16839502C22semidirect
348. C61#19021semidirect

Maximal subgroups

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

subgroupindex
#214570462
C2×#168401572
#214562122
#214451362
#214451252
#214444302
#214444172
#214444072
C61×#35582
#1902161