Finite Group Databasebeta

← #29661442

Subgroups — #29661442

Order 15616.

1012 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457037C211yesmaximal, characteristic, complemented
2#21456279C211yesmaximal, characteristic, complemented
3C2×#16840121C211yesmaximal, characteristic, complemented
4#21445136C214yesmaximal, complemented
5#21445125C214yesmaximal, complemented
6#21444430C211yesmaximal, characteristic, complemented
7#21444417C211yesmaximal, characteristic, complemented
8#21444407C211yesmaximal, characteristic, complemented
9C61×#3558C211yesmaximal, characteristic
10C2×#14519807—22no—
11C2×#14519777C2211yescharacteristic, complemented
12#16840172C2212yescomplemented
13#16840157C2214yescomplemented
14#16840141—21no—
15#16840134—21no—
16#16840121—22no—
17#16840121C2212yescomplemented
18#16840121C2214yescomplemented
19#16840090—21no—
20#16840073—21no—
21#16840057—21no—
22#16840052—21no—
23#16839819C2211yescharacteristic, complemented
24C2×#14519701—21no—
25C2×#14519698—21no—
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—
48#14519807—48no—
49C2×#11758202—24no—
50C2×#11758200—24no—
51C2×#11758194—21no—
52C22×#92418—21no—
53#14519777—21no—
54#14519777—21no—
55#14519777—21no—
56#14519777C2312yes—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61—21no—
59#14519710—22no—
60#14519708—22no—
61C4⋊#92418—21no—
62#14519701—24no—
63#14519701—22no—
64#14519698—24no—
65#14519698—22no—
66C22⋊#92418—21no—
67#14519688—21no—
68#14519688C2312yes—
69#14519688C2312yes—
70#14519688C2312yes—
71C4×#92418—21no—
72C2×#41×C61—21no—
73C2×#41×C61D412yes—
74C22×Q8×C61D411yescharacteristic
75C22×D4×C61—22no—
76C22×D4×C61—21no—
77C22×D4×C61D411yescharacteristic
78C4⋊Q8×C61D411yescharacteristic
79#127×C61—21no—
80#127×C61D411yescharacteristic
81#126×C61—22no—
82#126×C61—21no—
83#125×C61C2311yescharacteristic
84#124×C61—21no—
85#124×C61—22no—
86#124×C61—22no—
87#124×C61—22no—
88#124×C61—21no—
89#124×C61—21no—
90#124×C61D412yes—
91#124×C61C2312yes—
92#124×C61C2311yescharacteristic
93#124×C61C2312yes—
94#123×C61—22no—
95#123×C61—22no—
96#123×C61—21no—
97#123×C61—22no—
98#123×C61—22no—
99C22⋊Q8×C61—21no—
100C22⋊Q8×C61—22no—
101C22⋊Q8×C61—22no—
102C22⋊Q8×C61D411yescharacteristic
103#121×C61—22no—
104#121×C61—22no—
105#121×C61—22no—
106#121×C61—21no—
107#121×C61—22no—
108#121×C61—22no—
109#121×C61—21no—
110#121×C61—21no—
111#121×C61D411yescharacteristic
112#120×C61—22no—
113#120×C61—22no—
114#120×C61—22no—
115#120×C61—22no—
116#120×C61—22no—
117C4×Q8×C61—21no—
118C4×Q8×C61D411yescharacteristic
119C4×D4×C61—22no—
120C4×D4×C61—22no—
121C4×D4×C61—21no—
122C4×D4×C61—21no—
123C4×D4×C61—21no—
124C4×D4×C61—21no—
125C4×D4×C61D411yescharacteristic
126#117×C61—21no—
127C2×C22⋊C4×C61—22no—
128C2×C22⋊C4×C61—21no—
129C2×C22⋊C4×C61—21no—
130C2×C42×C61C2311yescharacteristic
131C24×C61—22no—
132C24×C61—21no—
133#41×C61—44no—
134#41×C61—82no—
135C2×Q8×C61—21no—
136C2×Q8×C61—42no—
137C2×Q8×C61C2×D411yescharacteristic
138C2×Q8×C61C2×D412yes—
139C2×Q8×C61C2×D412yes—
140C2×D4×C61—22no—
141C2×D4×C61—21no—
142C2×D4×C61—22no—
143C2×D4×C61—22no—
144C2×D4×C61—22no—
145C2×D4×C61—22no—
146C2×D4×C61—22no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—42no—
151C2×D4×C61—82no—
152C2×D4×C61—84no—
153C2×D4×C61C2×D411yescharacteristic
154C2×D4×C61C2×D412yes—
155C2×D4×C61C2×D412yes—
156C22×C4×C61—22no—
157C22×C4×C61—22no—
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61—22no—
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2×D411yescharacteristic
164C22×Dic61—41no—
165C2×Dic122—22no—
166C2×Dic122—44no—
167C2×Dic122—41no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61C2×D411yescharacteristic
177C4⋊C4×C61C2×D411yescharacteristic
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—21no—
188C22⋊C4×C61—22no—
189C22⋊C4×C61—22no—
190C22⋊C4×C61—21no—
191C22⋊C4×C61—21no—
192C22⋊C4×C61—22no—
193C22⋊C4×C61—22no—
194C42×C61—22no—
195C42×C61—21no—
196C42×C61C2×D411yescharacteristic
197C42×C61C2×D411yescharacteristic
198C42×C61C2×D411yescharacteristic
199C42×C61C2411yescharacteristic
200#11758202—48no—
201#11758200—48no—
202C4⋊Dic61—22no—
203C4⋊Dic61—22no—
204C4⋊Dic61—41no—
205C4⋊Dic61—41no—
206#11758194—24no—
207C2×#92418—22no—
208C2×#92418—22no—
209C2×#92418—22no—
210C2×#92418—42no—
211C23×C61—22no—
212C23×C61—21no—
213C23×C61—22no—
214C23×C61—42no—
215C23×C61—81no—
216C23×C61—81no—
217C23×C61—82no—
218C23×C61—82no—
219C23×C61—82no—
220C23×C61#12011yescharacteristic
221Q8×C61—42no—
222Q8×C61—42no—
223Q8×C61—41no—
224Q8×C61—42no—
225D4×C61—42no—
226D4×C61—42no—
227D4×C61—41no—
228D4×C61—42no—
229D4×C61—44no—
230D4×C61—42no—
231D4×C61—42no—
232D4×C61—81no—
233D4×C61—82no—
234D4×C61—82no—
235D4×C61—82no—
236D4×C61—82no—
237D4×C61—82no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—41no—
246C2×C4×C61—42no—
247C2×C4×C61—82no—
248C2×C4×C61—81no—
249C2×C4×C61—82no—
250C2×C4×C61—82no—
251C2×C4×C61#12011yescharacteristic
252C2×C4×C61#12011yescharacteristic
253C2×C4×C61#12011yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×Dic61—42no—
258C2×Dic61—82no—
259Dic122—44no—
260Dic122—82no—
261#92418—24no—
262#92418—42no—
263#92418—42no—
264#19015—611nomaximal, Sylow
265C22×C61—21no—
266C22×C61—21no—
267C22×C61—42no—
268C22×C61—42no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—41no—
274C22×C61—42no—
275C22×C61—42no—
276C22×C61—162no—
277C22×C61—164no—
278C22×C61#45312yes—
279C22×C61C2×#12011yescharacteristic
280C4×C61—21no—
281C4×C61—21no—
282C4×C61—21no—
283C4×C61—21no—
284C4×C61—21no—
285C4×C61—41no—
286C4×C61—42no—
287C4×C61—41no—
288C4×C61—42no—
289C4×C61—41no—
290C4×C61#45512yes—
291Dic61—82no—
292#3558D6111yescharacteristic
293#3238—611no—
294#3152—611no—
295C2×#489—611no—
296#1661—614no—
297#1659—614no—
298#1530—611no—
299#1527—611no—
300#1524—611no—
301C2×C61—22no—
302C2×C61—82no—
303C2×C61—82no—
304C2×C61—81no—
305C2×C61C2×#45311yescharacteristic
306C2×C61C2×#45511yescharacteristic
307C2×C61#302211yescharacteristic
308C2×#137—1222no—
309#562C2×D6112yes—
310H(3,𝔽4)C2×D6112yes—
311#559—21no—
312#548—21no—
313#546—22no—
314#546—21no—
315D42—21no—
316#542—21no—
317#538—22no—
318#538—21no—
319#535—21no—
320#534—21no—
321#532C2×D6111yescharacteristic
322C2×C4⋊Q8—611no—
323#529C2×D6111yescharacteristic
324C2×#124C2×D6111yescharacteristic
325#501—612no—
326#497—614no—
327#494—1221no—
328#492—1221no—
329#489—614no—
330#489—612no—
331#489—1222no—
332#483—1221no—
333#480—1221no—
334#477—1221no—
335#476—1221no—
336#432—611no—
337C2×#103—1221no—
338C2×#102—1221no—
339C4×M16—611no—
340C2×#97—611no—
341#330—612no—
342#329—612no—
343#329—614no—
344#329—614no—
345C61#1901511yescharacteristic, complemented, Sylow
346C2×#41—21no—
347C2×#41C22⋊D6112yes—
348C22×Q8—1221no—
349C22×Q8C22⋊D6111yescharacteristic
350C22×D4—22no—
351C22×D4—21no—
352C22×D4C22⋊D6111yescharacteristic
353#137—2448no—
354C2×Q16—1224no—
355C2×SD16—1224no—
356C2×M16—1221no—
357C22×C8—1221no—
358C4⋊Q8—612no—
359C4⋊Q8—1221no—
360C4⋊Q8—1221no—
361C4⋊Q8—1221no—
362C4⋊Q8C22⋊D6111yescharacteristic
363#127—21no—
364#127C22⋊D6111yescharacteristic
365#126—22no—
366#126—21no—
367#125C22×D6111yescharacteristic
368#124—21no—
369#124—22no—
370#124—22no—
371#124—22no—
372#124—21no—
373#124—21no—
374#124C22⋊D6112yes—
375#124C22×D6112yes—
376#124C22×D6111yescharacteristic
377#124C22×D6112yes—
378#123—22no—
379#123—22no—
380#123—21no—
381#123—22no—
382#123—22no—
383C22⋊Q8—21no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8C22⋊D6111yescharacteristic
387#121—22no—
388#121—22no—
389#121—22no—
390#121—21no—
391#121—22no—
392#121—22no—
393#121—21no—
394#121—21no—
395#121C22⋊D6111yescharacteristic
396#120—22no—
397#120—22no—
398#120—22no—
399#120—22no—
400#120—22no—
401C4×Q8—21no—
402C4×Q8C22⋊D6111yescharacteristic
403C4×D4—22no—
404C4×D4—22no—
405C4×D4—21no—
406C4×D4—21no—
407C4×D4—21no—
408C4×D4—21no—
409C4×D4C22⋊D6111yescharacteristic
410#117—21no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C22⋊C4—22no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C42C22×D6111yescharacteristic
417#107—1222no—
418#106—1222no—
419C4⋊C8—1221no—
420#103—1224no—
421#103—1222no—
422#102—1224no—
423#102—1222no—
424C22⋊C8—1221no—
425#97—612no—
426#97—612no—
427#97—612no—
428#97—1221no—
429C4×C8—1221no—
430C24—22no—
431C24—21no—
432#41—44no—
433#41—82no—
434C2×Q8—21no—
435C2×Q8—42no—
436C2×Q8—1222no—
437C2×Q8—2444no—
438C2×Q8—2441no—
439C2×Q8C2×C22⋊D6111yescharacteristic
440C2×Q8C2×C22⋊D6112yes—
441C2×Q8C2×C22⋊D6112yes—
442C2×D4—22no—
443C2×D4—21no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—42no—
453C2×D4—82no—
454C2×D4—84no—
455C2×D4C2×C22⋊D6111yescharacteristic
456C2×D4C2×C22⋊D6112yes—
457C2×D4C2×C22⋊D6112yes—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—2441no—
464C22×C4C2×C22⋊D6111yescharacteristic
465C22×C4C2×C22⋊D6111yescharacteristic
466C22×C4C2×C22⋊D6111yescharacteristic
467Q16—2448no—
468SD16—2448no—
469M16—1224no—
470C2×C8—1222no—
471C2×C8—1222no—
472C2×C8—1222no—
473C2×C8—2442no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C4⋊C4—2441no—
485C4⋊C4—2441no—
486C4⋊C4C2×C22⋊D6111yescharacteristic
487C4⋊C4C2×C22⋊D6111yescharacteristic
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—21no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—21no—
497C22⋊C4—21no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—21no—
502C22⋊C4—22no—
503C22⋊C4—21no—
504C42—22no—
505C42—21no—
506C42C2×C22⋊D6111yescharacteristic
507C42C2×C22⋊D6111yescharacteristic
508C42C2×C22⋊D6111yescharacteristic
509C42C23×D6111yescharacteristic
510C23—22no—
511C23—21no—
512C23—22no—
513C23—42no—
514C23—81no—
515C23—81no—
516C23—82no—
517C23—82no—
518C23—82no—
519C23#1451974611yescharacteristic
520Q8—42no—
521Q8—42no—
522Q8—41no—
523Q8—42no—
524Q8—4882no—
525Q8—2444no—
526D4—42no—
527D4—42no—
528D4—41no—
529D4—42no—
530D4—44no—
531D4—42no—
532D4—42no—
533D4—81no—
534D4—82no—
535D4—82no—
536D4—82no—
537D4—82no—
538D4—82no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—82no—
549C2×C4—81no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—4882no—
553C2×C4—2442no—
554C2×C4#1451974611yescharacteristic
555C2×C4#1451974611yescharacteristic
556C2×C4#1451974611yescharacteristic
557C2×C4C22×C22⋊D6111yescharacteristic
558C2×C4C22×C22⋊D6111yescharacteristic
559C2×C4C22×C22⋊D6111yescharacteristic
560C8—1224no—
561C8—2442no—
562C8—2442no—
563C22—21no—
564C22—21no—
565C22—42no—
566C22—42no—
567C22—41no—
568C22—41no—
569C22—42no—
570C22—42no—
571C22—41no—
572C22—42no—
573C22—42no—
574C22—162no—
575C22—164no—
576C22#1683993012yes—
577C22C2×#1451974611yescharacteristic
578C4—21no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—21no—
583C4—41no—
584C4—42no—
585C4—41no—
586C4—42no—
587C4—41no—
588C4—4882no—
589C4#1683994012yes—
590C2—22no—
591C2—82no—
592C2—81no—
593C2—82no—
594C2C2×#1683993011yescharacteristic
595C2C2×#1683994011yescharacteristic
596C2#2145519811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3558C2
socleC22×C61C2×#120
hypercenter#3558D61
hypocenterC61#19015
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. #21457037C2semidirect
2. #21456279C2semidirect
3. C2×#16840121C2semidirect
4. #21445136C2semidirect
5. #21445125C2semidirect
6. #21444430C2semidirect
7. #21444417C2semidirect
8. #21444407C2semidirect
11. C2×#14519777C22semidirect
12. #16840172C22semidirect
13. #16840157C22semidirect
17. #16840121C22semidirect
18. #16840121C22semidirect
23. #16839819C22semidirect
26. C4×#11758194C22semidirect
27. C2×#14519688C22semidirect
28. #16839504C22semidirect
29. #16839502C22semidirect
30. #16839502C22semidirect
31. #16839502C22semidirect
345. C61#19015semidirect

Maximal subgroups

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

subgroupindex
#214570372
#214562792
C2×#168401212
#214451362
#214451252
#214444302
#214444172
#214444072
C61×#35582
#1901561