Finite Group Databasebeta

← #29661491

Subgroups — #29661491

Order 15616.

960 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#21456843C211yesmaximal, characteristic, complemented
2#21456212C211yesmaximal, characteristic, complemented
3C2×#16840121C211yesmaximal, characteristic, complemented
4#21445142C214yesmaximal, complemented
5#21445125C214yesmaximal, complemented
6#21444430C211yesmaximal, characteristic, complemented
7#21444417C211yesmaximal, characteristic, complemented
8#21444407C211yesmaximal, characteristic, complemented
9C61×#3558C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic, complemented
11#16840157C2212yescomplemented
12#16840132C2212yescomplemented
13#16840121—22no—
14#16840121C2214yescomplemented
15#16840121C2214yescomplemented
16#16840112—21no—
17#16840108—21no—
18#16839972—21no—
19#16839967—21no—
20#16839917—21no—
21#16839900—21no—
22#16839819C2211yescharacteristic, complemented
23#16839763—22no—
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—
48C2×#11758202—22no—
49C2×#11758200—22no—
50C2×#11758194—21no—
51C22×#92418—21no—
52#14519777—21no—
53#14519777—21no—
54#14519777—21no—
55#14519777C2312yes—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—21no—
58C4⋊#92418—21no—
59#14519701—24no—
60#14519701—24no—
61#14519700—22no—
62#14519700—22no—
63#14519698—24no—
64#14519698—24no—
65C22⋊#92418—21no—
66#14519688—21no—
67#14519688C2312yes—
68#14519688C2312yes—
69#14519688C2312yes—
70C4×#92418—21no—
71C2×#41×C61—21no—
72C2×#41×C61D412yes—
73C22×Q8×C61D411yescharacteristic
74C22×D4×C61—22no—
75C22×D4×C61—21no—
76C22×D4×C61D411yescharacteristic
77C4⋊Q8×C61D411yescharacteristic
78#127×C61—21no—
79#127×C61D411yescharacteristic
80#126×C61—22no—
81#126×C61—21no—
82#125×C61C2311yescharacteristic
83#124×C61—21no—
84#124×C61—22no—
85#124×C61—22no—
86#124×C61—22no—
87#124×C61—21no—
88#124×C61—21no—
89#124×C61D412yes—
90#124×C61C2312yes—
91#124×C61C2311yescharacteristic
92#124×C61C2312yes—
93#123×C61—22no—
94#123×C61—22no—
95#123×C61—21no—
96#123×C61—22no—
97#123×C61—22no—
98C22⋊Q8×C61—21no—
99C22⋊Q8×C61—22no—
100C22⋊Q8×C61—22no—
101C22⋊Q8×C61D411yescharacteristic
102#121×C61—22no—
103#121×C61—22no—
104#121×C61—22no—
105#121×C61—21no—
106#121×C61—22no—
107#121×C61—22no—
108#121×C61—21no—
109#121×C61—21no—
110#121×C61D411yescharacteristic
111#120×C61—22no—
112#120×C61—22no—
113#120×C61—22no—
114#120×C61—22no—
115#120×C61—22no—
116C4×Q8×C61—21no—
117C4×Q8×C61D411yescharacteristic
118C4×D4×C61—22no—
119C4×D4×C61—22no—
120C4×D4×C61—21no—
121C4×D4×C61—21no—
122C4×D4×C61—21no—
123C4×D4×C61—21no—
124C4×D4×C61D411yescharacteristic
125#117×C61—21no—
126C2×C22⋊C4×C61—22no—
127C2×C22⋊C4×C61—21no—
128C2×C22⋊C4×C61—21no—
129C2×C42×C61C2311yescharacteristic
130C24×C61—22no—
131C24×C61—21no—
132#41×C61—44no—
133#41×C61—82no—
134C2×Q8×C61—21no—
135C2×Q8×C61—42no—
136C2×Q8×C61C2×D411yescharacteristic
137C2×Q8×C61C2×D412yes—
138C2×Q8×C61C2×D412yes—
139C2×D4×C61—22no—
140C2×D4×C61—21no—
141C2×D4×C61—22no—
142C2×D4×C61—22no—
143C2×D4×C61—22no—
144C2×D4×C61—22no—
145C2×D4×C61—22no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—42no—
150C2×D4×C61—82no—
151C2×D4×C61—84no—
152C2×D4×C61C2×D411yescharacteristic
153C2×D4×C61C2×D412yes—
154C2×D4×C61C2×D412yes—
155C22×C4×C61—22no—
156C22×C4×C61—22no—
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61—22no—
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61C2×D411yescharacteristic
163C22×Dic61—41no—
164C2×Dic122—22no—
165C2×Dic122—41no—
166C2×Dic122—82no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61C2×D411yescharacteristic
176C4⋊C4×C61C2×D411yescharacteristic
177C22⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—21no—
183C22⋊C4×C61—22no—
184C22⋊C4×C61—21no—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—21no—
187C22⋊C4×C61—22no—
188C22⋊C4×C61—22no—
189C22⋊C4×C61—21no—
190C22⋊C4×C61—21no—
191C22⋊C4×C61—22no—
192C22⋊C4×C61—22no—
193C42×C61—22no—
194C42×C61—21no—
195C42×C61C2×D411yescharacteristic
196C42×C61C2×D411yescharacteristic
197C42×C61C2×D411yescharacteristic
198C42×C61C2411yescharacteristic
199#11758202—82no—
200#11758200—82no—
201C4⋊Dic61—22no—
202C4⋊Dic61—22no—
203C4⋊Dic61—41no—
204C4⋊Dic61—41no—
205#11758194—42no—
206C2×#92418—22no—
207C2×#92418—22no—
208C2×#92418—22no—
209C2×#92418—42no—
210C23×C61—21no—
211C23×C61—22no—
212C23×C61—22no—
213C23×C61—42no—
214C23×C61—81no—
215C23×C61—81no—
216C23×C61—82no—
217C23×C61—82no—
218C23×C61—82no—
219C23×C61#12011yescharacteristic
220Q8×C61—41no—
221Q8×C61—42no—
222Q8×C61—42no—
223Q8×C61—42no—
224D4×C61—41no—
225D4×C61—42no—
226D4×C61—42no—
227D4×C61—42no—
228D4×C61—44no—
229D4×C61—42no—
230D4×C61—42no—
231D4×C61—81no—
232D4×C61—82no—
233D4×C61—82no—
234D4×C61—82no—
235D4×C61—82no—
236D4×C61—82no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—21no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—41no—
245C2×C4×C61—42no—
246C2×C4×C61—82no—
247C2×C4×C61—81no—
248C2×C4×C61—82no—
249C2×C4×C61—82no—
250C2×C4×C61#12011yescharacteristic
251C2×C4×C61#12011yescharacteristic
252C2×C4×C61#12011yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×Dic61—42no—
257C2×Dic61—82no—
258Dic122—42no—
259Dic122—42no—
260Dic122—82no—
261#92418—42no—
262#92418—42no—
263#92418—42no—
264#19020—611nomaximal, Sylow
265C22×C61—21no—
266C22×C61—21no—
267C22×C61—41no—
268C22×C61—42no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—42no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—42no—
275C22×C61—42no—
276C22×C61—162no—
277C22×C61—164no—
278C22×C61#45512yes—
279C22×C61C2×#12011yescharacteristic
280C4×C61—21no—
281C4×C61—21no—
282C4×C61—21no—
283C4×C61—21no—
284C4×C61—21no—
285C4×C61—41no—
286C4×C61—41no—
287C4×C61—42no—
288C4×C61—42no—
289C4×C61—41no—
290C4×C61#45512yes—
291Dic61—82no—
292#3558D6111yescharacteristic
293#3218—611no—
294#3141—611no—
295C2×#489—611no—
296#1662—614no—
297#1659—614no—
298#1530—611no—
299#1527—611no—
300#1524—611no—
301C2×C61—22no—
302C2×C61—82no—
303C2×C61—81no—
304C2×C61—82no—
305C2×C61C2×#45511yescharacteristic
306C2×C61C2×#45511yescharacteristic
307C2×C61#302111yescharacteristic
308#562C2×D6112yes—
309H(3,𝔽4)C2×D6112yes—
310#559—21no—
311#548—21no—
312#546—22no—
313#546—21no—
314D42—21no—
315#542—21no—
316#538—22no—
317#538—21no—
318#535—21no—
319#534—21no—
320#532C2×D6111yescharacteristic
321C2×C4⋊Q8—611no—
322#529C2×D6111yescharacteristic
323C2×#124C2×D6111yescharacteristic
324#497—612no—
325#491—612no—
326#489—614no—
327#489—614no—
328#489—1222no—
329#487—1221no—
330#486—1221no—
331#462—1221no—
332#461—1221no—
333#451—1221no—
334#448—1221no—
335#432—611no—
336#417—1222no—
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#1902011yescharacteristic, complemented, Sylow
346C2×#41—21no—
347C2×#41C22⋊D6112yes—
348C22×Q8—1221no—
349C22×Q8C22⋊D6111yescharacteristic
350C22×D4—22no—
351C22×D4—21no—
352C22×D4C22⋊D6111yescharacteristic
353C2×Q16—1222no—
354C2×SD16—1222no—
355C2×M16—1221no—
356C22×C8—1221no—
357C4⋊Q8—612no—
358C4⋊Q8—1221no—
359C4⋊Q8—1221no—
360C4⋊Q8—1221no—
361C4⋊Q8C22⋊D6111yescharacteristic
362#127—21no—
363#127C22⋊D6111yescharacteristic
364#126—22no—
365#126—21no—
366#125C22×D6111yescharacteristic
367#124—21no—
368#124—22no—
369#124—22no—
370#124—22no—
371#124—21no—
372#124—21no—
373#124C22⋊D6112yes—
374#124C22×D6112yes—
375#124C22×D6111yescharacteristic
376#124C22×D6112yes—
377#123—22no—
378#123—22no—
379#123—21no—
380#123—22no—
381#123—22no—
382C22⋊Q8—21no—
383C22⋊Q8—22no—
384C22⋊Q8—22no—
385C22⋊Q8C22⋊D6111yescharacteristic
386#121—22no—
387#121—22no—
388#121—22no—
389#121—21no—
390#121—22no—
391#121—22no—
392#121—21no—
393#121—21no—
394#121C22⋊D6111yescharacteristic
395#120—22no—
396#120—22no—
397#120—22no—
398#120—22no—
399#120—22no—
400C4×Q8—21no—
401C4×Q8C22⋊D6111yescharacteristic
402C4×D4—22no—
403C4×D4—22no—
404C4×D4—21no—
405C4×D4—21no—
406C4×D4—21no—
407C4×D4—21no—
408C4×D4C22⋊D6111yescharacteristic
409#117—21no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4—1221no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C42C22×D6111yescharacteristic
416C4⋊C8—1221no—
417#103—1224no—
418#103—1224no—
419#103—1222no—
420#103—1222no—
421#102—1224no—
422#102—1224no—
423C22⋊C8—1221no—
424#97—612no—
425#97—612no—
426#97—612no—
427#97—1221no—
428C4×C8—1221no—
429C24—22no—
430C24—21no—
431#41—44no—
432#41—82no—
433C2×Q8—21no—
434C2×Q8—42no—
435C2×Q8—1222no—
436C2×Q8—4882no—
437C2×Q8—2441no—
438C2×Q8C2×C22⋊D6112yes—
439C2×Q8C2×C22⋊D6112yes—
440C2×Q8C2×C22⋊D6111yescharacteristic
441C2×D4—22no—
442C2×D4—21no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—42no—
452C2×D4—82no—
453C2×D4—84no—
454C2×D4C2×C22⋊D6112yes—
455C2×D4C2×C22⋊D6112yes—
456C2×D4C2×C22⋊D6111yescharacteristic
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—2441no—
463C22×C4C2×C22⋊D6111yescharacteristic
464C22×C4C2×C22⋊D6111yescharacteristic
465C22×C4C2×C22⋊D6111yescharacteristic
466Q16—4882no—
467SD16—4882no—
468M16—2442no—
469C2×C8—1222no—
470C2×C8—1222no—
471C2×C8—1222no—
472C2×C8—2442no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—2441no—
484C4⋊C4—2441no—
485C4⋊C4C2×C22⋊D6111yescharacteristic
486C4⋊C4C2×C22⋊D6111yescharacteristic
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—21no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—21no—
496C22⋊C4—21no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—21no—
501C22⋊C4—22no—
502C22⋊C4—21no—
503C42—22no—
504C42—21no—
505C42C2×C22⋊D6111yescharacteristic
506C42C2×C22⋊D6111yescharacteristic
507C42C2×C22⋊D6111yescharacteristic
508C42C23×D6111yescharacteristic
509C23—21no—
510C23—22no—
511C23—22no—
512C23—42no—
513C23—81no—
514C23—81no—
515C23—82no—
516C23—82no—
517C23—82no—
518C23#1451974611yescharacteristic
519Q8—41no—
520Q8—42no—
521Q8—42no—
522Q8—42no—
523Q8—4882no—
524Q8—2442no—
525Q8—2442no—
526D4—41no—
527D4—42no—
528D4—42no—
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—2442no—
561C8—2442no—
562C8—2442no—
563C22—21no—
564C22—21no—
565C22—41no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—41no—
571C22—41no—
572C22—42no—
573C22—42no—
574C22—162no—
575C22—164no—
576C22#1683994012yes—
577C22C2×#1451974611yescharacteristic
578C4—21no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—21no—
583C4—41no—
584C4—41no—
585C4—42no—
586C4—42no—
587C4—41no—
588C4—4882no—
589C4#1683994012yes—
590C2—22no—
591C2—81no—
592C2—82no—
593C2—82no—
594C2C2×#1683994011yescharacteristic
595C2C2×#1683994011yescharacteristic
596C2#2145519111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3558C2
socleC22×C61C2×#120
hypercenter#3558D61
hypocenterC61#19020
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. #21456843C2semidirect
2. #21456212C2semidirect
3. C2×#16840121C2semidirect
4. #21445142C2semidirect
5. #21445125C2semidirect
6. #21444430C2semidirect
7. #21444417C2semidirect
8. #21444407C2semidirect
10. C2×#14519777C22semidirect
11. #16840157C22semidirect
12. #16840132C22semidirect
14. #16840121C22semidirect
15. #16840121C22semidirect
22. #16839819C22semidirect
26. C4×#11758194C22semidirect
27. C2×#14519688C22semidirect
28. #16839504C22semidirect
29. #16839502C22semidirect
30. #16839502C22semidirect
31. #16839502C22semidirect
345. C61#19020semidirect

Maximal subgroups

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

subgroupindex
#214568432
#214562122
C2×#168401212
#214451422
#214451252
#214444302
#214444172
#214444072
C61×#35582
#1902061