Finite Group Databasebeta

← #29670118

Subgroups — #29670118

Order 15616.

676 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458682C211yesmaximal, characteristic, complemented
2#21458170C211yesmaximal, characteristic, complemented
3#21457646C211yesmaximal, characteristic, complemented
4#21457414C211yesmaximal, characteristic
5#21445439C211yesmaximal, characteristic, complemented
6#21445341C211yesmaximal, characteristic, complemented
7#21445226C211yesmaximal, characteristic, complemented
8#21445197C211yesmaximal, characteristic, complemented
9#21445165C211yesmaximal, characteristic, complemented
10#21445029C211yesmaximal, characteristic, complemented
11#21444972C211yesmaximal, characteristic, complemented
12#21444933C211yesmaximal, characteristic, complemented
13#21444659C211yesmaximal, characteristic, complemented
14#21444640C211yesmaximal, characteristic, complemented
15C61×#1622C211yesmaximal, characteristic, complemented
16C2×#14519801—21no—
17#16840472C2211yescharacteristic
18#16840416C2211yescharacteristic
19#16840397C2211yescharacteristic
20#16840165C2211yescharacteristic, complemented
21#16840161—21no—
22#16840158C2211yescharacteristic, complemented
23#16840134C2211yescharacteristic
24#16840113C2211yescharacteristic, complemented
25#16840108C2211yescharacteristic
26#16840090—21no—
27#16840080—21no—
28#16840079—21no—
29#16840062—21no—
30#16840059—21no—
31#16840053C2211yescharacteristic
32#16840052C2211yescharacteristic, complemented
33#16840049C2211yescharacteristic
34#16840049C2211yescharacteristic
35#16840047C2211yescharacteristic, complemented
36#16840007—21no—
37#16839993—21no—
38#16839988—21no—
39#16839968C2211yescharacteristic
40#16839914—21no—
41#16839911—21no—
42#16839904—21no—
43Q8×#92418C2211yescharacteristic, complemented
44C4×#11758201C2211yescharacteristic
45C4×#11758199—21no—
46C4×#11758199C2211yescharacteristic
47#16839839C2211yescharacteristic
48D4×#92418C2211yescharacteristic, complemented
49#16839827C2211yescharacteristic
50C2×#14519708—21no—
51#102×D61—21no—
52#16839512C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839510C2211yescharacteristic, complemented
55#16839507C2211yescharacteristic, complemented
56#16839506C2211yescharacteristic
57#16839491C2211yescharacteristic
58#16839491C2211yescharacteristic
59#16839484C2211yescharacteristic, complemented
60C61×#554C2211yescharacteristic, complemented
61C61×#492C2211yescharacteristic
62C61×#460C2211yescharacteristic
63C61×#459C2211yescharacteristic
64C61×#333C2211yescharacteristic, complemented
65C61×Q8⋊C8C2211yescharacteristic, complemented
66C61×D4⋊C8C2211yescharacteristic, complemented
67C2×#11758224—21no—
68C2×D4×D61—21no—
69C2×D4×D61—21no—
70#14519801—44no—
71C2×#11758190—21no—
72C2×#11758200—21no—
73C2×#11758200—21no—
74C2×#11758201D411yescharacteristic
75C2×#11758199D411yescharacteristic
76C2×#11758189—21no—
77C22×#92418—21no—
78C2×C8×D61—21no—
79#14519777C2311yescharacteristic
80#14519777C2311yescharacteristic
81#14519769—21no—
82#14519761—41no—
83#14519757—21no—
84#14519755—41no—
85#14519753—21no—
86#14519751—21no—
87#14519748—21no—
88#14519748—21no—
89#14519745—41no—
90C4×D244—21no—
91C4×D244C2311yescharacteristic
92C4×D244C2311yescharacteristic
93C2×C4⋊Dic61—21no—
94C4⋊C4×D61—21no—
95#14519710—21no—
96#14519710—21no—
97#14519710D411yescharacteristic
98#14519709—21no—
99#14519708—21no—
100#14519708—21no—
101#14519708—21no—
102#14519708—21no—
103#14519708—21no—
104#14519708—21no—
105#14519708D411yescharacteristic
106C4⋊#92418—21no—
107C4⋊#92418C2311yescharacteristic
108C4⋊#92418C2311yescharacteristic
109#14519702—21no—
110#14519702—41no—
111#14519701—21no—
112#14519701D411yescharacteristic
113#14519699—21no—
114#14519698—21no—
115#14519698—21no—
116#14519698—21no—
117#14519698D411yescharacteristic
118#14519697—21no—
119#14519697—21no—
120#14519697D411yescharacteristic
121#14519697D411yescharacteristic
122C22⋊#92418—21no—
123#14519689—21no—
124#14519689—21no—
125C4×#92418—21no—
126C4×#92418C2311yescharacteristic
127C4×#92418C2311yescharacteristic
128C2×SD16×C61—21no—
129C2×SD16×C61—41no—
130C2×D8×C61—21no—
131C4⋊Q8×C61C2311yescharacteristic
132#127×C61C2311yescharacteristic
133#125×C61—21no—
134C22⋊Q8×C61—21no—
135C22⋊Q8×C61—41no—
136C4×Q8×C61C2311yescharacteristic
137C4×D4×C61—21no—
138C4×D4×C61C2311yescharacteristic
139C2×C4⋊C4×C61—21no—
140C4⋊C8×C61C2311yescharacteristic
141C4⋊C8×C61C2311yescharacteristic
142#102×C61—21no—
143#102×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—41no—
146C2×Q8×C61—21no—
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61C2×D411yescharacteristic
152C22×C4×C61—21no—
153C22×C4×C61—41no—
154C2×C22⋊D61—41no—
155C2×C22⋊D61—41no—
156C2×C22⋊D61—41no—
157C22×Dic61—41no—
158#11758224—42no—
159#11758224—81no—
160D4×D61—42no—
161D4×D61—42no—
162D4×D61—81no—
163D4×D61—81no—
164C2×D244C2×D411yescharacteristic
165C2×D244C2×D411yescharacteristic
166C2×C4×D61—21no—
167C2×C4×D61—21no—
168C2×C4×D61—41no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171SD16×C61—81no—
172SD16×C61—82no—
173D8×C61—42no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—41no—
181C4⋊C4×C61—41no—
182C4⋊C4×C61C2×D411yescharacteristic
183C4⋊C4×C61C2×D411yescharacteristic
184C22⋊C4×C61—21no—
185C22⋊C4×C61—41no—
186C42×C61—21no—
187C42×C61C2411yescharacteristic
188C22⋊Dic61—41no—
189#11758201—41no—
190#11758200—42no—
191#11758200—81no—
192#11758199SD1614yescomplemented
193#11758198—21no—
194#11758198—21no—
195#11758198—41no—
196#11758198—41no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—41no—
202C4⋊Dic61C2×D411yescharacteristic
203C4⋊Dic61C2×D411yescharacteristic
204#11758196—41no—
205#11758196—41no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—42no—
209C2×#92418C2×D411yescharacteristic
210C2×#92418C2×D411yescharacteristic
211C2×#92418C2×D411yescharacteristic
212C2×#92418C2×D411yescharacteristic
213#11758190—42no—
214#11758189—42no—
215C8×D61—81no—
216C23×C61—21no—
217C23×C61—41no—
218C22×D61—21no—
219C22×D61—21no—
220C22×D61—82no—
221C22×D61—82no—
222C22×D61—161no—
223Q8×C61—21no—
224Q8×C61—21no—
225Q8×C61—81no—
226D4×C61—21no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—81no—
232D4×C61C2×SD1612yes—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—161no—
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61H(5,𝔽2)11yescharacteristic
243C22⋊D61—82no—
244C22⋊D61—82no—
245C22⋊D61—82no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—82no—
251D244—21no—
252D244—21no—
253D244—21no—
254D244C2×SD1612yes—
255C4×D61—41no—
256C4×D61—41no—
257C4×D61—41no—
258C4×D61—41no—
259C4×D61—161no—
260Dic122—41no—
261Dic122—41no—
262Dic122—81no—
263C8×C61—41no—
264C8×C61—41no—
265C8×C61—41no—
266#92418—21no—
267#92418—21no—
268#92418—21no—
269#92418—41no—
270#92418—41no—
271#92418C2×SD1612yes—
272#19806—611nomaximal, Sylow
273C22×C61—22no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—81no—
277C22×C61—81no—
278C22×C61—81no—
279C22×C61D4211yescharacteristic
280C2×D61—22no—
281C2×D61—41no—
282C2×D61—41no—
283C2×D61—41no—
284C2×D61—41no—
285C2×D61—41no—
286C2×D61—164no—
287C4×C61—21no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61—41no—
291C4×C61—81no—
292C4×C61C22×SD1611yescharacteristic
293C4×C61#57511yescharacteristic
294C4×C61#57711yescharacteristic
295C4×C61#57711yescharacteristic
296Dic61—41no—
297Dic61—41no—
298Dic61—81no—
299Dic61—81no—
300#3380—611no—
301#3335—611no—
302#3295—611no—
303#3272—611no—
304#1714—611no—
305#1695—611no—
306#1677—611no—
307#1671—611no—
308#1666—611no—
309#1643—611no—
310#1635—611no—
311#1629—611no—
312#1622D6111yescharacteristic, complemented
313#1583—611no—
314#1580—611no—
315C2×C61—42no—
316C2×C61—161no—
317C2×C61#326711yescharacteristic
318C2×C61D4×SD1611yescharacteristic
319C2×C61#328811yescharacteristic
320D61—42no—
321D61—81no—
322C2×AGL(1,ℤ/8)—1221no—
323#554—611no—
324#554C2×D6111yescharacteristic, complemented
325#547—611no—
326D42—611no—
327#500—611no—
328#499—1221no—
329#498—611no—
330#492—611no—
331#492C2×D6111yescharacteristic
332#487—611no—
333#486—611no—
334#483—1221no—
335#481—1221no—
336#481—1221no—
337#478—1221no—
338#478—1221no—
339#476—611no—
340#476—611no—
341#476—611no—
342#476—611no—
343#475—611no—
344#468—1221no—
345#465—1221no—
346#465—1221no—
347#461—611no—
348#460C2×D6111yescharacteristic
349#459C2×D6111yescharacteristic
350#450—1221no—
351#450—1221no—
352#449—1221no—
353C8×Q8—611no—
354C4×SD16—611no—
355C4×D8—611no—
356C4×D8—1221no—
357#436—611no—
358C8×D4—611no—
359C8×D4—611no—
360C2×#106—1221no—
361C2×#102—1221no—
362#334—611no—
363#334—611no—
364#333—611no—
365#333C2×D6111yescharacteristic, complemented
366#332—611no—
367#331—611no—
368Q8⋊C8C2×D6111yescharacteristic, complemented
369D4⋊C8—611no—
370D4⋊C8—611no—
371D4⋊C8C2×D6111yescharacteristic, complemented
372#322—611no—
373C61#1980611yescharacteristic, complemented, Sylow
374C2×#41—1221no—
375C22×D4—1221no—
376C22×D4—1221no—
377AGL(1,ℤ/8)—2444no—
378C2×SD16—21no—
379C2×SD16—41no—
380C2×SD16—611no—
381C2×SD16—1221no—
382C2×SD16—1221no—
383C2×SD16—1221no—
384C2×D8—21no—
385C2×D8—611no—
386C2×M16—1221no—
387C22×C8—1221no—
388C22×C8—1221no—
389C4⋊Q8—611no—
390C4⋊Q8—611no—
391C4⋊Q8C22×D6111yescharacteristic
392#127C22×D6111yescharacteristic
393#125—21no—
394#125—1221no—
395#123—2441no—
396C22⋊Q8—21no—
397C22⋊Q8—41no—
398C22⋊Q8—1221no—
399C22⋊Q8—1221no—
400C22⋊Q8—2441no—
401#121—1221no—
402#121—1221no—
403#121—1221no—
404#120—2441no—
405C4×Q8C22×D6111yescharacteristic
406C4×D4—21no—
407C4×D4—611no—
408C4×D4—611no—
409C4×D4—1221no—
410C4×D4C22×D6111yescharacteristic
411C2×C4⋊C4—21no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414#107—611no—
415#107—1221no—
416#107—1221no—
417#106—611no—
418#106—1221no—
419#106—1221no—
420#106—1221no—
421#106—1221no—
422#106—1221no—
423#106—1221no—
424#106—1221no—
425C4⋊C8—611no—
426C4⋊C8—611no—
427C4⋊C8—1221no—
428C4⋊C8C22×D6111yescharacteristic
429C4⋊C8C22×D6111yescharacteristic
430#103—611no—
431#103—1221no—
432#103—1221no—
433#103—2441no—
434#102—21no—
435#102—21no—
436#102—611no—
437#102—611no—
438#102—611no—
439#102—1221no—
440#102—1221no—
441#102—1221no—
442#102—1221no—
443#102—1221no—
444#102—1221no—
445C22⋊C8—1221no—
446C22⋊C8—1221no—
447C22⋊C8—1221no—
448C4×C8—611no—
449C4×C8—611no—
450C4×C8—1221no—
451C4×C8C22×D6111yescharacteristic
452C24—2441no—
453#41—4881no—
454#41—2442no—
455C2×Q8—21no—
456C2×Q8—1221no—
457C2×Q8—1221no—
458C2×Q8D4×D6111yescharacteristic
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—611no—
463C2×D4—611no—
464C2×D4—4881no—
465C2×D4—4881no—
466C2×D4—2441no—
467C2×D4—2441no—
468C2×D4—2442no—
469C2×D4—2442no—
470C2×D4—2441no—
471C2×D4D4×D6111yescharacteristic
472C22×C4—21no—
473C22×C4—41no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—2441no—
477C22×C4—2441no—
478SD16—81no—
479SD16—82no—
480SD16—4881no—
481SD16—2442no—
482SD16—2442no—
483SD16—2441no—
484D8—42no—
485D8—614no—
486M16—2442no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—611no—
491C2×C8—611no—
492C2×C8—611no—
493C2×C8—611no—
494C2×C8—1221no—
495C2×C8—1221no—
496C2×C8—4881no—
497C2×C8—2442no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—611no—
504C4⋊C4—611no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—2441no—
510C4⋊C4—2441no—
511C4⋊C4—2441no—
512C4⋊C4D4×D6111yescharacteristic
513C4⋊C4D4×D6111yescharacteristic
514C22⋊C4—21no—
515C22⋊C4—41no—
516C22⋊C4—1221no—
517C22⋊C4—1221no—
518C22⋊C4—2441no—
519C22⋊C4—2441no—
520C22⋊C4—2441no—
521C42—21no—
522C42C23×D6111yescharacteristic
523C23—21no—
524C23—41no—
525C23—1221no—
526C23—1221no—
527C23—9761no—
528C23—4882no—
529C23—4882no—
530Q8—21no—
531Q8—21no—
532Q8—81no—
533Q8—4881no—
534Q8—2441no—
535Q8—2441no—
536D4—21no—
537D4—41no—
538D4—41no—
539D4—41no—
540D4—41no—
541D4—81no—
542D4—612no—
543D4—1221no—
544D4—1221no—
545D4—1221no—
546D4—4882no—
547D4—4882no—
548D4—4882no—
549D4SD16×D6112yes—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—161no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—9761no—
560C2×C4—4882no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4C2×D4×D6111yescharacteristic
568C2×C4C2×D4×D6111yescharacteristic
569C2×C4#1451982211yescharacteristic
570C8—41no—
571C8—41no—
572C8—41no—
573C8—612no—
574C8—1221no—
575C8—1221no—
576C8—1221no—
577C8—2441no—
578C8—2441no—
579C22—22no—
580C22—41no—
581C22—41no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—1222no—
586C22—9764no—
587C22—2441no—
588C22—2441no—
589C22—2441no—
590C22—2441no—
591C22—2441no—
592C22#1684039411yescharacteristic
593C4—21no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—81no—
598C4—4881no—
599C4—4881no—
600C4—2441no—
601C4—2441no—
602C4C2×SD16×D6111yescharacteristic
603C4#1684055911yescharacteristic
604C4#1684057111yescharacteristic
605C4#1684057211yescharacteristic
606C2—42no—
607C2—161no—
608C2—4881no—
609C2—2442no—
610C2#2145733711yescharacteristic
611C2#2145750811yescharacteristic
612C2#2145758011yescharacteristic
subgroupG/H
centerC22#16840394
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1622C2
socleC22×C61D42
prime radical#21457414C2
hypercenter#1622D61
hypocenterC61#19806
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1622

Complements

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

NH ≅ G/Ntype
1. #21458682C2semidirect
2. #21458170C2semidirect
3. #21457646C2semidirect
5. #21445439C2semidirect
6. #21445341C2semidirect
7. #21445226C2semidirect
8. #21445197C2semidirect
9. #21445165C2semidirect
10. #21445029C2semidirect
11. #21444972C2semidirect
12. #21444933C2semidirect
13. #21444659C2semidirect
14. #21444640C2semidirect
15. C61×#1622C2semidirect
20. #16840165C22semidirect
22. #16840158C22semidirect
24. #16840113C22semidirect
32. #16840052C22semidirect
35. #16840047C22semidirect
43. Q8×#92418C22semidirect
48. D4×#92418C22semidirect
52. #16839512C22semidirect
53. #16839512C22semidirect
54. #16839510C22semidirect
55. #16839507C22semidirect
59. #16839484C22semidirect
60. C61×#554C22semidirect
64. C61×#333C22semidirect
65. C61×Q8⋊C8C22semidirect
66. C61×D4⋊C8C22semidirect
192. #11758199SD16semidirect
312. #1622D61semidirect
324. #554C2×D61semidirect
365. #333C2×D61semidirect
368. Q8⋊C8C2×D61semidirect
371. D4⋊C8C2×D61semidirect
373. C61#19806semidirect

Maximal subgroups

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

subgroupindex
#214586822
#214581702
#214576462
#214574142
#214454392
#214453412
#214452262
#214451972
#214451652
#214450292
#214449722
#214449332
#214446592
#214446402
C61×#16222
#1980661