Finite Group Databasebeta

← #29670126

Subgroups — #29670126

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#21457645C211yesmaximal, characteristic, complemented
3#21457419C211yesmaximal, characteristic
4#21445444C211yesmaximal, characteristic, complemented
5#21445340C211yesmaximal, characteristic, complemented
6#21445228C211yesmaximal, characteristic, complemented
7#21445191C211yesmaximal, characteristic, complemented
8#21445169C211yesmaximal, characteristic, complemented
9#21445025C211yesmaximal, characteristic, complemented
10#21444970C211yesmaximal, characteristic, complemented
11#21444934C211yesmaximal, characteristic, complemented
12#21444891C211yesmaximal, characteristic, complemented
13#21444663C211yesmaximal, characteristic, complemented
14#21444641C211yesmaximal, characteristic, complemented
15C61×#3335C211yesmaximal, characteristic, complemented
16C2×#14519802—21no—
17#16840472C2211yescharacteristic
18D4×D244C2211yescharacteristic
19#16840165C2211yescharacteristic, complemented
20#16840161—21no—
21#16840136C2211yescharacteristic
22#16840136C2211yescharacteristic
23#16840114C2211yescharacteristic, complemented
24#16840109C2211yescharacteristic
25#16840082—21no—
26#16840062—21no—
27#16840059—21no—
28#16840052C2211yescharacteristic
29#16840052C2211yescharacteristic, complemented
30#16840049C2211yescharacteristic
31#16840010—21no—
32#16839994—21no—
33#16839964C2211yescharacteristic
34#16839962C2211yescharacteristic
35#16839957C2211yescharacteristic
36#16839912—21no—
37#16839911—21no—
38#16839907—21no—
39Q8×#92418C2211yescharacteristic, complemented
40C4×#11758199—21no—
41C4×#11758199C2211yescharacteristic
42#16839840C2211yescharacteristic
43C8×D244C2211yescharacteristic
44#102×D61—21no—
45#16839513C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839511C2211yescharacteristic, complemented
48#16839510C2211yescharacteristic, complemented
49#16839508C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic, complemented
54#16839489C2211yescharacteristic
55#16839484C2211yescharacteristic, complemented
56C61×#554C2211yescharacteristic, complemented
57C61×#547C2211yescharacteristic
58C61×#498C2211yescharacteristic, complemented
59C61×#483—21no—
60C61×#481—21no—
61C61×#476C2211yescharacteristic
62C61×#475C2211yescharacteristic, complemented
63C61×#465—21no—
64C4×SD16×C61C2211yescharacteristic
65C8×D4×C61C2211yescharacteristic, complemented
66C2×#106×C61—21no—
67C22×D244—21no—
68C2×D4×D61—21no—
69#14519802—44no—
70C2×#11758201—21no—
71C2×#11758201—21no—
72C2×#11758201—21no—
73C2×#11758201—21no—
74C2×#11758201—41no—
75C2×#11758199—21no—
76C2×#11758199D411yescharacteristic
77C2×#11758194—21no—
78C2×C8×D61—21no—
79#14519777C2311yescharacteristic
80#14519774C2311yescharacteristic
81#14519769—21no—
82#14519755—41no—
83#14519753—21no—
84#14519752—21no—
85#14519751—21no—
86#14519747—41no—
87C4×D244—21no—
88C4×D244C2311yescharacteristic
89C4⋊C4×D61—21no—
90#14519710—21no—
91#14519710—21no—
92#14519710D411yescharacteristic
93#14519708—21no—
94#14519708—21no—
95#14519708—21no—
96C4⋊#92418—21no—
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100#14519701—21no—
101#14519700—41no—
102#14519699—21no—
103#14519699—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698D411yescharacteristic
107#14519697—21no—
108#14519697—21no—
109#14519697—21no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112#14519689—21no—
113C4×#92418—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C2×#41×C61—21no—
117C2×SD16×C61D411yescharacteristic
118C22×C8×C61—21no—
119C4⋊Q8×C61C2311yescharacteristic
120C4⋊Q8×C61C2311yescharacteristic
121#125×C61—21no—
122#123×C61—41no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—41no—
126#121×C61—21no—
127C4×Q8×C61C2311yescharacteristic
128C4×D4×C61—21no—
129C4×D4×C61C2311yescharacteristic
130C4×D4×C61C2311yescharacteristic
131C2×C4⋊C4×C61—21no—
132C2×C4⋊C4×C61—21no—
133#106×C61—21no—
134#106×C61—21no—
135#106×C61—21no—
136#106×C61—21no—
137#106×C61D411yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—41no—
146#41×C61—42no—
147#41×C61—81no—
148C2×Q8×C61—21no—
149C2×Q8×C61—21no—
150C2×Q8×C61C2×D411yescharacteristic
151C2×D4×C61—41no—
152C2×D4×C61C2×D411yescharacteristic
153C2×D4×C61C2×D411yescharacteristic
154C22×C4×C61—21no—
155C22×C4×C61—21no—
156C22×C4×C61—41no—
157C22×C4×C61—41no—
158C2×C22⋊D61—41no—
159D4×D61—42no—
160D4×D61—81no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—42no—
165C2×D244—41no—
166C2×D244—81no—
167C2×D244C2×D411yescharacteristic
168C2×C4×D61—21no—
169C2×C4×D61—41no—
170C2×Dic122—21no—
171SD16×C61—41no—
172C2×C8×C61—21no—
173C2×C8×C61—42no—
174C2×C8×C61C2×D411yescharacteristic
175C2×C8×C61C2×D411yescharacteristic
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—41no—
182C4⋊C4×C61—41no—
183C4⋊C4×C61—41no—
184C4⋊C4×C61C2×D411yescharacteristic
185C4⋊C4×C61C2×D411yescharacteristic
186C4⋊C4×C61C2×D411yescharacteristic
187C22⋊C4×C61—21no—
188C22⋊C4×C61—21no—
189C22⋊C4×C61—41no—
190C22⋊C4×C61—41no—
191C42×C61—21no—
192C42×C61C2411yescharacteristic
193#11758201—42no—
194#11758201—42no—
195#11758201—81no—
196#11758201—81no—
197#11758201—82no—
198#11758199—42no—
199#11758199SD1614yescomplemented
200#11758198—21no—
201#11758198—41no—
202#11758198—41no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61C2×D411yescharacteristic
206#11758196—41no—
207#11758196—41no—
208#11758194—42no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418C2×D411yescharacteristic
214C2×#92418C2×D411yescharacteristic
215C8×D61—81no—
216C23×C61—21no—
217C23×C61—21no—
218C22×D61—21no—
219C22×D61—41no—
220C22×D61—82no—
221C22×D61—82no—
222C22×D61—161no—
223Q8×C61—21no—
224Q8×C61—21no—
225Q8×C61—41no—
226Q8×C61—41no—
227Q8×C61—81no—
228D4×C61—21no—
229D4×C61—21no—
230D4×C61—21no—
231D4×C61—82no—
232D4×C61C2×SD1612yes—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—82no—
244C2×C4×C61—161no—
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61H(5,𝔽2)11yescharacteristic
248C22⋊D61—82no—
249C2×Dic61—21no—
250C2×Dic61—41no—
251D244—21no—
252D244—41no—
253D244—41no—
254D244—41no—
255D244—41no—
256D244—81no—
257D244—82no—
258D244C2×SD1612yes—
259C4×D61—41no—
260C4×D61—41no—
261C4×D61—161no—
262Dic122—81no—
263C8×C61—21no—
264C8×C61—21no—
265C8×C61—41no—
266#92418—21no—
267#92418—41no—
268#92418—41no—
269#92418—41no—
270#92418—41no—
271#92418C2×SD1612yes—
272#19806—611nomaximal, Sylow
273C22×C61—22no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61D4211yescharacteristic
280C2×D61—22no—
281C2×D61—41no—
282C2×D61—41no—
283C2×D61—81no—
284C2×D61—81no—
285C2×D61—81no—
286C2×D61—164no—
287C4×C61—21no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61—41no—
291C4×C61—41no—
292C4×C61—81no—
293C4×C61—81no—
294C4×C61C22×SD1611yescharacteristic
295C4×C61#57511yescharacteristic
296C4×C61#57711yescharacteristic
297C4×C61#57711yescharacteristic
298Dic61—41no—
299Dic61—81no—
300#3380—611no—
301#3335D6111yescharacteristic, complemented
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#1622—611no—
313#1583—611no—
314#1580—611no—
315C2×C61—42no—
316C2×C61—81no—
317C2×C61#326711yescharacteristic
318C2×C61D4×SD1611yescharacteristic
319C2×C61#328811yescharacteristic
320D61—42no—
321D61—161no—
322C2×AGL(1,ℤ/8)—1221no—
323#554—611no—
324#554C2×D6111yescharacteristic, complemented
325#547C2×D6111yescharacteristic
326D42—611no—
327#500—611no—
328#499—1221no—
329#498C2×D6111yescharacteristic, complemented
330#492—611no—
331#492—611no—
332#487—611no—
333#486—611no—
334#483—21no—
335#481—21no—
336#481—1221no—
337#478—1221no—
338#478—1221no—
339#476—611no—
340#476—611no—
341#476—611no—
342#476C2×D6111yescharacteristic
343#475C2×D6111yescharacteristic, complemented
344#468—1221no—
345#465—21no—
346#465—1221no—
347#461—611no—
348#460—611no—
349#459—611no—
350#450—1221no—
351#450—1221no—
352#449—1221no—
353C8×Q8—611no—
354C4×SD16C2×D6111yescharacteristic
355C4×D8—611no—
356C4×D8—1221no—
357#436—611no—
358C8×D4—611no—
359C8×D4C2×D6111yescharacteristic, complemented
360C2×#106—21no—
361C2×#102—1221no—
362#334—611no—
363#334—611no—
364#333—611no—
365#333—611no—
366#332—611no—
367#331—611no—
368Q8⋊C8—611no—
369D4⋊C8—611no—
370D4⋊C8—611no—
371D4⋊C8—611no—
372#322—611no—
373C61#1980611yescharacteristic, complemented, Sylow
374C2×#41—21no—
375C22×D4—1221no—
376C22×D4—1221no—
377AGL(1,ℤ/8)—2444no—
378C2×SD16—1221no—
379C2×SD16—1221no—
380C2×SD16—1221no—
381C2×SD16—1221no—
382C2×SD16—2441no—
383C2×SD16C22⋊D6111yescharacteristic
384C2×D8—611no—
385C2×D8—1221no—
386C2×M16—1221no—
387C22×C8—21no—
388C22×C8—1221no—
389C4⋊Q8—611no—
390C4⋊Q8C22×D6111yescharacteristic
391C4⋊Q8C22×D6111yescharacteristic
392#127—611no—
393#125—21no—
394#125—1221no—
395#123—41no—
396C22⋊Q8—21no—
397C22⋊Q8—21no—
398C22⋊Q8—41no—
399C22⋊Q8—1221no—
400C22⋊Q8—2441no—
401#121—21no—
402#121—1221no—
403#121—1221no—
404#120—2441no—
405C4×Q8C22×D6111yescharacteristic
406C4×D4—21no—
407C4×D4—611no—
408C4×D4—1221no—
409C4×D4C22×D6111yescharacteristic
410C4×D4C22×D6111yescharacteristic
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—1221no—
414#107—611no—
415#107—1221no—
416#107—1221no—
417#106—21no—
418#106—21no—
419#106—21no—
420#106—21no—
421#106—1221no—
422#106—1221no—
423#106—1221no—
424#106C22⋊D6111yescharacteristic
425C4⋊C8—611no—
426C4⋊C8—611no—
427C4⋊C8—611no—
428C4⋊C8—1221no—
429C4⋊C8C22×D6111yescharacteristic
430#103—21no—
431#103—1221no—
432#103—2441no—
433#103C22⋊D6111yescharacteristic
434#102—21no—
435#102—611no—
436#102—611no—
437#102—1221no—
438#102—1221no—
439#102—1221no—
440#102—1221no—
441#102—1221no—
442#102—1221no—
443#102—1221no—
444#102C22⋊D6111yescharacteristic
445C22⋊C8—21no—
446C22⋊C8—1221no—
447C22⋊C8—1221no—
448C4×C8—611no—
449C4×C8—611no—
450C4×C8—1221no—
451C4×C8C22×D6111yescharacteristic
452C24—2441no—
453#41—42no—
454#41—81no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—1221no—
458C2×Q8C2×C22⋊D6111yescharacteristic
459C2×D4—41no—
460C2×D4—611no—
461C2×D4—1221no—
462C2×D4—1221no—
463C2×D4—1221no—
464C2×D4—4881no—
465C2×D4—4881no—
466C2×D4—2441no—
467C2×D4—2442no—
468C2×D4—2441no—
469C2×D4—2442no—
470C2×D4D4×D6111yescharacteristic
471C2×D4C2×C22⋊D6111yescharacteristic
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—1221no—
477C22×C4—2441no—
478SD16—41no—
479SD16—4881no—
480SD16—4882no—
481SD16—4881no—
482SD16—2442no—
483SD16—2442no—
484D8—614no—
485D8—2442no—
486M16—2442no—
487C2×C8—21no—
488C2×C8—42no—
489C2×C8—611no—
490C2×C8—611no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—4881no—
496C2×C8C2×C22⋊D6111yescharacteristic
497C2×C8C2×C22⋊D6111yescharacteristic
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—611no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—2441no—
510C4⋊C4—2441no—
511C4⋊C4D4×D6111yescharacteristic
512C4⋊C4C2×C22⋊D6111yescharacteristic
513C4⋊C4C2×C22⋊D6111yescharacteristic
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—41no—
517C22⋊C4—41no—
518C22⋊C4—1221no—
519C22⋊C4—2441no—
520C22⋊C4—2441no—
521C42—21no—
522C42C23×D6111yescharacteristic
523C23—21no—
524C23—21no—
525C23—1221no—
526C23—9761no—
527C23—4882no—
528C23—4882no—
529C23—2441no—
530Q8—21no—
531Q8—21no—
532Q8—41no—
533Q8—41no—
534Q8—81no—
535Q8—4881no—
536D4—21no—
537D4—21no—
538D4—21no—
539D4—82no—
540D4—612no—
541D4—1221no—
542D4—4881no—
543D4—4882no—
544D4—4882no—
545D4—2441no—
546D4—2441no—
547D4—2441no—
548D4—2441no—
549D4SD16×D6112yes—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—82no—
561C2×C4—161no—
562C2×C4—1221no—
563C2×C4—9761no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4C2×D4×D6111yescharacteristic
568C2×C4C22×C22⋊D6111yescharacteristic
569C2×C4#1451982211yescharacteristic
570C8—21no—
571C8—21no—
572C8—41no—
573C8—612no—
574C8—1221no—
575C8—2441no—
576C8—2441no—
577C8—2441no—
578C8—2441no—
579C22—22no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—1222no—
586C22—9764no—
587C22—4881no—
588C22—4881no—
589C22—4881no—
590C22—2441no—
591C22—2441no—
592C22D4×C22⋊D6111yescharacteristic
593C4—21no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—81no—
599C4—81no—
600C4—4881no—
601C4—2441no—
602C4C2×SD16×D6111yescharacteristic
603C4#1684056011yescharacteristic
604C4#1684056811yescharacteristic
605C4#1684057011yescharacteristic
606C2—42no—
607C2—81no—
608C2—9761no—
609C2—2442no—
610C2#2145734511yescharacteristic
611C2SD16×C22⋊D6111yescharacteristic
612C2#2145758111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3335C2
socleC22×C61D42
prime radical#21457419C2
hypercenter#3335D61
hypocenterC61#19806
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3335

Complements

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

NH ≅ G/Ntype
1. #21458682C2semidirect
2. #21457645C2semidirect
4. #21445444C2semidirect
5. #21445340C2semidirect
6. #21445228C2semidirect
7. #21445191C2semidirect
8. #21445169C2semidirect
9. #21445025C2semidirect
10. #21444970C2semidirect
11. #21444934C2semidirect
12. #21444891C2semidirect
13. #21444663C2semidirect
14. #21444641C2semidirect
15. C61×#3335C2semidirect
19. #16840165C22semidirect
23. #16840114C22semidirect
29. #16840052C22semidirect
39. Q8×#92418C22semidirect
45. #16839513C22semidirect
46. #16839512C22semidirect
47. #16839511C22semidirect
48. #16839510C22semidirect
49. #16839508C22semidirect
51. Q8⋊#92418C22semidirect
53. D4⋊#92418C22semidirect
55. #16839484C22semidirect
56. C61×#554C22semidirect
58. C61×#498C22semidirect
62. C61×#475C22semidirect
65. C8×D4×C61C22semidirect
199. #11758199SD16semidirect
301. #3335D61semidirect
324. #554C2×D61semidirect
329. #498C2×D61semidirect
343. #475C2×D61semidirect
359. C8×D4C2×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
#214576452
#214574192
#214454442
#214453402
#214452282
#214451912
#214451692
#214450252
#214449702
#214449342
#214448912
#214446632
#214446412
C61×#33352
#1980661