Finite Group Databasebeta

← #29669451

Subgroups — #29669451

Order 15616.

694 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458452C211yesmaximal, characteristic
2D4×#11758202C211yesmaximal, characteristic, complemented
3#21457481C211yesmaximal, characteristic, complemented
4#21445489C211yesmaximal, characteristic, complemented
5#21445306C211yesmaximal, characteristic, complemented
6#21445263C211yesmaximal, characteristic, complemented
7#21445210C211yesmaximal, characteristic, complemented
8#21445162C211yesmaximal, characteristic, complemented
9#21445068C211yesmaximal, characteristic, complemented
10#21445067C211yesmaximal, characteristic, complemented
11#21445011C211yesmaximal, characteristic, complemented
12#21444922C211yesmaximal, characteristic, complemented
13#21444649C211yesmaximal, characteristic, complemented
14#21444636C211yesmaximal, characteristic, complemented
15Q8×D8×C61C211yesmaximal, characteristic, complemented
16C2×#14519808—21no—
17C22×#11758202—21no—
18#16840433C2211yescharacteristic
19D4×Dic122C2211yescharacteristic
20#16840165C2211yescharacteristic
21#16840141C2211yescharacteristic
22#16840136C2211yescharacteristic
23#16840110C2211yescharacteristic
24#16840109C2211yescharacteristic
25#16840107—21no—
26#16840042C2211yescharacteristic
27#16840041—21no—
28#16840010—21no—
29#16840001—21no—
30#16839972C2211yescharacteristic
31#16839969C2211yescharacteristic
32#16839969C2211yescharacteristic
33#16839962C2211yescharacteristic
34#16839957C2211yescharacteristic
35#16839955—21no—
36#16839922—21no—
37#16839918—21no—
38#16839917—21no—
39#16839901—21no—
40C8×Dic122C2211yescharacteristic
41C4×#11758202C2211yescharacteristic
42#16839842—21no—
43#16839840C2211yescharacteristic
44D4×#92418C2211yescharacteristic
45#16839515C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839508C2211yescharacteristic
48#16839508C2211yescharacteristic
49#16839507C2211yescharacteristic
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52#16839492C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic
56D4×Q8×C61C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic
58C61×#500—21no—
59C61×#500C2211yescharacteristic
60C61×#474—21no—
61C61×#474—21no—
62C61×#474C2211yescharacteristic
63C61×#474C2211yescharacteristic
64C8×Q8×C61C2211yescharacteristic
65C4×D8×C61—21no—
66C4×D8×C61C2211yescharacteristic
67C2×#11758222—21no—
68C22×Dic122—21no—
69#14519808—44no—
70C2×#11758202—21no—
71C2×#11758202—21no—
72C2×#11758202—21no—
73C2×#11758202—21no—
74C2×#11758202—21no—
75C2×#11758202—44no—
76C2×#11758202D411yescharacteristic
77C2×#11758201—21no—
78C2×#11758201—21no—
79C2×#11758201—21no—
80C2×#11758199—21no—
81C2×#11758194—21no—
82C22×#92418—21no—
83#14519777C2311yescharacteristic
84#14519776—21no—
85#14519774C2311yescharacteristic
86#14519773—21no—
87#14519756—21no—
88#14519754—41no—
89#14519751—21no—
90#14519750—41no—
91C4×Dic122C2311yescharacteristic
92D4×Dic61—21no—
93#14519739—21no—
94#14519710—21no—
95#14519710—41no—
96#14519710D411yescharacteristic
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100#14519706—21no—
101#14519701—21no—
102#14519701—21no—
103#14519701D411yescharacteristic
104#14519700—21no—
105#14519700D411yescharacteristic
106#14519699—21no—
107#14519699—21no—
108#14519698—21no—
109#14519697—21no—
110#14519697—21no—
111#14519697—41no—
112C22⋊#92418—21no—
113C22⋊#92418—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C8×Dic61—21no—
117C22×Q8×C61—21no—
118C22×Q8×C61—21no—
119C2×D8×C61D411yescharacteristic
120C4⋊Q8×C61—21no—
121C4⋊Q8×C61—21no—
122C4⋊Q8×C61C2311yescharacteristic
123C4⋊Q8×C61C2311yescharacteristic
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—21no—
126C22⋊Q8×C61—41no—
127C22⋊Q8×C61—41no—
128C4×Q8×C61C2311yescharacteristic
129C4×D4×C61—21no—
130C4×D4×C61—21no—
131C4×D4×C61C2311yescharacteristic
132C4×D4×C61C2311yescharacteristic
133#107×C61—21no—
134#107×C61—21no—
135#107×C61—21no—
136#107×C61—21no—
137#107×C61D411yescharacteristic
138C4⋊C8×C61—21no—
139C4⋊C8×C61C2311yescharacteristic
140#102×C61—21no—
141#102×C61—21no—
142#102×C61D411yescharacteristic
143#102×C61D411yescharacteristic
144C4×C8×C61—21no—
145C4×C8×C61C2311yescharacteristic
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61—41no—
149C2×Q8×C61—41no—
150C2×Q8×C61—42no—
151C2×Q8×C61—42no—
152C2×Q8×C61—81no—
153C2×Q8×C61—81no—
154C2×Q8×C61C2×D411yescharacteristic
155C2×D4×C61C2×D411yescharacteristic
156C2×D4×C61C2×D411yescharacteristic
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61—41no—
160C22×C4×C61—41no—
161C2×C22⋊D61—41no—
162C2×C22⋊D61—41no—
163C22×Dic61—41no—
164#11758222—42no—
165#11758222—81no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×C4×D61—41no—
170C2×Dic122—21no—
171C2×Dic122—42no—
172C2×Dic122—41no—
173C2×Dic122—81no—
174C2×Dic122C2×D411yescharacteristic
175D8×C61—22no—
176C2×C8×C61—21no—
177C2×C8×C61C2×D411yescharacteristic
178C2×C8×C61C2×D411yescharacteristic
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—41no—
185C4⋊C4×C61—41no—
186C4⋊C4×C61C2×D411yescharacteristic
187C4⋊C4×C61C2×D411yescharacteristic
188C4⋊C4×C61C2×D411yescharacteristic
189C22⋊C4×C61—21no—
190C22⋊C4×C61—21no—
191C22⋊C4×C61—41no—
192C22⋊C4×C61—41no—
193C42×C61—21no—
194C42×C61C2411yescharacteristic
195C22⋊Dic61—41no—
196#11758202—42no—
197#11758202—42no—
198#11758202—42no—
199#11758202—42no—
200#11758202—81no—
201#11758202D814yescomplemented
202#11758201—42no—
203#11758201—42no—
204#11758201—81no—
205#11758199—42no—
206#11758198—41no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61C2×D411yescharacteristic
210#11758196—21no—
211#11758196—41no—
212C4×Dic61—21no—
213#11758194—42no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—42no—
219C2×#92418C2×D411yescharacteristic
220C2×#92418C2×D411yescharacteristic
221C23×C61—21no—
222C23×C61—21no—
223C22×D61—41no—
224Q8×C61—21no—
225Q8×C61—41no—
226Q8×C61—41no—
227Q8×C61—41no—
228Q8×C61—41no—
229Q8×C61—82no—
230Q8×C61—82no—
231Q8×C61C2×D812yes—
232D4×C61—21no—
233D4×C61—21no—
234D4×C61—21no—
235D4×C61#13712yescomplemented
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—82no—
247C2×C4×C61—82no—
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61H(5,𝔽2)11yescharacteristic
251C22⋊D61—82no—
252C22⋊D61—161no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—41no—
256C2×Dic61—82no—
257D244—41no—
258D244—41no—
259D244—41no—
260D244—41no—
261D244—81no—
262C4×D61—82no—
263Dic122—21no—
264Dic122—41no—
265Dic122—41no—
266Dic122—82no—
267Dic122C2×D812yes—
268C8×C61—21no—
269C8×C61—21no—
270C8×C61—41no—
271#92418—21no—
272#92418—41no—
273#92418—41no—
274#92418—41no—
275#92418—41no—
276#92418C2×D812yes—
277#19761—611nomaximal, Sylow
278C22×C61—22no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61D4211yescharacteristic
285C2×D61—81no—
286C2×D61—81no—
287C2×D61—81no—
288C4×C61—22no—
289C4×C61—21no—
290C4×C61—41no—
291C4×C61—41no—
292C4×C61—41no—
293C4×C61—81no—
294C4×C61—81no—
295C4×C61C22×D811yescharacteristic
296C4×C61C2×#13711yescharacteristic
297C4×C61#57611yescharacteristic
298C4×C61#57811yescharacteristic
299Dic61—22no—
300Dic61—41no—
301Dic61—81no—
302Q8×D8D6111yescharacteristic, complemented
303#3358—611no—
304D4×Q16—611no—
305#3278—611no—
306#1722—611no—
307#1689—611no—
308#1682—611no—
309#1674—611no—
310#1665—611no—
311#1649—611no—
312#1649—611no—
313#1641—611no—
314#1627—611no—
315#1581—611no—
316#1579—611no—
317C2×C61—42no—
318C2×C61—81no—
319C2×C61#327011yescharacteristic
320C2×C61D4×D811yescharacteristic
321C2×C61#328911yescharacteristic
322D61—161no—
323C2×#137—1221no—
324C22×Q16—1221no—
325#550—611no—
326D4×Q8—611no—
327D4×Q8C2×D6111yescharacteristic
328D4×Q8C2×D6111yescharacteristic
329#500—21no—
330#500—611no—
331#500C2×D6111yescharacteristic
332#494—611no—
333#492—611no—
334#486—611no—
335#486—611no—
336#486—1221no—
337#474—21no—
338#474—21no—
339#474—611no—
340#474—1221no—
341#474C2×D6111yescharacteristic
342#474C2×D6111yescharacteristic
343#468—1221no—
344#467—1221no—
345#462—611no—
346#462—611no—
347#462—611no—
348#460—611no—
349#459—611no—
350#459—1221no—
351#452—1221no—
352#451—1221no—
353#451—1221no—
354#448—1221no—
355C8×Q8—611no—
356C8×Q8C2×D6111yescharacteristic
357C4×Q16—611no—
358C4×D8—21no—
359C4×D8—1221no—
360C4×D8C2×D6111yescharacteristic
361#436—611no—
362C8×D4—611no—
363#335—611no—
364#334—611no—
365#332—611no—
366#332—611no—
367#332—611no—
368#331—611no—
369Q8⋊C8—611no—
370Q8⋊C8—611no—
371D4⋊C8—611no—
372D4⋊C8—611no—
373#322—611no—
374C61#1976111yescharacteristic, complemented, Sylow
375C2×#41—1221no—
376C22×Q8—21no—
377C22×Q8—21no—
378C22×Q8—1221no—
379#137—2444no—
380C2×Q16—611no—
381C2×Q16—1221no—
382C2×Q16—1221no—
383C2×Q16—1221no—
384C2×Q16—1221no—
385C2×Q16—1221no—
386C2×Q16—2444no—
387C2×SD16—1221no—
388C2×SD16—1221no—
389C2×SD16—1221no—
390C2×D8—1221no—
391C2×D8C22⋊D6111yescharacteristic
392C2×M16—1221no—
393C22×C8—1221no—
394C4⋊Q8—21no—
395C4⋊Q8—21no—
396C4⋊Q8—611no—
397C4⋊Q8—1221no—
398C4⋊Q8C22×D6111yescharacteristic
399C4⋊Q8C22×D6111yescharacteristic
400#127—611no—
401#127—1221no—
402C22⋊Q8—21no—
403C22⋊Q8—21no—
404C22⋊Q8—41no—
405C22⋊Q8—41no—
406C22⋊Q8—1221no—
407C22⋊Q8—2441no—
408#121—1221no—
409#121—2441no—
410C4×Q8—611no—
411C4×Q8C22×D6111yescharacteristic
412C4×D4—21no—
413C4×D4—21no—
414C4×D4—1221no—
415C4×D4—1221no—
416C4×D4C22×D6111yescharacteristic
417C4×D4C22×D6111yescharacteristic
418#107—21no—
419#107—21no—
420#107—21no—
421#107—21no—
422#107—611no—
423#107—1221no—
424#107—2441no—
425#107C22⋊D6111yescharacteristic
426C4⋊C8—21no—
427C4⋊C8—611no—
428C4⋊C8—611no—
429C4⋊C8—611no—
430C4⋊C8—1221no—
431C4⋊C8C22×D6111yescharacteristic
432#103—611no—
433#103—611no—
434#103—1221no—
435#103—1221no—
436#103—1221no—
437#102—21no—
438#102—21no—
439#102—1221no—
440#102—1221no—
441#102—1221no—
442#102—1221no—
443#102—1221no—
444#102—2441no—
445#102C22⋊D6111yescharacteristic
446#102C22⋊D6111yescharacteristic
447C22⋊C8—1221no—
448C22⋊C8—1221no—
449C4×C8—21no—
450C4×C8—611no—
451C4×C8—611no—
452C4×C8—1221no—
453C4×C8C22×D6111yescharacteristic
454#41—4881no—
455#41—2442no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—41no—
459C2×Q8—41no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×Q8—81no—
463C2×Q8—81no—
464C2×Q8—611no—
465C2×Q8—1221no—
466C2×Q8—4881no—
467C2×Q8—2441no—
468C2×Q8—2442no—
469C2×Q8D4×D6111yescharacteristic
470C2×D4—1221no—
471C2×D4—1221no—
472C2×D4—1221no—
473C2×D4—2441no—
474C2×D4—2441no—
475C2×D4C2×C22⋊D6111yescharacteristic
476C2×D4C2×C22⋊D6111yescharacteristic
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—2441no—
482C22×C4—2441no—
483Q16—614no—
484Q16—4881no—
485Q16—2442no—
486Q16—2442no—
487Q16—2442no—
488Q16—2442no—
489SD16—4881no—
490SD16—2442no—
491SD16—2442no—
492D8—22no—
493D8—2442no—
494M16—2442no—
495C2×C8—21no—
496C2×C8—611no—
497C2×C8—611no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—2442no—
503C2×C8C2×C22⋊D6111yescharacteristic
504C2×C8C2×C22⋊D6111yescharacteristic
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—611no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—2441no—
517C4⋊C4D4×D6111yescharacteristic
518C4⋊C4C2×C22⋊D6111yescharacteristic
519C4⋊C4C2×C22⋊D6111yescharacteristic
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—41no—
523C22⋊C4—41no—
524C22⋊C4—2441no—
525C22⋊C4—2441no—
526C42—21no—
527C42—1221no—
528C42C23×D6111yescharacteristic
529C23—21no—
530C23—21no—
531C23—2441no—
532Q8—21no—
533Q8—41no—
534Q8—41no—
535Q8—41no—
536Q8—41no—
537Q8—82no—
538Q8—82no—
539Q8—612no—
540Q8—1221no—
541Q8—4882no—
542Q8—2441no—
543Q8—2441no—
544Q8D8×D6112yes—
545D4—21no—
546D4—21no—
547D4—21no—
548D4—9761no—
549D4—4881no—
550D4—4882no—
551D4—2441no—
552D4—2441no—
553D4—2441no—
554D4—2441no—
555D4#1451980812yescomplemented
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—4882no—
571C2×C4—4882no—
572C2×C4—2441no—
573C2×C4C2×D4×D6111yescharacteristic
574C2×C4C22×C22⋊D6111yescharacteristic
575C2×C4#1451982211yescharacteristic
576C8—21no—
577C8—21no—
578C8—41no—
579C8—612no—
580C8—1221no—
581C8—2441no—
582C8—2441no—
583C8—2441no—
584C8—2441no—
585C22—22no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—4881no—
592C22—4881no—
593C22—4881no—
594C22D4×C22⋊D6111yescharacteristic
595C4—22no—
596C4—21no—
597C4—41no—
598C4—41no—
599C4—41no—
600C4—81no—
601C4—81no—
602C4—1222no—
603C4—4881no—
604C4—2441no—
605C4C2×D8×D6111yescharacteristic
606C4C2×#1451980811yescharacteristic
607C4#1684056311yescharacteristic
608C4#1684057711yescharacteristic
609C2—42no—
610C2—81no—
611C2—9761no—
612C2#2145739011yescharacteristic
613C2D8×C22⋊D6111yescharacteristic
614C2#2145759011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingQ8×D8×C61C2
socleC22×C61D42
prime radical#21458452C2
hypercenterQ8×D8D61
hypocenterC61#19761
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ Q8×D8

Complements

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

NH ≅ G/Ntype
2. D4×#11758202C2semidirect
3. #21457481C2semidirect
4. #21445489C2semidirect
5. #21445306C2semidirect
6. #21445263C2semidirect
7. #21445210C2semidirect
8. #21445162C2semidirect
9. #21445068C2semidirect
10. #21445067C2semidirect
11. #21445011C2semidirect
12. #21444922C2semidirect
13. #21444649C2semidirect
14. #21444636C2semidirect
15. Q8×D8×C61C2semidirect
201. #11758202D8semidirect
235. D4×C61#137semidirect
302. Q8×D8D61semidirect
374. C61#19761semidirect
555. D4#14519808semidirect

Maximal subgroups

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

subgroupindex
#214584522
D4×#117582022
#214574812
#214454892
#214453062
#214452632
#214452102
#214451622
#214450682
#214450672
#214450112
#214449222
#214446492
#214446362
Q8×D8×C612
#1976161