Finite Group Databasebeta

← #29650220

Subgroups — #29650220

Order 15616.

666 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455825C211yesmaximal, characteristic
2#21454750C211yesmaximal, characteristic, complemented
3#436×D61C211yesmaximal, characteristic
4#21444711C211yesmaximal, characteristic, complemented
5#21444709C211yesmaximal, characteristic, complemented
6#21444707C211yesmaximal, characteristic, complemented
7#21444665C211yesmaximal, characteristic, complemented
8#21444663C211yesmaximal, characteristic, complemented
9#21444661C211yesmaximal, characteristic, complemented
10#21444659C211yesmaximal, characteristic, complemented
11#21444588C211yesmaximal, characteristic, complemented
12#21444242C211yesmaximal, characteristic
13#21444220C211yesmaximal, characteristic, complemented
14#21444101C211yesmaximal, characteristic, complemented
15C61×#1590C211yesmaximal, characteristic, complemented
16C2×M16×D61—21no—
17#16840316C2211yescharacteristic
18C4×D4×D61C2211yescharacteristic
19#16840094C411yescharacteristic
20#16840091C411yescharacteristic
21#16840051C411yescharacteristic
22#16840050C411yescharacteristic
23#16839966C411yescharacteristic
24#16839965C411yescharacteristic
25#16839914C411yescharacteristic
26#16839911C411yescharacteristic
27#16839892C2211yescharacteristic, complemented
28#16839890—21no—
29Q8×#92418C2211yescharacteristic, complemented
30#16839886—21no—
31#16839886C2211yescharacteristic, complemented
32C4×#11758190C2211yescharacteristic
33C4×#11758201C2211yescharacteristic
34C4×#11758200C2211yescharacteristic
35#16839841C2211yescharacteristic
36#16839840C2211yescharacteristic
37#16839839C2211yescharacteristic
38#16839838—21no—
39#16839837—21no—
40#16839813C2211yescharacteristic
41#16839813C2211yescharacteristic
42#16839787—21no—
43#16839784C2211yescharacteristic
44C4⋊C8×D61C2211yescharacteristic
45C22⋊C8×D61—21no—
46#16839719—21no—
47C4×C8×D61C2211yescharacteristic
48#16839513C2211yescharacteristic, complemented
49#16839512C2211yescharacteristic, complemented
50#16839512C2211yescharacteristic, complemented
51#16839494C2211yescharacteristic, complemented
52Q8⋊#92418C2211yescharacteristic, complemented
53#16839492C2211yescharacteristic, complemented
54#16839491C2211yescharacteristic
55D4⋊#92418C2211yescharacteristic
56#16839489C2211yescharacteristic
57#16839484C2211yescharacteristic, complemented
58#16839484C2211yescharacteristic, complemented
59C8×#92418C2211yescharacteristic, complemented
60C61×#446C2211yescharacteristic, complemented
61C4×SD16×C61C2211yescharacteristic
62C61×#436C2211yescharacteristic
63C61×#334C2211yescharacteristic, complemented
64C61×Q8⋊C8C2211yescharacteristic, complemented
65C61×D4⋊C8C2211yescharacteristic
66C82×C61C2211yescharacteristic, complemented
67C2×D4×D61C2×C411yescharacteristic
68C22×C4×D61—21no—
69C2×#11758190C2×C411yescharacteristic
70C2×#11758200C2×C411yescharacteristic
71C2×#11758201C2×C411yescharacteristic
72C2×#11758194—21no—
73C2×#11758189—21no—
74C2×#11758189—21no—
75C2×#11758189—21no—
76M16×D61—44no—
77C2×C8×D61—21no—
78C2×C8×D61D411yescharacteristic
79C2×C8×D61D411yescharacteristic
80#14519776C2×C411yescharacteristic
81#14519775C2×C411yescharacteristic
82#14519767—21no—
83#14519757C2×C411yescharacteristic
84#14519755—21no—
85#14519753C2×C411yescharacteristic
86C4×Dic122—21no—
87C4×Dic122C2311yescharacteristic
88C4×D244C2311yescharacteristic
89D4×Dic61C2×C411yescharacteristic
90#14519739C2×C411yescharacteristic
91C4×C22⋊D61—21no—
92#14519737—21no—
93#14519734—21no—
94C4⋊C4×D61C2×C411yescharacteristic
95C22⋊C4×D61—21no—
96C42×D61C2311yescharacteristic
97#14519709C2×C411yescharacteristic
98#14519708C2×C411yescharacteristic
99#14519708C2×C411yescharacteristic
100C4⋊#92418—21no—
101C4⋊#92418C2311yescharacteristic
102C4⋊#92418C2311yescharacteristic
103#14519706—21no—
104#14519706—21no—
105#14519706—21no—
106#14519706—21no—
107#14519706—21no—
108#14519706C2×C411yescharacteristic
109#14519706C2×C411yescharacteristic
110#14519702C2×C411yescharacteristic
111#14519701C2×C411yescharacteristic
112#14519700C2×C411yescharacteristic
113#14519699C2×C411yescharacteristic
114#14519698C2×C411yescharacteristic
115#14519697C2×C411yescharacteristic
116C22⋊#92418—21no—
117#14519689—21no—
118#14519689—21no—
119#14519689—21no—
120#14519689C2×C411yescharacteristic
121#14519689C2×C411yescharacteristic
122#14519687—21no—
123C4×#92418—21no—
124C4×#92418C2311yescharacteristic
125C4×#92418C2311yescharacteristic
126C4×#92418C2311yescharacteristic
127C8×Dic61—21no—
128C8×Dic61D411yescharacteristic
129C8×Dic61D411yescharacteristic
130C2×SD16×C61C2×C411yescharacteristic
131C2×M16×C61—21no—
132C4×Q8×C61C2311yescharacteristic
133C4×D4×C61C2311yescharacteristic
134#106×C61C2×C411yescharacteristic
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61C2×C411yescharacteristic
139#102×C61C2×C411yescharacteristic
140C22⋊C8×C61—21no—
141#97×C61—21no—
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144C4×C8×C61C2311yescharacteristic
145C23×D61—21no—
146C2×Q8×C61C22×C411yescharacteristic
147C2×D4×C61C22×C411yescharacteristic
148C22×C4×C61—21no—
149C2×C22⋊D61—21no—
150C22×Dic61—21no—
151D4×D61—41no—
152D4×D61—41no—
153C2×D244C22×C411yescharacteristic
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61—21no—
157C2×C4×D61—42no—
158C2×C4×D61—42no—
159C2×C4×D61C22×C411yescharacteristic
160C2×C4×D61C2×D411yescharacteristic
161C2×C4×D61#4111yescharacteristic
162C2×Dic122—21no—
163C2×Dic122C22×C411yescharacteristic
164SD16×C61—41no—
165M16×C61—42no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61C22×C411yescharacteristic
170C2×C8×C61C22×C411yescharacteristic
171C2×C8×C61C2×D411yescharacteristic
172C2×C8×C61C2×D411yescharacteristic
173C4⋊C4×C61—21no—
174C4⋊C4×C61C22×C411yescharacteristic
175C4⋊C4×C61C22×C411yescharacteristic
176C22⋊C4×C61—21no—
177C42×C61—21no—
178C42×C61C2411yescharacteristic
179C22⋊Dic61—21no—
180#11758201—41no—
181#11758200—41no—
182#11758198—21no—
183#11758198—21no—
184#11758198—21no—
185#11758198—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61C22×C411yescharacteristic
188C4⋊Dic61C22×C411yescharacteristic
189#11758196—21no—
190#11758196—21no—
191#11758196—21no—
192#11758196—21no—
193#11758196—21no—
194#11758196—21no—
195C4×Dic61—21no—
196C4×Dic61C22×C411yescharacteristic
197C4×Dic61C2×D411yescharacteristic
198C4×Dic61#4111yescharacteristic
199#11758194—42no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418C22×C411yescharacteristic
204C2×#92418C22×C411yescharacteristic
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207#11758190—41no—
208#11758189—42no—
209#11758189—42no—
210#11758189—42no—
211C8×D61—22no—
212C8×D61—42no—
213C8×D61SD1614yescomplemented
214C23×C61—21no—
215C22×D61—21no—
216C22×D61—42no—
217C22×D61—42no—
218C22×D61—42no—
219C22×D61C4×D411yescharacteristic
220Q8×C61—21no—
221Q8×C61—21no—
222D4×C61—21no—
223D4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61C23×C411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C22⋊D61—42no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—42no—
237C2×Dic61C4×D411yescharacteristic
238C2×Dic61C4×D411yescharacteristic
239C2×Dic61C4×D411yescharacteristic
240D244—21no—
241D244—21no—
242C4×D61—21no—
243C4×D61—21no—
244C4×D61—41no—
245C4×D61—42no—
246C4×D61—41no—
247C4×D61—41no—
248C4×D61—42no—
249C4×D61C2×SD1612yes—
250C4×D61#13512yes—
251Dic122—21no—
252Dic122—21no—
253Dic122—42no—
254C8×C61—21no—
255C8×C61—21no—
256C8×C61—21no—
257C8×C61—41no—
258C8×C61—41no—
259C8×C61—41no—
260C8×C61C2×SD1612yes—
261#92418—21no—
262#92418—21no—
263#92418—21no—
264#92418—41no—
265#92418—41no—
266#92418—41no—
267#92418C2×SD1612yes—
268#17930—611nomaximal, Sylow
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61C2×C4×D411yescharacteristic
273C2×D61—21no—
274C2×D61—21no—
275C2×D61—41no—
276C2×D61—41no—
277C2×D61—41no—
278C2×D61—84no—
279C2×D61C4×SD1612yes—
280C4×C61—21no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61#56811yescharacteristic
285C4×C61#56811yescharacteristic
286C4×C61C22×SD1611yescharacteristic
287C4×C61C2×#13511yescharacteristic
288Dic61—21no—
289Dic61—21no—
290Dic61—41no—
291Dic61—41no—
292Dic61—41no—
293Dic61C4×SD1612yes—
294#3101—611no—
295#2960—611no—
296C2×#436—611no—
297#1590—611no—
298#1590—611no—
299#1590—611no—
300#1590D6111yescharacteristic, complemented
301#1583—611no—
302#1583—611no—
303#1583—611no—
304#1583—611no—
305#1570—611no—
306#1486—611no—
307#1481—611no—
308#1450—611no—
309C2×C61—81no—
310C2×C61#293011yescharacteristic
311C2×C61C2×C4×SD1611yescharacteristic
312C2×C61#295511yescharacteristic
313D61—22no—
314D61—81no—
315C22×M16—1221no—
316#533—611no—
317C2×C4×D4—611no—
318#483—611no—
319#483—611no—
320#476—611no—
321#476—611no—
322#461—611no—
323#461—611no—
324#450—611no—
325#450—611no—
326#446—611no—
327#446—1221no—
328#446C2×D6111yescharacteristic, complemented
329C8×Q8—611no—
330C8×Q8—611no—
331C8×Q8—1221no—
332C4×SD16—611no—
333C4×SD16—611no—
334C4×SD16—611no—
335C4×SD16C2×D6111yescharacteristic
336#436—611no—
337#436—611no—
338#436—611no—
339#436—1221no—
340#436—1221no—
341#436C2×D6111yescharacteristic
342#431—611no—
343#431—611no—
344#424—1221no—
345#423—611no—
346C2×C4⋊C8—611no—
347C2×C22⋊C8—1221no—
348#405—1221no—
349C2×C4×C8—611no—
350#334—611no—
351#334—611no—
352#334—611no—
353#334C2×D6111yescharacteristic, complemented
354Q8⋊C8—611no—
355Q8⋊C8—611no—
356Q8⋊C8—611no—
357Q8⋊C8C2×D6111yescharacteristic, complemented
358D4⋊C8—611no—
359D4⋊C8—611no—
360D4⋊C8—611no—
361D4⋊C8C2×D6111yescharacteristic
362#322—611no—
363#322—611no—
364C82—611no—
365C82C2×D6111yescharacteristic, complemented
366C61#1793011yescharacteristic, complemented, Sylow
367C22×D4—611no—
368C23×C4—1221no—
369C2×SD16—611no—
370C2×SD16—611no—
371C2×SD16—611no—
372C2×SD16C4×D6111yescharacteristic
373C2×M16—21no—
374C2×M16—1221no—
375C2×M16—1221no—
376C2×M16—1221no—
377C2×M16—1221no—
378C2×M16—2444no—
379C22×C8—611no—
380C22×C8—611no—
381C22×C8—1221no—
382C4⋊Q8—611no—
383C4⋊Q8—611no—
384#125—1221no—
385C22⋊Q8—611no—
386C22⋊Q8—611no—
387C22⋊Q8—1221no—
388C4×Q8—611no—
389C4×Q8—1221no—
390C4×Q8C22×D6111yescharacteristic
391C4×D4—611no—
392C4×D4—611no—
393C4×D4—611no—
394C4×D4—1221no—
395C4×D4—1221no—
396C4×D4C22×D6111yescharacteristic
397#117—1221no—
398C2×C4⋊C4—611no—
399C2×C22⋊C4—1221no—
400C2×C42—611no—
401#106—611no—
402#106—611no—
403#106—611no—
404#106C4×D6111yescharacteristic
405C4⋊C8—21no—
406C4⋊C8—611no—
407C4⋊C8—611no—
408C4⋊C8—611no—
409C4⋊C8—611no—
410C4⋊C8—1221no—
411C4⋊C8—1221no—
412C4⋊C8—1221no—
413C4⋊C8—1221no—
414C4⋊C8—1221no—
415C4⋊C8—1221no—
416C4⋊C8C22×D6111yescharacteristic
417C4⋊C8C22×D6111yescharacteristic
418#103—611no—
419#103—611no—
420#103—611no—
421#103C4×D6111yescharacteristic
422#102—611no—
423#102—611no—
424#102—611no—
425#102C4×D6111yescharacteristic
426C22⋊C8—21no—
427C22⋊C8—611no—
428C22⋊C8—611no—
429C22⋊C8—1221no—
430C22⋊C8—1221no—
431C22⋊C8—1221no—
432C22⋊C8—1221no—
433#97—21no—
434#97—1221no—
435C4×C8—611no—
436C4×C8—611no—
437C4×C8—611no—
438C4×C8—611no—
439C4×C8—611no—
440C4×C8—1221no—
441C4×C8—1221no—
442C4×C8C22×D6111yescharacteristic
443C4×C8C22×D6111yescharacteristic
444C4×C8C22×D6111yescharacteristic
445C24—1221no—
446C2×Q8—611no—
447C2×Q8—1221no—
448C2×Q8C2×C4×D6111yescharacteristic
449C2×D4—611no—
450C2×D4—1221no—
451C2×D4—2441no—
452C2×D4—2441no—
453C2×D4C2×C4×D6111yescharacteristic
454C22×C4—21no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—2442no—
463C22×C4—2442no—
464SD16—41no—
465SD16—2441no—
466SD16—2441no—
467SD16—2441no—
468M16—42no—
469M16—2442no—
470M16—2442no—
471M16—2442no—
472M16—2442no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—611no—
477C2×C8—614no—
478C2×C8—611no—
479C2×C8—611no—
480C2×C8—611no—
481C2×C8—1222no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—2442no—
486C2×C8C2×C4×D6111yescharacteristic
487C2×C8C2×C4×D6111yescharacteristic
488C2×C8D4×D6111yescharacteristic
489C2×C8D4×D6111yescharacteristic
490C4⋊C4—21no—
491C4⋊C4—611no—
492C4⋊C4—611no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4C2×C4×D6111yescharacteristic
501C4⋊C4C2×C4×D6111yescharacteristic
502C22⋊C4—21no—
503C22⋊C4—1221no—
504C22⋊C4—1221no—
505C22⋊C4—1221no—
506C22⋊C4—1221no—
507C22⋊C4—1221no—
508C42—21no—
509C42—611no—
510C42—611no—
511C42—611no—
512C42—1221no—
513C42C23×D6111yescharacteristic
514C23—21no—
515C23—611no—
516C23—1221no—
517C23—2442no—
518C23—2442no—
519C23—2442no—
520Q8—21no—
521Q8—21no—
522Q8—1221no—
523Q8—1221no—
524Q8—2442no—
525D4—21no—
526D4—21no—
527D4—1221no—
528D4—1221no—
529D4—2442no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—611no—
536C2×C4—612no—
537C2×C4—612no—
538C2×C4—611no—
539C2×C4—611no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—2441no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4C22×C4×D6111yescharacteristic
552C2×C4C2×D4×D6111yescharacteristic
553C2×C4#41×D6111yescharacteristic
554C8—21no—
555C8—21no—
556C8—21no—
557C8—41no—
558C8—41no—
559C8—41no—
560C8—612no—
561C8—1221no—
562C8—1221no—
563C8—1221no—
564C8—2441no—
565C8—2441no—
566C8—2441no—
567C8SD16×D6112yes—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—612no—
572C22—1221no—
573C22—1221no—
574C22—4884no—
575C22—2441no—
576C22—2441no—
577C22—2441no—
578C22C4×D4×D6111yescharacteristic
579C4—21no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—612no—
584C4—1221no—
585C4—1221no—
586C4—2441no—
587C4—2441no—
588C4—2441no—
589C4#1684052711yescharacteristic
590C4#1684052811yescharacteristic
591C4C2×SD16×D6111yescharacteristic
592C4#135×D6111yescharacteristic
593C2—81no—
594C2—1222no—
595C2—4881no—
596C2#2145451911yescharacteristic
597C2C4×SD16×D6111yescharacteristic
598C2#2145471711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1590C2
socleC22×C61C2×C4×D4
prime radicalC2×D4×D61C2×C4
cosocleC42×C61C24
hypercenter#1590D61
hypocenterC61#17930
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1590

Complements

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

NH ≅ G/Ntype
2. #21454750C2semidirect
4. #21444711C2semidirect
5. #21444709C2semidirect
6. #21444707C2semidirect
7. #21444665C2semidirect
8. #21444663C2semidirect
9. #21444661C2semidirect
10. #21444659C2semidirect
11. #21444588C2semidirect
13. #21444220C2semidirect
14. #21444101C2semidirect
15. C61×#1590C2semidirect
27. #16839892C22semidirect
29. Q8×#92418C22semidirect
31. #16839886C22semidirect
48. #16839513C22semidirect
49. #16839512C22semidirect
50. #16839512C22semidirect
51. #16839494C22semidirect
52. Q8⋊#92418C22semidirect
53. #16839492C22semidirect
57. #16839484C22semidirect
58. #16839484C22semidirect
59. C8×#92418C22semidirect
60. C61×#446C22semidirect
63. C61×#334C22semidirect
64. C61×Q8⋊C8C22semidirect
66. C82×C61C22semidirect
213. C8×D61SD16semidirect
300. #1590D61semidirect
328. #446C2×D61semidirect
353. #334C2×D61semidirect
357. Q8⋊C8C2×D61semidirect
365. C82C2×D61semidirect
366. C61#17930semidirect

Maximal subgroups

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

subgroupindex
#214558252
#214547502
#436×D612
#214447112
#214447092
#214447072
#214446652
#214446632
#214446612
#214446592
#214445882
#214442422
#214442202
#214441012
C61×#15902
#1793061