Finite Group Databasebeta

← #30560715

Subgroups — #30560715

Order 15616.

1706 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459645C211yesmaximal, characteristic, complemented
2#21459643C211yesmaximal, characteristic, complemented
3#21458786C214yesmaximal, complemented
4#21458777C214yesmaximal, complemented
5#21458697C214yesmaximal, complemented
6#21458689C214yesmaximal, complemented
7#21456349C212yesmaximal, complemented
8C2×#16840161C211yesmaximal, characteristic, complemented
9#21455674C212yesmaximal, complemented
10#21455669C212yesmaximal, complemented
11#21455642C211yesmaximal, characteristic, complemented
12#21454749C211yesmaximal, characteristic, complemented
13#21454698C212yesmaximal
14C4×#14519797C211yesmaximal, characteristic
15C61×#3080C211yesmaximal, characteristic, complemented
16C2×#14519797C2211yescharacteristic
17#16840489C2214yes—
18#16840488C2214yes—
19#16840471—24no—
20#16840471C2214yes—
21#16840468C2214yes—
22#16840467—24no—
23#16840393C2212yes—
24#16840392—22no—
25#16840390—22no—
26#16840390C2212yes—
27#16840319—21no—
28#16840319C2211yescharacteristic
29#16840318—21no—
30C2×#14519769—21no—
31C2×#14519769C2211yescharacteristic
32#125×D61—21no—
33#16840245C2212yes—
34#16840243C2212yes—
35C4×#11758222C2211yescharacteristic
36#16840170C2214yes—
37#16840169C2214yes—
38#16840163C2214yes—
39#16840161C2214yes—
40#16840161C2214yes—
41#16840159C2214yes—
42#16840072C2214yes—
43#16840070C2214yes—
44#16840067C2214yes—
45#16840064C2214yes—
46#16840063C2214yes—
47#16840061C2214yes—
48#16840056C2214yes—
49#16840055C2214yes—
50#16840041C2214yes—
51#16840040C2214yes—
52#16839889C2214yes—
53#16839886C2214yes—
54#16839871C2214yes—
55#16839866C2214yes—
56C4×#11758202C2212yes—
57#16839853C2214yes—
58C4×#11758201C2212yes—
59C4×#11758200C2212yes—
60#16839847C2214yes—
61C4×#11758199C2212yes—
62#16839804C2212yes—
63#16839798C2211yescharacteristic
64C2×#14519710C2212yes—
65C2×#14519708C2212yes—
66#16839790C2212yes—
67#16839764C2212yes—
68#16839758C2211yescharacteristic
69#16839756C2211yescharacteristic
70#16839720C2212yes—
71C2×C4×#92418C2211yescharacteristic
72C2×#125×C61C2211yescharacteristic
73C4×#41×C61C2211yescharacteristic
74C61×#479C2214yes—
75C61×#478C2214yes—
76C61×#428C2212yes—
77C61×#423C2211yescharacteristic
78C61×#417C2212yes—
79C2×#11758222C2311yescharacteristic
80#14519797Q818yescomplemented
81C2×#11758202C2311yescharacteristic
82C2×#11758200C2311yescharacteristic
83C2×#11758201C2311yescharacteristic
84C2×#11758199C2311yescharacteristic
85C22×#92418C2311yescharacteristic
86C22×#92418C2311yescharacteristic
87#14519777—22no—
88#14519777D412yes—
89#14519776—24no—
90#14519776—24no—
91#14519776—22no—
92#14519776C2314yes—
93#14519776C2314yes—
94#14519769—21no—
95#14519769—21no—
96#14519769—22no—
97#14519769—24no—
98#14519769D411yescharacteristic
99#14519769D411yescharacteristic
100#14519769C2314yes—
101#14519768—24no—
102#14519768—22no—
103#14519768C2314yes—
104#14519768C2314yes—
105#14519767—24no—
106#14519767—24no—
107#14519757—24no—
108#14519756—24no—
109#14519756—24no—
110#14519756D414yes—
111#14519755—24no—
112#14519755—24no—
113#14519754—24no—
114#14519754—24no—
115#14519754—24no—
116Q8×Dic61C2314yes—
117C4×Dic122—24no—
118C4×Dic122C2312yes—
119C4×Dic122C2314yes—
120#14519742—24no—
121#14519742C2314yes—
122C4×D244C2312yes—
123D4×Dic61—24no—
124D4×Dic61C2314yes—
125#14519739C2314yes—
126C4×C22⋊D61—22no—
127C4×C22⋊D61—24no—
128#14519736—22no—
129#14519736C2311yescharacteristic
130#14519736C2312yes—
131#14519735—22no—
132#14519734—21no—
133C2×C4⋊Dic61—22no—
134C2×C4⋊Dic61C2312yes—
135C2×#11758196—22no—
136C2×#11758196—24no—
137C4⋊C4×D61—22no—
138C4⋊C4×D61—24no—
139C42×D61—21no—
140#14519711C2314yes—
141#14519710C2314yes—
142#14519710C2314yes—
143#14519710C2312yes—
144#14519709C2314yes—
145#14519708C2314yes—
146#14519708C2314yes—
147#14519708C2312yes—
148#14519706C2314yes—
149#14519706C2314yes—
150#14519702C2314yes—
151#14519701C2312yes—
152#14519700C2312yes—
153#14519699C2314yes—
154#14519698C2312yes—
155#14519697C2312yes—
156#14519687C2314yes—
157C4×#92418C2312yes—
158C4×#92418C2312yes—
159C8×Dic61C2314yes—
160C2×#41×C61C2311yescharacteristic
161C2×M16×C61C2312yes—
162#125×C61—21no—
163#125×C61—21no—
164#125×C61C2314yes—
165C4×Q8×C61C2312yes—
166C4×D4×C61—22no—
167C4×D4×C61C2312yes—
168#117×C61—21no—
169#117×C61C2311yescharacteristic
170C2×C4⋊C4×C61—22no—
171C2×C4⋊C4×C61C2312yes—
172C2×C42×C61—21no—
173C2×C42×C61C2311yescharacteristic
174#107×C61C2314yes—
175#106×C61C2314yes—
176C4⋊C8×C61C2314yes—
177#103×C61C2314yes—
178#102×C61C2314yes—
179#41×C61—22no—
180#41×C61C2×Q814yes—
181C2×Q8×C61C2411yescharacteristic
182C2×D4×C61—21no—
183C2×D4×C61C2411yescharacteristic
184C22×C4×C61—21no—
185C22×C4×C61—22no—
186C22×C4×C61—22no—
187C22×C4×C61C2×D412yes—
188C22×C4×C61C2411yescharacteristic
189C2×C22⋊D61—21no—
190C22×Dic61—22no—
191C22×Dic61—24no—
192#11758222—22no—
193#11758222C2×Q814yes—
194C2×D244C2411yescharacteristic
195C2×C4×D61—21no—
196C2×C4×D61—22no—
197C2×C4×D61—24no—
198C2×Dic122—24no—
199C2×Dic122—24no—
200C2×Dic122C2×D414yes—
201C2×Dic122C2411yescharacteristic
202M16×C61—24no—
203C2×C8×C61C2414yes—
204C4⋊C4×C61—24no—
205C4⋊C4×C61—24no—
206C4⋊C4×C61—22no—
207C4⋊C4×C61—22no—
208C4⋊C4×C61C2414yes—
209C4⋊C4×C61C2412yes—
210C22⋊C4×C61—21no—
211C22⋊C4×C61—22no—
212C42×C61—21no—
213C42×C61—22no—
214C42×C61C2×D411yescharacteristic
215C42×C61C2×D411yescharacteristic
216C42×C61C2412yes—
217C22⋊Dic61—21no—
218C22⋊Dic61—24no—
219C22⋊Dic61C2×D414yes—
220#11758202C2×Q814yes—
221#11758201C2×Q814yes—
222#11758200C2×Q814yes—
223#11758199C2×Q814yes—
224#11758198—22no—
225#11758198—24no—
226C4⋊Dic61—22no—
227C4⋊Dic61—24no—
228C4⋊Dic61—24no—
229C4⋊Dic61—24no—
230C4⋊Dic61C2×D412yes—
231C4⋊Dic61C2×D412yes—
232C4⋊Dic61C2×D414yes—
233C4⋊Dic61C2412yes—
234C4⋊Dic61C2414yes—
235#11758196—24no—
236#11758196—24no—
237#11758196—22no—
238#11758196—24no—
239#11758196—24no—
240#11758196—24no—
241#11758196—24no—
242#11758196—24no—
243#11758196C2×D414yes—
244#11758196C2×D414yes—
245C4×Dic61—21no—
246C4×Dic61—22no—
247C4×Dic61—24no—
248C4×Dic61—24no—
249C4×Dic61C2414yes—
250C2×#92418—22no—
251C2×#92418C2×Q814yes—
252C2×#92418C2412yes—
253C2×#92418C2411yescharacteristic
254C2×#92418C2411yescharacteristic
255C23×C61—21no—
256C23×C61C22×D411yescharacteristic
257C22×D61—21no—
258Q8×C61—21no—
259Q8×C61C22×Q812yes—
260D4×C61—24no—
261D4×C61—21no—
262D4×C61C22×Q812yes—
263C2×C4×C61—21no—
264C2×C4×C61—24no—
265C2×C4×C61—24no—
266C2×C4×C61—22no—
267C2×C4×C61—42no—
268C2×C4×C61—82no—
269C2×C4×C61—82no—
270C2×C4×C61C22×D412yes—
271C2×C4×C61C22×D412yes—
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61C22×Q812yes—
274C2×C4×C612−1+412yes—
275C2×C4×C61C2511yescharacteristic
276C22⋊D61—24no—
277C2×Dic61—21no—
278C2×Dic61—22no—
279C2×Dic61—24no—
280C2×Dic61—84no—
281C2×Dic61—82no—
282C2×Dic61C22×D414yes—
283C2×Dic61C22×D414yes—
284D244—21no—
285D244C22×Q812yes—
286C4×D61—24no—
287C4×D61—84no—
288C4×D61—82no—
289Dic122—21no—
290Dic122—44no—
291Dic122—84no—
292Dic122—84no—
293Dic122C22×Q812yes—
294C8×C61—24no—
295#92418—22no—
296#92418C22×Q812yes—
297#92418C22×Q812yes—
298#61375—611nomaximal, Sylow
299C22×C61—22no—
300C22×C61—22no—
301C22×C61—42no—
302C22×C61D4×Q812yes—
303C22×C61C23×D411yescharacteristic
304C2×D61—22no—
305C2×D61—42no—
306C4×C61—22no—
307C4×C61—22no—
308C4×C61—22no—
309C4×C61—42no—
310C4×C61—44no—
311C4×C61D4×Q812yes—
312C4×C61C23×Q811yescharacteristic
313C4×C61C2×2−1+411yescharacteristic
314Dic61—24no—
315Dic61—24no—
316Dic61—22no—
317Dic61—42no—
318Dic61—44no—
319#3504—611no—
320#3504—611no—
321#3390—614no—
322#3389—614no—
323#3381—614no—
324#3381—614no—
325#3163—612no—
326C2×#499—611no—
327#3083—612no—
328#3083—612no—
329#3080—611no—
330#3080D6111yescharacteristic, complemented
331#2960—611no—
332#2952—612no—
333C4×#135—611no—
334C2×C61—22no—
335C2×C61—42no—
336C2×C61C2×D4×Q811yescharacteristic
337C2×C61#358412yes—
338D61—42no—
339C2×#135—611no—
340#557—614no—
341#557—614no—
342#554—614no—
343#554—614no—
344#554—1224no—
345#554—1224no—
346#544—612no—
347#544—612no—
348#544—1222no—
349#544—1222no—
350#533—611no—
351#533—1221no—
352#533—1221no—
353C2×#125—611no—
354C2×#125—1221no—
355C2×#125—1221no—
356C2×#125C2×D6111yescharacteristic
357#520—612no—
358#520—612no—
359C4×#41—611no—
360C4×#41C2×D6111yescharacteristic
361#501—614no—
362#501—614no—
363#499—614no—
364#499—614no—
365#499—614no—
366#498—614no—
367#479—614no—
368#479—614no—
369#479—614no—
370#479C2×D6114yes—
371#478—614no—
372#478—614no—
373#478—614no—
374#478C2×D6114yes—
375#477—614no—
376#477—614no—
377#474—614no—
378#474—614no—
379#446—614no—
380C8×Q8—614no—
381#442—614no—
382#441—614no—
383C4×Q16—612no—
384C4×SD16—612no—
385C4×SD16—612no—
386C4×SD16—614no—
387C4×SD16—614no—
388C4×D8—612no—
389#428—612no—
390#428C2×D6112yes—
391#427—611no—
392C2×#107—612no—
393C2×#106—612no—
394#424—612no—
395#423C2×D6111yescharacteristic
396#417—612no—
397#417C2×D6112yes—
398#416—611no—
399#416—611no—
400#405—612no—
401C2×C4×C8—611no—
402C61#6137511yescharacteristic, complemented, Sylow
403C2×#41—611no—
404C2×#41C22×D6111yescharacteristic
405#135—618no—
406C2×Q16—611no—
407C2×SD16—611no—
408C2×SD16—611no—
409C2×D8—611no—
410C2×M16C22×D6112yes—
411C22×C8—611no—
412C22×C8—611no—
413C4⋊Q8—614no—
414C4⋊Q8—614no—
415C4⋊Q8—612no—
416C4⋊Q8—1222no—
417C4⋊Q8—1222no—
418C4⋊Q8—1224no—
419C4⋊Q8—1224no—
420#125—21no—
421#125—21no—
422#125—614no—
423#125—614no—
424#125—611no—
425#125—611no—
426#125—614no—
427#125—1222no—
428#125—1224no—
429#125—1222no—
430#125—1224no—
431#125—1224no—
432#125—1224no—
433#125—1221no—
434#125—1221no—
435#125C22×D6114yes—
436C22⋊Q8—614no—
437C22⋊Q8—1224no—
438C22⋊Q8—1224no—
439C22⋊Q8—1224no—
440C22⋊Q8—1224no—
441C22⋊Q8—1224no—
442C22⋊Q8—1224no—
443C22⋊Q8—1224no—
444C22⋊Q8—1224no—
445C4×Q8—614no—
446C4×Q8—614no—
447C4×Q8—614no—
448C4×Q8—612no—
449C4×Q8—1224no—
450C4×Q8—1224no—
451C4×Q8C22×D6112yes—
452C4×D4—22no—
453C4×D4—614no—
454C4×D4—614no—
455C4×D4—612no—
456C4×D4—1224no—
457C4×D4—1224no—
458C4×D4—1222no—
459C4×D4C22×D6112yes—
460#117—21no—
461#117—612no—
462#117—611no—
463#117—1222no—
464#117—1222no—
465#117—1221no—
466#117C22×D6111yescharacteristic
467C2×C4⋊C4—22no—
468C2×C4⋊C4—612no—
469C2×C4⋊C4—1222no—
470C2×C4⋊C4—1224no—
471C2×C4⋊C4—1222no—
472C2×C4⋊C4—1224no—
473C2×C4⋊C4—1222no—
474C2×C4⋊C4C22×D6112yes—
475C2×C42—21no—
476C2×C42—1221no—
477C2×C42C22×D6111yescharacteristic
478#107—614no—
479#107—614no—
480#107—614no—
481#107—612no—
482#107C22×D6114yes—
483#106—614no—
484#106—614no—
485#106—614no—
486#106—612no—
487#106C22×D6114yes—
488C4⋊C8—614no—
489C4⋊C8—614no—
490C4⋊C8C22×D6114yes—
491#103—612no—
492#103—614no—
493#103—612no—
494#103C22×D6114yes—
495#102—612no—
496#102—614no—
497#102—612no—
498#102C22×D6114yes—
499#97—614no—
500C4×C8—614no—
501C4×C8—612no—
502C4×C8—612no—
503#41—22no—
504#41—614no—
505#41—1222no—
506#41Q8×D6114yes—
507C2×Q8—614no—
508C2×Q8—611no—
509C2×Q8—1224no—
510C2×Q8—1224no—
511C2×Q8C23×D6111yescharacteristic
512C2×D4—21no—
513C2×D4—611no—
514C2×D4—1221no—
515C2×D4C23×D6111yescharacteristic
516C22×C4—21no—
517C22×C4—22no—
518C22×C4—22no—
519C22×C4—1224no—
520C22×C4—1224no—
521C22×C4—1221no—
522C22×C4—1222no—
523C22×C4—1222no—
524C22×C4D4×D6112yes—
525C22×C4C23×D6111yescharacteristic
526Q16—614no—
527SD16—614no—
528SD16—614no—
529D8—614no—
530M16—24no—
531C2×C8—614no—
532C2×C8—611no—
533C2×C8—611no—
534C2×C8—612no—
535C2×C8—1222no—
536C2×C8C23×D6114yes—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—614no—
542C4⋊C4—614no—
543C4⋊C4—612no—
544C4⋊C4—612no—
545C4⋊C4—614no—
546C4⋊C4—614no—
547C4⋊C4—612no—
548C4⋊C4—1224no—
549C4⋊C4—1224no—
550C4⋊C4—1224no—
551C4⋊C4—1224no—
552C4⋊C4—1224no—
553C4⋊C4—1224no—
554C4⋊C4—1224no—
555C4⋊C4—1224no—
556C4⋊C4—1222no—
557C4⋊C4—1224no—
558C4⋊C4—1224no—
559C4⋊C4—1222no—
560C4⋊C4C23×D6112yes—
561C4⋊C4C23×D6114yes—
562C22⋊C4—21no—
563C22⋊C4—22no—
564C22⋊C4—614no—
565C22⋊C4—1224no—
566C22⋊C4—1224no—
567C22⋊C4—1221no—
568C22⋊C4—1222no—
569C42—21no—
570C42—22no—
571C42—614no—
572C42—1224no—
573C42—1224no—
574C42—1221no—
575C42—1222no—
576C42D4×D6111yescharacteristic
577C42D4×D6111yescharacteristic
578C42C23×D6112yes—
579C23—21no—
580C23—1221no—
581C23C2×D4×D6111yescharacteristic
582Q8—21no—
583Q8—612no—
584Q8—1221no—
585Q8—4884no—
586Q8—4884no—
587Q8—2444no—
588Q8C2×Q8×D6112yes—
589D4—24no—
590D4—21no—
591D4—612no—
592D4—1224no—
593D4—1221no—
594D4C2×Q8×D6112yes—
595C2×C4—21no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—42no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—614no—
603C2×C4—614no—
604C2×C4—1221no—
605C2×C4—1224no—
606C2×C4—1222no—
607C2×C4—1224no—
608C2×C4—4882no—
609C2×C4—4884no—
610C2×C4—4884no—
611C2×C4—4882no—
612C2×C4C2×D4×D6112yes—
613C2×C4C2×D4×D6112yes—
614C2×C4C2×D4×D6111yescharacteristic
615C2×C4C2×Q8×D6112yes—
616C2×C4#1451982412yes—
617C2×C4C24×D6111yescharacteristic
618C8—24no—
619C8—612no—
620C8—612no—
621C8—1222no—
622C22—22no—
623C22—22no—
624C22—42no—
625C22—1222no—
626C22—2442no—
627C22#1684042512yes—
628C22C22×D4×D6111yescharacteristic
629C4—22no—
630C4—22no—
631C4—22no—
632C4—44no—
633C4—42no—
634C4—1222no—
635C4—1224no—
636C4—1224no—
637C4—2444no—
638C4—2442no—
639C4#1684042512yes—
640C4C22×Q8×D6111yescharacteristic
641C4C2×#1451982411yescharacteristic
642C2—22no—
643C2—42no—
644C2—2442no—
645C2C2×#1684042511yescharacteristic
646C2#2146059212yes—
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3080C2
socleC22×C61C23×D4
prime radicalC2×#14519797C22
hypercenter#3080D61
hypocenterC61#61375
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3080

Complements

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

NH ≅ G/Ntype
1. #21459645C2semidirect
2. #21459643C2semidirect
3. #21458786C2semidirect
4. #21458777C2semidirect
5. #21458697C2semidirect
6. #21458689C2semidirect
7. #21456349C2semidirect
8. C2×#16840161C2semidirect
9. #21455674C2semidirect
10. #21455669C2semidirect
11. #21455642C2semidirect
12. #21454749C2semidirect
15. C61×#3080C2semidirect
80. #14519797Q8semidirect
330. #3080D61semidirect
402. C61#61375semidirect

Maximal subgroups

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

subgroupindex
#214596452
#214596432
#214587862
#214587772
#214586972
#214586892
#214563492
C2×#168401612
#214556742
#214556692
#214556422
#214547492
#214546982
C4×#145197972
C61×#30802
#6137561