Finite Group Databasebeta

← #29678590

Subgroups — #29678590

Order 15616.

1014 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839881C211yesmaximal, characteristic
2C2×#16839876C211yesmaximal, characteristic
3#21454617C211yesmaximal, characteristic
4#21454248C211yesmaximal, characteristic, complemented
5#21454247C211yesmaximal, characteristic, complemented
6#21445726C211yesmaximal, characteristic, complemented
7#21445702C211yesmaximal, characteristic, complemented
8#21445697C211yesmaximal, characteristic, complemented
9#21445664C211yesmaximal, characteristic, complemented
10#21445660C211yesmaximal, characteristic, complemented
11#21445657C211yesmaximal, characteristic, complemented
12#21445653C211yesmaximal, characteristic, complemented
13C4×#14519705C211yesmaximal, characteristic, complemented
14C4×#14519688C211yesmaximal, characteristic, complemented
15C4×#104×C61C211yesmaximal, characteristic, complemented
16C2×#14519797C2211yescharacteristic
17C2×#14519785C2211yescharacteristic
18C2×#14519785C2211yescharacteristic
19C2×#14519784C2211yescharacteristic
20C2×#14519784C2211yescharacteristic
21C4×#11758222C2211yescharacteristic
22#16839881C418yescomplemented
23#16839876C418yescomplemented
24#16839872C412yescomplemented
25#16839867C412yescomplemented
26#16839862C412yescomplemented
27#16839860C412yescomplemented
28C2×#14519712C2211yescharacteristic
29C2×#14519712C2211yescharacteristic
30#16839798C2211yescharacteristic
31C2×#14519705C2211yescharacteristic
32C2×#14519705C2211yescharacteristic
33#16839758C2211yescharacteristic
34#16839756C2211yescharacteristic
35#16839720C2211yescharacteristic
36#16839720C2211yescharacteristic
37#16839718—21no—
38#16839718—21no—
39#16839718C2211yescharacteristic
40#16839718C2211yescharacteristic
41#16839716—21no—
42#16839716—21no—
43C4×#11758194—21no—
44C4×#11758194C2211yescharacteristic
45#16839714—21no—
46C2×#14519688—21no—
47C2×#14519688C2211yescharacteristic
48C2×#14519688C2211yescharacteristic
49C2×#14519688C2211yescharacteristic
50C2×#14519688C2211yescharacteristic
51#97×D61—21no—
52C2×C4×#92418C2211yescharacteristic
53C2×C4×#92418C2211yescharacteristic
54#16839526C2211yescharacteristic
55#16839526C2211yescharacteristic
56#16839522C2211yescharacteristic
57#16839522C2211yescharacteristic
58C4×#41×C61C2211yescharacteristic
59C2×#104×C61C2211yescharacteristic
60C2×#104×C61C2211yescharacteristic
61C4×M16×C61C2211yescharacteristic
62C43×C61C2211yescharacteristic
63C61×#339C2211yescharacteristic
64C61×#339C2211yescharacteristic
65C2×#11758222C2311yescharacteristic
66#14519797C2×C414yes—
67#14519797C2×C414yes—
68C2×#11758202C2×C411yescharacteristic
69C2×#11758200C2×C411yescharacteristic
70C2×#11758201C2×C411yescharacteristic
71C2×#11758199C2×C411yescharacteristic
72#14519785—22no—
73#14519785—22no—
74#14519785C2×C414yescomplemented
75#14519785C2×C414yescomplemented
76#14519784—22no—
77#14519784—22no—
78#14519784C2×C414yescomplemented
79#14519784C2×C414yescomplemented
80C2×#11758194—21no—
81C2×#11758194—21no—
82C2×#11758194C2311yescharacteristic
83C2×#11758194C2311yescharacteristic
84C2×#11758189—21no—
85C2×#11758189—21no—
86C22×#92418—21no—
87C22×#92418—21no—
88C22×#92418C2311yescharacteristic
89C22×#92418C2311yescharacteristic
90C22×#92418C2311yescharacteristic
91C22×#92418C2311yescharacteristic
92C2×C8×D61—21no—
93C2×C8×D61—21no—
94C4×Dic122C2×C412yes—
95C4×D244C2×C412yes—
96C4×C22⋊D61—22no—
97#14519736C2311yescharacteristic
98#14519734—21no—
99C42×D61—21no—
100#14519712C2×C414yescomplemented
101#14519712C2×C414yescomplemented
102#14519710C2×C412yes—
103#14519708C2×C412yes—
104#14519705C2×C414yescomplemented
105#14519705C2×C414yescomplemented
106#14519701C2×C412yes—
107#14519700C2×C412yes—
108#14519698C2×C412yes—
109#14519697C2×C412yes—
110#14519688—21no—
111#14519688—21no—
112#14519688—21no—
113#14519688—22no—
114#14519688C2×C412yes—
115#14519688C2×C412yes—
116#14519688C2×C414yescomplemented
117#14519688D411yescharacteristic
118#14519688D411yescharacteristic
119#14519688D411yescharacteristic
120#14519688D411yescharacteristic
121#14519687—21no—
122#14519687—21no—
123C4×#92418—21no—
124C4×#92418—21no—
125C4×#92418—21no—
126C4×#92418—22no—
127C4×#92418C2×C414yescomplemented
128C8×Dic61—21no—
129C8×Dic61—21no—
130C2×#41×C61C2311yescharacteristic
131C2×M16×C61C2311yescharacteristic
132C2×M16×C61C2311yescharacteristic
133C4×Q8×C61C2×C412yes—
134C4×D4×C61—22no—
135C4×D4×C61C2×C412yes—
136#117×C61—21no—
137#117×C61C2311yescharacteristic
138C2×C42×C61—21no—
139C2×C42×C61—22no—
140C2×C42×C61C2311yescharacteristic
141C2×C42×C61C2311yescharacteristic
142C2×C42×C61C2311yescharacteristic
143#104×C61C2×C414yescomplemented
144#104×C61C2×C414yescomplemented
145#97×C61—21no—
146C4×C8×C61—21no—
147#41×C61—21no—
148#41×C61—21no—
149#41×C61C22×C412yes—
150#41×C61C22×C412yes—
151C2×Q8×C61C22×C411yescharacteristic
152C2×D4×C61—21no—
153C2×D4×C61C22×C411yescharacteristic
154C22×C4×C61—21no—
155C22×C4×C61—21no—
156C22×C4×C61—21no—
157C22×C4×C61—22no—
158C22×C4×C61#4112yes—
159C22×C4×C61C2411yescharacteristic
160C2×C22⋊D61—21no—
161#11758222—21no—
162#11758222—21no—
163#11758222C22×C412yes—
164#11758222C22×C412yes—
165C2×D244C22×C411yescharacteristic
166C2×C4×D61—21no—
167C2×C4×D61—22no—
168C2×Dic122C22×C411yescharacteristic
169M16×C61—21no—
170M16×C61—21no—
171M16×C61C22×C412yes—
172M16×C61C22×C412yes—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61C22×C412yes—
177C22⋊C4×C61—21no—
178C22⋊C4×C61—22no—
179C42×C61—22no—
180C42×C61—21no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—44no—
184C42×C61C22×C412yes—
185C42×C61C22×C412yes—
186C42×C61C22×C412yes—
187C42×C61C2×D411yescharacteristic
188C42×C61C2×D411yescharacteristic
189C22⋊Dic61—21no—
190#11758202C4214yescomplemented
191#11758201C4214yescomplemented
192#11758200C4214yescomplemented
193#11758199C4214yescomplemented
194#11758198—22no—
195C4⋊Dic61C22×C412yes—
196#11758196—22no—
197C4×Dic61—22no—
198C4×Dic61—21no—
199#11758194—21no—
200#11758194—21no—
201#11758194—24no—
202#11758194—24no—
203#11758194C22×C412yes—
204#11758194C22×C412yes—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—24no—
210C2×#92418—24no—
211C2×#92418C22×C412yes—
212C2×#92418C22×C412yes—
213C2×#92418C22×C411yescharacteristic
214C2×#92418C22×C411yescharacteristic
215C2×#92418C22×C412yes—
216C2×#92418C22×C412yes—
217C2×#92418C22×C412yes—
218C2×#92418C2×D411yescharacteristic
219C2×#92418C2×D411yescharacteristic
220C2×#92418C2×D411yescharacteristic
221C2×#92418C2×D411yescharacteristic
222C2×#92418#4112yes—
223C2×#92418#4112yes—
224#11758189—24no—
225#11758189—24no—
226C8×D61—24no—
227C8×D61—24no—
228C23×C61—21no—
229C23×C61C2×#4111yescharacteristic
230C22×D61—21no—
231Q8×C61—21no—
232Q8×C61C2×C4212yes—
233D4×C61—22no—
234D4×C61—22no—
235D4×C61—21no—
236D4×C61C2×C4212yes—
237C2×C4×C61—24no—
238C2×C4×C61—21no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—82no—
247C2×C4×C61C4×D412yes—
248C2×C4×C61C4×D412yes—
249C2×C4×C61C23×C411yescharacteristic
250C2×C4×C61C23×C411yescharacteristic
251C2×C4×C61C23×C411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C22⋊D61—22no—
256C22⋊D61—22no—
257C2×Dic61—21no—
258C2×Dic61—22no—
259D244—21no—
260D244C2×C4212yes—
261C4×D61—22no—
262C4×D61—22no—
263C4×D61—82no—
264Dic122—21no—
265Dic122C2×C4212yes—
266C8×C61—22no—
267C8×C61—22no—
268#92418—22no—
269#92418—22no—
270#92418—22no—
271#92418C2×C4212yes—
272#92418C2×C4212yes—
273#92418C4×D412yes—
274#92418C4×D412yes—
275#92418C4×D412yes—
276#92418C4×D412yes—
277#20512—611nomaximal, Sylow
278C22×C61—22no—
279C22×C61—22no—
280C22×C61—42no—
281C22×C61C2×C4×D411yescharacteristic
282C22×C61C4×#4111yescharacteristic
283C22×C61C4×#4111yescharacteristic
284C2×D61—22no—
285C2×D61—42no—
286C4×C61—24no—
287C4×C61—22no—
288C4×C61—22no—
289C4×C61—22no—
290C4×C61—42no—
291C4×C61—42no—
292C4×C61C22×C4211yescharacteristic
293C4×C61C2×C4×D411yescharacteristic
294C4×C61C2×C4×D411yescharacteristic
295C4×C61C4×#4111yescharacteristic
296Dic61—22no—
297Dic61—42no—
298C2×#444—611no—
299C2×#443—611no—
300#2943—611no—
301#2875—611no—
302#2875—611no—
303#1778—611no—
304#1773—611no—
305#1772—611no—
306#1765—611no—
307#1765—611no—
308#1764—611no—
309#1764—611no—
310C4×#104—611no—
311C4×#104D6111yescharacteristic, complemented
312C4×#97—611no—
313C2×C61—21no—
314C2×C61—21no—
315C2×C61—42no—
316C2×C61C42×D411yescharacteristic
317C2×C61#295612yes—
318D61—42no—
319C2×#135—611no—
320C2×#131—611no—
321C2×#131—611no—
322C2×#131—611no—
323C2×#131—611no—
324C4×#41—611no—
325C4×#41C2×D6111yescharacteristic
326#444—618no—
327#443—618no—
328#442—612no—
329#441—612no—
330#440—612no—
331#440—612no—
332C2×#108—611no—
333C2×#108—611no—
334#427—611no—
335C2×#104—611no—
336C2×#104—611no—
337C2×#104C2×D6111yescharacteristic
338C2×#104C2×D6111yescharacteristic
339#416—611no—
340#416—611no—
341#405—611no—
342#405—611no—
343#405—611no—
344#405—611no—
345#405—1221no—
346#405—1221no—
347#405—1221no—
348#405—1221no—
349C4×M16—611no—
350C4×M16—1221no—
351C4×M16—1221no—
352C4×M16C2×D6111yescharacteristic
353C2×#97—611no—
354C2×#97—611no—
355C2×#97—611no—
356C2×#97—611no—
357C2×#97—1221no—
358C2×#97—1221no—
359C2×C4×C8—611no—
360C2×C4×C8—611no—
361C43C2×D6111yescharacteristic
362#341—611no—
363#341—611no—
364#339—611no—
365#339—611no—
366#339C2×D6111yescharacteristic
367#339C2×D6111yescharacteristic
368C61#2051211yescharacteristic, complemented, Sylow
369C2×#41—611no—
370C2×#41C22×D6111yescharacteristic
371#135—614no—
372#135—614no—
373C2×Q16—611no—
374C2×SD16—611no—
375C2×SD16—611no—
376C2×D8—611no—
377#131—614no—
378#131—614no—
379#131—614no—
380#131—614no—
381#131—1222no—
382#131—1222no—
383#131—1222no—
384#131—1222no—
385C2×M16—611no—
386C2×M16—611no—
387C2×M16—1221no—
388C2×M16—1221no—
389C2×M16—1221no—
390C2×M16—1221no—
391C2×M16C22×D6111yescharacteristic
392C2×M16C22×D6111yescharacteristic
393C22×C8—611no—
394C22×C8—611no—
395C22×C8—611no—
396C22×C8—611no—
397C22×C8—1221no—
398C22×C8—1221no—
399C22×C8—1221no—
400C22×C8—1221no—
401C4×Q8—612no—
402C4×Q8C4×D6112yes—
403C4×D4—22no—
404C4×D4—612no—
405C4×D4—1222no—
406C4×D4C4×D6112yes—
407#117—21no—
408#117—611no—
409#117—1221no—
410#117C22×D6111yescharacteristic
411C2×C42—21no—
412C2×C42—22no—
413C2×C42—1221no—
414C2×C42C22×D6111yescharacteristic
415C2×C42C22×D6111yescharacteristic
416C2×C42C22×D6111yescharacteristic
417#108—614no—
418#108—614no—
419#107—612no—
420#106—612no—
421#104—614no—
422#104—614no—
423#104C4×D6114yescomplemented
424#104C4×D6114yescomplemented
425#103—612no—
426#103—612no—
427#102—612no—
428#102—612no—
429#97—21no—
430#97—612no—
431#97—612no—
432#97—614no—
433#97—611no—
434#97—611no—
435#97—611no—
436#97—611no—
437#97—1222no—
438#97—1221no—
439#97—1221no—
440#97—1221no—
441#97—1221no—
442#97—1221no—
443C4×C8—21no—
444C4×C8—614no—
445C4×C8—1222no—
446C4×C8—1221no—
447C4×C8—1221no—
448C4×C8—1221no—
449C4×C8—1221no—
450C4×C8—1221no—
451#41—21no—
452#41—21no—
453#41—612no—
454#41—612no—
455#41—1221no—
456#41—1221no—
457#41C2×C4×D6112yes—
458#41C2×C4×D6112yes—
459C2×Q8—611no—
460C2×Q8C2×C4×D6111yescharacteristic
461C2×D4—21no—
462C2×D4—611no—
463C2×D4—1221no—
464C2×D4C2×C4×D6111yescharacteristic
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—1221no—
470C22×C4—1222no—
471C22×C4#1175822412yes—
472C22×C4C23×D6111yescharacteristic
473Q16—614no—
474SD16—614no—
475SD16—614no—
476D8—614no—
477M16—21no—
478M16—21no—
479M16—612no—
480M16—612no—
481M16—1224no—
482M16—1224no—
483M16—1224no—
484M16—1224no—
485M16—1221no—
486M16—1221no—
487M16C2×C4×D6112yes—
488M16C2×C4×D6112yes—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—612no—
492C2×C8—612no—
493C2×C8—611no—
494C2×C8—611no—
495C2×C8—612no—
496C2×C8—612no—
497C2×C8—611no—
498C2×C8—611no—
499C2×C8—611no—
500C2×C8—611no—
501C2×C8—612no—
502C2×C8—612no—
503C2×C8—612no—
504C2×C8—1224no—
505C2×C8—1224no—
506C2×C8—1224no—
507C2×C8—1224no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C4⋊C4—22no—
513C4⋊C4—612no—
514C4⋊C4—1222no—
515C4⋊C4C2×C4×D6112yes—
516C22⋊C4—21no—
517C22⋊C4—22no—
518C22⋊C4—1221no—
519C22⋊C4—1222no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—22no—
524C42—44no—
525C42—1221no—
526C42—1222no—
527C42C2×C4×D6112yes—
528C42C2×C4×D6112yes—
529C42C2×C4×D6112yes—
530C42D4×D6111yescharacteristic
531C42D4×D6111yescharacteristic
532C23—21no—
533C23—1221no—
534C23C2×#1175822411yescharacteristic
535Q8—21no—
536Q8—612no—
537Q8—1221no—
538Q8C42×D6112yes—
539D4—22no—
540D4—22no—
541D4—21no—
542D4—612no—
543D4—1222no—
544D4—1222no—
545D4—1221no—
546D4C42×D6112yes—
547C2×C4—24no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—82no—
557C2×C4—1221no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—4882no—
562C2×C4#1451973712yes—
563C2×C4#1451973712yes—
564C2×C4C22×C4×D6111yescharacteristic
565C2×C4C22×C4×D6111yescharacteristic
566C2×C4C22×C4×D6111yescharacteristic
567C2×C4C2×D4×D6111yescharacteristic
568C2×C4#41×D6111yescharacteristic
569C2×C4#41×D6111yescharacteristic
570C8—22no—
571C8—22no—
572C8—612no—
573C8—612no—
574C8—612no—
575C8—612no—
576C8—612no—
577C8—612no—
578C8—1222no—
579C8—1222no—
580C8—1222no—
581C22—22no—
582C22—22no—
583C22—42no—
584C22—1222no—
585C22—2442no—
586C22C2×#1451973711yescharacteristic
587C22C4×#1175822411yescharacteristic
588C22C4×#1175822411yescharacteristic
589C4—24no—
590C4—22no—
591C4—22no—
592C4—22no—
593C4—42no—
594C4—42no—
595C4—1222no—
596C4—2442no—
597C4C2×C42×D6111yescharacteristic
598C4C4×D4×D6111yescharacteristic
599C4C4×D4×D6111yescharacteristic
600C4#1684023011yescharacteristic
601C2—21no—
602C2—21no—
603C2—42no—
604C2—2442no—
605C2C4×#1451973711yescharacteristic
606C2#2145472512yes—
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC4×C61C22×C42
FrattiniC22×C4C23×D61
FittingC4×#104×C61C2
socleC22×C61C2×C4×D4
prime radicalC2×#14519797C22
cosocleC22×C4×C61C24
hypercenterC4×#104D61
hypocenterC61#20512
LCSG ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#104

Complements

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

NH ≅ G/Ntype
4. #21454248C2semidirect
5. #21454247C2semidirect
6. #21445726C2semidirect
7. #21445702C2semidirect
8. #21445697C2semidirect
9. #21445664C2semidirect
10. #21445660C2semidirect
11. #21445657C2semidirect
12. #21445653C2semidirect
13. C4×#14519705C2semidirect
14. C4×#14519688C2semidirect
15. C4×#104×C61C2semidirect
22. #16839881C4semidirect
23. #16839876C4semidirect
24. #16839872C4semidirect
25. #16839867C4semidirect
26. #16839862C4semidirect
27. #16839860C4semidirect
74. #14519785C2×C4semidirect
75. #14519785C2×C4semidirect
78. #14519784C2×C4semidirect
79. #14519784C2×C4semidirect
100. #14519712C2×C4semidirect
101. #14519712C2×C4semidirect
104. #14519705C2×C4semidirect
105. #14519705C2×C4semidirect
116. #14519688C2×C4semidirect
127. C4×#92418C2×C4semidirect
143. #104×C61C2×C4semidirect
144. #104×C61C2×C4semidirect
190. #11758202C42semidirect
191. #11758201C42semidirect
192. #11758200C42semidirect
193. #11758199C42semidirect
311. C4×#104D61semidirect
368. C61#20512semidirect
423. #104C4×D61semidirect
424. #104C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398812
C2×#168398762
#214546172
#214542482
#214542472
#214457262
#214457022
#214456972
#214456642
#214456602
#214456572
#214456532
C4×#145197052
C4×#145196882
C4×#104×C612
#2051261