Finite Group Databasebeta

← #29678610

Subgroups — #29678610

Order 15616.

998 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839877C211yesmaximal, characteristic
2C2×#16839876C211yesmaximal, characteristic
3#21454587C211yesmaximal, characteristic
4#21454246C211yesmaximal, characteristic, complemented
5#21454245C211yesmaximal, characteristic, complemented
6#21445720C211yesmaximal, characteristic, complemented
7#21445699C211yesmaximal, characteristic, complemented
8#21445697C211yesmaximal, characteristic, complemented
9#21445666C211yesmaximal, characteristic, complemented
10#21445663C211yesmaximal, characteristic, complemented
11#21445662C211yesmaximal, characteristic, complemented
12#21445631C211yesmaximal, characteristic, complemented
13#21445628C211yesmaximal, characteristic, complemented
14#21445507C211yesmaximal, characteristic, complemented
15C61×#1765C211yesmaximal, characteristic, complemented
16C2×#14519797C2211yescharacteristic
17C2×#14519785C2211yescharacteristic
18C2×#14519784C2211yescharacteristic
19C2×#14519783C2211yescharacteristic
20#41×Dic61C2211yescharacteristic
21#16840230C2211yescharacteristic
22#16839877C418yescomplemented
23#16839876C418yescomplemented
24#16839854C412yescomplemented
25#16839853C412yescomplemented
26#16839847C412yescomplemented
27#16839842C412yescomplemented
28C2×#14519712C2211yescharacteristic
29C2×#14519712C2211yescharacteristic
30#16839798C2211yescharacteristic
31C2×#14519705C2211yescharacteristic
32C2×#14519704C2211yescharacteristic
33C2×#14519703C2211yescharacteristic
34#16839759C2211yescharacteristic
35#16839720—21no—
36#16839720C2211yescharacteristic
37#16839719—21no—
38#16839718C2211yescharacteristic
39#16839717—21no—
40#16839717C2211yescharacteristic
41#16839716—21no—
42M16×Dic61—21no—
43M16×Dic61C2211yescharacteristic
44M16×Dic61C2211yescharacteristic
45C4×#11758189—21no—
46C2×#14519687—21no—
47C2×#14519687C2211yescharacteristic
48C2×#14519687C2211yescharacteristic
49#97×D61—21no—
50C2×C4×#92418C2211yescharacteristic
51C2×C8×Dic61C2211yescharacteristic
52C2×C8×Dic61C2211yescharacteristic
53#16839608C2211yescharacteristic
54#16839527C2211yescharacteristic
55#16839523C2211yescharacteristic
56#16839522C2211yescharacteristic
57#16839521C2211yescharacteristic
58C2×#131×C61C2211yescharacteristic
59C2×#104×C61C2211yescharacteristic
60C61×#416C2211yescharacteristic
61C61×#405C2211yescharacteristic
62C2×#97×C61C2211yescharacteristic
63C61×#341C2211yescharacteristic
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#14519785C2×C414yescomplemented
74#14519784—22no—
75#14519784C2×C414yescomplemented
76#14519783—22no—
77#14519783C2×C414yescomplemented
78C2×#11758194—21no—
79C2×#11758194C2311yescharacteristic
80C2×#11758194C2311yescharacteristic
81C2×#11758189—21no—
82C2×#11758189—21no—
83C22×#92418—21no—
84C22×#92418C2311yescharacteristic
85C22×#92418C2311yescharacteristic
86C2×C8×D61—21no—
87C2×C8×D61—21no—
88Q8×Dic61C2×C412yes—
89#14519742C2×C412yes—
90D4×Dic61—22no—
91D4×Dic61C2×C412yes—
92#14519739C2×C412yes—
93#14519737—22no—
94#14519736—21no—
95#14519736C2311yescharacteristic
96#14519734—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61C2311yescharacteristic
99C2×C4×Dic61C2311yescharacteristic
100C42×D61—21no—
101#14519712C2×C414yescomplemented
102#14519712C2×C414yescomplemented
103#14519710C2×C412yes—
104#14519708C2×C412yes—
105#14519705C2×C414yescomplemented
106#14519704C2×C414yescomplemented
107#14519703C2×C414yescomplemented
108#14519702C2×C412yes—
109#14519699C2×C412yes—
110#14519688—21no—
111#14519687—21no—
112#14519687—21no—
113#14519687—21no—
114#14519687—21no—
115#14519687—22no—
116#14519687D411yescharacteristic
117#14519687D411yescharacteristic
118#14519687D411yescharacteristic
119#14519687D411yescharacteristic
120C4×#92418—21no—
121C4×#92418C2×C414yescomplemented
122C8×Dic61—21no—
123C8×Dic61—21no—
124C8×Dic61—21no—
125C8×Dic61—21no—
126C8×Dic61C2×C412yes—
127C8×Dic61C2×C412yes—
128C8×Dic61C2×C414yescomplemented
129#14519684—22no—
130C2×#41×C61C2311yescharacteristic
131#131×C61—22no—
132#131×C61C2×C414yescomplemented
133C2×M16×C61—21no—
134C2×M16×C61C2311yescharacteristic
135C2×M16×C61C2311yescharacteristic
136C22×C8×C61—21no—
137C22×C8×C61C2311yescharacteristic
138C22×C8×C61C2311yescharacteristic
139#117×C61C2311yescharacteristic
140C2×C42×C61C2311yescharacteristic
141#104×C61C2×C414yescomplemented
142#103×C61C2×C412yes—
143#102×C61C2×C412yes—
144#97×C61—21no—
145#97×C61—22no—
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×C61C2411yescharacteristic
157C2×C22⋊D61—21no—
158C22×Dic61—21no—
159C22×Dic61—22no—
160C22×Dic61#4112yes—
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×C61—24no—
172M16×C61C22×C412yes—
173M16×C61C22×C412yes—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—24no—
179C2×C8×C61C22×C412yes—
180C2×C8×C61C22×C412yes—
181C2×C8×C61C2×D411yescharacteristic
182C2×C8×C61C2×D411yescharacteristic
183C2×C8×C61#4112yes—
184C4⋊C4×C61C22×C412yes—
185C22⋊C4×C61—21no—
186C42×C61—21no—
187C42×C61—21no—
188C42×C61C22×C412yes—
189C22⋊Dic61—21no—
190C22⋊Dic61—22no—
191#11758202C4214yes—
192#11758201C4214yes—
193#11758200C4214yes—
194#11758199C4214yes—
195#11758198—22no—
196C4⋊Dic61—22no—
197C4⋊Dic61C22×C412yes—
198#11758196—22no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61—22no—
202C4×Dic61—22no—
203C4×Dic61C22×C412yes—
204C4×Dic61C22×C412yes—
205C4×Dic61C2×D411yescharacteristic
206C4×Dic61C2×D411yescharacteristic
207#11758194—21no—
208#11758194—21no—
209#11758194—24no—
210#11758194C22×C412yes—
211#11758194C22×C412yes—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—24no—
215C2×#92418C22×C411yescharacteristic
216C2×#92418C22×C411yescharacteristic
217C2×#92418C22×C412yes—
218C2×#92418C22×C412yes—
219C2×#92418C22×C412yes—
220C2×#92418C2×D411yescharacteristic
221C2×#92418C2×D411yescharacteristic
222C2×#92418#4112yes—
223#11758189—24no—
224#11758189—24no—
225C8×D61—24no—
226C8×D61—24no—
227C23×C61—21no—
228C23×C61C2×#4111yescharacteristic
229C22×D61—21no—
230Q8×C61—21no—
231Q8×C61C2×C4212yes—
232D4×C61—22no—
233D4×C61—22no—
234D4×C61—21no—
235D4×C61C2×C4212yes—
236C2×C4×C61—21no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—42no—
242C2×C4×C61C23×C411yescharacteristic
243C2×C4×C61C23×C411yescharacteristic
244C2×C4×C61C23×C411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C22⋊D61—22no—
249C22⋊D61—22no—
250C2×Dic61—21no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—82no—
256C2×Dic61C4×D412yes—
257C2×Dic61C4×D412yes—
258D244—21no—
259D244C2×C4212yes—
260C4×D61—22no—
261C4×D61—22no—
262C4×D61—82no—
263Dic122—21no—
264Dic122C2×C4212yes—
265C8×C61—22no—
266C8×C61—22no—
267C8×C61—22no—
268C8×C61C4×D412yes—
269C8×C61C4×D412yes—
270#92418—22no—
271#92418—22no—
272#92418C2×C4212yes—
273#92418C2×C4212yes—
274#92418C4×D412yes—
275#92418C4×D412yes—
276#20514—611nomaximal, Sylow
277C22×C61—22no—
278C22×C61—22no—
279C22×C61—42no—
280C22×C61C2×C4×D411yescharacteristic
281C22×C61C4×#4111yescharacteristic
282C22×C61C4×#4111yescharacteristic
283C2×D61—22no—
284C2×D61—42no—
285C4×C61—22no—
286C4×C61—42no—
287C4×C61—42no—
288C4×C61C22×C4211yescharacteristic
289C4×C61C2×C4×D411yescharacteristic
290C4×C61C2×C4×D411yescharacteristic
291C4×C61C4×#4111yescharacteristic
292Dic61—22no—
293Dic61—22no—
294Dic61—22no—
295Dic61—42no—
296Dic61—42no—
297C2×#443—611no—
298C2×#443—611no—
299C4×#135—611no—
300#2875—611no—
301#2875—611no—
302C4×#108—611no—
303#1772—611no—
304#1772—611no—
305#1765—611no—
306#1765—611no—
307#1765—611no—
308#1765D6111yescharacteristic, complemented
309#1759—611no—
310#1759—611no—
311#1726—611no—
312C2×C61—21no—
313C2×C61—21no—
314C2×C61—42no—
315C2×C61C42×D411yescharacteristic
316C2×C61#295712yes—
317D61—42no—
318C2×#135—611no—
319C2×#131—611no—
320C2×#131—611no—
321C2×#131—611no—
322C2×#131C2×D6111yescharacteristic
323C4×#41—611no—
324C4×#41—611no—
325#443—618no—
326#443—618no—
327C4×Q16—612no—
328C4×SD16—612no—
329C4×SD16—612no—
330C4×D8—612no—
331C2×#108—611no—
332C2×#108—611no—
333#427—611no—
334C2×#104—611no—
335C2×#104—611no—
336C2×#104—611no—
337C2×#104C2×D6111yescharacteristic
338#416—611no—
339#416C2×D6111yescharacteristic
340#405—611no—
341#405—611no—
342#405—611no—
343#405—1221no—
344#405—1221no—
345#405—1221no—
346#405—1221no—
347#405C2×D6111yescharacteristic
348C4×M16—611no—
349C4×M16—611no—
350C4×M16—1221no—
351C4×M16—1221no—
352C2×#97—611no—
353C2×#97—611no—
354C2×#97—1221no—
355C2×#97—1221no—
356C2×#97C2×D6111yescharacteristic
357C2×C4×C8—611no—
358C2×C4×C8—611no—
359C2×C4×C8—611no—
360#376—611no—
361#341—611no—
362#341C2×D6111yescharacteristic
363#339—611no—
364#339—611no—
365#339—611no—
366#339C2×D6111yescharacteristic
367C61#2051411yescharacteristic, complemented, Sylow
368C2×#41—611no—
369C2×#41C22×D6111yescharacteristic
370#135—614no—
371#135—614no—
372C2×Q16—611no—
373C2×SD16—611no—
374C2×SD16—611no—
375C2×D8—611no—
376#131—22no—
377#131—614no—
378#131—614no—
379#131—614no—
380#131—1222no—
381#131—1222no—
382#131—1222no—
383#131C4×D6114yescomplemented
384C2×M16—21no—
385C2×M16—611no—
386C2×M16—611no—
387C2×M16—1221no—
388C2×M16—1221no—
389C2×M16—1221no—
390C2×M16C22×D6111yescharacteristic
391C2×M16C22×D6111yescharacteristic
392C22×C8—21no—
393C22×C8—611no—
394C22×C8—611no—
395C22×C8—1221no—
396C22×C8—1221no—
397C22×C8—1221no—
398C22×C8C22×D6111yescharacteristic
399C22×C8C22×D6111yescharacteristic
400C4×Q8—612no—
401C4×Q8—612no—
402C4×D4—612no—
403C4×D4—612no—
404C4×D4—1222no—
405C4×D4—1222no—
406#117—611no—
407#117—1221no—
408#117—1221no—
409#117C22×D6111yescharacteristic
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—1221no—
413C2×C42—1221no—
414C2×C42C22×D6111yescharacteristic
415#108—614no—
416#108—614no—
417#107—612no—
418#106—612no—
419#104—614no—
420#104—614no—
421#104—614no—
422#104C4×D6114yescomplemented
423#103—612no—
424#103C4×D6112yes—
425#102—612no—
426#102C4×D6112yes—
427#97—21no—
428#97—22no—
429#97—611no—
430#97—611no—
431#97—611no—
432#97—611no—
433#97—1221no—
434#97—1221no—
435#97—1221no—
436#97—1222no—
437#97—1221no—
438#97—1221no—
439C4×C8—21no—
440C4×C8—612no—
441C4×C8—612no—
442C4×C8—614no—
443C4×C8—614no—
444C4×C8—1221no—
445C4×C8—1221no—
446C4×C8—1221no—
447C4×C8—1221no—
448C4×C8—1221no—
449#95—1222no—
450#41—21no—
451#41—21no—
452#41—612no—
453#41—612no—
454#41—1221no—
455#41—1221no—
456#41C2×C4×D6112yes—
457#41C2×C4×D6112yes—
458C2×Q8—611no—
459C2×Q8C2×C4×D6111yescharacteristic
460C2×D4—21no—
461C2×D4—611no—
462C2×D4—1221no—
463C2×D4C2×C4×D6111yescharacteristic
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—612no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4C23×D6111yescharacteristic
472Q16—614no—
473SD16—614no—
474SD16—614no—
475D8—614no—
476M16—21no—
477M16—21no—
478M16—24no—
479M16—612no—
480M16—612no—
481M16—1224no—
482M16—1224no—
483M16—1221no—
484M16—1221no—
485M16—1224no—
486M16C2×C4×D6112yes—
487M16C2×C4×D6112yes—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—24no—
493C2×C8—611no—
494C2×C8—611no—
495C2×C8—612no—
496C2×C8—612no—
497C2×C8—612no—
498C2×C8—611no—
499C2×C8—611no—
500C2×C8—612no—
501C2×C8—1224no—
502C2×C8—1224no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1224no—
506C2×C8C2×C4×D6112yes—
507C2×C8C2×C4×D6112yes—
508C2×C8D4×D6111yescharacteristic
509C2×C8D4×D6111yescharacteristic
510C2×C8#1175822412yes—
511C4⋊C4—612no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4C2×C4×D6112yes—
515C22⋊C4—21no—
516C22⋊C4—1222no—
517C22⋊C4—1222no—
518C22⋊C4—1221no—
519C42—21no—
520C42—21no—
521C42—612no—
522C42—611no—
523C42—611no—
524C42—612no—
525C42—1222no—
526C42—1222no—
527C42—1221no—
528C42—1221no—
529C42C2×C4×D6112yes—
530C23—21no—
531C23—1221no—
532C23#41×D6111yescharacteristic
533Q8—21no—
534Q8—612no—
535Q8—1221no—
536Q8C42×D6112yes—
537D4—22no—
538D4—22no—
539D4—21no—
540D4—612no—
541D4—1222no—
542D4—1222no—
543D4—1221no—
544D4C42×D6112yes—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—42no—
551C2×C4—612no—
552C2×C4—612no—
553C2×C4—1221no—
554C2×C4—1222no—
555C2×C4—1222no—
556C2×C4—1222no—
557C2×C4—1222no—
558C2×C4—4882no—
559C2×C4—4882no—
560C2×C4—2442no—
561C2×C4—2442no—
562C2×C4C22×C4×D6111yescharacteristic
563C2×C4C22×C4×D6111yescharacteristic
564C2×C4C22×C4×D6111yescharacteristic
565C2×C4C2×D4×D6111yescharacteristic
566C2×C4#41×D6111yescharacteristic
567C2×C4C2×#1175822411yescharacteristic
568C8—22no—
569C8—22no—
570C8—22no—
571C8—612no—
572C8—612no—
573C8—612no—
574C8—612no—
575C8—1222no—
576C8—1222no—
577C8#1451973712yes—
578C8#1451973712yes—
579C22—22no—
580C22—22no—
581C22—42no—
582C22—1222no—
583C22—2442no—
584C22C4×D4×D6111yescharacteristic
585C22C4×#1175822411yescharacteristic
586C22#1684023011yescharacteristic
587C4—22no—
588C4—42no—
589C4—42no—
590C4—1222no—
591C4—1222no—
592C4—1222no—
593C4—2442no—
594C4—2442no—
595C4C2×C42×D6111yescharacteristic
596C4C4×D4×D6111yescharacteristic
597C4C2×#1451973711yescharacteristic
598C4C4×#1175822411yescharacteristic
599C2—21no—
600C2—21no—
601C2—42no—
602C2—2442no—
603C2C4×#1451973711yescharacteristic
604C2#2145473312yes—
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC4×C61C22×C42
FrattiniC22×C4C23×D61
FittingC61×#1765C2
socleC22×C61C2×C4×D4
prime radicalC2×#14519797C22
cosocleC22×C4×C61C24
hypercenter#1765D61
hypocenterC61#20514
LCSG ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1765

Complements

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

NH ≅ G/Ntype
4. #21454246C2semidirect
5. #21454245C2semidirect
6. #21445720C2semidirect
7. #21445699C2semidirect
8. #21445697C2semidirect
9. #21445666C2semidirect
10. #21445663C2semidirect
11. #21445662C2semidirect
12. #21445631C2semidirect
13. #21445628C2semidirect
14. #21445507C2semidirect
15. C61×#1765C2semidirect
22. #16839877C4semidirect
23. #16839876C4semidirect
24. #16839854C4semidirect
25. #16839853C4semidirect
26. #16839847C4semidirect
27. #16839842C4semidirect
73. #14519785C2×C4semidirect
75. #14519784C2×C4semidirect
77. #14519783C2×C4semidirect
101. #14519712C2×C4semidirect
102. #14519712C2×C4semidirect
105. #14519705C2×C4semidirect
106. #14519704C2×C4semidirect
107. #14519703C2×C4semidirect
121. C4×#92418C2×C4semidirect
128. C8×Dic61C2×C4semidirect
132. #131×C61C2×C4semidirect
141. #104×C61C2×C4semidirect
308. #1765D61semidirect
367. C61#20514semidirect
383. #131C4×D61semidirect
422. #104C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398772
C2×#168398762
#214545872
#214542462
#214542452
#214457202
#214456992
#214456972
#214456662
#214456632
#214456622
#214456312
#214456282
#214455072
C61×#17652
#2051461