Finite Group Databasebeta

← #29686809

Subgroups — #29686809

Order 15616.

958 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455397C211yesmaximal, characteristic
2C2×#16839881C211yesmaximal, characteristic, complemented
3#21454514C211yesmaximal, characteristic, complemented
4#21454509C211yesmaximal, characteristic, complemented
5C22×#14519688C211yesmaximal, characteristic, complemented
6#21446709C211yesmaximal, characteristic, complemented
7#21446571C211yesmaximal, characteristic, complemented
8#21446531C211yesmaximal, characteristic, complemented
9#21446527C211yesmaximal, characteristic, complemented
10#21446262C211yesmaximal, characteristic, complemented
11#21445855C211yesmaximal, characteristic, complemented
12#21445702C211yesmaximal, characteristic, complemented
13#21445664C211yesmaximal, characteristic, complemented
14#21445660C211yesmaximal, characteristic, complemented
15C61×#1896C211yesmaximal, characteristic, complemented
16C2×#14519797C2211yescharacteristic
17C2×#14519785C2211yescharacteristic
18C2×#14519784C2211yescharacteristic
19C23×#92418C2211yescharacteristic
20C23×#92418C2211yescharacteristic
21#16840273C2211yescharacteristic
22#16840001C412yescomplemented
23#16839996C412yescomplemented
24#16839993C412yescomplemented
25#16839989C412yescomplemented
26#16839881—24no—
27#16839834—22no—
28#16839830—22no—
29D4×#92418—22no—
30#16839825—22no—
31C2×#14519712C2211yescharacteristic
32C2×#14519712C2211yescharacteristic
33#16839798C2211yescharacteristic
34#16839790C2211yescharacteristic
35#16839789C2211yescharacteristic
36C2×#14519705C2211yescharacteristic
37C2×#14519705C2211yescharacteristic
38#16839758C2211yescharacteristic
39#16839756C2211yescharacteristic
40#16839730C2211yescharacteristic
41#16839729C2211yescharacteristic
42#16839726C2211yescharacteristic
43#16839720C2211yescharacteristic
44#16839718C2211yescharacteristic
45C2×#14519688—24no—
46C2×#14519688C2211yescharacteristic
47C2×#14519688C2211yescharacteristic
48C2×#14519688C2211yescharacteristic
49C2×#14519688C2211yescharacteristic
50#16839526C2211yescharacteristic
51#16839526C2211yescharacteristic
52#16839522C2211yescharacteristic
53#16839522C2211yescharacteristic
54C61×#525C2211yescharacteristic
55C22×C42×C61C2211yescharacteristic
56C2×#104×C61C2211yescharacteristic
57C2×#104×C61C2211yescharacteristic
58C61×#407C2211yescharacteristic
59C61×#339C2211yescharacteristic
60C61×#339C2211yescharacteristic
61C2×#11758222C2311yescharacteristic
62#14519797—22no—
63#14519797—22no—
64C2×#11758202C2×C411yescharacteristic, complemented
65C2×#11758200C2×C411yescharacteristic, complemented
66C2×#11758201C2×C411yescharacteristic, complemented
67C2×#11758199C2×C411yescharacteristic, complemented
68#14519785—22no—
69#14519785—22no—
70#14519784—22no—
71#14519784—22no—
72C2×#11758194—21no—
73C2×#11758194C2311yescharacteristic
74C2×#11758194C2311yescharacteristic
75C2×#11758189—21no—
76C22×#92418—22no—
77C22×#92418—22no—
78C22×#92418—22no—
79C22×#92418—22no—
80C22×#92418—21no—
81C22×#92418C2×C412yes—
82C22×#92418C2×C412yes—
83C22×#92418D412yes—
84C22×#92418D412yes—
85C22×#92418C2311yescharacteristic
86C22×#92418C2311yescharacteristic
87C22×#92418C2311yescharacteristic
88C22×#92418C2311yescharacteristic
89C2×C8×D61—21no—
90#14519760—21no—
91#14519756C2×C412yes—
92#14519751C2×C412yes—
93#14519746—21no—
94C4×C22⋊D61—22no—
95#14519736C2311yescharacteristic
96#14519712—22no—
97#14519712—22no—
98#14519710C2×C412yescomplemented
99#14519708C2×C412yescomplemented
100C4⋊#92418—22no—
101#14519706—22no—
102#14519705—22no—
103#14519705—22no—
104#14519701C2×C412yescomplemented
105#14519700C2×C412yescomplemented
106#14519698C2×C412yescomplemented
107#14519697C2×C412yescomplemented
108C22⋊#92418—22no—
109C22⋊#92418—22no—
110#14519689—22no—
111#14519688—22no—
112#14519688—22no—
113#14519688—22no—
114#14519688—21no—
115#14519688D414yescomplemented
116#14519687—21no—
117C4×#92418—21no—
118C8×Dic61—21no—
119C2×#41×C61C2311yescharacteristic
120C23×C4×C61—21no—
121C23×C4×C61C2311yescharacteristic
122C2×M16×C61C2311yescharacteristic
123C2×M16×C61C2311yescharacteristic
124#123×C61—21no—
125C22⋊Q8×C61C2×C412yes—
126#121×C61C2×C412yes—
127#120×C61—21no—
128C4×D4×C61—22no—
129#117×C61C2311yescharacteristic
130C2×C42×C61—21no—
131C2×C42×C61—22no—
132C2×C42×C61—22no—
133C2×C42×C61C2311yescharacteristic
134C2×C42×C61C2311yescharacteristic
135#104×C61—22no—
136#104×C61—22no—
137C22⋊C8×C61—22no—
138C24×C61#4111yescharacteristic
139#41×C61—21no—
140#41×C61—21no—
141#41×C61—21no—
142#41×C61—21no—
143C2×Q8×C61C22×C411yescharacteristic
144C2×D4×C61—22no—
145C2×D4×C61—21no—
146C2×D4×C61C22×C411yescharacteristic
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—24no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—44no—
156C22×C4×C61C22×C412yes—
157C22×C4×C61C2×D412yes—
158C22×C4×C61#4111yescharacteristic
159C22×C4×C61C2411yescharacteristic
160C2×C22⋊D61—22no—
161C2×C22⋊D61—21no—
162#11758222—21no—
163#11758222—21no—
164#11758222—21no—
165#11758222—21no—
166C2×D244C22×C411yescharacteristic
167C2×C4×D61—21no—
168C2×Dic122C22×C411yescharacteristic
169M16×C61—21no—
170M16×C61—21no—
171M16×C61—21no—
172M16×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61C22×C412yes—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—21no—
179C22⋊C4×C61—22no—
180C42×C61—24no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61C2×D412yes—
185C42×C61#4112yes—
186C22⋊Dic61—22no—
187C22⋊Dic61—21no—
188#11758202—22no—
189#11758201—22no—
190#11758200—22no—
191#11758199—22no—
192#11758198—22no—
193C4⋊Dic61C22×C412yes—
194#11758196—22no—
195C4×Dic61—21no—
196#11758194—21no—
197#11758194—21no—
198#11758194—21no—
199#11758194—21no—
200#11758194—42no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—24no—
206C2×#92418—24no—
207C2×#92418—24no—
208C2×#92418—24no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—42no—
212C2×#92418C22×C412yes—
213C2×#92418C22×C411yescharacteristic
214C2×#92418C22×C411yescharacteristic
215C2×#92418C2×D412yes—
216C2×#92418C2×D412yes—
217C2×#92418C2×D411yescharacteristic
218C2×#92418C2×D411yescharacteristic
219C2×#92418C2×D412yes—
220C2×#92418#4112yes—
221C2×#92418#4112yes—
222#11758189—42no—
223C8×D61—42no—
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—24no—
227C23×C61—21no—
228C23×C61C4×D412yes—
229C23×C61C2×#4111yescharacteristic
230C22×D61—21no—
231Q8×C61—21no—
232Q8×C61—21no—
233D4×C61—21no—
234D4×C61—21no—
235D4×C61—41no—
236D4×C61—41no—
237D4×C61—44no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—21no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—48no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61C4×D412yes—
254C2×C4×C61C23×C411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C22⋊D61—41no—
261C22⋊D61—41no—
262C22⋊D61—44no—
263C2×Dic61—22no—
264C2×Dic61—21no—
265D244—21no—
266D244—21no—
267C4×D61—41no—
268C4×D61—41no—
269Dic122—21no—
270Dic122—21no—
271C8×C61—41no—
272C8×C61—41no—
273#92418—22no—
274#92418—22no—
275#92418—41no—
276#92418—41no—
277#92418C4×D412yes—
278#92418C4×D412yes—
279#92418#12112yes—
280#92418#12112yes—
281#21546—611nomaximal, Sylow
282C22×C61—24no—
283C22×C61—24no—
284C22×C61—22no—
285C22×C61—22no—
286C22×C61—22no—
287C22×C61—42no—
288C22×C61—41no—
289C22×C61#44414yescomplemented
290C22×C61C2×C4×D411yescharacteristic
291C22×C61#52511yescharacteristic
292C22×C61#52911yescharacteristic
293C2×D61—42no—
294C2×D61—41no—
295C4×C61—22no—
296C4×C61—42no—
297C4×C61—42no—
298C4×C61—42no—
299C4×C61—41no—
300C4×C61C2×C4×D411yescharacteristic
301C4×C61C4×#4111yescharacteristic
302C4×C61C2×#12111yescharacteristic
303C4×C61#53211yescharacteristic
304Dic61—42no—
305Dic61—41no—
306#3050—611no—
307C2×#444—611no—
308#2929—611no—
309#2929—611no—
310C22×#97—611no—
311#1975—611no—
312#1946—611no—
313#1938—611no—
314#1938—611no—
315#1896—611no—
316#1896D6111yescharacteristic, complemented
317#1810—611no—
318#1773—611no—
319#1765—611no—
320#1765—611no—
321C2×C61—21no—
322C2×C61—21no—
323C2×C61—24no—
324C2×C61—81no—
325C2×C61C4×#12111yescharacteristic
326C2×C61C2×#44412yes—
327D61—81no—
328C2×#135—611no—
329C2×#131—611no—
330C2×#131—611no—
331C23×C8—611no—
332C23×C8—611no—
333#525—611no—
334#525C2×D6111yescharacteristic
335C22×C42C2×D6111yescharacteristic
336#467—612no—
337#466—612no—
338#465—612no—
339#465—612no—
340#444—1224no—
341#435—1222no—
342#435—1222no—
343C8×D4—1222no—
344C8×D4—1222no—
345C2×#108—611no—
346C2×#108—611no—
347#427—611no—
348#424—611no—
349#424—611no—
350C2×#104—611no—
351C2×#104—611no—
352C2×#104C2×D6111yescharacteristic
353C2×#104C2×D6111yescharacteristic
354#416—611no—
355#416—611no—
356#408—611no—
357#408—611no—
358#407—611no—
359#407C2×D6111yescharacteristic
360#405—611no—
361#405—611no—
362C2×#97—611no—
363C2×#97—611no—
364C2×#97—611no—
365C2×#97—611no—
366C2×#97—1224no—
367#341—611no—
368#341—611no—
369#339—611no—
370#339—611no—
371#339C2×D6111yescharacteristic
372#339C2×D6111yescharacteristic
373C61#2154611yescharacteristic, complemented, Sylow
374C2×#41—611no—
375C2×#41C22×D6111yescharacteristic
376C23×C4—21no—
377C23×C4C22×D6111yescharacteristic
378#135—1222no—
379#135—1222no—
380C2×Q16—611no—
381C2×SD16—611no—
382C2×SD16—611no—
383C2×D8—611no—
384#131—1222no—
385#131—1222no—
386#131—1222no—
387#131—1222no—
388C2×M16—611no—
389C2×M16—611no—
390C2×M16—1221no—
391C2×M16—1221no—
392C2×M16C22×D6111yescharacteristic
393C2×M16C22×D6111yescharacteristic
394C22×C8—611no—
395C22×C8—611no—
396C22×C8—612no—
397C22×C8—612no—
398C22×C8—612no—
399C22×C8—612no—
400C22×C8—611no—
401C22×C8—611no—
402C22×C8—1221no—
403C22×C8—1222no—
404C22×C8—1222no—
405C22×C8—1222no—
406C22×C8—1222no—
407C22×C8—1221no—
408#123—21no—
409#123—1221no—
410C22⋊Q8—612no—
411C22⋊Q8C4×D6112yes—
412#121—612no—
413#121C4×D6112yes—
414#120—21no—
415#120—1221no—
416C4×D4—22no—
417C4×D4—1222no—
418#117—611no—
419#117C22×D6111yescharacteristic
420C2×C42—21no—
421C2×C42—22no—
422C2×C42—22no—
423C2×C42C22×D6111yescharacteristic
424C2×C42C22×D6111yescharacteristic
425#108—1222no—
426#108—1222no—
427#107—612no—
428#106—612no—
429C4⋊C8—1222no—
430C4⋊C8—1222no—
431#104—22no—
432#104—22no—
433#104—1222no—
434#104—1222no—
435#103—612no—
436#103—612no—
437#102—612no—
438#102—612no—
439C22⋊C8—22no—
440C22⋊C8—1222no—
441C22⋊C8—1222no—
442C22⋊C8—1222no—
443#97—614no—
444#97—1222no—
445#97—1222no—
446#97—1221no—
447#97—1221no—
448#97—1222no—
449C4×C8—1221no—
450C4×C8—1221no—
451C24#1175822411yescharacteristic
452#41—21no—
453#41—21no—
454#41—21no—
455#41—21no—
456#41—1221no—
457#41—1221no—
458#41—1221no—
459#41—1221no—
460C2×Q8—611no—
461C2×Q8C2×C4×D6111yescharacteristic
462C2×D4—21no—
463C2×D4—22no—
464C2×D4—611no—
465C2×D4—1221no—
466C2×D4—1222no—
467C2×D4C2×C4×D6111yescharacteristic
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—24no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—44no—
477C22×C4—1221no—
478C22×C4C2×C4×D6112yes—
479C22×C4D4×D6112yes—
480C22×C4#1175822411yescharacteristic
481C22×C4C23×D6111yescharacteristic
482Q16—1222no—
483SD16—1222no—
484SD16—1222no—
485D8—1222no—
486M16—21no—
487M16—21no—
488M16—21no—
489M16—21no—
490M16—1221no—
491M16—1221no—
492M16—1221no—
493M16—1221no—
494M16—2442no—
495M16—2442no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—611no—
499C2×C8—611no—
500C2×C8—612no—
501C2×C8—612no—
502C2×C8—611no—
503C2×C8—611no—
504C2×C8—612no—
505C2×C8—612no—
506C2×C8—612no—
507C2×C8—612no—
508C2×C8—1224no—
509C2×C8—1224no—
510C2×C8—1224no—
511C2×C8—1224no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—2442no—
519C2×C8—2442no—
520C4⋊C4—22no—
521C4⋊C4—612no—
522C4⋊C4—1222no—
523C4⋊C4C2×C4×D6112yes—
524C22⋊C4—21no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—1221no—
528C22⋊C4—1222no—
529C22⋊C4—1222no—
530C42—24no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—1221no—
535C42D4×D6112yes—
536C42#1175822412yes—
537C23—21no—
538C23—21no—
539C23—24no—
540C23—21no—
541C23—1221no—
542C23#1451973712yes—
543C23C2×#1175822411yescharacteristic
544Q8—21no—
545Q8—21no—
546Q8—1221no—
547Q8—1221no—
548D4—21no—
549D4—21no—
550D4—41no—
551D4—41no—
552D4—44no—
553D4—1221no—
554D4—1221no—
555D4—2441no—
556D4—2441no—
557D4—2444no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—48no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—1222no—
574C2×C4—1221no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4#1451973712yes—
578C2×C4C22×C4×D6111yescharacteristic
579C2×C4C2×D4×D6111yescharacteristic
580C2×C4C2×D4×D6111yescharacteristic
581C2×C4#41×D6111yescharacteristic
582C2×C4#41×D6111yescharacteristic
583C2×C4C2×#1175822411yescharacteristic
584C8—41no—
585C8—41no—
586C8—612no—
587C8—612no—
588C8—612no—
589C8—612no—
590C8—1222no—
591C8—1222no—
592C8—2441no—
593C8—2441no—
594C22—24no—
595C22—24no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—42no—
600C22—41no—
601C22—2442no—
602C22—2441no—
603C22#1683988114yescomplemented
604C22C2×#1451973711yescharacteristic
605C22#1684026811yescharacteristic
606C22#1684028911yescharacteristic
607C4—22no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—41no—
612C4—2442no—
613C4—2441no—
614C4C4×D4×D6111yescharacteristic
615C4C4×#1175822411yescharacteristic
616C4#121×D6111yescharacteristic
617C4#1684030711yescharacteristic
618C2—21no—
619C2—21no—
620C2—24no—
621C2—81no—
622C2—4881no—
623C2#2144832911yescharacteristic
624C2C2×#1683988112yes—
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1896C2
socleC22×C61C2×C4×D4
prime radical#21455397C2
cosocleC22×C4×C61C24
hypercenter#1896D61
hypocenterC61#21546
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1896

Complements

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

NH ≅ G/Ntype
2. C2×#16839881C2semidirect
3. #21454514C2semidirect
4. #21454509C2semidirect
5. C22×#14519688C2semidirect
6. #21446709C2semidirect
7. #21446571C2semidirect
8. #21446531C2semidirect
9. #21446527C2semidirect
10. #21446262C2semidirect
11. #21445855C2semidirect
12. #21445702C2semidirect
13. #21445664C2semidirect
14. #21445660C2semidirect
15. C61×#1896C2semidirect
22. #16840001C4semidirect
23. #16839996C4semidirect
24. #16839993C4semidirect
25. #16839989C4semidirect
64. C2×#11758202C2×C4semidirect
65. C2×#11758200C2×C4semidirect
66. C2×#11758201C2×C4semidirect
67. C2×#11758199C2×C4semidirect
98. #14519710C2×C4semidirect
99. #14519708C2×C4semidirect
104. #14519701C2×C4semidirect
105. #14519700C2×C4semidirect
106. #14519698C2×C4semidirect
107. #14519697C2×C4semidirect
115. #14519688D4semidirect
289. C22×C61#444semidirect
316. #1896D61semidirect
373. C61#21546semidirect
603. C22#16839881semidirect

Maximal subgroups

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

subgroupindex
#214553972
C2×#168398812
#214545142
#214545092
C22×#145196882
#214467092
#214465712
#214465312
#214465272
#214462622
#214458552
#214457022
#214456642
#214456602
C61×#18962
#2154661