Finite Group Databasebeta

← #30547668

Subgroups — #30547668

Order 15616.

2090 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459632C211yesmaximal, characteristic, complemented
2#21459025C211yesmaximal, characteristic
3#21455675C213yesmaximal, complemented
4#21455669C213yesmaximal, complemented
5#21455653C213yesmaximal, complemented
6#21455648C213yesmaximal, complemented
7#21455632C213yesmaximal, complemented
8#21455615C213yesmaximal, complemented
9#21455605C213yesmaximal, complemented
10C2×#16840041C213yesmaximal, complemented
11#21454417C211yesmaximal, characteristic, complemented
12#21454385C211yesmaximal, characteristic, complemented
13#21454346C211yesmaximal, characteristic
14#21454342C211yesmaximal, characteristic
15C61×#2913C211yesmaximal, characteristic, complemented
16C2×#14519823C2211yescharacteristic
17C2×#14519784C2211yescharacteristic
18Q8×Dic122—21no—
19#16840488—21no—
20#16840471—23no—
21D4×Dic122—23no—
22#16840393C2213yescomplemented
23#16840390C2213yescomplemented
24#16840315C2213yescomplemented
25C2×#14519776C2213yescomplemented
26#16840245C2211yescharacteristic
27#16840244—21no—
28#16840243C2213yes—
29#16840236C2213yes—
30#16840234—23no—
31#41×Dic61C2211yescharacteristic
32C4×#11758226—21no—
33#16840230C2213yes—
34C2×#14519739C2213yes—
35C4×D4×D61—23no—
36#16840072C2213yescomplemented
37#16840068C2213yescomplemented
38#16840065C2216yescomplemented
39#16840065C2213yescomplemented
40#16840061C2213yescomplemented
41#16840061C2216yescomplemented
42#16840056C2216yescomplemented
43#16840056C2211yescharacteristic, complemented
44#16840051C2216yescomplemented
45#16840051C2213yescomplemented
46#16840046C2213yescomplemented
47#16840041C2216yescomplemented
48#16840041C2216yescomplemented
49#16840041C2216yescomplemented
50#16840041C2213yescomplemented
51#16839805C2213yescomplemented
52#16839802C2213yescomplemented
53#16839798C2213yescomplemented
54C2×#14519710C2213yescomplemented
55#16839790C2213yescomplemented
56#16839788C2213yescomplemented
57#16839785C2213yescomplemented
58C2×#14519706C2213yescomplemented
59#16839768C2213yes—
60#16839766C2213yes—
61#16839764C2213yes—
62#16839761C2213yes—
63#16839759C2213yes—
64#16839756C2213yes—
65C2×#14519699C2213yes—
66C2×#14519697C2213yes—
67C2×#131×C61C2211yescharacteristic
68C61×#520C2211yescharacteristic
69C61×#520C2211yescharacteristic
70C61×#428C2213yescomplemented
71C61×#428C2213yescomplemented
72C61×#427C2213yescomplemented
73C2×#107×C61C2213yescomplemented
74#14519823—24no—
75#14519823Q818yescomplemented
76C2×#11758226—21no—
77C2×#11758222C2313yes—
78C22×Dic122—23no—
79C22×D244C2313yes—
80C2×D4×D61—23no—
81C22×C4×D61—23no—
82#14519784—24no—
83C2×#11758194C2313yes—
84C22×#92418C2313yes—
85#14519777—23no—
86#14519776—23no—
87#14519776—23no—
88#14519776—23no—
89#14519776C2313yes—
90#14519776C2316yes—
91#14519776C2311yescharacteristic
92#14519769—23no—
93#14519768—23no—
94#14519768C2313yes—
95#14519768C2313yes—
96#14519756—26no—
97#14519754—23no—
98#14519754—23no—
99#14519754—26no—
100Q8×Dic61—21no—
101Q8×Dic61D411yescharacteristic
102Q8×Dic61D411yescharacteristic
103C4×Dic122—23no—
104C4×Dic122—23no—
105#14519742C2311yescharacteristic
106#14519742C2313yes—
107C4×D244—23no—
108C4×D244—23no—
109D4×Dic61—23no—
110D4×Dic61D413yes—
111D4×Dic61D413yes—
112#14519739—23no—
113#14519739—23no—
114#14519739C2313yes—
115#14519739C2316yes—
116#14519739C2313yes—
117C4×C22⋊D61—26no—
118#14519737—23no—
119#14519737—23no—
120#14519736C2313yes—
121#14519736C2313yes—
122#14519735—23no—
123#14519734—23no—
124C2×C4⋊Dic61C2313yes—
125C2×#11758196—23no—
126C4⋊C4×D61—23no—
127C2×#11758198—23no—
128C22⋊C4×D61—26no—
129C2×C4×Dic61C2313yes—
130C42×D61—23no—
131#14519710C2313yes—
132#14519710C2316yes—
133#14519710C2311yescharacteristic
134#14519708C2313yes—
135#14519708C2313yes—
136#14519706C2311yescharacteristic
137#14519706C2313yes—
138#14519706C2313yes—
139#14519706C2313yes—
140#14519706C2316yes—
141#14519702C2311yescharacteristic
142#14519702C2313yes—
143#14519700C2311yescharacteristic
144#14519700C2313yes—
145#14519699C2313yes—
146#14519699C2316yes—
147#14519699C2313yes—
148#14519697C2313yes—
149#14519697C2316yes—
150#14519697C2313yes—
151C2×#41×C61C2311yescharacteristic
152#131×C61—24no—
153C2×M16×C61C2313yes—
154C22×C8×C61C2313yes—
155C4×Q8×C61—21no—
156C4×Q8×C61—21no—
157C4×D4×C61—23no—
158C4×D4×C61—23no—
159#117×C61C2313yes—
160#117×C61C2313yes—
161C2×C4⋊C4×C61C2313yes—
162C2×C4⋊C4×C61C2313yes—
163#107×C61C2313yes—
164#107×C61C2316yes—
165#107×C61C2311yescharacteristic
166#106×C61C2313yes—
167#106×C61C2313yes—
168C23×D61—23no—
169#41×C61C2×Q814yes—
170#41×C61#4114yes—
171C2×Q8×C61C2×D411yescharacteristic
172C2×D4×C61C2×D413yes—
173C22×C4×C61—23no—
174C22×C4×C61—23no—
175C22×C4×C61C2413yes—
176C2×C22⋊D61—23no—
177C22×Dic61—23no—
178C22×Dic61C2×D413yes—
179C22×Dic61C2×D413yes—
180#11758226—28no—
181D4×D61—224no—
182#11758222—26no—
183#11758222C2×Q8112yescomplemented
184C2×D244—23no—
185C2×D244—26no—
186C2×D244C2×Q8112yescomplemented
187C2×D244C2413yes—
188C2×C4×D61—23no—
189C2×C4×D61—23no—
190C2×C4×D61—26no—
191C2×C4×D61—23no—
192C2×C4×D61—86no—
193C2×Dic122—23no—
194C2×Dic122—21no—
195C2×Dic122—26no—
196C2×Dic122—86no—
197C2×Dic122C2411yescharacteristic
198M16×C61—26no—
199C2×C8×C61—26no—
200C2×C8×C61C2413yes—
201C2×C8×C61C2411yescharacteristic
202C4⋊C4×C61—23no—
203C4⋊C4×C61—23no—
204C4⋊C4×C61C2411yescharacteristic
205C4⋊C4×C61C2411yescharacteristic
206C4⋊C4×C61C2413yes—
207C4⋊C4×C61C2413yes—
208C22⋊C4×C61—23no—
209C22⋊C4×C61—23no—
210C42×C61—23no—
211C42×C61—23no—
212C22⋊Dic61—23no—
213C22⋊Dic61C2×D416yes—
214#11758198—23no—
215#11758198—26no—
216#11758198—23no—
217C4⋊Dic61—23no—
218C4⋊Dic61—23no—
219C4⋊Dic61—23no—
220C4⋊Dic61C2×D413yes—
221C4⋊Dic61C2×D413yes—
222C4⋊Dic61C2411yescharacteristic
223C4⋊Dic61C2413yes—
224#11758196—23no—
225#11758196—23no—
226#11758196—26no—
227#11758196—21no—
228#11758196—21no—
229#11758196—23no—
230#11758196—23no—
231C4×Dic61—23no—
232C4×Dic61—21no—
233C4×Dic61—23no—
234C4×Dic61C2×D413yes—
235C4×Dic61C2×D413yes—
236C4×Dic61C2411yescharacteristic
237C4×Dic61C2413yes—
238#11758194—26no—
239C2×#92418—26no—
240C2×#92418C2413yes—
241C2×#92418C2411yescharacteristic
242C23×C61C22×D413yes—
243C22×D61—23no—
244C22×D61—212no—
245C22×D61—83no—
246C22×D61—83no—
247Q8×C61C22⋊Q814yes—
248D4×C61C22⋊Q8112yes—
249C2×C4×C61—21no—
250C2×C4×C61—23no—
251C2×C4×C61—21no—
252C2×C4×C61—23no—
253C2×C4×C61—83no—
254C2×C4×C61—83no—
255C2×C4×C61C22×D413yes—
256C2×C4×C61C22×Q816yes—
257C2×C4×C61C2×#4116yes—
258C2×C4×C61C2511yescharacteristic
259C22⋊D61—212no—
260C2×Dic61—21no—
261C2×Dic61—26no—
262C2×Dic61—21no—
263C2×Dic61—23no—
264C2×Dic61—43no—
265C2×Dic61—83no—
266C2×Dic61C22×D413yes—
267C2×Dic61C22×D411yescharacteristic
268C2×Dic61C22×D413yes—
269C2×Dic61C22×D411yescharacteristic
270D244—212no—
271D244—23no—
272D244C22×Q816yes—
273C4×D61—212no—
274C4×D61—46no—
275C4×D61—83no—
276C4×D61—86no—
277Dic122—21no—
278Dic122—42no—
279Dic122—46no—
280Dic122—86no—
281Dic122C22×Q812yes—
282C8×C61—21no—
283C8×C61—23no—
284#92418—21no—
285#92418—23no—
286#60494—611nomaximal, Sylow
287C22×C61—23no—
288C22×C61—23no—
289C22×C61C2×C22⋊Q816yes—
290C22×C61C23×D411yescharacteristic
291C2×D61—26no—
292C2×D61—43no—
293C2×D61—43no—
294C2×D61—812no—
295C4×C61—41no—
296C4×C61—41no—
297C4×C61—43no—
298C4×C61—43no—
299C4×C61C2×C22⋊Q816yes—
300C4×C61C23×Q811yescharacteristic
301C4×C61C22×#4111yescharacteristic
302Dic61—21no—
303Dic61—23no—
304Dic61—22no—
305Dic61—23no—
306Dic61—41no—
307Dic61—43no—
308Dic61#57612yes—
309#3503—611no—
310C4×H(5,𝔽2)—611no—
311#3083—613no—
312#3083—613no—
313#3081—613no—
314#3081—613no—
315#3079—613no—
316#3077—613no—
317#3076—613no—
318C2×#474—613no—
319#2913—611no—
320#2913D6111yescharacteristic, complemented
321#2906—611no—
322#2897—611no—
323#2897—611no—
324C2×C61—26no—
325C2×C61C22×C22⋊Q811yescharacteristic
326C2×C61C2×#57611yescharacteristic
327C2×C61#358611yescharacteristic
328D61—46no—
329C2×H(5,𝔽2)—611no—
330C2×#131—611no—
331C2×#131C2×D6111yescharacteristic
332Q82—1221no—
333#557—1221no—
334#554—1223no—
335D4×Q8—1223no—
336#544—613no—
337#544—613no—
338#533—613no—
339C2×C4⋊Q8—613no—
340#520—611no—
341#520—613no—
342#520—1221no—
343#520C2×D6111yescharacteristic
344#520C2×D6111yescharacteristic
345#518—613no—
346#518—1223no—
347C4×#41—611no—
348C4×#41—613no—
349C4×#41—1221no—
350C2×C4×D4—613no—
351C2×C4×D4—1223no—
352#479—613no—
353#479—613no—
354#478—613no—
355#478—616no—
356#478—613no—
357#478—616no—
358#477—611no—
359#477—616no—
360#476—613no—
361#476—616no—
362#475—613no—
363#474—613no—
364#474—616no—
365#474—616no—
366#474—616no—
367#428—613no—
368#428—613no—
369#428C2×D6113yescomplemented
370#428C2×D6113yescomplemented
371#427—613no—
372#427C2×D6113yescomplemented
373C2×#107—613no—
374C2×#107C2×D6113yescomplemented
375#424—613no—
376#424—613no—
377#423—613no—
378C2×C4⋊C8—613no—
379#418—613no—
380#418—613no—
381#417—613no—
382#417—613no—
383#416—613no—
384#416—613no—
385C2×#102—613no—
386C2×#102—613no—
387C61#6049411yescharacteristic, complemented, Sylow
388H(5,𝔽2)—618no—
389H(5,𝔽2)—1224no—
390C2×#41—613no—
391C2×#41—1221no—
392C2×#41C22×D6111yescharacteristic
393C22×Q8—1223no—
394C22×D4—613no—
395C22×D4—1223no—
396C23×C4—1223no—
397#131—24no—
398#131—1224no—
399C2×M16—613no—
400C2×M16C22×D6113yes—
401C22×C8—613no—
402C22×C8C22×D6113yes—
403C4⋊Q8—613no—
404C4⋊Q8—616no—
405C4⋊Q8—611no—
406C4⋊Q8—1223no—
407C4⋊Q8—1223no—
408C4⋊Q8—1223no—
409C4⋊Q8—1223no—
410#125—613no—
411#125—613no—
412#125—1223no—
413#125—1223no—
414C22⋊Q8—1226no—
415C22⋊Q8—1223no—
416C22⋊Q8—1223no—
417C22⋊Q8—1226no—
418C4×Q8—21no—
419C4×Q8—21no—
420C4×Q8—611no—
421C4×Q8—611no—
422C4×Q8—611no—
423C4×Q8—613no—
424C4×Q8—1223no—
425C4×Q8—1223no—
426C4×Q8—1221no—
427C4×D4—23no—
428C4×D4—23no—
429C4×D4—613no—
430C4×D4—613no—
431C4×D4—613no—
432C4×D4—616no—
433C4×D4—613no—
434C4×D4—1223no—
435C4×D4—1223no—
436C4×D4—1223no—
437C4×D4—1223no—
438C4×D4—1223no—
439C4×D4—1223no—
440C4×D4—1226no—
441C4×D4—1223no—
442#117—613no—
443#117—613no—
444#117—1223no—
445#117—1223no—
446#117C22×D6113yes—
447#117C22×D6113yes—
448C2×C4⋊C4—613no—
449C2×C4⋊C4—1223no—
450C2×C4⋊C4—1223no—
451C2×C4⋊C4C22×D6113yes—
452C2×C4⋊C4C22×D6113yes—
453C2×C22⋊C4—1226no—
454C2×C22⋊C4—1223no—
455C2×C42—613no—
456C2×C42—1223no—
457#107—613no—
458#107—616no—
459#107—611no—
460#107C22×D6113yes—
461#107C22×D6116yes—
462#107C22×D6111yescharacteristic
463#106—613no—
464#106—613no—
465#106C22×D6113yes—
466#106C22×D6113yes—
467C4⋊C8—611no—
468C4⋊C8—613no—
469C4⋊C8—613no—
470C4⋊C8—613no—
471C4⋊C8—616no—
472#103—611no—
473#103—613no—
474#103—611no—
475#103—613no—
476#102—613no—
477#102—616no—
478#102—613no—
479#102—613no—
480#102—616no—
481#102—613no—
482C24—1223no—
483#41—6112no—
484#41—1228no—
485#41—1226no—
486#41#1175822214yes—
487#41Q8×D6114yes—
488C2×Q8—611no—
489C2×Q8—1226no—
490C2×Q8—1221no—
491C2×Q8—1223no—
492C2×Q8—4886no—
493C2×Q8D4×D6111yescharacteristic
494C2×D4—6112no—
495C2×D4—613no—
496C2×D4—1223no—
497C2×D4—12224no—
498C2×D4—1223no—
499C2×D4—1226no—
500C2×D4D4×D6113yes—
501C22×C4—23no—
502C22×C4—23no—
503C22×C4—613no—
504C22×C4—613no—
505C22×C4—1223no—
506C22×C4—1226no—
507C22×C4—1223no—
508C22×C4—1223no—
509C22×C4—1223no—
510C22×C4—4886no—
511C22×C4C23×D6113yes—
512M16—26no—
513M16—1226no—
514C2×C8—26no—
515C2×C8—613no—
516C2×C8—611no—
517C2×C8—1226no—
518C2×C8C23×D6113yes—
519C2×C8C23×D6111yescharacteristic
520C4⋊C4—23no—
521C4⋊C4—23no—
522C4⋊C4—613no—
523C4⋊C4—613no—
524C4⋊C4—611no—
525C4⋊C4—613no—
526C4⋊C4—1223no—
527C4⋊C4—1223no—
528C4⋊C4—1226no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1223no—
532C4⋊C4—1223no—
533C4⋊C4—1223no—
534C4⋊C4—1223no—
535C4⋊C4—1223no—
536C4⋊C4C23×D6111yescharacteristic
537C4⋊C4C23×D6113yes—
538C4⋊C4C23×D6111yescharacteristic
539C4⋊C4C23×D6113yes—
540C22⋊C4—23no—
541C22⋊C4—23no—
542C22⋊C4—616no—
543C22⋊C4—1223no—
544C22⋊C4—1223no—
545C22⋊C4—1226no—
546C22⋊C4—1223no—
547C42—23no—
548C42—23no—
549C42—613no—
550C42—613no—
551C42—611no—
552C42—613no—
553C42—1221no—
554C42—1223no—
555C42—1223no—
556C23—12212no—
557C23—1223no—
558C23—4883no—
559C23—4883no—
560C23C2×D4×D6113yes—
561Q8—612no—
562Q8—1221no—
563Q8—4886no—
564Q8—2446no—
565Q8—2442no—
566Q8#1451975514yes—
567D4—616no—
568D4—12212no—
569D4—12212no—
570D4—1223no—
571D4#14519755112yes—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—23no—
575C2×C4—23no—
576C2×C4—83no—
577C2×C4—83no—
578C2×C4—611no—
579C2×C4—611no—
580C2×C4—613no—
581C2×C4—613no—
582C2×C4—1226no—
583C2×C4—1221no—
584C2×C4—12212no—
585C2×C4—1221no—
586C2×C4—1223no—
587C2×C4—4883no—
588C2×C4—4886no—
589C2×C4—4883no—
590C2×C4—2446no—
591C2×C4—2443no—
592C2×C4C2×D4×D6113yes—
593C2×C4C2×Q8×D6116yes—
594C2×C4C2×#1175822216yes—
595C2×C4C24×D6111yescharacteristic
596C8—21no—
597C8—23no—
598C8—1221no—
599C8—1223no—
600C22—23no—
601C22—23no—
602C22—1226no—
603C22—48812no—
604C22—2443no—
605C22—2443no—
606C22C2×#1451975516yes—
607C22C22×D4×D6111yescharacteristic
608C4—43no—
609C4—43no—
610C4—41no—
611C4—41no—
612C4—612no—
613C4—1223no—
614C4—1222no—
615C4—1223no—
616C4—1221no—
617C4—2443no—
618C4—2441no—
619C4C2×#1451975516yes—
620C4C22×Q8×D6111yescharacteristic
621C4C22×#1175822211yescharacteristic
622C2—26no—
623C2—2446no—
624C2C22×#1451975511yescharacteristic
625C2#576×D6111yescharacteristic
626C2#2146060411yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#2913C2
socleC22×C61C23×D4
prime radicalC2×#14519823C22
hypercenter#2913D61
hypocenterC61#60494
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×#41 ◁ #2913

Complements

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

NH ≅ G/Ntype
1. #21459632C2semidirect
3. #21455675C2semidirect
4. #21455669C2semidirect
5. #21455653C2semidirect
6. #21455648C2semidirect
7. #21455632C2semidirect
8. #21455615C2semidirect
9. #21455605C2semidirect
10. C2×#16840041C2semidirect
11. #21454417C2semidirect
12. #21454385C2semidirect
15. C61×#2913C2semidirect
22. #16840393C22semidirect
23. #16840390C22semidirect
24. #16840315C22semidirect
25. C2×#14519776C22semidirect
36. #16840072C22semidirect
37. #16840068C22semidirect
38. #16840065C22semidirect
39. #16840065C22semidirect
40. #16840061C22semidirect
41. #16840061C22semidirect
42. #16840056C22semidirect
43. #16840056C22semidirect
44. #16840051C22semidirect
45. #16840051C22semidirect
46. #16840046C22semidirect
47. #16840041C22semidirect
48. #16840041C22semidirect
49. #16840041C22semidirect
50. #16840041C22semidirect
51. #16839805C22semidirect
52. #16839802C22semidirect
53. #16839798C22semidirect
54. C2×#14519710C22semidirect
55. #16839790C22semidirect
56. #16839788C22semidirect
57. #16839785C22semidirect
58. C2×#14519706C22semidirect
70. C61×#428C22semidirect
71. C61×#428C22semidirect
72. C61×#427C22semidirect
73. C2×#107×C61C22semidirect
75. #14519823Q8semidirect
183. #11758222C2×Q8semidirect
186. C2×D244C2×Q8semidirect
320. #2913D61semidirect
369. #428C2×D61semidirect
370. #428C2×D61semidirect
372. #427C2×D61semidirect
374. C2×#107C2×D61semidirect
387. C61#60494semidirect

Maximal subgroups

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

subgroupindex
#214596322
#214590252
#214556752
#214556692
#214556532
#214556482
#214556322
#214556152
#214556052
C2×#168400412
#214544172
#214543852
#214543462
#214543422
C61×#29132
#6049461