Finite Group Databasebeta

← #30547695

Subgroups — #30547695

Order 15616.

2054 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459641C211yesmaximal, characteristic, complemented
2H(5,𝔽2)×Dic61C211yesmaximal, characteristic
3#21455679C213yesmaximal, complemented
4#21455674C213yesmaximal, complemented
5#21455652C213yesmaximal, complemented
6#21455646C213yesmaximal, complemented
7#21455639C213yesmaximal, complemented
8#21455613C213yesmaximal, complemented
9#21455603C213yesmaximal, complemented
10C2×#16840064C213yesmaximal, complemented
11#21454417C211yesmaximal, characteristic, complemented
12#21454414C211yesmaximal, characteristic, complemented
13#21454385C211yesmaximal, characteristic, complemented
14#21454345C211yesmaximal, characteristic
15C61×#2897C211yesmaximal, characteristic
16C2×#14519784C2211yescharacteristic
17#16840491—21no—
18#16840488—21no—
19#16840471—23no—
20#16840469—23no—
21#16840393C2213yescomplemented
22#16840390C2213yescomplemented
23#16840315C2213yescomplemented
24C2×#14519768C2213yescomplemented
25#16840245—21no—
26#16840245C2213yes—
27#16840245C2211yescharacteristic
28#16840245C2211yescharacteristic
29#16840237—23no—
30#16840237C2213yes—
31#41×Dic61—21no—
32#41×Dic61C2213yes—
33#41×Dic61C2211yescharacteristic
34C2×D4×Dic61—23no—
35C2×D4×Dic61C2213yes—
36#16840071C2213yescomplemented
37#16840067C2216yescomplemented
38#16840067C2211yescharacteristic, complemented
39#16840064C2216yescomplemented
40#16840064C2216yescomplemented
41#16840064C2216yescomplemented
42#16840064C2213yescomplemented
43#16840060C2216yescomplemented
44#16840060C2213yescomplemented
45#16840055C2213yescomplemented
46#16840050C2213yescomplemented
47#16840050C2216yescomplemented
48#16840045C2213yescomplemented
49#16840040C2216yescomplemented
50#16840040C2213yescomplemented
51#16839805C2213yescomplemented
52#16839804C2213yescomplemented
53#16839804C2213yescomplemented
54#16839802C2213yescomplemented
55#16839800C2213yescomplemented
56#16839798C2213yescomplemented
57C2×#14519710C2213yescomplemented
58C2×#14519709C2213yescomplemented
59#16839790C2213yescomplemented
60#16839788C2213yescomplemented
61#16839785C2213yescomplemented
62C2×#14519706C2213yescomplemented
63#16839767C2213yes—
64#16839763C2213yes—
65#16839758C2213yes—
66C2×#14519698C2213yes—
67C2×H(5,𝔽2)×C61C2211yescharacteristic
68C2×#131×C61C2211yescharacteristic
69C61×#520C2211yescharacteristic
70C61×#418C2213yes—
71C61×#417C2213yes—
72C61×#416C2213yes—
73C2×#102×C61C2213yes—
74C23×Dic61—23no—
75#14519784—24no—
76C2×#11758194C2313yes—
77C22×#92418C2313yes—
78#14519777—23no—
79#14519776—23no—
80#14519776C2313yes—
81#14519776C2313yes—
82#14519769—23no—
83#14519768—23no—
84#14519768—23no—
85#14519768—23no—
86#14519768C2313yes—
87#14519768C2316yes—
88#14519768C2311yescharacteristic
89#14519756—26no—
90#14519754—23no—
91#14519754—23no—
92#14519754—26no—
93Q8×Dic61—21no—
94Q8×Dic61—21no—
95Q8×Dic61D411yescharacteristic
96Q8×Dic61D411yescharacteristic
97Q8×Dic61C2311yescharacteristic
98Q8×Dic61C2313yes—
99C4×Dic122—23no—
100C4×Dic122—23no—
101D4×Dic61—23no—
102D4×Dic61—23no—
103D4×Dic61—23no—
104D4×Dic61—23no—
105D4×Dic61—26no—
106D4×Dic61—23no—
107D4×Dic61—23no—
108D4×Dic61—23no—
109D4×Dic61—23no—
110D4×Dic61D413yes—
111D4×Dic61D413yes—
112D4×Dic61C2313yes—
113D4×Dic61C2316yes—
114D4×Dic61C2313yes—
115#14519736—23no—
116#14519736—23no—
117#14519736C2313yes—
118#14519736C2313yes—
119#14519736C2313yes—
120C2×C4⋊Dic61—23no—
121C2×C4⋊Dic61—23no—
122C2×C4⋊Dic61C2313yes—
123C2×C4⋊Dic61C2313yes—
124C2×#11758196—23no—
125C2×C22⋊Dic61—23no—
126C2×C22⋊Dic61—26no—
127C2×C4×Dic61—23no—
128C2×C4×Dic61C2313yes—
129#14519711C2313yes—
130#14519711C2313yes—
131#14519710C2311yescharacteristic
132#14519710C2316yes—
133#14519710C2313yes—
134#14519709C2313yes—
135#14519709C2316yes—
136#14519709C2311yescharacteristic
137#14519708C2313yes—
138#14519708C2313yes—
139#14519706C2311yescharacteristic
140#14519706C2313yes—
141#14519706C2313yes—
142#14519706C2313yes—
143#14519706C2316yes—
144#14519701C2311yescharacteristic
145#14519701C2313yes—
146#14519698C2313yes—
147#14519698C2316yes—
148#14519698C2313yes—
149H(5,𝔽2)×C61—24no—
150H(5,𝔽2)×C61Q818yescomplemented
151C2×#41×C61—21no—
152C2×#41×C61C2313yes—
153C2×#41×C61C2311yescharacteristic
154C22×D4×C61—23no—
155C22×D4×C61C2313yes—
156#131×C61—24no—
157C2×M16×C61C2313yes—
158C22×C8×C61C2313yes—
159C4×Q8×C61—21no—
160C4×D4×C61—23no—
161#117×C61C2313yes—
162C2×C4⋊C4×C61C2313yes—
163#103×C61C2311yescharacteristic
164#103×C61C2313yes—
165#102×C61C2313yes—
166#102×C61C2316yes—
167#102×C61C2313yes—
168C24×C61—23no—
169#41×C61—28no—
170#41×C61—26no—
171#41×C61C2×Q8112yes—
172#41×C61C2×Q814yes—
173#41×C61#4114yes—
174C2×Q8×C61C2×D411yescharacteristic
175C2×Q8×C61C2411yescharacteristic
176C2×D4×C61—23no—
177C2×D4×C61—224no—
178C2×D4×C61—23no—
179C2×D4×C61—26no—
180C2×D4×C61C2×D413yes—
181C2×D4×C61C2×Q8112yes—
182C2×D4×C61C2413yes—
183C22×C4×C61—23no—
184C22×C4×C61—23no—
185C22×C4×C61C2413yes—
186C22×Dic61—23no—
187C22×Dic61—23no—
188C22×Dic61—23no—
189C22×Dic61—26no—
190C22×Dic61—23no—
191C22×Dic61—86no—
192C22×Dic61C2×D413yes—
193C22×Dic61C2×D413yes—
194C2×Dic122—23no—
195C2×Dic122—23no—
196M16×C61—26no—
197C2×C8×C61—26no—
198C2×C8×C61C2411yescharacteristic
199C2×C8×C61C2413yes—
200C4⋊C4×C61—23no—
201C4⋊C4×C61C2413yes—
202C4⋊C4×C61C2411yescharacteristic
203C22⋊C4×C61—23no—
204C42×C61—23no—
205C22⋊Dic61—23no—
206C22⋊Dic61—23no—
207C22⋊Dic61—23no—
208C22⋊Dic61—26no—
209C22⋊Dic61—23no—
210C22⋊Dic61C2×D416yes—
211C4⋊Dic61—23no—
212C4⋊Dic61—23no—
213C4⋊Dic61—23no—
214C4⋊Dic61—23no—
215C4⋊Dic61—26no—
216C4⋊Dic61—21no—
217C4⋊Dic61—21no—
218C4⋊Dic61C2×D413yes—
219C4⋊Dic61C2×D413yes—
220C4⋊Dic61C2413yes—
221C4⋊Dic61C2411yescharacteristic
222C4⋊Dic61C2411yescharacteristic
223C4⋊Dic61C2413yes—
224#11758196—23no—
225#11758196—26no—
226#11758196—21no—
227#11758196—23no—
228#11758196—23no—
229C4×Dic61—23no—
230C4×Dic61—23no—
231C4×Dic61—21no—
232C4×Dic61—23no—
233C4×Dic61C2×D413yes—
234C4×Dic61C2×D413yes—
235C4×Dic61C2411yescharacteristic
236C4×Dic61C2413yes—
237#11758194—26no—
238C2×#92418—26no—
239C2×#92418C2413yes—
240C2×#92418C2411yescharacteristic
241C23×C61—23no—
242C23×C61—212no—
243C23×C61—83no—
244C23×C61—83no—
245C23×C61C22×D413yes—
246Q8×C61—21no—
247Q8×C61C22⋊Q814yes—
248Q8×C61C22×Q812yes—
249D4×C61—212no—
250D4×C61—212no—
251D4×C61—23no—
252D4×C61C22⋊Q8112yes—
253D4×C61C22×Q816yes—
254C2×C4×C61—21no—
255C2×C4×C61—212no—
256C2×C4×C61—23no—
257C2×C4×C61—21no—
258C2×C4×C61—83no—
259C2×C4×C61C22×D413yes—
260C2×C4×C61C22×Q816yes—
261C2×C4×C61C2×#4116yes—
262C2×C4×C61C2511yescharacteristic
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—23no—
266C2×Dic61—23no—
267C2×Dic61—43no—
268C2×Dic61—43no—
269C2×Dic61—46no—
270C2×Dic61—83no—
271C2×Dic61—86no—
272C2×Dic61—83no—
273C2×Dic61—83no—
274C2×Dic61C22×D411yescharacteristic
275C2×Dic61C22×D413yes—
276C2×Dic61C22×D411yescharacteristic
277C2×Dic61C22×D413yes—
278Dic122—83no—
279Dic122—83no—
280C8×C61—21no—
281C8×C61—23no—
282#92418—21no—
283#92418—23no—
284#60496—611nomaximal, Sylow
285C22×C61—23no—
286C22×C61—23no—
287C22×C61—26no—
288C22×C61—43no—
289C22×C61—43no—
290C22×C61—812no—
291C22×C61C2×C22⋊Q816yes—
292C22×C61C23×D411yescharacteristic
293C4×C61—22no—
294C4×C61—41no—
295C4×C61—43no—
296C4×C61C2×C22⋊Q816yes—
297C4×C61C23×Q811yescharacteristic
298C4×C61C22×#4111yescharacteristic
299Dic61—23no—
300Dic61—21no—
301Dic61—23no—
302Dic61—21no—
303Dic61—43no—
304Dic61—41no—
305Dic61—41no—
306Dic61—43no—
307#3504—611no—
308C4×H(5,𝔽2)—611no—
309#3084—613no—
310#3083—613no—
311#3081—613no—
312#3081—613no—
313#3080—613no—
314#3077—613no—
315#3076—613no—
316C2×#478—613no—
317#2913—611no—
318#2912—611no—
319#2906—611no—
320#2897—611no—
321#2897D6111yescharacteristic
322C2×C61—26no—
323C2×C61—46no—
324C2×C61C22×C22⋊Q811yescharacteristic
325C2×C61#358411yescharacteristic
326C2×C61#358511yescharacteristic
327C2×H(5,𝔽2)C2×D6111yescharacteristic
328C2×#131—611no—
329C2×#131C2×D6111yescharacteristic
330#557—1221no—
331#557—1221no—
332#554—1223no—
333#554—1223no—
334#544—613no—
335#544—613no—
336#533—613no—
337C2×#125—613no—
338#520—611no—
339#520—611no—
340#520—613no—
341#520—1221no—
342#520C2×D6111yescharacteristic
343#518—613no—
344#518—1223no—
345C4×#41—611no—
346C4×#41—613no—
347C4×#41—1221no—
348C2×C4×D4—613no—
349C2×C4×D4—1223no—
350#479—611no—
351#479—613no—
352#479—616no—
353#478—613no—
354#478—616no—
355#478—616no—
356#478—616no—
357#478—613no—
358#478—616no—
359#477—613no—
360#476—613no—
361#476—616no—
362#475—613no—
363#474—613no—
364#474—616no—
365#428—613no—
366#428—613no—
367#428—613no—
368#428—613no—
369#427—613no—
370#427—613no—
371C2×#107—613no—
372C2×#106—613no—
373#424—613no—
374#424—613no—
375#423—613no—
376C2×C4⋊C8—613no—
377#418—613no—
378#418C2×D6113yes—
379#417—613no—
380#417C2×D6113yes—
381#416—613no—
382#416C2×D6113yes—
383C2×#102—613no—
384C2×#102C2×D6113yes—
385C61#6049611yescharacteristic, complemented, Sylow
386H(5,𝔽2)—24no—
387H(5,𝔽2)Dic12218yescomplemented
388C2×#41—21no—
389C2×#41C22×D6113yes—
390C2×#41C22×D6111yescharacteristic
391C22×D4—23no—
392C22×D4C22×D6113yes—
393C23×C4—1223no—
394#131—24no—
395#131—1224no—
396C2×M16—613no—
397C2×M16C22×D6113yes—
398C22×C8—613no—
399C22×C8C22×D6113yes—
400C4⋊Q8—613no—
401C4⋊Q8—613no—
402C4⋊Q8—1223no—
403C4⋊Q8—1223no—
404#125—613no—
405#125—616no—
406#125—611no—
407#125—1223no—
408#125—1223no—
409#125—1223no—
410#125—1223no—
411C22⋊Q8—1226no—
412C22⋊Q8—1223no—
413C22⋊Q8—1223no—
414C22⋊Q8—1226no—
415C4×Q8—21no—
416C4×Q8—611no—
417C4×Q8—611no—
418C4×Q8—611no—
419C4×Q8—613no—
420C4×Q8—1223no—
421C4×Q8—1223no—
422C4×Q8—1221no—
423C4×Q8—1221no—
424C4×D4—23no—
425C4×D4—613no—
426C4×D4—613no—
427C4×D4—613no—
428C4×D4—616no—
429C4×D4—613no—
430C4×D4—1223no—
431C4×D4—1223no—
432C4×D4—1226no—
433C4×D4—1223no—
434C4×D4—1223no—
435C4×D4—1223no—
436C4×D4—1223no—
437C4×D4—1223no—
438C4×D4—1223no—
439#117—613no—
440#117—613no—
441#117—613no—
442#117—1223no—
443#117—1223no—
444#117C22×D6113yes—
445C2×C4⋊C4—613no—
446C2×C4⋊C4—613no—
447C2×C4⋊C4—1223no—
448C2×C4⋊C4—1223no—
449C2×C4⋊C4—1223no—
450C2×C4⋊C4C22×D6113yes—
451C2×C22⋊C4—1226no—
452C2×C22⋊C4—1223no—
453C2×C42—613no—
454C2×C42—1223no—
455#107—613no—
456#107—613no—
457#107—613no—
458#107—616no—
459#107—611no—
460#106—613no—
461#106—616no—
462#106—611no—
463#106—613no—
464#106—613no—
465C4⋊C8—611no—
466C4⋊C8—613no—
467C4⋊C8—613no—
468C4⋊C8—613no—
469C4⋊C8—616no—
470#103—611no—
471#103—613no—
472#103C22×D6111yescharacteristic
473#103C22×D6113yes—
474#102—613no—
475#102—616no—
476#102—613no—
477#102C22×D6113yes—
478#102C22×D6116yes—
479#102C22×D6113yes—
480C24—23no—
481#41—28no—
482#41—26no—
483#41C2×Dic122112yes—
484#41C2×Dic12214yes—
485#41#1175822414yes—
486C2×Q8—1223no—
487C2×Q8—1223no—
488C2×Q8D4×D6111yescharacteristic
489C2×Q8C23×D6111yescharacteristic
490C2×D4—23no—
491C2×D4—224no—
492C2×D4—23no—
493C2×D4—26no—
494C2×D4C2×Dic122112yes—
495C2×D4D4×D6113yes—
496C2×D4C23×D6113yes—
497C22×C4—23no—
498C22×C4—23no—
499C22×C4—613no—
500C22×C4—613no—
501C22×C4—1223no—
502C22×C4—1226no—
503C22×C4—1223no—
504C22×C4—1223no—
505C22×C4—1223no—
506C22×C4—4886no—
507C22×C4C23×D6113yes—
508M16—26no—
509M16—1226no—
510C2×C8—26no—
511C2×C8—613no—
512C2×C8—611no—
513C2×C8—1226no—
514C2×C8C23×D6111yescharacteristic
515C2×C8C23×D6113yes—
516C4⋊C4—23no—
517C4⋊C4—613no—
518C4⋊C4—613no—
519C4⋊C4—613no—
520C4⋊C4—611no—
521C4⋊C4—613no—
522C4⋊C4—611no—
523C4⋊C4—1223no—
524C4⋊C4—1223no—
525C4⋊C4—1223no—
526C4⋊C4—1226no—
527C4⋊C4—1226no—
528C4⋊C4—1221no—
529C4⋊C4—1223no—
530C4⋊C4—1223no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1223no—
534C4⋊C4—1223no—
535C4⋊C4C23×D6111yescharacteristic
536C4⋊C4C23×D6113yes—
537C22⋊C4—23no—
538C22⋊C4—616no—
539C22⋊C4—1223no—
540C22⋊C4—1226no—
541C22⋊C4—1223no—
542C22⋊C4—1223no—
543C22⋊C4—1223no—
544C42—23no—
545C42—613no—
546C42—613no—
547C42—611no—
548C42—613no—
549C42—1221no—
550C42—1223no—
551C42—1223no—
552C42—1223no—
553C23—23no—
554C23—212no—
555C23—83no—
556C23—83no—
557C23C2×D4×D6113yes—
558Q8—21no—
559Q8—4883no—
560Q8—4883no—
561Q8#1451975414yes—
562Q8C22×Dic12212yes—
563D4—212no—
564D4—212no—
565D4—23no—
566D4#14519754112yes—
567D4C22×Dic12216yes—
568C2×C4—21no—
569C2×C4—212no—
570C2×C4—21no—
571C2×C4—23no—
572C2×C4—83no—
573C2×C4—611no—
574C2×C4—611no—
575C2×C4—613no—
576C2×C4—613no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1223no—
580C2×C4—1223no—
581C2×C4—4883no—
582C2×C4—4886no—
583C2×C4—4883no—
584C2×C4—4883no—
585C2×C4—2443no—
586C2×C4—2443no—
587C2×C4—2446no—
588C2×C4C2×D4×D6113yes—
589C2×C4C22×Dic12216yes—
590C2×C4C2×#1175822416yes—
591C2×C4C24×D6111yescharacteristic
592C8—21no—
593C8—23no—
594C8—1221no—
595C8—1223no—
596C22—23no—
597C22—23no—
598C22—26no—
599C22—43no—
600C22—43no—
601C22—812no—
602C22C2×#1451975416yes—
603C22C22×D4×D6111yescharacteristic
604C4—22no—
605C4—41no—
606C4—43no—
607C4—1223no—
608C4—1221no—
609C4—1223no—
610C4—1221no—
611C4—2441no—
612C4—2443no—
613C4—2443no—
614C4—2441no—
615C4C2×#1451975416yes—
616C4C23×Dic12211yescharacteristic
617C4C22×#1175822411yescharacteristic
618C2—26no—
619C2—46no—
620C2C22×#1451975411yescharacteristic
621C2#2146059111yescharacteristic
622C2#2146059411yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#2897C2
socleC22×C61C23×D4
prime radicalC2×H(5,𝔽2)×C61C22
hypercenter#2897D61
hypocenterC61#60496
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×#41 ◁ #2897

Complements

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

NH ≅ G/Ntype
1. #21459641C2semidirect
3. #21455679C2semidirect
4. #21455674C2semidirect
5. #21455652C2semidirect
6. #21455646C2semidirect
7. #21455639C2semidirect
8. #21455613C2semidirect
9. #21455603C2semidirect
10. C2×#16840064C2semidirect
11. #21454417C2semidirect
12. #21454414C2semidirect
13. #21454385C2semidirect
21. #16840393C22semidirect
22. #16840390C22semidirect
23. #16840315C22semidirect
24. C2×#14519768C22semidirect
36. #16840071C22semidirect
37. #16840067C22semidirect
38. #16840067C22semidirect
39. #16840064C22semidirect
40. #16840064C22semidirect
41. #16840064C22semidirect
42. #16840064C22semidirect
43. #16840060C22semidirect
44. #16840060C22semidirect
45. #16840055C22semidirect
46. #16840050C22semidirect
47. #16840050C22semidirect
48. #16840045C22semidirect
49. #16840040C22semidirect
50. #16840040C22semidirect
51. #16839805C22semidirect
52. #16839804C22semidirect
53. #16839804C22semidirect
54. #16839802C22semidirect
55. #16839800C22semidirect
56. #16839798C22semidirect
57. C2×#14519710C22semidirect
58. C2×#14519709C22semidirect
59. #16839790C22semidirect
60. #16839788C22semidirect
61. #16839785C22semidirect
62. C2×#14519706C22semidirect
150. H(5,𝔽2)×C61Q8semidirect
385. C61#60496semidirect
387. H(5,𝔽2)Dic122semidirect

Maximal subgroups

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

subgroupindex
#214596412
H(5,𝔽2)×Dic612
#214556792
#214556742
#214556522
#214556462
#214556392
#214556132
#214556032
C2×#168400642
#214544172
#214544142
#214543852
#214543452
C61×#28972
#6049661