Finite Group Databasebeta

← #30558411

Subgroups — #30558411

Order 15616.

2020 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459743C211yesmaximal, characteristic, complemented
2#21459604C211yesmaximal, characteristic, complemented
3#21457987C214yesmaximal, complemented
4#21457912C214yesmaximal, complemented
5#21457867C214yesmaximal, complemented
6#21457792C214yesmaximal, complemented
7C2×#16840132C211yesmaximal, characteristic, complemented
8#21455790C212yesmaximal, complemented
9#21455511C212yesmaximal, complemented
10#21455307C211yesmaximal, characteristic, complemented
11#21455136C212yesmaximal, complemented
12#21454617C211yesmaximal, characteristic, complemented
13#21454536C211yesmaximal, characteristic, complemented
14#21454350C212yesmaximal, complemented
15C61×#3080C211yesmaximal, characteristic, complemented
16C2×#14519824C2212yes—
17C2×#14519810C2212yes—
18C2×#14519797C2211yescharacteristic
19C2×#14519785C2212yes—
20#16840465C2214yes—
21#16840460—24no—
22#16840450C2214yes—
23Q8×D244C2214yes—
24#16840416—24no—
25#16840416C2214yes—
26#16840375—21no—
27#16840368—21no—
28#16840343C2212yes—
29#16840338—22no—
30#16840338C2212yes—
31#16840312—21no—
32#16840312C2211yescharacteristic
33C2×#14519777—21no—
34C2×#14519777C2211yescharacteristic
35#16840297—22no—
36#16840243C2212yes—
37C4×#11758222C2211yescharacteristic
38#16840132C2214yes—
39#16840132C2214yes—
40#16840106C2214yes—
41#16840105C2214yes—
42#16840099C2214yes—
43#16840094C2214yes—
44#16840021C2214yes—
45#16840009C2214yes—
46#16840004C2214yes—
47#16839992C2214yes—
48#16839973C2214yes—
49#16839968C2214yes—
50#16839925C2214yes—
51#16839919C2214yes—
52#16839908C2214yes—
53#16839902C2214yes—
54#16839872C2212yes—
55#16839867C2212yes—
56#16839862C2212yes—
57#16839860C2212yes—
58#16839832C2214yes—
59#16839832C2214yes—
60#16839798C2211yescharacteristic
61#16839772C2212yes—
62#16839769C2214yes—
63#16839760C2214yes—
64#16839758C2211yescharacteristic
65#16839756C2211yescharacteristic
66#16839755C2214yes—
67C2×#14519700C2212yes—
68C2×#14519700C2212yes—
69#103×D61C2214yes—
70#16839730C2212yes—
71C2×#14519688C2211yescharacteristic
72C2×#125×C61C2211yescharacteristic
73C4×#41×C61C2211yescharacteristic
74C61×#479C2214yes—
75C61×#478C2214yes—
76C61×#428C2212yes—
77C61×#423C2211yescharacteristic
78C61×#417C2212yes—
79#14519824—28no—
80#14519824—44no—
81C2×#11758222—22no—
82C2×#11758222C2312yes—
83C2×#11758222C2311yescharacteristic
84C2×#11758224—24no—
85C2×#11758224C2314yes—
86C22×Dic122—22no—
87C22×Dic122C2312yes—
88C2×Q8×D61C2314yes—
89#14519810—44no—
90#14519797—24no—
91C2×Dic244C2314yes—
92C2×#11758202C2311yescharacteristic
93C2×#11758190C2314yes—
94C2×#11758200C2311yescharacteristic
95C2×#11758201C2311yescharacteristic
96C2×#11758199C2311yescharacteristic
97#14519785—44no—
98C2×#11758189C2314yes—
99C22×#92418C2311yescharacteristic
100C22×#92418C2311yescharacteristic
101C2×C8×D61C2314yes—
102#14519777—22no—
103#14519777—21no—
104#14519777—21no—
105#14519777—24no—
106#14519777D411yescharacteristic
107#14519777C2314yes—
108#14519774D411yescharacteristic
109#14519772—24no—
110#14519772—24no—
111#14519768—24no—
112#14519765—22no—
113#14519765D412yes—
114#14519762—24no—
115#14519762—24no—
116#14519761—24no—
117#14519761—24no—
118#14519761—24no—
119#14519757—22no—
120#14519757—24no—
121#14519757—24no—
122#14519757C2314yes—
123#14519757C2314yes—
124#14519757C2314yes—
125#14519752—24no—
126#14519752—22no—
127#14519752C2314yes—
128#14519751—24no—
129#14519751—24no—
130#14519751D414yes—
131C4×Dic122C2312yes—
132#14519742—22no—
133C4×D244—24no—
134C4×D244C2314yes—
135C4×D244C2312yes—
136#14519739—22no—
137#14519739—24no—
138C4×C22⋊D61—22no—
139C4×C22⋊D61—24no—
140#14519736C2311yescharacteristic
141#14519735—24no—
142#14519735C2314yes—
143#14519734—21no—
144C2×C4⋊Dic61—22no—
145C2×C4⋊Dic61—24no—
146C4⋊C4×D61C2314yes—
147C42×D61—21no—
148#14519710C2312yes—
149#14519708C2312yes—
150#14519702C2314yes—
151#14519702C2314yes—
152#14519701C2312yes—
153#14519700C2314yes—
154#14519700C2312yes—
155#14519700C2314yes—
156#14519700C2314yes—
157#14519700C2314yes—
158#14519698C2312yes—
159#14519697C2312yes—
160#14519689C2314yes—
161#14519689C2314yes—
162#14519688C2312yes—
163#14519688C2312yes—
164C2×#41×C61C2311yescharacteristic
165C2×M16×C61C2312yes—
166#125×C61—21no—
167#125×C61—21no—
168#125×C61C2314yes—
169C4×Q8×C61C2312yes—
170C4×D4×C61—22no—
171C4×D4×C61C2312yes—
172#117×C61—21no—
173#117×C61C2311yescharacteristic
174C2×C4⋊C4×C61—22no—
175C2×C4⋊C4×C61C2312yes—
176C2×C42×C61—21no—
177C2×C42×C61C2311yescharacteristic
178#107×C61C2314yes—
179#106×C61C2314yes—
180C4⋊C8×C61C2314yes—
181#103×C61C2314yes—
182#102×C61C2314yes—
183#41×C61—22no—
184#41×C61#4114yes—
185C2×Q8×C61C2411yescharacteristic
186C2×D4×C61—21no—
187C2×D4×C61C2411yescharacteristic
188C22×C4×C61—21no—
189C22×C4×C61—22no—
190C22×C4×C61—22no—
191C22×C4×C61C2×D412yes—
192C22×C4×C61C2411yescharacteristic
193C2×C22⋊D61—21no—
194C2×C22⋊D61—24no—
195C2×C22⋊D61C2×D414yes—
196C22×Dic61—22no—
197C22×Dic61—24no—
198Q8×D61—28no—
199Q8×D61—44no—
200#11758224—28no—
201#11758224—44no—
202#11758224—416no—
203#11758222—24no—
204#11758222—24no—
205#11758222—22no—
206#11758222—22no—
207#11758222—48no—
208C2×D244—24no—
209C2×D244C2×D412yes—
210C2×D244C2×D414yes—
211C2×D244C2411yescharacteristic
212C2×C4×D61—24no—
213C2×C4×D61—21no—
214C2×C4×D61—24no—
215C2×C4×D61—22no—
216C2×C4×D61C2414yes—
217C2×Dic122—24no—
218C2×Dic122—24no—
219C2×Dic122—22no—
220C2×Dic122—24no—
221C2×Dic122—48no—
222C2×Dic122C2×D412yes—
223C2×Dic122C2414yes—
224C2×Dic122C2411yescharacteristic
225M16×C61—24no—
226C2×C8×C61C2414yes—
227C4⋊C4×C61—24no—
228C4⋊C4×C61—24no—
229C4⋊C4×C61—22no—
230C4⋊C4×C61—22no—
231C4⋊C4×C61C2414yes—
232C4⋊C4×C61C2412yes—
233C22⋊C4×C61—21no—
234C22⋊C4×C61—22no—
235C42×C61—21no—
236C42×C61—22no—
237C42×C61C2×D411yescharacteristic
238C42×C61C2×D411yescharacteristic
239C42×C61C2412yes—
240C22⋊Dic61—21no—
241#11758202—22no—
242#11758201—22no—
243#11758200—22no—
244#11758199—22no—
245#11758198—24no—
246#11758198—24no—
247#11758198—24no—
248#11758198—24no—
249#11758198—22no—
250#11758198—24no—
251#11758198C2×D414yes—
252#11758198C2×D414yes—
253C4⋊Dic61—24no—
254C4⋊Dic61—24no—
255C4⋊Dic61—24no—
256C4⋊Dic61—24no—
257C4⋊Dic61—24no—
258C4⋊Dic61C2×D414yes—
259C4⋊Dic61C2412yes—
260#11758196—22no—
261#11758196—24no—
262C4×Dic61—21no—
263C4×Dic61—22no—
264C4×Dic61—24no—
265C2×#92418—22no—
266C2×#92418—22no—
267C2×#92418C2411yescharacteristic
268C2×#92418C2411yescharacteristic
269C2×#92418C2412yes—
270Dic244—44no—
271#11758190—44no—
272#11758189—44no—
273C8×D61—44no—
274C23×C61—21no—
275C23×C61C22×D411yescharacteristic
276C22×D61—21no—
277C22×D61C22×D414yes—
278Q8×C61—21no—
279Q8×C61C2×#4112yes—
280D4×C61—24no—
281D4×C61—21no—
282D4×C61C2×#4112yes—
283C2×C4×C61—21no—
284C2×C4×C61—24no—
285C2×C4×C61—24no—
286C2×C4×C61—22no—
287C2×C4×C61—42no—
288C2×C4×C61—44no—
289C2×C4×C61—82no—
290C2×C4×C61C22×D412yes—
291C2×C4×C61C22×D412yes—
292C2×C4×C61C22×D411yescharacteristic
293C2×C4×C61C2×#4112yes—
294C2×C4×C61H(5,𝔽2)12yes—
295C2×C4×C61C2511yescharacteristic
296C22⋊D61—28no—
297C22⋊D61—48no—
298C22⋊D61—42no—
299C2×Dic61—22no—
300C2×Dic61—21no—
301C2×Dic61—24no—
302C2×Dic61—48no—
303C2×Dic61—44no—
304C2×Dic61C22×D414yes—
305D244—24no—
306D244—21no—
307D244—21no—
308D244—44no—
309D244—84no—
310C4×D61—24no—
311C4×D61—24no—
312C4×D61—48no—
313C4×D61—42no—
314C4×D61—82no—
315C4×D61—84no—
316Dic122—24no—
317Dic122—21no—
318Dic122—24no—
319Dic122—24no—
320Dic122—21no—
321Dic122—48no—
322Dic122—44no—
323C8×C61—24no—
324#92418—21no—
325#92418—21no—
326#92418—22no—
327#61224—611nomaximal, Sylow
328C22×C61—22no—
329C22×C61—22no—
330C22×C61—42no—
331C22×C61#54612yes—
332C22×C61C23×D411yescharacteristic
333C2×D61—24no—
334C2×D61—24no—
335C2×D61—24no—
336C2×D61—42no—
337C2×D61—41no—
338C4×C61—22no—
339C4×C61—22no—
340C4×C61—22no—
341C4×C61—44no—
342C4×C61—42no—
343C4×C61#54612yes—
344C4×C61C22×#4111yescharacteristic
345C4×C61C2×H(5,𝔽2)11yescharacteristic
346Dic61—24no—
347Dic61—42no—
348Dic61—44no—
349Dic61—41no—
350#3512—611no—
351#3499—611no—
352#3320—614no—
353#3315—614no—
354#3312—614no—
355#3307—614no—
356C2×#491—611no—
357#3097—612no—
358#3080D6111yescharacteristic, complemented
359#3063—612no—
360#3039—611no—
361#3013—612no—
362#2943—611no—
363#2932—611no—
364#2898—612no—
365C2×C61—22no—
366C2×C61—42no—
367C2×C61C2×#54611yescharacteristic
368C2×C61#358612yes—
369D61—44no—
370D61—81no—
371C2×2−1+4—612no—
372C2×#137—612no—
373C2×#135—611no—
374C2×#131—612no—
375#553—614no—
376#553—1224no—
377#552—614no—
378D4×Q8—614no—
379#547—614no—
380#547—1224no—
381#541—1221no—
382#540—1221no—
383#537—612no—
384#536—612no—
385#536—1222no—
386#532—611no—
387#532—1221no—
388C2×C4⋊Q8—611no—
389C2×C4⋊Q8—1221no—
390#529—1222no—
391C2×#125C2×D6111yescharacteristic
392#520—612no—
393C4×#41—611no—
394C4×#41C2×D6111yescharacteristic
395#491—614no—
396#491—614no—
397#485—614no—
398#485—614no—
399#484—614no—
400#483—614no—
401#479C2×D6114yes—
402#478C2×D6114yes—
403#470—614no—
404#468—614no—
405#467—614no—
406#465—614no—
407#462—614no—
408#461—614no—
409#452—614no—
410#451—614no—
411#449—614no—
412#448—614no—
413#442—612no—
414#441—612no—
415#440—612no—
416#440—612no—
417#435—614no—
418#435—614no—
419#428C2×D6112yes—
420#427—611no—
421#423C2×D6111yescharacteristic
422#419—612no—
423#419—614no—
424#417—614no—
425#417C2×D6112yes—
426#416—611no—
427#416—611no—
428#416—614no—
429C2×#103—612no—
430C2×#103—612no—
431C2×#103—614no—
432#408—612no—
433C2×#97—611no—
434C61#6122411yescharacteristic, complemented, Sylow
4352−1+4—1228no—
4362−1+4—2444no—
437C2×#41—612no—
438C2×#41—614no—
439C2×#41—611no—
440C2×#41—1222no—
441C2×#41—1224no—
442C2×#41C22×D6111yescharacteristic
443C22×Q8—614no—
444C22×Q8—612no—
445C22×Q8—1222no—
446#137—2444no—
447#135—1224no—
448C2×Q16—611no—
449C2×Q16—614no—
450C2×SD16—611no—
451C2×SD16—611no—
452C2×SD16—614no—
453C2×D8—611no—
454#131—2444no—
455C2×M16—614no—
456C2×M16C22×D6112yes—
457C22×C8—611no—
458C22×C8—611no—
459C22×C8—614no—
460C4⋊Q8—611no—
461C4⋊Q8—614no—
462C4⋊Q8—1224no—
463C4⋊Q8—1222no—
464C4⋊Q8—1221no—
465C4⋊Q8—1221no—
466#127—611no—
467#126—1224no—
468#126—1224no—
469#125—21no—
470#125—21no—
471#125—1224no—
472#125C22×D6114yes—
473#124—612no—
474#124—1222no—
475#123—1224no—
476#123—1224no—
477#123—1224no—
478#123—1224no—
479#123—1224no—
480C22⋊Q8—614no—
481C22⋊Q8—614no—
482C22⋊Q8—614no—
483C22⋊Q8—1224no—
484C22⋊Q8—1224no—
485C22⋊Q8—1222no—
486#121—614no—
487#121—614no—
488#121—1222no—
489#121—1224no—
490#121—1224no—
491#121—1224no—
492C4×Q8—612no—
493C4×Q8—1222no—
494C4×Q8C22×D6112yes—
495C4×D4—22no—
496C4×D4—614no—
497C4×D4—612no—
498C4×D4—1224no—
499C4×D4—1224no—
500C4×D4—1222no—
501C4×D4—1224no—
502C4×D4—1222no—
503C4×D4C22×D6112yes—
504#117—21no—
505#117—611no—
506#117—614no—
507#117—1224no—
508#117—1221no—
509#117C22×D6111yescharacteristic
510C2×C4⋊C4—22no—
511C2×C4⋊C4—614no—
512C2×C4⋊C4—1224no—
513C2×C4⋊C4—1222no—
514C2×C4⋊C4C22×D6112yes—
515C2×C42—21no—
516C2×C42—1221no—
517C2×C42C22×D6111yescharacteristic
518#107—612no—
519#107C22×D6114yes—
520#106—612no—
521#106C22×D6114yes—
522C4⋊C8C22×D6114yes—
523#103—612no—
524#103—614no—
525#103—614no—
526#103—614no—
527#103—614no—
528#103—614no—
529#103—614no—
530#103—612no—
531#103C22×D6114yes—
532#102—612no—
533#102—612no—
534#102C22×D6114yes—
535C22⋊C8—614no—
536C22⋊C8—614no—
537#97—612no—
538#97—612no—
539#41—22no—
540#41—1224no—
541#41—1224no—
542#41—1228no—
543#41—1222no—
544#41—1222no—
545#41—2448no—
546#41—2444no—
547#41—24416no—
548#41#1175822614yes—
549C2×Q8—612no—
550C2×Q8—614no—
551C2×Q8—611no—
552C2×Q8—1224no—
553C2×Q8—1222no—
554C2×Q8—1228no—
555C2×Q8—1224no—
556C2×Q8—1224no—
557C2×Q8—2448no—
558C2×Q8—2444no—
559C2×Q8C23×D6111yescharacteristic
560C2×D4—21no—
561C2×D4—614no—
562C2×D4—612no—
563C2×D4—614no—
564C2×D4—611no—
565C2×D4—1224no—
566C2×D4—1224no—
567C2×D4—1221no—
568C2×D4C23×D6111yescharacteristic
569C22×C4—21no—
570C22×C4—22no—
571C22×C4—22no—
572C22×C4—614no—
573C22×C4—1224no—
574C22×C4—1224no—
575C22×C4—1224no—
576C22×C4—1222no—
577C22×C4—1222no—
578C22×C4—1221no—
579C22×C4D4×D6112yes—
580C22×C4C23×D6111yescharacteristic
581Q16—1222no—
582Q16—2444no—
583SD16—1222no—
584SD16—1222no—
585SD16—2444no—
586D8—1222no—
587M16—24no—
588M16—2444no—
589C2×C8—611no—
590C2×C8—611no—
591C2×C8—612no—
592C2×C8—1222no—
593C2×C8—1222no—
594C2×C8—2444no—
595C2×C8C23×D6114yes—
596C4⋊C4—24no—
597C4⋊C4—24no—
598C4⋊C4—22no—
599C4⋊C4—22no—
600C4⋊C4—614no—
601C4⋊C4—612no—
602C4⋊C4—1224no—
603C4⋊C4—1224no—
604C4⋊C4—1224no—
605C4⋊C4—1224no—
606C4⋊C4—1224no—
607C4⋊C4—1224no—
608C4⋊C4—1222no—
609C4⋊C4C23×D6112yes—
610C4⋊C4C23×D6114yes—
611C22⋊C4—21no—
612C22⋊C4—22no—
613C22⋊C4—614no—
614C22⋊C4—614no—
615C22⋊C4—1224no—
616C22⋊C4—1224no—
617C22⋊C4—1224no—
618C22⋊C4—1224no—
619C22⋊C4—1224no—
620C22⋊C4—1221no—
621C22⋊C4—1222no—
622C42—21no—
623C42—22no—
624C42—1224no—
625C42—1221no—
626C42—1222no—
627C42D4×D6111yescharacteristic
628C42D4×D6111yescharacteristic
629C42C23×D6112yes—
630C23—21no—
631C23—614no—
632C23—1221no—
633C23C2×D4×D6111yescharacteristic
634Q8—21no—
635Q8—1224no—
636Q8—1221no—
637Q8—1224no—
638Q8—1224no—
639Q8—1221no—
640Q8—2448no—
641Q8—2444no—
642Q8C2×#1175822612yes—
643D4—24no—
644D4—21no—
645D4—1228no—
646D4—1221no—
647D4—1221no—
648D4—1224no—
649D4—4884no—
650D4—2444no—
651D4—2448no—
652D4—2442no—
653D4C2×#1175822612yes—
654C2×C4—21no—
655C2×C4—24no—
656C2×C4—22no—
657C2×C4—24no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—82no—
661C2×C4—614no—
662C2×C4—1224no—
663C2×C4—1224no—
664C2×C4—1222no—
665C2×C4—1221no—
666C2×C4—1224no—
667C2×C4—4882no—
668C2×C4—4884no—
669C2×C4—2448no—
670C2×C4—2448no—
671C2×C4—2444no—
672C2×C4—2442no—
673C2×C4C2×D4×D6112yes—
674C2×C4C2×D4×D6112yes—
675C2×C4C2×D4×D6111yescharacteristic
676C2×C4C2×#1175822612yes—
677C2×C4#1451982312yes—
678C2×C4C24×D6111yescharacteristic
679C8—24no—
680C8—1221no—
681C8—1221no—
682C8—1222no—
683C22—22no—
684C22—22no—
685C22—42no—
686C22—1224no—
687C22—1224no—
688C22—1224no—
689C22—2441no—
690C22—2442no—
691C22#1684040312yes—
692C22C22×D4×D6111yescharacteristic
693C4—22no—
694C4—22no—
695C4—22no—
696C4—42no—
697C4—44no—
698C4—1224no—
699C4—2444no—
700C4—2441no—
701C4—2442no—
702C4#1684040312yes—
703C4C22×#1175822611yescharacteristic
704C4C2×#1451982311yescharacteristic
705C2—22no—
706C2—42no—
707C2—4881no—
708C2—2444no—
709C2C2×#1684040311yescharacteristic
710C2#2146060312yes—
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3080C2
socleC22×C61C23×D4
hypercenter#3080D61
hypocenterC61#61224
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3080

Complements

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

NH ≅ G/Ntype
1. #21459743C2semidirect
2. #21459604C2semidirect
3. #21457987C2semidirect
4. #21457912C2semidirect
5. #21457867C2semidirect
6. #21457792C2semidirect
7. C2×#16840132C2semidirect
8. #21455790C2semidirect
9. #21455511C2semidirect
10. #21455307C2semidirect
11. #21455136C2semidirect
12. #21454617C2semidirect
13. #21454536C2semidirect
14. #21454350C2semidirect
15. C61×#3080C2semidirect
358. #3080D61semidirect
434. C61#61224semidirect

Maximal subgroups

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

subgroupindex
#214597432
#214596042
#214579872
#214579122
#214578672
#214577922
C2×#168401322
#214557902
#214555112
#214553072
#214551362
#214546172
#214545362
#214543502
C61×#30802
#6122461