Finite Group Databasebeta

← #30570411

Subgroups — #30570411

Order 15616.

1704 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21460219C211yesmaximal, characteristic
2#21458758C214yesmaximal, complemented
3#21458725C212yesmaximal, complemented
4#21458690C212yesmaximal, complemented
5#21458548C212yesmaximal, complemented
6#21458548C214yesmaximal, complemented
7#21458490C212yesmaximal, complemented
8#21457072C214yesmaximal, complemented
9#21457017C211yesmaximal, characteristic, complemented
10#21456955C211yesmaximal, characteristic, complemented
11#21456110C214yesmaximal, complemented
12#21455941C211yesmaximal, characteristic, complemented
13#21455850C211yesmaximal, characteristic, complemented
14#21454914C211yesmaximal, characteristic, complemented
15C61×#3502C211yesmaximal, characteristic, complemented
16#16840510—22no—
17#16840472C2212yes—
18#16840472C2212yes—
19#16840434C2212yes—
20#16840434C2212yes—
21#16840415—21no—
22#16840415—21no—
23#16840367C2214yes—
24#16840342—22no—
25#16840342—22no—
26#16840312C2211yescharacteristic
27#16840312C2211yescharacteristic
28#16840172C2212yes—
29#16840168C2214yes—
30#16840165C2211yescharacteristic
31#16840161C2214yes—
32#16840158C2211yescharacteristic
33#16840128C2212yes—
34#16840128C2214yes—
35#16840110C2211yescharacteristic
36#16840110C2214yes—
37#16840110C2211yescharacteristic
38#16840086C2214yes—
39#16840086C2214yes—
40#16840080C2212yes—
41#16840075C2212yes—
42#16840069C2214yes—
43#16840069C2214yes—
44#16840059C2214yes—
45#16840059C2214yes—
46#16840059C2214yes—
47#16840057C2212yes—
48#16840049C2212yes—
49#16840049C2214yes—
50#16840047C2212yes—
51#16840039C2212yes—
52#16840039C2214yes—
53#16839962C2212yes—
54#16839962C2214yes—
55#16839962C2212yes—
56#16839962C2214yes—
57#16839901C2212yes—
58#16839901C2212yes—
59#16839893C2212yes—
60#16839893C2214yes—
61Q8×#92418C2212yes—
62#16839860C2214yes—
63#16839860C2214yes—
64#16839860C2214yes—
65C4×#11758201C2212yes—
66C4×#11758201C2214yes—
67C4×#11758201C2212yes—
68C4×#11758201C2214yes—
69#16839823C2214yes—
70#16839819C2211yescharacteristic
71#16839815C2211yescharacteristic
72Q82×C61—21no—
73Q82×C61C2212yes—
74C61×#557—21no—
75C61×#557—22no—
76C61×#557C2212yes—
77C61×#557C2214yes—
78C61×#544—22no—
79C61×#544C2212yes—
80C61×#533—22no—
81C61×#533C2211yescharacteristic
82C61×#533C2211yescharacteristic
83C61×#519C2212yes—
84C2×C4×Q8×C61C2211yescharacteristic
85C2×#11758226—22no—
86C2×#11758226—22no—
87C2×#11758222—21no—
88C2×#11758222—21no—
89C2×#11758201C2312yes—
90C2×#11758201C2314yes—
91C2×#11758201C2312yes—
92#14519777C2311yescharacteristic
93#14519777C2311yescharacteristic
94#14519774C2311yescharacteristic
95#14519774C2311yescharacteristic
96#14519774C2314yes—
97#14519769C2314yes—
98#14519765—21no—
99#14519765—21no—
100#14519762—24no—
101#14519762—24no—
102#14519755—24no—
103#14519755—24no—
104#14519753—22no—
105#14519753—22no—
106#14519752—24no—
107#14519752—24no—
108#14519752—22no—
109#14519752—22no—
110#14519751—21no—
111#14519751—21no—
112#14519751C2312yes—
113#14519751C2312yes—
114#14519750—24no—
115#14519750—24no—
116C4×D244—21no—
117C4×D244—21no—
118C4×D244C2312yes—
119C4×D244C2314yes—
120C4×D244C2314yes—
121C4×D244C2312yes—
122C4⋊C4×D61—22no—
123C4⋊C4×D61—22no—
124#14519710C2314yes—
125#14519710C2314yes—
126#14519710C2314yes—
127#14519708C2312yes—
128#14519708C2312yes—
129#14519708C2314yes—
130#14519708C2314yes—
131C4⋊#92418C2312yes—
132C4⋊#92418C2314yes—
133C4⋊#92418C2312yes—
134C4⋊#92418C2314yes—
135#14519701C2312yes—
136#14519701C2312yes—
137#14519701C2314yes—
138#14519697C2312yes—
139#14519697C2314yes—
140#14519697C2314yes—
141#14519697C2314yes—
142#14519697C2314yes—
143#14519697C2312yes—
144#14519697C2314yes—
145C22⋊#92418C2312yes—
146C22⋊#92418C2312yes—
147#14519688C2312yes—
148#14519688C2314yes—
149C4×#92418C2311yescharacteristic
150C4×#92418C2311yescharacteristic
151C4×#92418C2314yes—
152C22×Q8×C61C2311yescharacteristic
153C4⋊Q8×C61—22no—
154C4⋊Q8×C61—24no—
155C4⋊Q8×C61—22no—
156C4⋊Q8×C61—24no—
157C4⋊Q8×C61—22no—
158C4⋊Q8×C61D411yescharacteristic
159C4⋊Q8×C61D411yescharacteristic
160C4⋊Q8×C61C2311yescharacteristic
161C4⋊Q8×C61C2311yescharacteristic
162C4⋊Q8×C61C2314yes—
163#125×C61—24no—
164#125×C61—24no—
165#125×C61—24no—
166#125×C61—22no—
167#125×C61D412yes—
168#125×C61C2312yes—
169#125×C61C2314yes—
170C22⋊Q8×C61—21no—
171C22⋊Q8×C61—21no—
172C22⋊Q8×C61—24no—
173C22⋊Q8×C61D412yes—
174C22⋊Q8×C61D412yes—
175C4×Q8×C61—22no—
176C4×Q8×C61—24no—
177C4×Q8×C61—22no—
178C4×Q8×C61—22no—
179C4×Q8×C61—22no—
180C4×Q8×C61—21no—
181C4×Q8×C61—21no—
182C4×Q8×C61—21no—
183C4×Q8×C61—21no—
184C4×Q8×C61C2312yes—
185C4×Q8×C61C2314yes—
186C4×Q8×C61C2312yes—
187C4×Q8×C61C2314yes—
188C4×Q8×C61C2312yes—
189C4×Q8×C61C2312yes—
190#117×C61—24no—
191#117×C61C2312yes—
192#117×C61C2312yes—
193C2×C4⋊C4×C61—21no—
194C2×C4⋊C4×C61C2311yescharacteristic
195C2×C42×C61—21no—
196C2×C42×C61C2311yescharacteristic
197C2×Q8×C61—24no—
198C2×Q8×C61—21no—
199C2×Q8×C61—21no—
200C2×Q8×C61—21no—
201C2×Q8×C61—24no—
202C2×Q8×C61C2×D412yes—
203C2×Q8×C61C2×D412yes—
204C2×Q8×C61C2412yes—
205C22×C4×C61—21no—
206C22×C4×C61—21no—
207C22×C4×C61C2×D411yescharacteristic
208C22×C4×C61C2×D411yescharacteristic
209C22×C4×C61C2411yescharacteristic
210C2×C22⋊D61—22no—
211C2×C22⋊D61—22no—
212#11758226—84no—
213#11758226—84no—
214#11758222—82no—
215#11758222—82no—
216C2×D244—21no—
217C2×D244—21no—
218C2×D244—21no—
219C2×D244—21no—
220C2×D244—24no—
221C2×D244—24no—
222C2×D244C2412yes—
223C2×D244C2412yes—
224C2×C4×D61—22no—
225C2×C4×D61—22no—
226C2×C4×D61—24no—
227C2×C4×D61—24no—
228C2×Dic122—21no—
229C2×Dic122—21no—
230C4⋊C4×C61—22no—
231C4⋊C4×C61—22no—
232C4⋊C4×C61—21no—
233C4⋊C4×C61—24no—
234C4⋊C4×C61—24no—
235C4⋊C4×C61—22no—
236C4⋊C4×C61—21no—
237C4⋊C4×C61—21no—
238C4⋊C4×C61—24no—
239C4⋊C4×C61—24no—
240C4⋊C4×C61—22no—
241C4⋊C4×C61—24no—
242C4⋊C4×C61C2×D412yes—
243C4⋊C4×C61C2×D412yes—
244C4⋊C4×C61C2×D414yes—
245C4⋊C4×C61C2×D412yes—
246C4⋊C4×C61C2×D412yes—
247C4⋊C4×C61C2×D414yes—
248C4⋊C4×C61C2412yes—
249C4⋊C4×C61C2412yes—
250C4⋊C4×C61C2412yes—
251C22⋊C4×C61—22no—
252C22⋊C4×C61C2×D414yes—
253C42×C61—22no—
254C42×C61—22no—
255C42×C61—24no—
256C42×C61—24no—
257C42×C61—22no—
258C42×C61C2×D411yescharacteristic
259C42×C61C2×D411yescharacteristic
260C42×C61C2411yescharacteristic
261C42×C61C2411yescharacteristic
262C42×C61C2414yes—
263#11758201—24no—
264#11758201—28no—
265#11758201—24no—
266#11758198—22no—
267#11758198—22no—
268#11758198—24no—
269#11758198—24no—
270#11758198—22no—
271#11758198—22no—
272C4⋊Dic61—21no—
273C4⋊Dic61—21no—
274C4⋊Dic61C2412yes—
275C4⋊Dic61C2412yes—
276#11758196—24no—
277#11758196—24no—
278C2×#92418C2412yes—
279C2×#92418C2412yes—
280C2×#92418C2414yes—
281C23×C61C22×D411yescharacteristic
282C22×D61—22no—
283C22×D61—22no—
284Q8×C61—24no—
285Q8×C61—24no—
286Q8×C61—22no—
287Q8×C61—42no—
288Q8×C61—42no—
289Q8×C61—81no—
290Q8×C61—84no—
291Q8×C612−1+414yes—
292C2×C4×C61—24no—
293C2×C4×C61—22no—
294C2×C4×C61—22no—
295C2×C4×C61—22no—
296C2×C4×C61—22no—
297C2×C4×C61—22no—
298C2×C4×C61—22no—
299C2×C4×C61—41no—
300C2×C4×C61—81no—
301C2×C4×C61C22×D414yes—
302C2×C4×C61C22×D414yes—
303C2×C4×C61C22×D411yescharacteristic
304C2×C4×C61C22×D411yescharacteristic
305C2×C4×C61C22×D411yescharacteristic
306C2×C4×C61C22×D411yescharacteristic
307C2×C4×C61C22×D411yescharacteristic
308C2×C4×C61C2511yescharacteristic
309C22⋊D61—82no—
310C22⋊D61—82no—
311C2×Dic61—22no—
312C2×Dic61—22no—
313D244—22no—
314D244—22no—
315D244—24no—
316D244—42no—
317D244—42no—
318D244—81no—
319D244—81no—
320D244—84no—
321D244—84no—
322C4×D61—44no—
323C4×D61—44no—
324C4×D61—84no—
325C4×D61—84no—
326Dic122—81no—
327Dic122—81no—
328#92418—22no—
329#92418—22no—
330#92418—24no—
331#61851—611nomaximal, Sylow
332C22×C61—21no—
333C22×C61—21no—
334C22×C61—21no—
335C22×C61C23×D411yescharacteristic
336C2×D61—42no—
337C2×D61—42no—
338C2×D61—42no—
339C2×D61—42no—
340C2×D61—42no—
341C2×D61—42no—
342C4×C61—21no—
343C4×C61—21no—
344C4×C61—24no—
345C4×C61—24no—
346C4×C61—21no—
347C4×C61—21no—
348C4×C61—24no—
349C4×C61—42no—
350C4×C61—42no—
351C4×C61—42no—
352C4×C61#57712yes—
353C4×C61C2×2−1+412yes—
354Dic61—42no—
355Dic61—42no—
356#3548—611no—
357#3502D6111yescharacteristic, complemented
358#3388—614no—
359#3385—612no—
360#3381—612no—
361#3368—612no—
362#3368—614no—
363#3362—612no—
364#3241—614no—
365#3235—611no—
366#3228—611no—
367#3126—614no—
368#3112—611no—
369#3103—611no—
370#2985—611no—
371C2×C61—41no—
372C2×C61#348611yescharacteristic
373C2×C61C2×#57711yescharacteristic
374C2×C61#358511yescharacteristic
375D61—82no—
376D61—82no—
377#562—1222no—
378Q82—21no—
379Q82C2×D6112yes—
380#557—21no—
381#557—22no—
382#557C2×D6112yes—
383#557C2×D6114yes—
384#554—612no—
385#554—612no—
386#550—612no—
387#550—612no—
388#547—1221no—
389#547—1221no—
390#544—22no—
391#544C2×D6112yes—
392#540—614no—
393#537—1222no—
394#537—1222no—
395#533—22no—
396#533C2×D6111yescharacteristic
397#533C2×D6111yescharacteristic
398#532—611no—
399#532—611no—
400#519C2×D6112yes—
401C2×C4×Q8C2×D6111yescharacteristic
402#501—614no—
403#501—612no—
404#500—611no—
405#499—614no—
406#498—611no—
407#490—612no—
408#490—614no—
409#486—611no—
410#486—614no—
411#486—611no—
412#482—614no—
413#482—614no—
414#481—612no—
415#480—612no—
416#479—614no—
417#479—614no—
418#478—614no—
419#478—614no—
420#478—614no—
421#477—612no—
422#476—612no—
423#476—614no—
424#475—612no—
425#474—612no—
426#474—614no—
427#460—612no—
428#460—612no—
429#460—614no—
430#460—614no—
431#448—612no—
432#448—612no—
433#446—612no—
434#446—614no—
435C8×Q8—612no—
436#440—614no—
437#440—614no—
438#440—614no—
439C4×SD16—612no—
440C4×SD16—612no—
441C4×SD16—614no—
442C4×SD16—614no—
443#433—614no—
444#432—611no—
445#431—611no—
446C61#6185111yescharacteristic, complemented, Sylow
447C2×#41—1222no—
448C2×#41—1222no—
449C2×#41—1221no—
450C2×#41—1221no—
451C22×Q8C22×D6111yescharacteristic
452C2×SD16—612no—
453C2×SD16—612no—
454C2×SD16—614no—
455C4⋊Q8—22no—
456C4⋊Q8—24no—
457C4⋊Q8—22no—
458C4⋊Q8—24no—
459C4⋊Q8—22no—
460C4⋊Q8—611no—
461C4⋊Q8—611no—
462C4⋊Q8C22⋊D6111yescharacteristic
463C4⋊Q8C22⋊D6111yescharacteristic
464C4⋊Q8C22×D6111yescharacteristic
465C4⋊Q8C22×D6111yescharacteristic
466C4⋊Q8C22×D6114yes—
467#127—611no—
468#127—611no—
469#127—614no—
470#125—24no—
471#125—24no—
472#125—24no—
473#125—22no—
474#125—614no—
475#125C22⋊D6112yes—
476#125C22×D6112yes—
477#125C22×D6114yes—
478#124—1221no—
479#124—1221no—
480#123—1224no—
481#123—1224no—
482C22⋊Q8—21no—
483C22⋊Q8—21no—
484C22⋊Q8—24no—
485C22⋊Q8—1222no—
486C22⋊Q8—1222no—
487C22⋊Q8—1224no—
488C22⋊Q8—1224no—
489C22⋊Q8C22⋊D6112yes—
490C22⋊Q8C22⋊D6112yes—
491#121—612no—
492#121—612no—
493#121—1221no—
494#121—1221no—
495#121—1222no—
496#121—1222no—
497#121—1224no—
498#121—1224no—
499#121—1224no—
500#121—1224no—
501C4×Q8—22no—
502C4×Q8—22no—
503C4×Q8—22no—
504C4×Q8—24no—
505C4×Q8—22no—
506C4×Q8—21no—
507C4×Q8—21no—
508C4×Q8—21no—
509C4×Q8—21no—
510C4×Q8C22×D6112yes—
511C4×Q8C22×D6112yes—
512C4×Q8C22×D6114yes—
513C4×Q8C22×D6114yes—
514C4×Q8C22×D6112yes—
515C4×Q8C22×D6112yes—
516C4×D4—612no—
517C4×D4—612no—
518C4×D4—614no—
519C4×D4—614no—
520C4×D4—1221no—
521C4×D4—1221no—
522#117—24no—
523#117C22×D6112yes—
524#117C22×D6112yes—
525C2×C4⋊C4—21no—
526C2×C4⋊C4—1222no—
527C2×C4⋊C4—1222no—
528C2×C4⋊C4C22×D6111yescharacteristic
529C2×C42—21no—
530C2×C42C22×D6111yescharacteristic
531#107—614no—
532#107—614no—
533#107—614no—
534#106—612no—
535#106—612no—
536#106—614no—
537#106—614no—
538C4⋊C8—612no—
539C4⋊C8—612no—
540C4⋊C8—614no—
541C4⋊C8—614no—
542#103—612no—
543#103—612no—
544#103—614no—
545#102—612no—
546#102—614no—
547#102—614no—
548#102—612no—
549#102—614no—
550#102—614no—
551#102—614no—
552C22⋊C8—612no—
553C22⋊C8—612no—
554#97—612no—
555#97—614no—
556C4×C8—611no—
557C4×C8—611no—
558C4×C8—614no—
559#41—4882no—
560#41—4882no—
561#41—4884no—
562#41—4884no—
563C2×Q8—24no—
564C2×Q8—21no—
565C2×Q8—21no—
566C2×Q8—21no—
567C2×Q8—24no—
568C2×Q8—1221no—
569C2×Q8—1221no—
570C2×Q8C2×C22⋊D6112yes—
571C2×Q8C2×C22⋊D6112yes—
572C2×Q8C23×D6112yes—
573C2×D4—612no—
574C2×D4—612no—
575C2×D4—1222no—
576C2×D4—1222no—
577C2×D4—1224no—
578C2×D4—1224no—
579C2×D4—1221no—
580C2×D4—1221no—
581C2×D4—1221no—
582C2×D4—1221no—
583C22×C4—21no—
584C22×C4—21no—
585C22×C4—1224no—
586C22×C4—1224no—
587C22×C4—1222no—
588C22×C4—1222no—
589C22×C4C2×C22⋊D6111yescharacteristic
590C22×C4C2×C22⋊D6111yescharacteristic
591C22×C4C23×D6111yescharacteristic
592SD16—1224no—
593SD16—1224no—
594SD16—1228no—
595C2×C8—612no—
596C2×C8—612no—
597C2×C8—614no—
598C4⋊C4—22no—
599C4⋊C4—22no—
600C4⋊C4—21no—
601C4⋊C4—24no—
602C4⋊C4—24no—
603C4⋊C4—22no—
604C4⋊C4—21no—
605C4⋊C4—21no—
606C4⋊C4—24no—
607C4⋊C4—24no—
608C4⋊C4—22no—
609C4⋊C4—24no—
610C4⋊C4—612no—
611C4⋊C4—612no—
612C4⋊C4—1221no—
613C4⋊C4—1221no—
614C4⋊C4—1224no—
615C4⋊C4—1224no—
616C4⋊C4C2×C22⋊D6112yes—
617C4⋊C4C2×C22⋊D6112yes—
618C4⋊C4C2×C22⋊D6112yes—
619C4⋊C4C2×C22⋊D6112yes—
620C4⋊C4C2×C22⋊D6114yes—
621C4⋊C4C2×C22⋊D6114yes—
622C4⋊C4C23×D6112yes—
623C4⋊C4C23×D6112yes—
624C4⋊C4C23×D6112yes—
625C22⋊C4—22no—
626C22⋊C4—1224no—
627C22⋊C4—1224no—
628C22⋊C4—1222no—
629C22⋊C4—1222no—
630C22⋊C4—1222no—
631C22⋊C4—1222no—
632C22⋊C4C2×C22⋊D6114yes—
633C42—22no—
634C42—22no—
635C42—24no—
636C42—24no—
637C42—22no—
638C42C2×C22⋊D6111yescharacteristic
639C42C2×C22⋊D6111yescharacteristic
640C42C23×D6111yescharacteristic
641C42C23×D6114yes—
642C42C23×D6111yescharacteristic
643C23—1222no—
644C23—1222no—
645C23C22×C22⋊D6111yescharacteristic
646Q8—24no—
647Q8—24no—
648Q8—22no—
649Q8—42no—
650Q8—42no—
651Q8—81no—
652Q8—84no—
653Q8—4881no—
654Q8—4881no—
655Q8#1451982514yes—
656D4—1222no—
657D4—1222no—
658D4—1224no—
659D4—4881no—
660D4—4884no—
661D4—4881no—
662D4—4884no—
663D4—4882no—
664D4—4882no—
665D4—2442no—
666D4—2442no—
667C2×C4—24no—
668C2×C4—22no—
669C2×C4—22no—
670C2×C4—22no—
671C2×C4—22no—
672C2×C4—22no—
673C2×C4—22no—
674C2×C4—41no—
675C2×C4—81no—
676C2×C4—1222no—
677C2×C4—1222no—
678C2×C4—4884no—
679C2×C4—4884no—
680C2×C4—2444no—
681C2×C4—2444no—
682C2×C4C22×C22⋊D6111yescharacteristic
683C2×C4C22×C22⋊D6111yescharacteristic
684C2×C4C22×C22⋊D6114yes—
685C2×C4C22×C22⋊D6114yes—
686C2×C4C22×C22⋊D6111yescharacteristic
687C2×C4C22×C22⋊D6111yescharacteristic
688C2×C4C22×C22⋊D6111yescharacteristic
689C2×C4C24×D6111yescharacteristic
690C8—1222no—
691C8—1222no—
692C8—1224no—
693C22—21no—
694C22—21no—
695C22—21no—
696C22—2442no—
697C22—2442no—
698C22—2442no—
699C22—2442no—
700C22—2442no—
701C22—2442no—
702C22C23×C22⋊D6111yescharacteristic
703C4—24no—
704C4—21no—
705C4—21no—
706C4—24no—
707C4—21no—
708C4—21no—
709C4—24no—
710C4—42no—
711C4—42no—
712C4—42no—
713C4—2442no—
714C4—2442no—
715C4#1684057012yes—
716C4C2×#1451982512yes—
717C2—41no—
718C2—4882no—
719C2—4882no—
720C2#2145947011yescharacteristic
721C2C2×#1684057011yescharacteristic
722C2#2146059811yescharacteristic
subgroupG/H
centerC22C23×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3502C2
socleC22×C61C23×D4
prime radical#21460219C2
hypercenter#3502D61
hypocenterC61#61851
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ #533 ◁ #3502

Complements

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

NH ≅ G/Ntype
2. #21458758C2semidirect
3. #21458725C2semidirect
4. #21458690C2semidirect
5. #21458548C2semidirect
6. #21458548C2semidirect
7. #21458490C2semidirect
8. #21457072C2semidirect
9. #21457017C2semidirect
10. #21456955C2semidirect
11. #21456110C2semidirect
12. #21455941C2semidirect
13. #21455850C2semidirect
14. #21454914C2semidirect
15. C61×#3502C2semidirect
357. #3502D61semidirect
446. C61#61851semidirect

Maximal subgroups

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

subgroupindex
#214602192
#214587582
#214587252
#214586902
#214585482
#214584902
#214570722
#214570172
#214569552
#214561102
#214559412
#214558502
#214549142
C61×#35022
#6185161