Finite Group Databasebeta

← #30570452

Subgroups — #30570452

Order 15616.

1648 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21460230C211yesmaximal, characteristic
2#21459629C211yesmaximal, characteristic, complemented
3#21458749C212yesmaximal, complemented
4#21458743C214yesmaximal, complemented
5Q16×Dic122C212yesmaximal, complemented
6#21458654C214yesmaximal, complemented
7#21458631C212yesmaximal, complemented
8#21458513C212yesmaximal, complemented
9#21457106C214yesmaximal, complemented
10#21457029C211yesmaximal, characteristic, complemented
11#21456957C211yesmaximal, characteristic
12#21456049C214yesmaximal, complemented
13#21455941C211yesmaximal, characteristic
14#21454913C211yesmaximal, characteristic, complemented
15C61×#3104C211yesmaximal, characteristic, complemented
16#16840516—22no—
17#16840497C2212yes—
18Q8×Dic122—21no—
19Q8×Dic122C2212yes—
20Q8×Dic122C2212yes—
21#16840491—21no—
22#16840491C2212yes—
23#16840488—22no—
24#16840488C2214yes—
25#16840486C2212yes—
26#16840485—21no—
27Q8×D244—21no—
28#16840427C2212yes—
29#16840393—22no—
30#16840393C2212yes—
31#16840385C2214yes—
32#16840383—22no—
33#16840335—22no—
34#16840319—22no—
35#16840319C2211yescharacteristic
36#16840319C2211yescharacteristic
37#16840312C2211yescharacteristic
38#16840240C2212yes—
39#16840170C2214yes—
40#16840168C2212yes—
41#16840167C2211yescharacteristic
42#16840166C2214yes—
43#16840160C2211yescharacteristic
44#16840130C2212yes—
45#16840123C2214yes—
46#16840118C2214yes—
47#16840109C2211yescharacteristic
48#16840109C2211yescharacteristic
49#16840100C2214yes—
50#16840087C2214yes—
51#16840075C2212yes—
52#16840067C2214yes—
53#16840065C2214yes—
54#16840063C2214yes—
55#16840057C2212yes—
56#16840057C2212yes—
57#16840055C2214yes—
58#16840055C2214yes—
59#16840047C2212yes—
60#16840045C2214yes—
61#16840041C2214yes—
62#16839986C2214yes—
63#16839986C2214yes—
64#16839962C2212yes—
65#16839901C2212yes—
66#16839892C2212yes—
67#16839891C2214yes—
68C8×Dic122C2212yes—
69#16839869C2214yes—
70#16839865C2214yes—
71#16839860C2214yes—
72Q16×Dic61C2214yes—
73#16839847C2214yes—
74#16839820C2214yes—
75#16839819C2211yescharacteristic
76C61×#533C2211yescharacteristic
77C2×C4×Q8×C61C2211yescharacteristic
78C61×#484C2212yes—
79C61×#477C2212yes—
80C61×#462C2212yes—
81C61×#451C2212yes—
82C4×Q16×C61C2212yes—
83C4×Q16×C61C2212yes—
84C61×#431C2211yescharacteristic
85C2×#11758222—21no—
86C2×Q8×D61—22no—
87C2×#11758201C2314yes—
88#14519777—22no—
89#14519777—22no—
90#14519777—22no—
91#14519777D411yescharacteristic
92#14519777D411yescharacteristic
93#14519777C2311yescharacteristic
94#14519777C2311yescharacteristic
95#14519777C2311yescharacteristic
96#14519776—24no—
97#14519776—22no—
98#14519776—24no—
99#14519776—24no—
100#14519776C2314yes—
101#14519776C2314yes—
102#14519774C2311yescharacteristic
103#14519772—24no—
104#14519772—24no—
105#14519769—24no—
106#14519769D412yes—
107#14519769C2312yes—
108#14519768—22no—
109#14519768—24no—
110#14519768—24no—
111#14519768C2314yes—
112#14519766C2314yes—
113#14519765—22no—
114#14519765—21no—
115#14519763—24no—
116#14519762—24no—
117#14519759—24no—
118#14519757—22no—
119#14519757—24no—
120#14519756—21no—
121#14519756—21no—
122#14519756—24no—
123#14519756D412yes—
124#14519756D412yes—
125#14519755—24no—
126#14519753—22no—
127#14519751—21no—
128#14519751C2312yes—
129Q8×Dic61—22no—
130Q8×Dic61—22no—
131Q8×Dic61C2314yes—
132Q8×Dic61C2314yes—
133Q8×Dic61C2314yes—
134C4×Dic122—22no—
135C4×Dic122—22no—
136C4×Dic122—21no—
137C4×Dic122—21no—
138C4×Dic122—22no—
139C4×Dic122—24no—
140C4×Dic122C2312yes—
141C4×Dic122C2312yes—
142C4×D244—21no—
143C4×D244—22no—
144C4×D244C2312yes—
145#14519739C2314yes—
146#14519736—24no—
147#14519736C2312yes—
148#14519736C2312yes—
149#14519735—22no—
150#14519711C2314yes—
151#14519710C2314yes—
152#14519710C2314yes—
153#14519709C2314yes—
154#14519708C2314yes—
155C4⋊#92418C2312yes—
156#14519706C2314yes—
157#14519706C2314yes—
158#14519701C2314yes—
159#14519701C2314yes—
160#14519701C2314yes—
161#14519699C2314yes—
162#14519699C2314yes—
163#14519697C2314yes—
164C22⋊#92418C2312yes—
165#14519688C2312yes—
166#14519687C2314yes—
167C8×Dic61C2314yes—
168C22×Q8×C61C2311yescharacteristic
169C2×Q16×C61C2312yes—
170C2×Q16×C61C2312yes—
171C4⋊Q8×C61C2311yescharacteristic
172C4⋊Q8×C61C2311yescharacteristic
173#125×C61—21no—
174C22⋊Q8×C61—21no—
175C22⋊Q8×C61C2312yes—
176C4×Q8×C61—21no—
177C4×Q8×C61—21no—
178C4×Q8×C61—21no—
179C4×Q8×C61C2312yes—
180C4×Q8×C61C2312yes—
181C4×Q8×C61C2312yes—
182#117×C61—21no—
183C2×C4⋊C4×C61—21no—
184C2×C4⋊C4×C61C2311yescharacteristic
185C2×C42×C61—21no—
186C2×C42×C61C2311yescharacteristic
187#107×C61C2312yes—
188#107×C61C2312yes—
189C4⋊C8×C61C2312yes—
190#103×C61C2312yes—
191#103×C61C2312yes—
192#103×C61C2312yes—
193#103×C61C2312yes—
194C22⋊C8×C61C2312yes—
195C4×C8×C61C2311yescharacteristic
196C4×C8×C61C2311yescharacteristic
197C2×Q8×C61—24no—
198C2×Q8×C61—21no—
199C2×Q8×C61—21no—
200C2×Q8×C61C2412yes—
201C2×Q8×C61C2412yes—
202C22×C4×C61—21no—
203C22×C4×C61—21no—
204C22×C4×C61C2×D411yescharacteristic
205C22×C4×C61C2×D411yescharacteristic
206C22×C4×C61C2411yescharacteristic
207C2×C22⋊D61—22no—
208Q8×D61—84no—
209#11758222—82no—
210C2×D244—21no—
211C2×D244—21no—
212C2×D244C2412yes—
213C2×C4×D61—22no—
214C2×C4×D61—24no—
215C2×Dic122—21no—
216C2×Dic122—21no—
217C2×Dic122—21no—
218C2×Dic122—24no—
219C2×Dic122—24no—
220C2×Dic122C2×D412yes—
221C2×Dic122C2×D412yes—
222Q16×C61—24no—
223Q16×C61—24no—
224C2×C8×C61C2412yes—
225C2×C8×C61C2412yes—
226C4⋊C4×C61—22no—
227C4⋊C4×C61—22no—
228C4⋊C4×C61—21no—
229C4⋊C4×C61—21no—
230C4⋊C4×C61—21no—
231C4⋊C4×C61—21no—
232C4⋊C4×C61—22no—
233C4⋊C4×C61—22no—
234C4⋊C4×C61C2412yes—
235C4⋊C4×C61C2412yes—
236C22⋊C4×C61—22no—
237C42×C61—22no—
238C42×C61—22no—
239C42×C61—22no—
240C42×C61C2×D411yescharacteristic
241C42×C61C2×D411yescharacteristic
242C42×C61C2411yescharacteristic
243C42×C61C2411yescharacteristic
244C22⋊Dic61—22no—
245C22⋊Dic61C2×D414yes—
246#11758201—28no—
247#11758198—22no—
248#11758198—22no—
249#11758198—24no—
250#11758198—24no—
251#11758198—24no—
252C4⋊Dic61—21no—
253C4⋊Dic61—21no—
254C4⋊Dic61—21no—
255C4⋊Dic61—24no—
256C4⋊Dic61—24no—
257C4⋊Dic61—24no—
258C4⋊Dic61—24no—
259C4⋊Dic61C2×D412yes—
260C4⋊Dic61C2×D412yes—
261C4⋊Dic61C2×D412yes—
262C4⋊Dic61C2×D412yes—
263C4⋊Dic61C2412yes—
264C4⋊Dic61C2412yes—
265C4⋊Dic61C2412yes—
266#11758196—22no—
267#11758196—22no—
268#11758196—24no—
269#11758196—24no—
270#11758196—24no—
271#11758196C2×D414yes—
272#11758196C2×D414yes—
273C4×Dic61—22no—
274C4×Dic61—24no—
275C4×Dic61—24no—
276C4×Dic61—24no—
277C4×Dic61C2414yes—
278C2×#92418C2414yes—
279C23×C61C22×D411yescharacteristic
280C22×D61—22no—
281Q8×C61—24no—
282Q8×C61—22no—
283Q8×C61—22no—
284Q8×C61—22no—
285Q8×C61—42no—
286Q8×C612−1+414yes—
287C2×C4×C61—24no—
288C2×C4×C61—22no—
289C2×C4×C61—22no—
290C2×C4×C61—22no—
291C2×C4×C61—22no—
292C2×C4×C61—22no—
293C2×C4×C61—22no—
294C2×C4×C61—41no—
295C2×C4×C61—81no—
296C2×C4×C61C22×D411yescharacteristic
297C2×C4×C61C22×D411yescharacteristic
298C2×C4×C61C22×D411yescharacteristic
299C2×C4×C61C22×D411yescharacteristic
300C2×C4×C61C22×D411yescharacteristic
301C2×C4×C61C2511yescharacteristic
302C22⋊D61—82no—
303C2×Dic61—22no—
304C2×Dic61—22no—
305C2×Dic61—22no—
306C2×Dic61C22×D414yes—
307C2×Dic61C22×D414yes—
308D244—24no—
309D244—42no—
310D244—81no—
311C4×D61—44no—
312C4×D61—84no—
313Dic122—24no—
314Dic122—42no—
315Dic122—42no—
316Dic122—84no—
317Dic122—84no—
318Dic122—81no—
319Dic122—81no—
320C8×C61—22no—
321C8×C61—22no—
322#92418—24no—
323#61854—611nomaximal, Sylow
324C22×C61—21no—
325C22×C61—21no—
326C22×C61—21no—
327C22×C61C23×D411yescharacteristic
328C2×D61—42no—
329C2×D61—42no—
330C2×D61—42no—
331C4×C61—21no—
332C4×C61—21no—
333C4×C61—24no—
334C4×C61—21no—
335C4×C61—21no—
336C4×C61—42no—
337C4×C61—42no—
338C4×C61—42no—
339C4×C61#57812yes—
340C4×C61C2×2−1+412yes—
341Dic61—24no—
342Dic61—24no—
343Dic61—42no—
344Dic61—42no—
345Dic61—42no—
346#3550—611no—
347#3502—611no—
348#3387—612no—
349#3386—614no—
350Q8×Q16—612no—
351#3377—614no—
352#3375—612no—
353#3365—612no—
354#3243—614no—
355#3237—611no—
356#3228—611no—
357#3121—614no—
358#3112—611no—
359#3104D6111yescharacteristic, complemented
360#2985—611no—
361C2×C61—41no—
362C2×C61#348611yescharacteristic
363C2×C61C2×#57811yescharacteristic
364C2×C61#358511yescharacteristic
365D61—82no—
366#563—1222no—
367#559—612no—
368Q82—612no—
369Q82—612no—
370Q82—1221no—
371#557—612no—
372#557—614no—
373#557—1222no—
374#557—1221no—
375#556—612no—
376#556—1221no—
377D4×Q8—612no—
378D4×Q8—1221no—
379#544—612no—
380#544—1222no—
381#542—614no—
382#542—1222no—
383#536—1222no—
384#533—611no—
385#533—611no—
386#533—1222no—
387#533C2×D6111yescharacteristic
388#532—611no—
389#519—612no—
390C2×C4×Q8C2×D6111yescharacteristic
391#501—612no—
392#501—614no—
393#500—614no—
394#500—611no—
395#498—611no—
396#491—612no—
397#489—614no—
398#488—614no—
399#486—611no—
400#486—611no—
401#485—614no—
402#484C2×D6112yes—
403#482—614no—
404#480—612no—
405#479—614no—
406#478—614no—
407#478—614no—
408#477—612no—
409#477—612no—
410#477—614no—
411#477—614no—
412#477C2×D6112yes—
413#475—612no—
414#475—614no—
415#474—614no—
416#464—614no—
417#464—614no—
418#462C2×D6112yes—
419#460—612no—
420#451C2×D6112yes—
421#448—612no—
422#446—614no—
423#446—612no—
424C8×Q8—612no—
425#441—614no—
426#440—614no—
427#440—614no—
428C4×Q16—614no—
429C4×Q16C2×D6112yes—
430C4×Q16C2×D6112yes—
431C4×SD16—614no—
432#433—614no—
433#432—611no—
434#431C2×D6111yescharacteristic
435C61#6185411yescharacteristic, complemented, Sylow
436C2×#41—1221no—
437C22×Q8—1222no—
438C22×Q8C22×D6111yescharacteristic
439C2×Q16C22×D6112yes—
440C2×Q16C22×D6112yes—
441C2×SD16—614no—
442C4⋊Q8—614no—
443C4⋊Q8—614no—
444C4⋊Q8—611no—
445C4⋊Q8—611no—
446C4⋊Q8—611no—
447C4⋊Q8—611no—
448C4⋊Q8—611no—
449C4⋊Q8—1222no—
450C4⋊Q8—1222no—
451C4⋊Q8—1222no—
452C4⋊Q8—1224no—
453C4⋊Q8—1222no—
454C4⋊Q8—1224no—
455C4⋊Q8—1224no—
456C4⋊Q8C22×D6111yescharacteristic
457C4⋊Q8C22×D6111yescharacteristic
458#127—611no—
459#126—1224no—
460#126—1224no—
461#125—21no—
462#125—614no—
463#125—612no—
464#125—612no—
465#125—1224no—
466#125—1222no—
467#125—1224no—
468#125—1224no—
469#124—614no—
470#124—1222no—
471#124—1224no—
472#124—1221no—
473#123—1224no—
474#123—1224no—
475C22⋊Q8—21no—
476C22⋊Q8—612no—
477C22⋊Q8—612no—
478C22⋊Q8—1224no—
479C22⋊Q8—1224no—
480C22⋊Q8—1224no—
481C22⋊Q8—1222no—
482C22⋊Q8—1222no—
483C22⋊Q8—1221no—
484C22⋊Q8—1221no—
485C22⋊Q8C22×D6112yes—
486#121—612no—
487#121—1221no—
488C4×Q8—21no—
489C4×Q8—21no—
490C4×Q8—21no—
491C4×Q8—612no—
492C4×Q8—612no—
493C4×Q8—614no—
494C4×Q8—614no—
495C4×Q8—614no—
496C4×Q8—1224no—
497C4×Q8—1222no—
498C4×Q8—1222no—
499C4×Q8—1222no—
500C4×Q8—1222no—
501C4×Q8—1222no—
502C4×Q8—1221no—
503C4×Q8—1221no—
504C4×Q8C22×D6112yes—
505C4×Q8C22×D6112yes—
506C4×Q8C22×D6112yes—
507C4×D4—614no—
508C4×D4—612no—
509C4×D4—1222no—
510C4×D4—1221no—
511#117—21no—
512#117—612no—
513#117—612no—
514#117—1224no—
515#117—1222no—
516C2×C4⋊C4—21no—
517C2×C4⋊C4C22×D6111yescharacteristic
518C2×C42—21no—
519C2×C42C22×D6111yescharacteristic
520#107—614no—
521#107—614no—
522#107—614no—
523#107C22×D6112yes—
524#107C22×D6112yes—
525#106—614no—
526#106—614no—
527C4⋊C8—614no—
528C4⋊C8—614no—
529C4⋊C8—612no—
530C4⋊C8C22×D6112yes—
531#103—614no—
532#103—614no—
533#103—614no—
534#103C22×D6112yes—
535#103C22×D6112yes—
536#103C22×D6112yes—
537#103C22×D6112yes—
538#102—614no—
539#102—614no—
540#102—614no—
541C22⋊C8—612no—
542C22⋊C8C22×D6112yes—
543#97—614no—
544#97—612no—
545C4×C8—614no—
546C4×C8C22×D6111yescharacteristic
547C4×C8C22×D6111yescharacteristic
548#41—4882no—
549C2×Q8—24no—
550C2×Q8—21no—
551C2×Q8—21no—
552C2×Q8—612no—
553C2×Q8—612no—
554C2×Q8—1224no—
555C2×Q8—1224no—
556C2×Q8—1221no—
557C2×Q8—1221no—
558C2×Q8—1221no—
559C2×Q8—4884no—
560C2×Q8C23×D6112yes—
561C2×Q8C23×D6112yes—
562C2×D4—612no—
563C2×D4—1222no—
564C2×D4—1221no—
565C2×D4—1221no—
566C22×C4—21no—
567C22×C4—21no—
568C22×C4—1224no—
569C22×C4—1222no—
570C22×C4D4×D6111yescharacteristic
571C22×C4D4×D6111yescharacteristic
572C22×C4C23×D6111yescharacteristic
573Q16—24no—
574Q16—24no—
575SD16—1228no—
576C2×C8—614no—
577C2×C8C23×D6112yes—
578C2×C8C23×D6112yes—
579C4⋊C4—22no—
580C4⋊C4—22no—
581C4⋊C4—21no—
582C4⋊C4—21no—
583C4⋊C4—21no—
584C4⋊C4—21no—
585C4⋊C4—22no—
586C4⋊C4—22no—
587C4⋊C4—614no—
588C4⋊C4—614no—
589C4⋊C4—612no—
590C4⋊C4—612no—
591C4⋊C4—612no—
592C4⋊C4—612no—
593C4⋊C4—612no—
594C4⋊C4—612no—
595C4⋊C4—612no—
596C4⋊C4—1224no—
597C4⋊C4—1224no—
598C4⋊C4—1224no—
599C4⋊C4—1224no—
600C4⋊C4—1224no—
601C4⋊C4—1224no—
602C4⋊C4—1224no—
603C4⋊C4—1222no—
604C4⋊C4—1222no—
605C4⋊C4—1221no—
606C4⋊C4—1221no—
607C4⋊C4—1221no—
608C4⋊C4C23×D6112yes—
609C4⋊C4C23×D6112yes—
610C22⋊C4—22no—
611C22⋊C4—614no—
612C22⋊C4—1224no—
613C22⋊C4—1224no—
614C22⋊C4—1224no—
615C22⋊C4—1222no—
616C22⋊C4—1222no—
617C22⋊C4—1222no—
618C42—22no—
619C42—22no—
620C42—22no—
621C42—614no—
622C42—1224no—
623C42—1224no—
624C42—1224no—
625C42—1222no—
626C42D4×D6111yescharacteristic
627C42D4×D6111yescharacteristic
628C42C23×D6111yescharacteristic
629C42C23×D6111yescharacteristic
630C23—1222no—
631C23C2×D4×D6111yescharacteristic
632Q8—24no—
633Q8—22no—
634Q8—22no—
635Q8—22no—
636Q8—42no—
637Q8—1224no—
638Q8—4881no—
639Q8—4881no—
640Q8—4884no—
641Q8—4884no—
642Q8—2442no—
643Q8—2442no—
644Q8#1451982414yes—
645D4—1224no—
646D4—4881no—
647D4—4882no—
648D4—2442no—
649C2×C4—24no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—41no—
657C2×C4—81no—
658C2×C4—614no—
659C2×C4—614no—
660C2×C4—1222no—
661C2×C4—1222no—
662C2×C4—1222no—
663C2×C4—4884no—
664C2×C4—2444no—
665C2×C4C2×D4×D6111yescharacteristic
666C2×C4C2×D4×D6111yescharacteristic
667C2×C4C2×D4×D6111yescharacteristic
668C2×C4C2×D4×D6111yescharacteristic
669C2×C4C2×D4×D6111yescharacteristic
670C2×C4C24×D6111yescharacteristic
671C8—22no—
672C8—22no—
673C8—1224no—
674C22—21no—
675C22—21no—
676C22—21no—
677C22—2442no—
678C22—2442no—
679C22—2442no—
680C22C22×D4×D6111yescharacteristic
681C4—24no—
682C4—21no—
683C4—21no—
684C4—21no—
685C4—21no—
686C4—42no—
687C4—42no—
688C4—42no—
689C4—1224no—
690C4—1224no—
691C4—2442no—
692C4—2442no—
693C4—2442no—
694C4#1684057512yes—
695C4C2×#1451982412yes—
696C2—41no—
697C2—4882no—
698C2#2145946611yescharacteristic
699C2C2×#1684057511yescharacteristic
700C2#2146059611yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3104C2
socleC22×C61C23×D4
prime radical#16840312C22
hypercenter#3104D61
hypocenterC61#61854
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3104

Complements

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

NH ≅ G/Ntype
2. #21459629C2semidirect
3. #21458749C2semidirect
4. #21458743C2semidirect
5. Q16×Dic122C2semidirect
6. #21458654C2semidirect
7. #21458631C2semidirect
8. #21458513C2semidirect
9. #21457106C2semidirect
10. #21457029C2semidirect
12. #21456049C2semidirect
14. #21454913C2semidirect
15. C61×#3104C2semidirect
359. #3104D61semidirect
435. C61#61854semidirect

Maximal subgroups

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

subgroupindex
#214602302
#214596292
#214587492
#214587432
Q16×Dic1222
#214586542
#214586312
#214585132
#214571062
#214570292
#214569572
#214560492
#214559412
#214549132
C61×#31042
#6185461