Finite Group Databasebeta

← #30570369

Subgroups — #30570369

Order 15616.

1660 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21460314C211yesmaximal, characteristic, complemented
2#21459622C211yesmaximal, characteristic
3#21458765C214yesmaximal, complemented
4#21458744C212yesmaximal, complemented
5#21458698C212yesmaximal, complemented
6#21458604C212yesmaximal, complemented
7#21458601C214yesmaximal, complemented
8#21458456C212yesmaximal, complemented
9#21457091C214yesmaximal, complemented
10#21457035C211yesmaximal, characteristic, complemented
11#21456974C211yesmaximal, characteristic, complemented
12#21456065C214yesmaximal
13#21455827C211yesmaximal, characteristic
14#21454916C211yesmaximal, characteristic, complemented
15C61×#3110C211yesmaximal, characteristic
16C61⋊#564—22no—
17#16840491—21no—
18#16840491C2212yescomplemented
19#16840490—21no—
20#16840490C2212yescomplemented
21#16840487C2212yescomplemented
22#16840487C2212yescomplemented
23#16840471—22no—
24#16840471C2214yes—
25#16840469—21no—
26#16840469C2212yes—
27D4×Dic122—21no—
28D4×Dic122C2212yes—
29#16840393C2212yes—
30#16840388—22no—
31#16840374—22no—
32#16840372—22no—
33#16840372C2214yes—
34#16840319C2211yescharacteristic
35#16840319C2211yescharacteristic
36#16840319C2211yescharacteristic
37C2×#14519756—22no—
38#16840237C2212yes—
39#16840171C2212yescomplemented
40#16840170C2214yescomplemented
41#16840165C2211yescharacteristic, complemented
42#16840163C2214yescomplemented
43#16840158C2211yescharacteristic, complemented
44#16840131C2212yescomplemented
45#16840129C2214yescomplemented
46#16840112C2211yescharacteristic, complemented
47#16840112C2211yescharacteristic, complemented
48#16840111C2214yescomplemented
49#16840104C2214yes—
50#16840100C2214yes—
51#16840092C2212yes—
52#16840071C2214yescomplemented
53#16840069C2214yescomplemented
54#16840068C2214yescomplemented
55#16840064C2214yes—
56#16840060C2214yes—
57#16840058C2212yescomplemented
58#16840056C2214yescomplemented
59#16840052C2212yes—
60#16840048C2212yescomplemented
61#16840045C2214yescomplemented
62#16839967C2212yes—
63#16839965C2214yes—
64#16839965C2214yes—
65#16839913C2212yes—
66#16839890C2214yescomplemented
67#16839889C2212yescomplemented
68#16839886C2212yescomplemented
69#16839863C2214yes—
70#16839859C2214yes—
71#16839853C2214yes—
72SD16×Dic61C2214yes—
73C4×#11758200C2212yes—
74C4×#11758200C2212yes—
75#16839820C2214yes—
76#16839815C2211yescharacteristic
77C61×#533C2211yescharacteristic
78C2×C4×D4×C61C2211yescharacteristic
79C61×#484C2212yes—
80C61×#474C2212yes—
81C61×#461C2212yes—
82C61×#450C2212yes—
83C61×#440C2214yes—
84C61×#432C2211yescharacteristic
85C22×Dic122—22no—
86C2×#11758200C2312yes—
87C2×#11758200C2312yes—
88#14519777—22no—
89#14519777D411yescharacteristic
90#14519777D411yescharacteristic
91#14519777C2311yescharacteristic
92#14519777C2311yescharacteristic
93#14519777C2311yescharacteristic
94#14519777C2311yescharacteristic
95#14519776—22no—
96#14519776C2314yes—
97#14519775—22no—
98#14519775C2314yes—
99#14519770—24no—
100#14519770—24no—
101#14519769—21no—
102#14519769—22no—
103#14519769D412yes—
104#14519769C2312yes—
105#14519768—22no—
106#14519768—22no—
107#14519768—24no—
108#14519768—24no—
109#14519768C2314yes—
110#14519768C2314yes—
111#14519767—24no—
112#14519767—24no—
113#14519767—24no—
114#14519767—24no—
115#14519756—21no—
116#14519756—21no—
117#14519756—21no—
118#14519756—22no—
119#14519756—24no—
120#14519756—22no—
121#14519756—24no—
122#14519756D412yes—
123#14519756D412yes—
124#14519756C2312yes—
125#14519754—24no—
126#14519754—24no—
127#14519754—24no—
128#14519754—24no—
129Q8×Dic61—22no—
130Q8×Dic61C2314yes—
131C4×Dic122—22no—
132C4×Dic122—21no—
133C4×Dic122—21no—
134C4×Dic122—21no—
135C4×Dic122—22no—
136C4×Dic122C2312yes—
137C4×Dic122C2312yes—
138C4×Dic122C2312yes—
139#14519742—22no—
140#14519742C2314yes—
141D4×Dic61—22no—
142D4×Dic61—22no—
143D4×Dic61C2314yes—
144D4×Dic61C2314yes—
145#14519736—21no—
146#14519736C2312yes—
147#14519736C2312yes—
148C2×C4⋊Dic61—22no—
149C2×#11758196—24no—
150C2×C22⋊Dic61—24no—
151#14519711C2314yes—
152#14519710C2314yes—
153#14519709C2314yes—
154#14519708C2312yes—
155#14519708C2312yes—
156#14519708C2314yes—
157C4⋊#92418C2312yes—
158#14519706C2314yes—
159#14519706C2314yes—
160#14519702C2314yes—
161#14519702C2314yes—
162#14519701C2314yes—
163#14519701C2314yes—
164#14519700C2312yes—
165#14519700C2312yes—
166#14519698C2312yes—
167#14519698C2312yes—
168C22⋊#92418C2312yes—
169#14519687C2314yes—
170C4×#92418C2311yescharacteristic
171C4×#92418C2311yescharacteristic
172C8×Dic61C2314yes—
173C22×D4×C61C2311yescharacteristic
174C23×C4×C61—21no—
175C2×SD16×C61C2314yes—
176C4⋊Q8×C61C2311yescharacteristic
177C4⋊Q8×C61C2311yescharacteristic
178#125×C61—21no—
179C22⋊Q8×C61—21no—
180C22⋊Q8×C61C2312yes—
181C4×Q8×C61—21no—
182C4×Q8×C61C2312yes—
183C4×D4×C61—21no—
184C4×D4×C61—21no—
185C4×D4×C61C2312yes—
186C4×D4×C61C2312yes—
187#117×C61—21no—
188C2×C4⋊C4×C61C2311yescharacteristic
189C2×C22⋊C4×C61—21no—
190C2×C42×C61C2311yescharacteristic
191#107×C61C2314yes—
192C4⋊C8×C61C2312yes—
193#103×C61C2314yes—
194#102×C61C2314yes—
195C22⋊C8×C61C2312yes—
196#97×C61C2312yes—
197C24×C61—21no—
198C2×Q8×C61—21no—
199C2×Q8×C61C2412yes—
200C2×D4×C61—24no—
201C2×D4×C61—21no—
202C2×D4×C61C2412yes—
203C22×C4×C61—21no—
204C22×C4×C61—22no—
205C22×C4×C61—22no—
206C22×C4×C61—82no—
207C22×C4×C61C2×D411yescharacteristic
208C22×C4×C61C2×D411yescharacteristic
209C22×C4×C61C2411yescharacteristic
210C22×Dic61—24no—
211C22×Dic61—24no—
212C2×Dic122—21no—
213C2×Dic122—21no—
214C2×Dic122—21no—
215C2×Dic122—24no—
216C2×Dic122—84no—
217C2×Dic122C2×D412yes—
218C2×Dic122C2×D412yes—
219C2×Dic122C2412yes—
220SD16×C61—28no—
221C2×C8×C61C2414yes—
222C4⋊C4×C61—21no—
223C4⋊C4×C61—21no—
224C4⋊C4×C61—21no—
225C4⋊C4×C61—21no—
226C4⋊C4×C61—22no—
227C4⋊C4×C61—22no—
228C4⋊C4×C61C2412yes—
229C4⋊C4×C61C2412yes—
230C22⋊C4×C61—22no—
231C22⋊C4×C61—22no—
232C22⋊C4×C61—22no—
233C42×C61—22no—
234C42×C61C2×D411yescharacteristic
235C42×C61C2×D411yescharacteristic
236C42×C61C2411yescharacteristic
237C42×C61C2411yescharacteristic
238C22⋊Dic61—22no—
239C22⋊Dic61—22no—
240C22⋊Dic61—24no—
241C22⋊Dic61—24no—
242C22⋊Dic61C2×D414yes—
243#11758200—24no—
244#11758200—24no—
245C4⋊Dic61—21no—
246C4⋊Dic61—21no—
247C4⋊Dic61—21no—
248C4⋊Dic61—21no—
249C4⋊Dic61—21no—
250C4⋊Dic61—24no—
251C4⋊Dic61—24no—
252C4⋊Dic61—24no—
253C4⋊Dic61C2×D412yes—
254C4⋊Dic61C2×D412yes—
255C4⋊Dic61C2×D412yes—
256C4⋊Dic61C2×D412yes—
257C4⋊Dic61C2412yes—
258C4⋊Dic61C2412yes—
259C4⋊Dic61C2412yes—
260#11758196—22no—
261#11758196—22no—
262#11758196—22no—
263#11758196—22no—
264#11758196—24no—
265#11758196—24no—
266#11758196—24no—
267#11758196—24no—
268#11758196—24no—
269#11758196—24no—
270#11758196C2×D414yes—
271#11758196C2×D414yes—
272C4×Dic61—22no—
273C4×Dic61—22no—
274C4×Dic61—24no—
275C4×Dic61—24no—
276C4×Dic61C2414yes—
277C2×#92418C2412yes—
278C2×#92418C2412yes—
279C23×C61—24no—
280C23×C61—22no—
281C23×C61—81no—
282C23×C61—81no—
283C23×C61C22×D411yescharacteristic
284Q8×C61—24no—
285Q8×C61—42no—
286D4×C61—24no—
287D4×C61—22no—
288D4×C612−1+414yes—
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—44no—
296C2×C4×C61—82no—
297C2×C4×C61—81no—
298C2×C4×C61C22×D411yescharacteristic
299C2×C4×C61C22×D411yescharacteristic
300C2×C4×C61C22×D411yescharacteristic
301C2×C4×C61C22×D411yescharacteristic
302C2×C4×C61C22×D411yescharacteristic
303C2×C4×C61C2511yescharacteristic
304C2×Dic61—22no—
305C2×Dic61—22no—
306C2×Dic61—22no—
307C2×Dic61—22no—
308C2×Dic61—84no—
309C2×Dic61—84no—
310C2×Dic61C22×D414yes—
311C2×Dic61C22×D414yes—
312Dic122—24no—
313Dic122—22no—
314Dic122—22no—
315Dic122—42no—
316Dic122—42no—
317Dic122—42no—
318Dic122—84no—
319Dic122—81no—
320C8×C61—24no—
321#92418—22no—
322#92418—22no—
323#61848—611nomaximal, Sylow
324C22×C61—21no—
325C22×C61—21no—
326C22×C61—24no—
327C22×C61—21no—
328C22×C61—42no—
329C22×C61—42no—
330C22×C61—84no—
331C22×C61C23×D411yescharacteristic
332C4×C61—21no—
333C4×C61—21no—
334C4×C61—21no—
335C4×C61—21no—
336C4×C61—42no—
337C4×C61—42no—
338C4×C61—42no—
339C4×C61#57712yes—
340C4×C61C2×2−1+412yes—
341Dic61—24no—
342Dic61—24no—
343Dic61—42no—
344Dic61—42no—
345Dic61—42no—
346Dic61—42no—
347#3556—611no—
348#3501—611no—
349#3388—614no—
350#3386—612no—
351#3381—612no—
352#3372—612no—
353#3372—614no—
354#3359—612no—
355#3242—614no—
356#3237—611no—
357#3231—611no—
358#3123—614no—
359#3110D6111yescharacteristic
360#3101—611no—
361#2985—611no—
362C2×C61—41no—
363C2×C61—44no—
364C2×C61#348611yescharacteristic
365C2×C61C2×#57711yescharacteristic
366C2×C61#358611yescharacteristic
367#564—1222no—
368#557—612no—
369#557—612no—
370#557—612no—
371#557—612no—
372#557—1221no—
373#557—1221no—
374#554—612no—
375#554—614no—
376#554—1222no—
377#554—1221no—
378D4×Q8—612no—
379D4×Q8—1221no—
380#544—612no—
381#543—1222no—
382#541—614no—
383#541—1222no—
384#541—1222no—
385#533—611no—
386#533—611no—
387#533—611no—
388#533C2×D6111yescharacteristic
389C2×C22⋊Q8—1222no—
390#518—612no—
391C2×C4×D4C2×D6111yescharacteristic
392#501—614no—
393#501—612no—
394#500—611no—
395#499—614no—
396#498—611no—
397#491—614no—
398#491—612no—
399#487—611no—
400#487—611no—
401#487—614no—
402#485—614no—
403#485—614no—
404#484C2×D6112yes—
405#483—612no—
406#479—614no—
407#479—614no—
408#479—614no—
409#478—614no—
410#478—614no—
411#477—614no—
412#477—612no—
413#476—612no—
414#475—612no—
415#475—614no—
416#474C2×D6112yes—
417#461—614no—
418#461—614no—
419#461—612no—
420#461C2×D6112yes—
421#450—612no—
422#450C2×D6112yes—
423#446—612no—
424#446—614no—
425C8×Q8—612no—
426#440—614no—
427#440—614no—
428#440C2×D6114yes—
429C4×SD16—614no—
430C4×SD16—612no—
431C4×SD16—612no—
432C4×SD16—614no—
433#433—614no—
434#432C2×D6111yescharacteristic
435#431—611no—
436C61#6184811yescharacteristic, complemented, Sylow
437C22×Q8—1222no—
438C22×D4C22×D6111yescharacteristic
439C23×C4—21no—
440C2×SD16—612no—
441C2×SD16—612no—
442C2×SD16C22×D6114yes—
443C4⋊Q8—614no—
444C4⋊Q8—614no—
445C4⋊Q8—611no—
446C4⋊Q8—611no—
447C4⋊Q8—611no—
448C4⋊Q8—611no—
449C4⋊Q8—611no—
450C4⋊Q8—611no—
451C4⋊Q8—1222no—
452C4⋊Q8—1222no—
453C4⋊Q8—1222no—
454C4⋊Q8C22×D6111yescharacteristic
455C4⋊Q8C22×D6111yescharacteristic
456#126—1224no—
457#126—1224no—
458#125—21no—
459#125—614no—
460#125—614no—
461#125—612no—
462#125—612no—
463#125—1222no—
464#125—1224no—
465#125—1224no—
466#125—1222no—
467#125—1224no—
468#125—1224no—
469#125—1224no—
470#125—1224no—
471#125—1222no—
472#125—1221no—
473C22⋊Q8—21no—
474C22⋊Q8—612no—
475C22⋊Q8—612no—
476C22⋊Q8—612no—
477C22⋊Q8—1224no—
478C22⋊Q8—1224no—
479C22⋊Q8—1224no—
480C22⋊Q8—1224no—
481C22⋊Q8—1224no—
482C22⋊Q8—1222no—
483C22⋊Q8—1224no—
484C22⋊Q8—1222no—
485C22⋊Q8—1221no—
486C22⋊Q8—1221no—
487C22⋊Q8—1221no—
488C22⋊Q8C22×D6112yes—
489C4×Q8—21no—
490C4×Q8—612no—
491C4×Q8—612no—
492C4×Q8—614no—
493C4×Q8—614no—
494C4×Q8—612no—
495C4×Q8—1222no—
496C4×Q8—1222no—
497C4×Q8—1222no—
498C4×Q8—1222no—
499C4×Q8—1221no—
500C4×Q8—1221no—
501C4×Q8—1221no—
502C4×Q8C22×D6112yes—
503C4×D4—21no—
504C4×D4—21no—
505C4×D4—614no—
506C4×D4—614no—
507C4×D4—1222no—
508C4×D4—1222no—
509C4×D4C22×D6112yes—
510C4×D4C22×D6112yes—
511#117—21no—
512#117—612no—
513#117—612no—
514#117—1221no—
515C2×C4⋊C4—1224no—
516C2×C4⋊C4—1222no—
517C2×C4⋊C4C22×D6111yescharacteristic
518C2×C22⋊C4—21no—
519C2×C22⋊C4—1224no—
520C2×C42C22×D6111yescharacteristic
521#107—614no—
522#107—614no—
523#107C22×D6114yes—
524#106—614no—
525#106—612no—
526#106—612no—
527#106—614no—
528C4⋊C8—614no—
529C4⋊C8—614no—
530C4⋊C8—612no—
531C4⋊C8C22×D6112yes—
532#103—614no—
533#103—614no—
534#103—614no—
535#103—614no—
536#103—612no—
537#103—612no—
538#103C22×D6114yes—
539#102—612no—
540#102—612no—
541#102C22×D6114yes—
542C22⋊C8—612no—
543C22⋊C8C22×D6112yes—
544#97—614no—
545#97C22×D6112yes—
546C4×C8—614no—
547C4×C8—611no—
548C4×C8—611no—
549C24—21no—
550C2×Q8—21no—
551C2×Q8—612no—
552C2×Q8—612no—
553C2×Q8—612no—
554C2×Q8—1224no—
555C2×Q8—1221no—
556C2×Q8—1221no—
557C2×Q8—1221no—
558C2×Q8—4884no—
559C2×Q8C23×D6112yes—
560C2×D4—24no—
561C2×D4—21no—
562C2×D4C23×D6112yes—
563C22×C4—21no—
564C22×C4—22no—
565C22×C4—22no—
566C22×C4—82no—
567C22×C4—1224no—
568C22×C4—1224no—
569C22×C4D4×D6111yescharacteristic
570C22×C4D4×D6111yescharacteristic
571C22×C4C23×D6111yescharacteristic
572SD16—28no—
573SD16—1224no—
574SD16—1224no—
575C2×C8—612no—
576C2×C8—612no—
577C2×C8C23×D6114yes—
578C4⋊C4—21no—
579C4⋊C4—21no—
580C4⋊C4—21no—
581C4⋊C4—21no—
582C4⋊C4—22no—
583C4⋊C4—22no—
584C4⋊C4—614no—
585C4⋊C4—612no—
586C4⋊C4—614no—
587C4⋊C4—612no—
588C4⋊C4—612no—
589C4⋊C4—612no—
590C4⋊C4—612no—
591C4⋊C4—612no—
592C4⋊C4—612no—
593C4⋊C4—1224no—
594C4⋊C4—1224no—
595C4⋊C4—1224no—
596C4⋊C4—1224no—
597C4⋊C4—1224no—
598C4⋊C4—1224no—
599C4⋊C4—1224no—
600C4⋊C4—1224no—
601C4⋊C4—1224no—
602C4⋊C4—1222no—
603C4⋊C4—1221no—
604C4⋊C4—1222no—
605C4⋊C4—1222no—
606C4⋊C4—1222no—
607C4⋊C4—1221no—
608C4⋊C4—1221no—
609C4⋊C4—1221no—
610C4⋊C4—1221no—
611C4⋊C4C23×D6112yes—
612C4⋊C4C23×D6112yes—
613C22⋊C4—22no—
614C22⋊C4—22no—
615C22⋊C4—22no—
616C22⋊C4—614no—
617C22⋊C4—1224no—
618C22⋊C4—1224no—
619C22⋊C4—1222no—
620C22⋊C4—1222no—
621C42—22no—
622C42—614no—
623C42—1224no—
624C42—1224no—
625C42—1222no—
626C42—1222no—
627C42D4×D6111yescharacteristic
628C42D4×D6111yescharacteristic
629C42C23×D6111yescharacteristic
630C42C23×D6111yescharacteristic
631C23—24no—
632C23—22no—
633C23—81no—
634C23—81no—
635C23C2×D4×D6111yescharacteristic
636Q8—24no—
637Q8—42no—
638Q8—1224no—
639Q8—1222no—
640Q8—1222no—
641Q8—4881no—
642Q8—4884no—
643Q8—2442no—
644Q8—2442no—
645Q8—2442no—
646D4—24no—
647D4—22no—
648D4#1451982414yes—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—41no—
655C2×C4—44no—
656C2×C4—82no—
657C2×C4—81no—
658C2×C4—614no—
659C2×C4—614no—
660C2×C4—1222no—
661C2×C4—1222no—
662C2×C4—1222no—
663C2×C4—1222no—
664C2×C4—4884no—
665C2×C4—4884no—
666C2×C4C2×D4×D6111yescharacteristic
667C2×C4C2×D4×D6111yescharacteristic
668C2×C4C2×D4×D6111yescharacteristic
669C2×C4C2×D4×D6111yescharacteristic
670C2×C4C2×D4×D6111yescharacteristic
671C2×C4C24×D6111yescharacteristic
672C8—24no—
673C8—1222no—
674C8—1222no—
675C22—21no—
676C22—24no—
677C22—21no—
678C22—21no—
679C22—42no—
680C22—42no—
681C22—84no—
682C22C22×D4×D6111yescharacteristic
683C4—21no—
684C4—21no—
685C4—21no—
686C4—21no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—1224no—
691C4—1224no—
692C4—2442no—
693C4—2442no—
694C4—2442no—
695C4—2442no—
696C4#1684057212yes—
697C4C2×#1451982412yes—
698C2—41no—
699C2—44no—
700C2#2145946611yescharacteristic
701C2C2×#1684057211yescharacteristic
702C2#2146060211yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3110C2
socleC22×C61C23×D4
prime radicalC22×D4×C61C23
hypercenter#3110D61
hypocenterC61#61848
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
1. #21460314C2semidirect
3. #21458765C2semidirect
4. #21458744C2semidirect
5. #21458698C2semidirect
6. #21458604C2semidirect
7. #21458601C2semidirect
8. #21458456C2semidirect
9. #21457091C2semidirect
10. #21457035C2semidirect
11. #21456974C2semidirect
14. #21454916C2semidirect
18. #16840491C22semidirect
20. #16840490C22semidirect
21. #16840487C22semidirect
22. #16840487C22semidirect
39. #16840171C22semidirect
40. #16840170C22semidirect
41. #16840165C22semidirect
42. #16840163C22semidirect
43. #16840158C22semidirect
44. #16840131C22semidirect
45. #16840129C22semidirect
46. #16840112C22semidirect
47. #16840112C22semidirect
48. #16840111C22semidirect
52. #16840071C22semidirect
53. #16840069C22semidirect
54. #16840068C22semidirect
57. #16840058C22semidirect
58. #16840056C22semidirect
60. #16840048C22semidirect
61. #16840045C22semidirect
66. #16839890C22semidirect
67. #16839889C22semidirect
68. #16839886C22semidirect
436. C61#61848semidirect

Maximal subgroups

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

subgroupindex
#214603142
#214596222
#214587652
#214587442
#214586982
#214586042
#214586012
#214584562
#214570912
#214570352
#214569742
#214560652
#214558272
#214549162
C61×#31102
#6184861