Finite Group Databasebeta

← #29647079

Subgroups — #29647079

Order 15616.

990 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459445C211yesmaximal, characteristic
2#21454940C211yesmaximal, characteristic, complemented
3#21454461C211yesmaximal, characteristic, complemented
4#21444286C212yesmaximal, complemented
5#21444271C211yesmaximal, characteristic, complemented
6#21444258C211yesmaximal, characteristic, complemented
7#21444252C212yesmaximal, complemented
8#21444235C211yesmaximal, characteristic, complemented
9#21444216C212yesmaximal, complemented
10#21444205C212yesmaximal, complemented
11C61×#1476C211yesmaximal, characteristic, complemented
12Q8×Dic122C411yescharacteristic
13#16840477—21no—
14#16840477C411yescharacteristic
15#16840475—22no—
16#16840475C412yes—
17#16840434C411yescharacteristic
18#16840433—21no—
19#16840433C411yescharacteristic
20#16840431C412yes—
21#16840312—21no—
22#16840312C2211yescharacteristic
23#16840309—22no—
24#16840309C2212yes—
25C4×#11758226C2212yes—
26C4×#11758222—21no—
27C4×#11758222C2211yescharacteristic
28#16839892—21no—
29#16839836—22no—
30C8×D244—21no—
31#16839821—22no—
32#16839818C2212yes—
33#16839813C2212yes—
34#16839813C2212yes—
35#16839788—21no—
36#16839786C2211yescharacteristic
37C4⋊C8×D61C2212yes—
38#16839728—21no—
39C4×#11758194C2211yescharacteristic
40Q8⋊#92418C2212yes—
41#16839491C2212yes—
42#16839491C2212yes—
43D4⋊#92418C2212yes—
44#16839489C2212yes—
45#16839489C2212yes—
46C4×#41×C61C2211yescharacteristic
47C61×#431C2211yescharacteristic
48C2×C4×C8×C61C2211yescharacteristic
49C61×Q8⋊C8C2211yescharacteristic
50C61×Q8⋊C8C2211yescharacteristic
51C61×D4⋊C8C2211yescharacteristic
52C61×D4⋊C8C2211yescharacteristic
53C2×#11758226C2×C412yes—
54C2×#11758222—21no—
55C2×#11758222—22no—
56C2×#11758222—21no—
57C2×#11758222D411yescharacteristic
58C2×#11758194—21no—
59C2×#11758194—21no—
60C2×C8×D61—22no—
61#14519777—21no—
62#14519777C2×C411yescharacteristic
63#14519776—22no—
64#14519776C2×C412yes—
65#14519774—21no—
66#14519774C2×C411yescharacteristic
67#14519773—22no—
68#14519773C2×C412yes—
69#14519765—21no—
70#14519765—22no—
71#14519763—22no—
72#14519763—22no—
73#14519763—22no—
74#14519752—22no—
75#14519752C2×C412yes—
76#14519751—21no—
77#14519751—21no—
78#14519751—22no—
79#14519751—21no—
80#14519751C2×C412yes—
81#14519750—22no—
82#14519750—22no—
83#14519750—22no—
84#14519748—22no—
85#14519748C2×C412yes—
86Q8×Dic61C2×C412yes—
87C4×Dic122—21no—
88C4×Dic122C2×C411yescharacteristic
89C4×Dic122D411yescharacteristic
90C4×D244—22no—
91C4×D244—21no—
92C4×D244—21no—
93C4×D244C2×C411yescharacteristic
94C4×D244D411yescharacteristic
95C4×D244C2312yes—
96#14519739—22no—
97#14519739C2×C412yes—
98C4×C22⋊D61—22no—
99C4×C22⋊D61—22no—
100C4×C22⋊D61C2×C412yes—
101C4×C22⋊D61D412yes—
102#14519736—21no—
103#14519736D411yescharacteristic
104#14519735—22no—
105#14519735C2×C412yes—
106#14519734—22no—
107#14519734D412yes—
108C42×D61—22no—
109C42×D61C2312yes—
110C4⋊#92418—21no—
111C4⋊#92418C2312yes—
112#14519706—22no—
113#14519706—22no—
114#14519706—22no—
115#14519689—22no—
116#14519689—22no—
117#14519689—22no—
118#14519688—21no—
119#14519687—22no—
120C4×#92418C2312yes—
121C2×#41×C61C2×C411yescharacteristic
122C22×C8×C61—21no—
123C4×Q8×C61C2×C411yescharacteristic
124C4×Q8×C61C2311yescharacteristic
125C4×D4×C61—21no—
126C4×D4×C61—21no—
127C4×D4×C61C2×C411yescharacteristic
128C4×D4×C61C2311yescharacteristic
129#117×C61—21no—
130#117×C61C2×C411yescharacteristic
131C2×C42×C61—21no—
132C2×C42×C61C2311yescharacteristic
133C4⋊C8×C61C2311yescharacteristic
134C4⋊C8×C61C2311yescharacteristic
135C22⋊C8×C61—21no—
136C4×C8×C61—21no—
137C4×C8×C61—21no—
138C4×C8×C61C2311yescharacteristic
139C4×C8×C61C2311yescharacteristic
140#41×C61—22no—
141#41×C61—41no—
142C2×Q8×C61C22×C411yescharacteristic
143C2×D4×C61—21no—
144C2×D4×C61C22×C411yescharacteristic
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C22×C411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C2×C22⋊D61—22no—
152C2×C22⋊D61—22no—
153C2×C22⋊D61—22no—
154C2×C22⋊D61—22no—
155C2×C22⋊D61C22⋊C412yes—
156#11758226—24no—
157#11758226—42no—
158#11758222—42no—
159#11758222—44no—
160#11758222—82no—
161#11758222—82no—
162#11758222—81no—
163#11758222—81no—
164C2×D244—22no—
165C2×D244—21no—
166C2×D244—22no—
167C2×D244—21no—
168C2×D244C22⋊C411yescharacteristic
169C2×D244C22×C412yes—
170C2×C4×D61—22no—
171C2×C4×D61—22no—
172C2×C4×D61—22no—
173C2×C4×D61C22×C412yes—
174C2×C4×D61C2×D412yes—
175C2×C4×D61C2×D412yes—
176C2×Dic122—21no—
177C2×Dic122—22no—
178C2×Dic122C22⋊C411yescharacteristic
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—42no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61C22×C411yescharacteristic
187C4⋊C4×C61C22×C411yescharacteristic
188C22⋊C4×C61—21no—
189C22⋊C4×C61—21no—
190C22⋊C4×C61—21no—
191C42×C61—21no—
192C42×C61—21no—
193C42×C61—21no—
194C42×C61C22×C411yescharacteristic
195C42×C61C2×D411yescharacteristic
196C42×C61C2×D411yescharacteristic
197C42×C61C2411yescharacteristic
198C22⋊Dic61—22no—
199C22⋊Dic61C22⋊C412yes—
200#11758198—22no—
201#11758198—22no—
202#11758198—22no—
203#11758198—22no—
204#11758198—22no—
205#11758198C22⋊C412yes—
206#11758198C22⋊C412yes—
207C4⋊Dic61—21no—
208C4⋊Dic61—22no—
209C4⋊Dic61C22⋊C411yescharacteristic
210C4⋊Dic61C22⋊C411yescharacteristic
211C4⋊Dic61C22×C412yes—
212#11758196—22no—
213#11758196C22⋊C412yes—
214#11758196C22⋊C412yes—
215C4×Dic61—22no—
216C4×Dic61—22no—
217C4×Dic61C22×C412yes—
218C4×Dic61C2×D412yes—
219C4×Dic61C2×D412yes—
220#11758194—42no—
221#11758194—81no—
222C2×#92418—22no—
223C2×#92418—22no—
224C8×D61—82no—
225C23×C61—21no—
226C23×C61C2×C22⋊C411yescharacteristic
227C22×D61—22no—
228C22×D61C2×C22⋊C412yes—
229Q8×C61—21no—
230Q8×C61#13112yes—
231D4×C61—21no—
232D4×C61—21no—
233D4×C61—42no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—22no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—42no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—42no—
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C23×C411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C22⋊D61—42no—
253C22⋊D61—44no—
254C22⋊D61—82no—
255C22⋊D61—82no—
256C22⋊D61—82no—
257C2×Dic61—22no—
258C2×Dic61C2×C22⋊C412yes—
259C2×Dic61C2×C22⋊C412yes—
260C2×Dic61C2×C22⋊C412yes—
261D244—22no—
262D244—22no—
263D244—22no—
264D244—44no—
265D244—42no—
266D244—81no—
267D244—82no—
268C4×D61—22no—
269C4×D61—22no—
270C4×D61—22no—
271C4×D61—22no—
272C4×D61—42no—
273C4×D61—44no—
274C4×D61—44no—
275C4×D61—42no—
276C4×D61—42no—
277Dic122—22no—
278Dic122—82no—
279Dic122—81no—
280C8×C61—41no—
281C8×C61—41no—
282C8×C61—41no—
283C8×C61—41no—
284#92418—42no—
285#92418—42no—
286#17526—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—21no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61C22×C22⋊C411yescharacteristic
294C2×D61—22no—
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61—42no—
300C4×C61—22no—
301C4×C61—21no—
302C4×C61—21no—
303C4×C61—21no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61#42112yes—
307C4×C61C2×#13111yescharacteristic
308C4×C61#56811yescharacteristic
309C4×C61#57611yescharacteristic
310C4×C61#57711yescharacteristic
311Dic61—22no—
312Dic61—22no—
313Dic61—22no—
314Dic61—42no—
315#3483—611no—
316#2990—611no—
317#2922—611no—
318#1496—612no—
319#1492—611no—
320#1489—611no—
321#1488—612no—
322#1484—611no—
323#1480—612no—
324#1477—612no—
325#1476D6111yescharacteristic, complemented
326C2×C61—41no—
327C2×C61—81no—
328C2×C61#288011yescharacteristic
329C2×C61#289711yescharacteristic
330C2×C61C2×#42111yescharacteristic
331D61—42no—
332D61—82no—
333Q82—611no—
334#555—611no—
335#555—612no—
336#555—1222no—
337#555—1221no—
338#550—611no—
339#550—612no—
340#550—611no—
341#550—1221no—
342#532—612no—
343#532—611no—
344#532—1221no—
345#532—1222no—
346C4×#41—611no—
347C4×#41—612no—
348C4×#41—1221no—
349C4×#41C2×D6111yescharacteristic
350#446—1221no—
351#435—1222no—
352C8×D4—1221no—
353#433—1222no—
354#432—612no—
355#431—612no—
356#431—612no—
357#431C2×D6111yescharacteristic
358#424—1221no—
359#423—611no—
360C2×C4⋊C8—612no—
361#408—1221no—
362C4×M16—611no—
363C2×C4×C8C2×D6111yescharacteristic
364Q8⋊C8—612no—
365Q8⋊C8C2×D6111yescharacteristic
366Q8⋊C8C2×D6111yescharacteristic
367D4⋊C8—612no—
368D4⋊C8—612no—
369D4⋊C8—612no—
370D4⋊C8—612no—
371D4⋊C8—612no—
372D4⋊C8C2×D6111yescharacteristic
373D4⋊C8C2×D6111yescharacteristic
374C61#1752611yescharacteristic, complemented, Sylow
375C2×#41—611no—
376C2×#41—612no—
377C2×#41—1222no—
378C2×#41—1221no—
379C2×#41—1221no—
380C2×#41C4×D6111yescharacteristic
381C2×M16—1221no—
382C2×M16—1221no—
383C22×C8—21no—
384C22×C8—1222no—
385C4⋊Q8—612no—
386C4⋊Q8—611no—
387C4⋊Q8—1221no—
388C4⋊Q8—1222no—
389#127—612no—
390#127—611no—
391#127—1221no—
392#127—1222no—
393#124—1222no—
394#124—1222no—
395#124—1222no—
396#124—1222no—
397#124—1221no—
398#121—612no—
399#121—612no—
400#121—612no—
401#121—1222no—
402#121—1222no—
403#121—1222no—
404#121—1222no—
405#121—1222no—
406#121—1222no—
407#121—1221no—
408#121—1221no—
409#121—1221no—
410C4×Q8—611no—
411C4×Q8—611no—
412C4×Q8—612no—
413C4×Q8—1221no—
414C4×Q8C4×D6111yescharacteristic
415C4×Q8C22×D6111yescharacteristic
416C4×D4—21no—
417C4×D4—21no—
418C4×D4—612no—
419C4×D4—612no—
420C4×D4—612no—
421C4×D4—611no—
422C4×D4—611no—
423C4×D4—612no—
424C4×D4—1221no—
425C4×D4—1222no—
426C4×D4—1222no—
427C4×D4—1222no—
428C4×D4—1222no—
429C4×D4—1221no—
430C4×D4C4×D6111yescharacteristic
431C4×D4C22×D6111yescharacteristic
432#117—21no—
433#117—611no—
434#117—612no—
435#117—612no—
436#117—1222no—
437#117—1222no—
438#117—1221no—
439#117C4×D6111yescharacteristic
440C2×C42—21no—
441C2×C42—612no—
442C2×C42—1222no—
443C2×C42C22×D6111yescharacteristic
444C4⋊C8—612no—
445C4⋊C8—1222no—
446C4⋊C8—1222no—
447C4⋊C8—1222no—
448C4⋊C8—1221no—
449C4⋊C8C22×D6111yescharacteristic
450C4⋊C8C22×D6111yescharacteristic
451C22⋊C8—21no—
452C22⋊C8—1222no—
453C22⋊C8—1222no—
454C22⋊C8—1222no—
455#97—1222no—
456#97—1221no—
457C4×C8—21no—
458C4×C8—21no—
459C4×C8—612no—
460C4×C8C22×D6111yescharacteristic
461C4×C8C22×D6111yescharacteristic
462#41—22no—
463#41—41no—
464#41—1224no—
465#41—4881no—
466#41—4881no—
467#41—4882no—
468#41—4882no—
469#41—2442no—
470#41—2444no—
471#41—2442no—
472C2×Q8—611no—
473C2×Q8—1222no—
474C2×Q8—1221no—
475C2×Q8C2×C4×D6111yescharacteristic
476C2×D4—21no—
477C2×D4—612no—
478C2×D4—612no—
479C2×D4—611no—
480C2×D4—1222no—
481C2×D4—1222no—
482C2×D4—1222no—
483C2×D4—1222no—
484C2×D4—1222no—
485C2×D4—1222no—
486C2×D4—1221no—
487C2×D4—1221no—
488C2×D4C2×C4×D6111yescharacteristic
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—612no—
493C22×C4—612no—
494C22×C4—612no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—1222no—
498C22×C4C2×C4×D6111yescharacteristic
499C22×C4D4×D6111yescharacteristic
500C22×C4D4×D6111yescharacteristic
501M16—4881no—
502M16—2442no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—42no—
508C2×C8—1222no—
509C2×C8—1222no—
510C2×C8—4882no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—612no—
514C4⋊C4—612no—
515C4⋊C4—611no—
516C4⋊C4—611no—
517C4⋊C4—612no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4—1221no—
521C4⋊C4C2×C4×D6111yescharacteristic
522C4⋊C4C2×C4×D6111yescharacteristic
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—21no—
526C22⋊C4—612no—
527C22⋊C4—612no—
528C22⋊C4—612no—
529C22⋊C4—1222no—
530C22⋊C4—1222no—
531C22⋊C4—1222no—
532C22⋊C4—1222no—
533C22⋊C4—1222no—
534C22⋊C4—1222no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—612no—
539C42—612no—
540C42—612no—
541C42—1222no—
542C42—1222no—
543C42C2×C4×D6111yescharacteristic
544C42D4×D6111yescharacteristic
545C42D4×D6111yescharacteristic
546C42C23×D6111yescharacteristic
547C23—21no—
548C23—612no—
549C23—1222no—
550C23C22⋊C4×D6111yescharacteristic
551Q8—21no—
552Q8—1222no—
553Q8—4881no—
554Q8—4882no—
555Q8#1451978312yes—
556D4—21no—
557D4—21no—
558D4—42no—
559D4—1222no—
560D4—1222no—
561D4—1222no—
562D4—4881no—
563D4—4882no—
564D4—4882no—
565D4—4882no—
566D4—4882no—
567D4—2444no—
568D4—2442no—
569D4—2444no—
570D4—2442no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—612no—
584C2×C4—612no—
585C2×C4—612no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—2442no—
592C2×C4—2444no—
593C2×C4—2444no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4C22⋊C4×D6111yescharacteristic
597C2×C4C22⋊C4×D6111yescharacteristic
598C2×C4C22⋊C4×D6111yescharacteristic
599C2×C4C22×C4×D6111yescharacteristic
600C2×C4C2×D4×D6111yescharacteristic
601C2×C4C2×D4×D6111yescharacteristic
602C8—41no—
603C8—41no—
604C8—41no—
605C8—41no—
606C8—2442no—
607C8—2442no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—1222no—
615C22—1222no—
616C22—1222no—
617C22—2442no—
618C22—2442no—
619C22—2442no—
620C22C2×C22⋊C4×D6111yescharacteristic
621C4—21no—
622C4—21no—
623C4—21no—
624C4—22no—
625C4—41no—
626C4—41no—
627C4—1222no—
628C4—1222no—
629C4—1222no—
630C4—2442no—
631C4#1683977712yes—
632C4C2×#1451978311yescharacteristic
633C4#1684052811yescharacteristic
634C4#1684056411yescharacteristic
635C4#1684056911yescharacteristic
636C2—41no—
637C2—81no—
638C2—4882no—
639C2—2442no—
640C2#2145426511yescharacteristic
641C2#2145434111yescharacteristic
642C2C2×#1683977711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1476C2
socleC22×C61C22×C22⋊C4
prime radical#21459445C2
cosocleC42×C61C24
hypercenter#1476D61
hypocenterC61#17526
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1476

Complements

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

NH ≅ G/Ntype
2. #21454940C2semidirect
3. #21454461C2semidirect
4. #21444286C2semidirect
5. #21444271C2semidirect
6. #21444258C2semidirect
7. #21444252C2semidirect
8. #21444235C2semidirect
9. #21444216C2semidirect
10. #21444205C2semidirect
11. C61×#1476C2semidirect
325. #1476D61semidirect
374. C61#17526semidirect

Maximal subgroups

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

subgroupindex
#214594452
#214549402
#214544612
#214442862
#214442712
#214442582
#214442522
#214442352
#214442162
#214442052
C61×#14762
#1752661