Finite Group Databasebeta

← #30610838

Subgroups — #30610838

Order 15616.

1096 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462749C211yesmaximal, characteristic, complemented
2C4×#14519865C211yesmaximal, characteristic, complemented
3#21461171C212yesmaximal, complemented
4#21461165C211yesmaximal, characteristic, complemented
5#21461159C212yesmaximal, complemented
6#21461157C211yesmaximal, characteristic, complemented
7#21461139C211yesmaximal, characteristic, complemented
8#21461135C212yesmaximal, complemented
9#21461113C211yesmaximal, characteristic, complemented
10D61⋊(D4⋊C8)C212yesmaximal, complemented
11#21459428C211yesmaximal, characteristic
12C2×#14519865—21no—
13#16840813—21no—
14#16840792—22no—
15#16840790—21no—
16#16840784C2212yescomplemented
17#16840781C2211yescharacteristic, complemented
18#16840766C2211yescharacteristic, complemented
19D61⋊(C4⋊C8)C2212yescomplemented
20#16840727—21no—
21#16840727—22no—
22C4×#11758213C2212yescomplemented
23C4×#11758213C2211yescharacteristic, complemented
24C4×D61⋊C8C2212yescomplemented
25C4×D61⋊C8C2211yescharacteristic, complemented
26Q8⋊#92420C2211yescharacteristic, complemented
27Q8⋊#92420C2211yescharacteristic, complemented
28Q8⋊#92420C2212yescomplemented
29#16840610C2211yescharacteristic
30#16840610C2212yes—
31#16840610C2211yescharacteristic
32D4⋊#92420C2212yescomplemented
33D4⋊#92420C2212yescomplemented
34D4⋊#92420C2212yescomplemented
35D4⋊#92420C2212yescomplemented
36#16840472C411yescharacteristic
37#16840472C411yescharacteristic
38#16840462—21no—
39#16840462C412yes—
40#16840440—22no—
41#16840405—21no—
42#16840405C412yes—
43#16840397C411yescharacteristic
44#16840310C2211yescharacteristic
45#16840291—21no—
46#16840291C2212yes—
47#16840270—22no—
48C4×#11758226C2211yescharacteristic
49C4×#11758226C2211yescharacteristic
50C4×D4×D61—21no—
51C4×D4×D61C2212yes—
52C61×#559C411yescharacteristic
53#14519865—28no—
54C2×#11758213—22no—
55C2×#11758213—21no—
56C2×#11758213—21no—
57C2×D61⋊C8—22no—
58C2×D61⋊C8—21no—
59C2×D61⋊C8—22no—
60#14519844—22no—
61C4⋊#92420—21no—
62C4⋊#92420C2311yescharacteristic
63C4⋊#92420C2312yes—
64C4⋊#92420C2311yescharacteristic
65#14519834—21no—
66#14519834—22no—
67#14519832—21no—
68#14519832—22no—
69#14519832—22no—
70C4×#92420—21no—
71C4×#92420—22no—
72C4×#92420—21no—
73C4×#92420C2311yescharacteristic
74C4×#92420C2312yes—
75C4×#92420C2311yescharacteristic
76C2×#11758226C2×C411yescharacteristic
77C2×#11758226D411yescharacteristic
78C2×#11758224—21no—
79C2×D4×D61—21no—
80C2×D4×D61C2×C412yes—
81C22×C4×D61—22no—
82#14519777C2×C411yescharacteristic
83#14519770—22no—
84#14519770—22no—
85#14519769—21no—
86#14519769C2×C412yes—
87#14519764—21no—
88#14519760—22no—
89#14519759—22no—
90#14519759—22no—
91#14519755—22no—
92#14519755—22no—
93#14519753—21no—
94#14519753C2×C412yes—
95#14519750—22no—
96#14519750—22no—
97#14519748—21no—
98#14519748—21no—
99#14519748C2×C412yes—
100#14519748C2×C412yes—
101#14519747—22no—
102#14519745—22no—
103Q8×Dic61D411yescharacteristic
104Q8×Dic61C2311yescharacteristic
105C4×D244—21no—
106C4×D244—21no—
107C4×D244C2×C412yes—
108C4×D244C2×C411yescharacteristic
109C4×D244C2×C412yes—
110C4×D244C2×C411yescharacteristic
111D4×Dic61—21no—
112D4×Dic61C2312yes—
113#14519739—21no—
114#14519739D411yescharacteristic
115#14519739D412yes—
116#14519739C2311yescharacteristic
117C4×C22⋊D61—22no—
118C4×C22⋊D61—22no—
119#14519737—22no—
120#14519737—22no—
121#14519735—21no—
122#14519735C2×C411yescharacteristic
123#14519735D411yescharacteristic
124#14519735D412yes—
125#14519733—22no—
126C4⋊C4×D61—21no—
127C4⋊C4×D61C2×C412yes—
128C22⋊C4×D61—22no—
129C22⋊C4×D61—22no—
130C42×D61—21no—
131C42×D61C2312yes—
132C42×D61C2311yescharacteristic
133#127×C61C2×C411yescharacteristic
134#124×C61—21no—
135#124×C61C2×C412yes—
136#123×C61—22no—
137C4×Q8×C61C2×C411yescharacteristic
138C4×Q8×C61C2×C411yescharacteristic
139C4×D4×C61—21no—
140C4×D4×C61C2×C412yes—
141C23×D61—22no—
142C2×Q8×C61C22⋊C411yescharacteristic
143C2×Q8×C61C22×C411yescharacteristic
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61C22×C412yes—
147C22×C4×C61—22no—
148C2×C22⋊D61—22no—
149C2×C22⋊D61—22no—
150C2×C22⋊D61—22no—
151C22×Dic61—22no—
152#11758226—24no—
153#11758226—22no—
154#11758226—41no—
155#11758224—81no—
156#11758224—81no—
157D4×D61—24no—
158D4×D61—42no—
159D4×D61—44no—
160C2×D244—21no—
161C2×D244C22⋊C412yes—
162C2×D244C22⋊C411yescharacteristic
163C2×D244C22×C411yescharacteristic
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—22no—
168C2×C4×D61—22no—
169C2×C4×D61—82no—
170C2×C4×D61—82no—
171C2×C4×D61C22×C412yes—
172C2×C4×D61C22×C411yescharacteristic
173C2×C4×D61C2×D412yes—
174C2×C4×D61C2×D411yescharacteristic
175C2×C4×D61C2×D412yes—
176C2×C4×D61C2×D411yescharacteristic
177C2×Dic122—21no—
178C2×#92420—21no—
179C2×#92420—21no—
180C2×#92420—22no—
181C2×#92420—21no—
182C2×#92420—21no—
183C2×#92420—22no—
184#11758213—28no—
185#11758213—24no—
186#11758213—81no—
187D61⋊C8—28no—
188D61⋊C8—24no—
189D61⋊C8—82no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61C22⋊C411yescharacteristic
192C4⋊C4×C61C22⋊C412yes—
193C4⋊C4×C61C22×C411yescharacteristic
194C22⋊C4×C61—22no—
195C22⋊C4×C61—22no—
196C22⋊C4×C61—22no—
197C42×C61—21no—
198C42×C61C22×C412yes—
199C42×C61C22×C411yescharacteristic
200C22⋊Dic61—22no—
201C22⋊Dic61—22no—
202C22⋊Dic61—22no—
203#11758198—22no—
204#11758198—21no—
205#11758198—22no—
206#11758198—21no—
207#11758198—21no—
208#11758198C22⋊C412yes—
209#11758198C22⋊C412yes—
210#11758198C22⋊C412yes—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61C22⋊C411yescharacteristic
214C4⋊Dic61C22⋊C412yes—
215C4⋊Dic61C22×C412yes—
216C4⋊Dic61C22×C411yescharacteristic
217#11758196—22no—
218#11758196—22no—
219#11758196—22no—
220#11758196—22no—
221C4×Dic61—21no—
222C4×Dic61—21no—
223C4×Dic61C2×D411yescharacteristic
224C4×Dic61C2×D412yes—
225C4×Dic61C2×D411yescharacteristic
226C4×Dic61C2×D412yes—
227C4×Dic61C2411yescharacteristic
228C23×C61—22no—
229C22×D61—21no—
230C22×D61—44no—
231C22×D61—44no—
232C22×D61—82no—
233C22×D61C2×C22⋊C411yescharacteristic
234C22×D61C2×C22⋊C412yes—
235Q8×C61—22no—
236Q8×C61—21no—
237Q8×C61—21no—
238D4×C61—22no—
239D4×C61—22no—
240D4×C61—42no—
241D4×C61—81no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—82no—
245C2×C4×C61C2×C22⋊C412yes—
246C2×C4×C61C2×C22⋊C412yes—
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C23×C411yescharacteristic
250C22⋊D61—44no—
251C22⋊D61—44no—
252C22⋊D61—82no—
253C2×Dic61—21no—
254C2×Dic61—22no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61C2×C22⋊C411yescharacteristic
258C2×Dic61C2×C22⋊C412yes—
259C2×Dic61C22×D411yescharacteristic
260C2×Dic61C22×D411yescharacteristic
261D244—24no—
262D244—21no—
263D244—42no—
264D244#10414yes—
265D244#13112yescomplemented
266C4×D61—24no—
267C4×D61—22no—
268C4×D61—21no—
269C4×D61—22no—
270C4×D61—21no—
271C4×D61—22no—
272C4×D61—22no—
273C4×D61—21no—
274C4×D61—22no—
275C4×D61—21no—
276C4×D61—42no—
277C4×D61—42no—
278C4×D61—41no—
279C4×D61—41no—
280C4×D61—82no—
281C4×D61—82no—
282Dic122—81no—
283#92420—22no—
284#92420—24no—
285#92420—22no—
286#92420—41no—
287#92420—42no—
288#92420—41no—
289#17517—611nomaximal, Sylow
290C22×C61—42no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61C22×C22⋊C411yescharacteristic
294C2×D61—22no—
295C2×D61—22no—
296C2×D61—21no—
297C2×D61—22no—
298C2×D61—21no—
299C2×D61—21no—
300C2×D61—41no—
301C2×D61—41no—
302C2×D61—88no—
303C2×D61C2×#10414yes—
304C4×C61—21no—
305C4×C61—22no—
306C4×C61—22no—
307C4×C61—41no—
308C4×C61—41no—
309C4×C61C2×#10412yes—
310C4×C61C2×#13111yescharacteristic
311C4×C61#56811yescharacteristic
312Dic61—21no—
313Dic61—22no—
314Dic61—21no—
315Dic61—42no—
316Dic61—41no—
317Dic61#57611yescharacteristic
318Dic61#57711yescharacteristic
319#3481—611no—
320#2990—611no—
321C4×#131—611no—
322#1494—612no—
323#1492—611no—
324#1490—612no—
325#1489—611no—
326#1484—611no—
327#1483—612no—
328#1476—611no—
329C2×D4⋊C8—612no—
330C2×C61—82no—
331C2×C61#288011yescharacteristic
332C2×C61#289711yescharacteristic
333C2×C61C22×#10411yescharacteristic
334D61—24no—
335D61—41no—
336D61—42no—
337C2×#131—1221no—
338#559#692311yescharacteristic
339#554—611no—
340#554—611no—
341#553—612no—
342#553—1221no—
343#551—1222no—
344#546—612no—
345#546—1221no—
346D42—611no—
347#532—611no—
348#529—612no—
349#529—1221no—
350#525—1222no—
351C4×#41—611no—
352C4×#41—611no—
353C2×C4×D4—612no—
354C2×C4×D4—1221no—
355#446—1221no—
356#435—1222no—
357C8×D4—1221no—
358#432—612no—
359#431—611no—
360#423—611no—
361C2×C4⋊C8—612no—
362#405—1222no—
363#405—1221no—
364C4×M16—612no—
365C4×M16—611no—
366C2×C4×C8—612no—
367C2×C4×C8—611no—
368Q8⋊C8—611no—
369Q8⋊C8—611no—
370Q8⋊C8—612no—
371D4⋊C8—612no—
372D4⋊C8—612no—
373D4⋊C8—612no—
374D4⋊C8—612no—
375D4⋊C8—611no—
376D4⋊C8—612no—
377D4⋊C8—611no—
378C61#1751711yescharacteristic, complemented, Sylow
379C2×#41—611no—
380C2×#41—611no—
381C2×#41—1221no—
382C22×D4—612no—
383C22×D4—1221no—
384C23×C4—1222no—
385#131—1228no—
386C2×M16—1222no—
387C2×M16—1221no—
388C2×M16—1221no—
389C22×C8—1222no—
390C22×C8—1221no—
391C22×C8—1222no—
392C4⋊Q8—611no—
393#127C2×#692311yescharacteristic
394#126—1222no—
395#126—1222no—
396#125—612no—
397#125—1221no—
398#124—21no—
399#124—1221no—
400#124C2×#692312yes—
401#123—22no—
402#123—1222no—
403#123—1222no—
404#123—1222no—
405C22⋊Q8—612no—
406C22⋊Q8—1222no—
407C22⋊Q8—1221no—
408C22⋊Q8—1222no—
409#121—612no—
410#121—612no—
411#121—1222no—
412#121—1222no—
413#121—1221no—
414#121—1221no—
415#120—1222no—
416#120—1222no—
417C4×Q8—611no—
418C4×Q8—611no—
419C4×Q8C2×#692311yescharacteristic
420C4×Q8C2×#692311yescharacteristic
421C4×D4—21no—
422C4×D4—611no—
423C4×D4—612no—
424C4×D4—612no—
425C4×D4—611no—
426C4×D4—611no—
427C4×D4—612no—
428C4×D4—612no—
429C4×D4—611no—
430C4×D4—1221no—
431C4×D4—1221no—
432C4×D4—1222no—
433C4×D4—1222no—
434C4×D4—1221no—
435C4×D4—1222no—
436C4×D4—1222no—
437C4×D4—1221no—
438C4×D4C2×#692312yes—
439#117—612no—
440#117—611no—
441#117—611no—
442#117—1222no—
443#117—1221no—
444C2×C4⋊C4—612no—
445C2×C4⋊C4—1221no—
446C2×C22⋊C4—1222no—
447C2×C22⋊C4—1222no—
448C2×C42—612no—
449C2×C42—611no—
450C2×C42—1221no—
451C4⋊C8—611no—
452C4⋊C8—612no—
453C4⋊C8—611no—
454C4⋊C8—1221no—
455C4⋊C8—1222no—
456C22⋊C8—1221no—
457C22⋊C8—1222no—
458#97—1221no—
459#97—1222no—
460#97—1222no—
461C4×C8—611no—
462C4×C8—612no—
463C4×C8—611no—
464C4×C8—1221no—
465C4×C8—1222no—
466C4×C8—1221no—
467C24—1222no—
468#41—1224no—
469#41—1222no—
470#41—4881no—
471#41—4881no—
472#41—2441no—
473C2×Q8—1221no—
474C2×Q8C22⋊#692311yescharacteristic
475C2×Q8C22×#692311yescharacteristic
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—611no—
479C2×D4—612no—
480C2×D4—611no—
481C2×D4—1221no—
482C2×D4—1222no—
483C2×D4—1224no—
484C2×D4—1222no—
485C2×D4—1222no—
486C2×D4—2442no—
487C2×D4—2444no—
488C2×D4C22×#692312yes—
489C22×C4—22no—
490C22×C4—612no—
491C22×C4—611no—
492C22×C4—612no—
493C22×C4—611no—
494C22×C4—612no—
495C22×C4—611no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1222no—
499C22×C4—1222no—
500C22×C4—1222no—
501C22×C4—1221no—
502C22×C4—4882no—
503C22×C4—4882no—
504M16—1228no—
505M16—1224no—
506M16—4881no—
507C2×C8—1221no—
508C2×C8—1222no—
509C2×C8—1221no—
510C2×C8—1228no—
511C2×C8—1224no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1222no—
515C2×C8—4882no—
516C4⋊C4—21no—
517C4⋊C4—611no—
518C4⋊C4—612no—
519C4⋊C4—612no—
520C4⋊C4—611no—
521C4⋊C4—1222no—
522C4⋊C4—1222no—
523C4⋊C4—1221no—
524C4⋊C4—1222no—
525C4⋊C4—1221no—
526C4⋊C4—1222no—
527C4⋊C4C22⋊#692311yescharacteristic
528C4⋊C4C22⋊#692312yes—
529C4⋊C4C22×#692311yescharacteristic
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—612no—
534C22⋊C4—612no—
535C22⋊C4—612no—
536C22⋊C4—1221no—
537C22⋊C4—1222no—
538C22⋊C4—1221no—
539C22⋊C4—1222no—
540C22⋊C4—1222no—
541C22⋊C4—1222no—
542C22⋊C4—1222no—
543C22⋊C4—1221no—
544C42—21no—
545C42—611no—
546C42—612no—
547C42—611no—
548C42—612no—
549C42—611no—
550C42—1221no—
551C42—1221no—
552C42C22×#692312yes—
553C42C22×#692311yescharacteristic
554C23—22no—
555C23—612no—
556C23—611no—
557C23—1221no—
558C23—4882no—
559C23—2444no—
560C23—2444no—
561Q8—22no—
562Q8—21no—
563Q8—21no—
564Q8—4881no—
565D4—22no—
566D4—22no—
567D4—42no—
568D4—81no—
569D4—614no—
570D4—612no—
571D4—1224no—
572D4—1221no—
573D4—4882no—
574D4—2442no—
575D4—2444no—
576D4—2444no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—82no—
580C2×C4—611no—
581C2×C4—611no—
582C2×C4—612no—
583C2×C4—611no—
584C2×C4—1222no—
585C2×C4—1221no—
586C2×C4—1222no—
587C2×C4—1221no—
588C2×C4—1224no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—1221no—
592C2×C4—1222no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1222no—
596C2×C4—4882no—
597C2×C4—4882no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4—2442no—
603C2×C4—2442no—
604C2×C4C2×C22⋊#692312yes—
605C2×C4C2×C22⋊#692312yes—
606C2×C4C2×C22⋊#692311yescharacteristic
607C2×C4C2×C22⋊#692311yescharacteristic
608C2×C4C23×#692311yescharacteristic
609C8—1222no—
610C8—1224no—
611C8—1222no—
612C8—2441no—
613C8—2441no—
614C8—2442no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—614no—
619C22—1222no—
620C22—1222no—
621C22—1221no—
622C22—1222no—
623C22—1221no—
624C22—1221no—
625C22—4888no—
626C22—2441no—
627C22—2441no—
628C22C22×C22⋊#692311yescharacteristic
629C4—22no—
630C4—21no—
631C4—22no—
632C4—41no—
633C4—41no—
634C4—611no—
635C4—611no—
636C4—1221no—
637C4—1222no—
638C4—1221no—
639C4—2442no—
640C4—2441no—
641C4D61⋊#10412yes—
642C4D61⋊#13111yescharacteristic
643C4#1684084111yescharacteristic
644C2—82no—
645C2—1224no—
646C2—2441no—
647C2—2442no—
648C2#2146241011yescharacteristic
649C2#2146244011yescharacteristic
650C2C2×D61⋊#10411yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#559C4
socleC22×C61C22×C22⋊C4
prime radical#21459428C2
cosocleC4×Dic61C24
hypercenter#559#6923
hypocenterC61#17517
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #559

Complements

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

NH ≅ G/Ntype
1. #21462749C2semidirect
2. C4×#14519865C2semidirect
3. #21461171C2semidirect
4. #21461165C2semidirect
5. #21461159C2semidirect
6. #21461157C2semidirect
7. #21461139C2semidirect
8. #21461135C2semidirect
9. #21461113C2semidirect
10. D61⋊(D4⋊C8)C2semidirect
16. #16840784C22semidirect
17. #16840781C22semidirect
18. #16840766C22semidirect
19. D61⋊(C4⋊C8)C22semidirect
22. C4×#11758213C22semidirect
23. C4×#11758213C22semidirect
24. C4×D61⋊C8C22semidirect
25. C4×D61⋊C8C22semidirect
26. Q8⋊#92420C22semidirect
27. Q8⋊#92420C22semidirect
28. Q8⋊#92420C22semidirect
32. D4⋊#92420C22semidirect
33. D4⋊#92420C22semidirect
34. D4⋊#92420C22semidirect
35. D4⋊#92420C22semidirect
265. D244#131semidirect
378. C61#17517semidirect

Maximal subgroups

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

subgroupindex
#214627492
C4×#145198652
#214611712
#214611652
#214611592
#214611572
#214611392
#214611352
#214611132
D61⋊(D4⋊C8)2
#214594282
#1751761