Finite Group Databasebeta

← #30610866

Subgroups — #30610866

Order 15616.

1008 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462753C211yesmaximal, characteristic, complemented
2#21462504C211yesmaximal, characteristic, complemented
3#21461175C212yesmaximal, complemented
4#21461165C211yesmaximal, characteristic, complemented
5#21461157C211yesmaximal, characteristic, complemented
6#21461145C212yesmaximal, complemented
7#21461139C211yesmaximal, characteristic, complemented
8#21461131C212yesmaximal, complemented
9#21461119C212yesmaximal, complemented
10#21461113C211yesmaximal, characteristic, complemented
11#21459428C211yesmaximal, characteristic
12Q8×#92420—21no—
13#16840796—21no—
14#16840796—22no—
15#16840787—22no—
16#16840781C2211yescharacteristic, complemented
17#16840781C2212yescomplemented
18#16840781C2212yescomplemented
19#16840780C2212yescomplemented
20#16840769—21no—
21#16840766C2212yescomplemented
22#16840766C2211yescharacteristic, complemented
23#16840737—21no—
24C4×#11758213C2211yescharacteristic, complemented
25C4×D61⋊C8C2211yescharacteristic, complemented
26Q8⋊#92420C2211yescharacteristic, complemented
27Q8⋊#92420C2211yescharacteristic, complemented
28Q8⋊#92420C2212yescomplemented
29#16840610C2211yescharacteristic
30#16840610C2211yescharacteristic
31#16840610C2212yes—
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
53C2×#11758213—21no—
54C2×#11758213—22no—
55C2×#11758213—21no—
56C2×D61⋊C8—21no—
57#14519844—22no—
58C4⋊#92420—21no—
59C4⋊#92420—22no—
60C4⋊#92420—22no—
61C4⋊#92420C2311yescharacteristic
62C4⋊#92420C2312yes—
63C4⋊#92420C2311yescharacteristic
64#14519834—22no—
65#14519834—22no—
66#14519834—22no—
67#14519834—21no—
68#14519832—21no—
69C4×#92420—21no—
70C4×#92420—21no—
71C4×#92420—22no—
72C4×#92420C2311yescharacteristic
73C4×#92420C2311yescharacteristic
74C4×#92420C2312yes—
75C2×#11758226C2×C411yescharacteristic
76C2×#11758226D411yescharacteristic
77C2×#11758224—21no—
78C2×D4×D61—21no—
79C2×D4×D61C2×C412yes—
80C22×C4×D61—22no—
81#14519777C2×C411yescharacteristic
82#14519770—22no—
83#14519770—22no—
84#14519769—21no—
85#14519769C2×C412yes—
86#14519764—21no—
87#14519760—22no—
88#14519759—22no—
89#14519759—22no—
90#14519755—22no—
91#14519755—22no—
92#14519753—21no—
93#14519753C2×C412yes—
94#14519750—22no—
95#14519750—22no—
96#14519748—21no—
97#14519748—21no—
98#14519748C2×C412yes—
99#14519748C2×C412yes—
100#14519747—22no—
101#14519745—22no—
102Q8×Dic61D411yescharacteristic
103Q8×Dic61C2311yescharacteristic
104C4×D244—21no—
105C4×D244—21no—
106C4×D244C2×C412yes—
107C4×D244C2×C411yescharacteristic
108C4×D244C2×C412yes—
109C4×D244C2×C411yescharacteristic
110D4×Dic61—21no—
111D4×Dic61C2312yes—
112#14519739—21no—
113#14519739D411yescharacteristic
114#14519739D412yes—
115#14519739C2311yescharacteristic
116C4×C22⋊D61—22no—
117C4×C22⋊D61—22no—
118#14519737—22no—
119#14519737—22no—
120#14519735—21no—
121#14519735C2×C411yescharacteristic
122#14519735D411yescharacteristic
123#14519735D412yes—
124#14519733—22no—
125C4⋊C4×D61—21no—
126C4⋊C4×D61C2×C412yes—
127C22⋊C4×D61—22no—
128C22⋊C4×D61—22no—
129C42×D61—21no—
130C42×D61C2311yescharacteristic
131C42×D61C2312yes—
132#127×C61C2×C411yescharacteristic
133#124×C61—21no—
134#124×C61C2×C412yes—
135#123×C61—22no—
136C4×Q8×C61C2×C411yescharacteristic
137C4×Q8×C61C2×C411yescharacteristic
138C4×D4×C61—21no—
139C4×D4×C61C2×C412yes—
140C23×D61—22no—
141C2×Q8×C61C22⋊C411yescharacteristic
142C2×Q8×C61C22×C411yescharacteristic
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61C22×C412yes—
146C22×C4×C61—22no—
147C2×C22⋊D61—22no—
148C2×C22⋊D61—22no—
149C2×C22⋊D61—22no—
150C22×Dic61—22no—
151#11758226—22no—
152#11758226—41no—
153#11758226—42no—
154#11758224—81no—
155#11758224—81no—
156D4×D61—24no—
157D4×D61—42no—
158D4×D61—44no—
159C2×D244—21no—
160C2×D244C22⋊C411yescharacteristic
161C2×D244C22⋊C412yes—
162C2×D244C22×C411yescharacteristic
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—22no—
167C2×C4×D61—22no—
168C2×C4×D61—82no—
169C2×C4×D61—82no—
170C2×C4×D61C22×C412yes—
171C2×C4×D61C22×C411yescharacteristic
172C2×C4×D61C2×D412yes—
173C2×C4×D61C2×D411yescharacteristic
174C2×C4×D61C2×D411yescharacteristic
175C2×C4×D61C2×D412yes—
176C2×Dic122—21no—
177C2×#92420—21no—
178C2×#92420—21no—
179C2×#92420—21no—
180C2×#92420—21no—
181C2×#92420—22no—
182C2×#92420—22no—
183#11758213—42no—
184#11758213—82no—
185#11758213—81no—
186D61⋊C8—42no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61C22⋊C411yescharacteristic
189C4⋊C4×C61C22⋊C412yes—
190C4⋊C4×C61C22×C411yescharacteristic
191C22⋊C4×C61—22no—
192C22⋊C4×C61—22no—
193C22⋊C4×C61—22no—
194C42×C61—21no—
195C42×C61C22×C412yes—
196C42×C61C22×C411yescharacteristic
197C22⋊Dic61—22no—
198C22⋊Dic61—22no—
199C22⋊Dic61—22no—
200#11758198—22no—
201#11758198—21no—
202#11758198—22no—
203#11758198—21no—
204#11758198—21no—
205#11758198C22⋊C412yes—
206#11758198C22⋊C412yes—
207#11758198C22⋊C412yes—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61C22⋊C411yescharacteristic
211C4⋊Dic61C22⋊C412yes—
212C4⋊Dic61C22×C412yes—
213C4⋊Dic61C22×C411yescharacteristic
214#11758196—22no—
215#11758196—22no—
216#11758196—22no—
217#11758196—22no—
218C4×Dic61—21no—
219C4×Dic61—21no—
220C4×Dic61C2×D411yescharacteristic
221C4×Dic61C2×D411yescharacteristic
222C4×Dic61C2×D412yes—
223C4×Dic61C2×D412yes—
224C4×Dic61C2411yescharacteristic
225C23×C61—22no—
226C22×D61—21no—
227C22×D61—44no—
228C22×D61—44no—
229C22×D61—82no—
230C22×D61C2×C22⋊C412yes—
231C22×D61C2×C22⋊C411yescharacteristic
232Q8×C61—22no—
233Q8×C61—21no—
234Q8×C61—21no—
235D4×C61—22no—
236D4×C61—22no—
237D4×C61—42no—
238D4×C61—81no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—82no—
242C2×C4×C61C2×C22⋊C412yes—
243C2×C4×C61C2×C22⋊C412yes—
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C23×C411yescharacteristic
247C22⋊D61—44no—
248C22⋊D61—44no—
249C22⋊D61—82no—
250C2×Dic61—21no—
251C2×Dic61—22no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61C2×C22⋊C411yescharacteristic
255C2×Dic61C2×C22⋊C412yes—
256C2×Dic61C22×D411yescharacteristic
257C2×Dic61C22×D411yescharacteristic
258D244—21no—
259D244—22no—
260D244—42no—
261D244—42no—
262D244#13112yes—
263C4×D61—22no—
264C4×D61—21no—
265C4×D61—22no—
266C4×D61—21no—
267C4×D61—22no—
268C4×D61—22no—
269C4×D61—21no—
270C4×D61—22no—
271C4×D61—21no—
272C4×D61—42no—
273C4×D61—42no—
274C4×D61—42no—
275C4×D61—41no—
276C4×D61—41no—
277C4×D61—82no—
278C4×D61—82no—
279Dic122—81no—
280#92420—41no—
281#92420—41no—
282#92420—42no—
283#92420—41no—
284#92420—42no—
285#92420—41no—
286#17522—611nomaximal, Sylow
287C22×C61—42no—
288C22×C61—42no—
289C22×C61—42no—
290C22×C61C22×C22⋊C411yescharacteristic
291C2×D61—21no—
292C2×D61—22no—
293C2×D61—22no—
294C2×D61—22no—
295C2×D61—21no—
296C2×D61—21no—
297C2×D61—22no—
298C2×D61—41no—
299C2×D61—41no—
300C2×D61—88no—
301C4×C61—21no—
302C4×C61—22no—
303C4×C61—22no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61#42112yes—
307C4×C61C2×#13111yescharacteristic
308C4×C61#56811yescharacteristic
309Dic61—21no—
310Dic61—22no—
311Dic61—21no—
312Dic61—42no—
313Dic61—41no—
314Dic61#57611yescharacteristic
315Dic61#57711yescharacteristic
316#3481—611no—
317#2991—611no—
318#2922—611no—
319#1495—612no—
320#1492—611no—
321#1489—611no—
322#1486—612no—
323#1484—611no—
324#1482—612no—
325#1478—612no—
326#1476—611no—
327C2×C61—82no—
328C2×C61#288011yescharacteristic
329C2×C61#289711yescharacteristic
330C2×C61C2×#42111yescharacteristic
331D61—41no—
332D61—42no—
333D61—42no—
334#559#692311yescharacteristic
335#554—611no—
336#554—611no—
337#553—612no—
338#553—1221no—
339#551—1222no—
340#546—612no—
341#546—1221no—
342D42—611no—
343#532—611no—
344#529—612no—
345#529—1221no—
346#525—1222no—
347C4×#41—611no—
348C4×#41—611no—
349C2×C4×D4—612no—
350C2×C4×D4—1221no—
351C8×Q8—1221no—
352#436—1221no—
353#436—1222no—
354#433—1222no—
355#431—612no—
356#431—612no—
357#431—611no—
358#431—612no—
359#424—1221no—
360#423—612no—
361#423—611no—
362#408—1221no—
363C4×M16—611no—
364C2×C4×C8—611no—
365Q8⋊C8—611no—
366Q8⋊C8—611no—
367Q8⋊C8—612no—
368D4⋊C8—612no—
369D4⋊C8—612no—
370D4⋊C8—612no—
371D4⋊C8—612no—
372D4⋊C8—611no—
373D4⋊C8—611no—
374D4⋊C8—612no—
375C61#1752211yescharacteristic, complemented, Sylow
376C2×#41—611no—
377C2×#41—611no—
378C2×#41—1221no—
379C22×D4—612no—
380C22×D4—1221no—
381C23×C4—1222no—
382C2×M16—1222no—
383C2×M16—1221no—
384C2×M16—1221no—
385C22×C8—1221no—
386C4⋊Q8—611no—
387#127C2×#692311yescharacteristic
388#126—1222no—
389#126—1222no—
390#125—612no—
391#125—1221no—
392#124—21no—
393#124—1221no—
394#124C2×#692312yes—
395#123—22no—
396#123—1222no—
397#123—1222no—
398#123—1222no—
399C22⋊Q8—612no—
400C22⋊Q8—1222no—
401C22⋊Q8—1221no—
402C22⋊Q8—1222no—
403#121—612no—
404#121—612no—
405#121—1222no—
406#121—1222no—
407#121—1221no—
408#121—1221no—
409#120—1222no—
410#120—1222no—
411C4×Q8—611no—
412C4×Q8—611no—
413C4×Q8C2×#692311yescharacteristic
414C4×Q8C2×#692311yescharacteristic
415C4×D4—21no—
416C4×D4—611no—
417C4×D4—612no—
418C4×D4—612no—
419C4×D4—611no—
420C4×D4—611no—
421C4×D4—612no—
422C4×D4—612no—
423C4×D4—611no—
424C4×D4—1221no—
425C4×D4—1221no—
426C4×D4—1222no—
427C4×D4—1222no—
428C4×D4—1221no—
429C4×D4—1222no—
430C4×D4—1222no—
431C4×D4—1221no—
432C4×D4C2×#692312yes—
433#117—612no—
434#117—611no—
435#117—611no—
436#117—1222no—
437#117—1221no—
438C2×C4⋊C4—612no—
439C2×C4⋊C4—1221no—
440C2×C22⋊C4—1222no—
441C2×C22⋊C4—1222no—
442C2×C42—612no—
443C2×C42—611no—
444C2×C42—1221no—
445C4⋊C8—612no—
446C4⋊C8—611no—
447C4⋊C8—611no—
448C4⋊C8—1221no—
449C4⋊C8—1222no—
450C4⋊C8—1222no—
451C4⋊C8—1222no—
452C22⋊C8—1222no—
453C22⋊C8—1222no—
454C22⋊C8—1222no—
455C22⋊C8—1221no—
456#97—1221no—
457C4×C8—611no—
458C4×C8—611no—
459C4×C8—612no—
460C4×C8—1221no—
461C4×C8—1221no—
462C4×C8—1222no—
463C24—1222no—
464#41—1222no—
465#41—4881no—
466#41—4881no—
467#41—2442no—
468#41—2441no—
469C2×Q8—1221no—
470C2×Q8C22⋊#692311yescharacteristic
471C2×Q8C22×#692311yescharacteristic
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—611no—
475C2×D4—612no—
476C2×D4—611no—
477C2×D4—1221no—
478C2×D4—1222no—
479C2×D4—1224no—
480C2×D4—1222no—
481C2×D4—1222no—
482C2×D4—2442no—
483C2×D4—2444no—
484C2×D4C22×#692312yes—
485C22×C4—22no—
486C22×C4—612no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—612no—
490C22×C4—611no—
491C22×C4—612no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—1221no—
498C22×C4—4882no—
499C22×C4—4882no—
500M16—4882no—
501M16—4881no—
502M16—2442no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1222no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1222no—
509C2×C8—2442no—
510C4⋊C4—21no—
511C4⋊C4—611no—
512C4⋊C4—612no—
513C4⋊C4—612no—
514C4⋊C4—611no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1221no—
518C4⋊C4—1222no—
519C4⋊C4—1221no—
520C4⋊C4—1222no—
521C4⋊C4C22⋊#692311yescharacteristic
522C4⋊C4C22⋊#692312yes—
523C4⋊C4C22×#692311yescharacteristic
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—612no—
528C22⋊C4—612no—
529C22⋊C4—612no—
530C22⋊C4—1221no—
531C22⋊C4—1222no—
532C22⋊C4—1221no—
533C22⋊C4—1222no—
534C22⋊C4—1222no—
535C22⋊C4—1222no—
536C22⋊C4—1222no—
537C22⋊C4—1221no—
538C42—21no—
539C42—611no—
540C42—611no—
541C42—612no—
542C42—612no—
543C42—611no—
544C42—1221no—
545C42—1221no—
546C42C22×#692312yes—
547C42C22×#692311yescharacteristic
548C23—22no—
549C23—612no—
550C23—611no—
551C23—1221no—
552C23—4882no—
553C23—2444no—
554C23—2444no—
555Q8—22no—
556Q8—21no—
557Q8—21no—
558Q8—4881no—
559D4—22no—
560D4—22no—
561D4—42no—
562D4—81no—
563D4—612no—
564D4—1222no—
565D4—1221no—
566D4—4882no—
567D4—2442no—
568D4—2442no—
569D4—2444no—
570D4—2444no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—82no—
574C2×C4—611no—
575C2×C4—611no—
576C2×C4—612no—
577C2×C4—611no—
578C2×C4—1222no—
579C2×C4—1221no—
580C2×C4—1222no—
581C2×C4—1221no—
582C2×C4—1222no—
583C2×C4—1222no—
584C2×C4—1221no—
585C2×C4—1222no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1222no—
589C2×C4—4882no—
590C2×C4—4882no—
591C2×C4—2442no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2442no—
598C2×C4C2×C22⋊#692312yes—
599C2×C4C2×C22⋊#692312yes—
600C2×C4C2×C22⋊#692311yescharacteristic
601C2×C4C2×C22⋊#692311yescharacteristic
602C2×C4C23×#692311yescharacteristic
603C8—2441no—
604C8—2441no—
605C8—2442no—
606C8—2441no—
607C8—2441no—
608C8—2442no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—1221no—
613C22—1222no—
614C22—1222no—
615C22—1222no—
616C22—1221no—
617C22—1221no—
618C22—1222no—
619C22—4888no—
620C22—2441no—
621C22—2441no—
622C22C22×C22⋊#692311yescharacteristic
623C4—22no—
624C4—21no—
625C4—22no—
626C4—41no—
627C4—41no—
628C4—611no—
629C4—611no—
630C4—1221no—
631C4—1222no—
632C4—1221no—
633C4—2442no—
634C4—2441no—
635C4#1684075912yes—
636C4D61⋊#13111yescharacteristic
637C4#1684084111yescharacteristic
638C2—82no—
639C2—2442no—
640C2—2441no—
641C2—2442no—
642C2#2146241011yescharacteristic
643C2#2146244011yescharacteristic
644C2C2×#1684075911yescharacteristic
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#17522
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. #21462753C2semidirect
2. #21462504C2semidirect
3. #21461175C2semidirect
4. #21461165C2semidirect
5. #21461157C2semidirect
6. #21461145C2semidirect
7. #21461139C2semidirect
8. #21461131C2semidirect
9. #21461119C2semidirect
10. #21461113C2semidirect
16. #16840781C22semidirect
17. #16840781C22semidirect
18. #16840781C22semidirect
19. #16840780C22semidirect
21. #16840766C22semidirect
22. #16840766C22semidirect
24. C4×#11758213C22semidirect
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
375. C61#17522semidirect

Maximal subgroups

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

subgroupindex
#214627532
#214625042
#214611752
#214611652
#214611572
#214611452
#214611392
#214611312
#214611192
#214611132
#214594282
#1752261