Finite Group Databasebeta

← #30611361

Subgroups — #30611361

Order 15616.

918 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462713C211yesmaximal, characteristic, complemented
2#21462339C211yesmaximal, characteristic, complemented
3#21461262C212yesmaximal, complemented
4#21461261C212yesmaximal, complemented
5#21461251C211yesmaximal, characteristic, complemented
6#21461249C211yesmaximal, characteristic
7#21461202C211yesmaximal, characteristic, complemented
8#21461198C211yesmaximal, characteristic
9#21461193C212yesmaximal, complemented
10#21461190C212yesmaximal, complemented
11#21459682C211yesmaximal, characteristic
12#16840811—22no—
13#16840811—21no—
14Q8×#92420—21no—
15#16840781C2211yescharacteristic
16#16840781C2212yes—
17#16840780C2211yescharacteristic
18#16840780C2212yes—
19(C2×#6923)⋊Q8—21no—
20#16840711—22no—
21#16840707C2212yes—
22#16840706C2211yescharacteristic
23C4⋊C4×#6923—21no—
24C4×C4⋊#6923C2211yescharacteristic
25#16840698C2212yes—
26#16840615C2212yes—
27#16840615C2212yes—
28#16840615C2212yes—
29#16840615C2212yes—
30#16840614C2212yes—
31#16840614C2211yescharacteristic
32#16840614C2211yescharacteristic
33#16840613C2212yes—
34#16840613C2211yescharacteristic
35#16840613C2211yescharacteristic
36#16840495C412yescomplemented
37#16840489C412yescomplemented
38#16840485C411yescharacteristic, complemented
39#16840483—22no—
40#16840482C411yescharacteristic, complemented
41#16840473C411yescharacteristic, complemented
42#16840447—21no—
43#16840446—21no—
44#16840358—22no—
45#16840335—21no—
46#16840317C2211yescharacteristic
47#16840309C2211yescharacteristic
48#16840297C2212yes—
49#16840294C2212yes—
50#16840243—21no—
51C4×Q8×D61C2211yescharacteristic
52C61×#554C411yescharacteristic, complemented
53C2×C4⋊#6923—21no—
54C2×C4⋊#6923—21no—
55C2×C4×#6923—22no—
56C2×C4×#6923—21no—
57#14519844—22no—
58#14519844—21no—
59C4⋊#92420—22no—
60C4⋊#92420—21no—
61#14519834C2311yescharacteristic
62#14519834C2311yescharacteristic
63#14519834C2312yes—
64C4×#92420—22no—
65C4×#92420—21no—
66C4×#92420—22no—
67C4×#92420—21no—
68(C2×#6923)⋊C4—21no—
69(C2×#6923)⋊C4—22no—
70(C2×#6923)⋊C4—22no—
71(C2×#6923)⋊C4—22no—
72(C2×#6923)⋊C4C2312yes—
73(C2×#6923)⋊C4C2311yescharacteristic
74(C2×#6923)⋊C4C2311yescharacteristic
75C2×#11758222—21no—
76C2×Q8×D61D411yescharacteristic
77#14519777C2×C411yescharacteristic
78#14519776—22no—
79#14519776C2×C411yescharacteristic
80#14519773C2×C411yescharacteristic
81#14519772—22no—
82#14519772—22no—
83#14519770—22no—
84#14519770—22no—
85#14519768—21no—
86#14519768C2×C412yes—
87#14519767—22no—
88#14519767—22no—
89#14519765C2×C412yes—
90#14519764C2×C412yes—
91#14519763—22no—
92#14519763—21no—
93#14519762—22no—
94#14519760—22no—
95#14519759—22no—
96#14519759—22no—
97#14519757—21no—
98#14519757C2312yes—
99#14519756—21no—
100#14519753—21no—
101#14519753—21no—
102#14519753—22no—
103#14519752C2311yescharacteristic
104#14519750—21no—
105#14519748C2311yescharacteristic
106Q8×Dic61—21no—
107Q8×Dic61D411yescharacteristic
108C4×Dic122C2×C412yes—
109C4×Dic122C2×C411yescharacteristic
110C4×Dic122C2×C412yes—
111#14519742—22no—
112#14519742C2×C412yes—
113#14519742C2×C412yes—
114#14519742D412yes—
115#14519742D411yescharacteristic
116C4×D244C2×C411yescharacteristic
117#14519739—22no—
118#14519739C2×C412yes—
119C4×C22⋊D61—22no—
120C4×C22⋊D61—22no—
121C4×C22⋊D61—21no—
122#14519736—21no—
123#14519736—22no—
124#14519735—21no—
125#14519735—21no—
126#14519735—22no—
127C4⋊C4×D61—21no—
128C4⋊C4×D61D412yes—
129C4⋊C4×D61D411yescharacteristic
130C42×D61C2311yescharacteristic
131C42×D61C2312yes—
132C4⋊Q8×C61C2×C411yescharacteristic
133#125×C61C2×C412yes—
134C22⋊Q8×C61—21no—
135C22⋊Q8×C61—22no—
136C4×Q8×C61C2×C411yescharacteristic
137C4×D4×C61C2×C411yescharacteristic
138C4×D4×C61C2×C412yes—
139C2×C4⋊C4×C61—21no—
140C22×#6923—21no—
141C22×#6923—22no—
142C22×#6923—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61C22⋊C411yescharacteristic
145C2×D4×C61C22×C411yescharacteristic
146C22×C4×C61—21no—
147C22×C4×C61—22no—
148C2×C22⋊D61—21no—
149C2×C22⋊D61—21no—
150C2×C22⋊D61—21no—
151Q8×D61—22no—
152Q8×D61—22no—
153#11758222—42no—
154#11758222—42no—
155C2×D244—21no—
156C2×D244C22×C411yescharacteristic
157C2×C4×D61—21no—
158C2×C4×D61—22no—
159C2×C4×D61C2×D412yes—
160C2×C4×D61C2×D412yes—
161C2×C4×D61C2×D411yescharacteristic
162C2×C4×D61C2×D411yescharacteristic
163C2×C4×D61C2411yescharacteristic
164C2×Dic122—21no—
165C2×Dic122C22⋊C411yescharacteristic
166C2×Dic122C22⋊C412yes—
167C2×Dic122C22×C412yes—
168C2×#92420—22no—
169C2×#92420—21no—
170C2×#92420—21no—
171C4⋊#6923—42no—
172C4⋊#6923—42no—
173C4×#6923—44no—
174C4×#6923—42no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61C22⋊C411yescharacteristic
179C4⋊C4×C61C22⋊C412yes—
180C4⋊C4×C61C22×C411yescharacteristic
181C4⋊C4×C61C22×C412yes—
182C22⋊C4×C61—21no—
183C22⋊C4×C61—22no—
184C42×C61C22×C412yes—
185C42×C61C22×C411yescharacteristic
186C22⋊Dic61—22no—
187C22⋊Dic61—22no—
188C22⋊Dic61—21no—
189#11758198—22no—
190#11758198—22no—
191#11758198—22no—
192#11758198—21no—
193#11758198—22no—
194#11758198—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—22no—
198C4⋊Dic61—22no—
199C4⋊Dic61C22⋊C411yescharacteristic
200C4⋊Dic61C22⋊C412yes—
201C4⋊Dic61C22×C411yescharacteristic
202#11758196—22no—
203#11758196—22no—
204#11758196—21no—
205#11758196—22no—
206#11758196—22no—
207#11758196C22⋊C412yes—
208#11758196C22⋊C412yes—
209#11758196C22⋊C412yes—
210C4×Dic61—22no—
211C4×Dic61—22no—
212C4×Dic61—22no—
213C4×Dic61C22×C412yes—
214C4×Dic61C22×C411yescharacteristic
215C4×Dic61C2×D412yes—
216C4×Dic61C2×D411yescharacteristic
217C4×Dic61C2×D412yes—
218C4×Dic61C2×D411yescharacteristic
219C23×C61—21no—
220C22×D61—21no—
221C22×D61C22×D411yescharacteristic
222C2×#6923—22no—
223C2×#6923—41no—
224C2×#6923—41no—
225C2×#6923—42no—
226C2×#6923—41no—
227C2×#6923—41no—
228C2×#6923—42no—
229C2×#6923—41no—
230C2×#6923—42no—
231Q8×C61—22no—
232Q8×C61—81no—
233D4×C61—41no—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
236C2×C4×C61—42no—
237C2×C4×C61—82no—
238C2×C4×C61C2×C22⋊C412yes—
239C2×C4×C61C2×C22⋊C412yes—
240C2×C4×C61C2×C22⋊C411yescharacteristic
241C2×C4×C61C2×C22⋊C411yescharacteristic
242C2×C4×C61C23×C411yescharacteristic
243C22⋊D61—41no—
244C22⋊D61—41no—
245C22⋊D61—42no—
246C22⋊D61—81no—
247C2×Dic61—22no—
248C2×Dic61—21no—
249C2×Dic61C2×C22⋊C412yes—
250C2×Dic61C2×C22⋊C411yescharacteristic
251C2×Dic61C2×C22⋊C412yes—
252C2×Dic61C2×C22⋊C411yescharacteristic
253C2×Dic61C22×D411yescharacteristic
254D244—41no—
255D244—41no—
256D244—41no—
257C4×D61—22no—
258C4×D61—22no—
259C4×D61—42no—
260C4×D61—41no—
261C4×D61—41no—
262C4×D61—42no—
263C4×D61—82no—
264C4×D61#13512yes—
265C4×D61#13512yes—
266Dic122—22no—
267Dic122—24no—
268Dic122—42no—
269Dic122—42no—
270#92420—42no—
271#92420—41no—
272#92420—41no—
273#17616—611nomaximal, Sylow
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61C22×C22⋊C411yescharacteristic
278C2×D61—21no—
279C2×D61—21no—
280C2×D61—41no—
281C2×D61—41no—
282C2×D61—41no—
283C2×D61C2×#13511yescharacteristic
284C2×D61C2×#13511yescharacteristic
285#6923—42no—
286#6923—81no—
287#6923—82no—
288C4×C61—21no—
289C4×C61—22no—
290C4×C61—22no—
291C4×C61—21no—
292C4×C61—41no—
293C4×C61—42no—
294C4×C61#41612yes—
295Dic61—21no—
296Dic61—22no—
297Dic61—22no—
298Dic61—21no—
299Dic61—42no—
300Dic61—41no—
301Dic61#41612yes—
302#3507—611no—
303#2982—611no—
304#2356—611no—
305#1521—612no—
306#1521—612no—
307#1518—611no—
308#1517—611no—
309#1503—611no—
310#1502—611no—
311#1500—612no—
312#1500—612no—
313C2×C61—81no—
314C2×C61#288411yescharacteristic
315C2×C61C2×#41611yescharacteristic
316C2×C61#290111yescharacteristic
317D61—21no—
318D61—21no—
319D61—81no—
320#559—612no—
321#557—612no—
322#556—611no—
323#556—611no—
324#556—1222no—
325#554—611no—
326#554#692311yescharacteristic, complemented
327#552—1221no—
328#552—1221no—
329#539—1222no—
330#536—1221no—
331#533—611no—
332#532—611no—
333#529—612no—
334#529—612no—
335#520—1221no—
336C2×C4×Q8—611no—
337C8×Q8—1221no—
338C8×Q8—1221no—
339C8×Q8—1222no—
340#431—612no—
341#431—611no—
342#431—612no—
343#431—611no—
344#391—1221no—
345#387—1222no—
346#383—612no—
347#382—611no—
348C4×C4⋊C4—611no—
349C4×C4⋊C4—1221no—
350#376—612no—
351#328—612no—
352#328—612no—
353#328—612no—
354#328—612no—
355#327—611no—
356#327—612no—
357#327—611no—
358#327—611no—
359#327—612no—
360#327—611no—
361C61#1761611yescharacteristic, complemented, Sylow
362C2×#41—1221no—
363C22×Q8—611no—
364C4⋊Q8—611no—
365C4⋊Q8—611no—
366C4⋊Q8—1222no—
367C4⋊Q8C2×#692311yescharacteristic
368#127—611no—
369#126—1222no—
370#126—1222no—
371#126—1222no—
372#126—1222no—
373#125—612no—
374#125—1221no—
375#125—1222no—
376#125—1222no—
377#125C2×#692312yes—
378#124—612no—
379#124—612no—
380#124—1221no—
381#124—1222no—
382#123—1222no—
383#123—1222no—
384#123—1222no—
385#123—1222no—
386C22⋊Q8—21no—
387C22⋊Q8—22no—
388C22⋊Q8—612no—
389C22⋊Q8—1221no—
390C22⋊Q8—1222no—
391C22⋊Q8—1221no—
392C22⋊Q8—1221no—
393C22⋊Q8—1221no—
394#121—611no—
395#121—611no—
396#121—1221no—
397C4×Q8—611no—
398C4×Q8—611no—
399C4×Q8—612no—
400C4×Q8—611no—
401C4×Q8—612no—
402C4×Q8—612no—
403C4×Q8—612no—
404C4×Q8—612no—
405C4×Q8—1221no—
406C4×Q8—1222no—
407C4×Q8C2×#692311yescharacteristic
408C4×D4—612no—
409C4×D4—611no—
410C4×D4—1221no—
411C4×D4—1222no—
412C4×D4—1222no—
413C4×D4—1222no—
414C4×D4C2×#692312yes—
415C4×D4C2×#692311yescharacteristic
416#117—1222no—
417#117—1222no—
418#117—1221no—
419#117—1221no—
420#117—1221no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—612no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C42—612no—
428C2×C42—611no—
429C2×C42—1222no—
430C2×C42—1221no—
431C4⋊C8—1222no—
432C4⋊C8—1221no—
433C4⋊C8—1222no—
434C4⋊C8—1221no—
435C22⋊C8—611no—
436C22⋊C8—611no—
437C22⋊C8—612no—
438C4×C8—1222no—
439C4×C8—1222no—
440C4×C8—1221no—
441C4×C8—1221no—
442#95—611no—
443#95—612no—
444#95—611no—
445#95—1222no—
446#95—1222no—
447#95—1221no—
448#95—1222no—
449#41—2442no—
450#41—2442no—
451C2×Q8—21no—
452C2×Q8—611no—
453C2×Q8—612no—
454C2×Q8—612no—
455C2×Q8—1221no—
456C2×Q8—1222no—
457C2×Q8—1222no—
458C2×Q8C22⋊#692311yescharacteristic
459C2×D4—611no—
460C2×D4—1221no—
461C2×D4—1221no—
462C2×D4—1221no—
463C2×D4—1221no—
464C2×D4C22×#692311yescharacteristic
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—611no—
468C22×C4—612no—
469C22×C4—612no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—1222no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1222no—
477C2×C8—1222no—
478C2×C8—1221no—
479C2×C8—1221no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—611no—
484C4⋊C4—612no—
485C4⋊C4—612no—
486C4⋊C4—612no—
487C4⋊C4—612no—
488C4⋊C4—611no—
489C4⋊C4—1222no—
490C4⋊C4—1221no—
491C4⋊C4—1222no—
492C4⋊C4—1222no—
493C4⋊C4—1222no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—2442no—
499C4⋊C4—2442no—
500C4⋊C4C22⋊#692311yescharacteristic
501C4⋊C4C22⋊#692312yes—
502C4⋊C4C22×#692312yes—
503C4⋊C4C22×#692311yescharacteristic
504C22⋊C4—21no—
505C22⋊C4—22no—
506C22⋊C4—1221no—
507C22⋊C4—1222no—
508C22⋊C4—1222no—
509C22⋊C4—1221no—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C22⋊C4—1222no—
513C22⋊C4—1222no—
514C22⋊C4—1221no—
515C42—612no—
516C42—612no—
517C42—611no—
518C42—611no—
519C42—612no—
520C42—611no—
521C42—1222no—
522C42—1222no—
523C42—1222no—
524C42—2442no—
525C42—2444no—
526C42C22×#692311yescharacteristic
527C42C22×#692312yes—
528C23—21no—
529C23—611no—
530C23—1221no—
531Q8—22no—
532Q8—81no—
533Q8—1224no—
534Q8—1222no—
535Q8—2442no—
536Q8—2442no—
537D4—41no—
538D4—4881no—
539D4—2441no—
540D4—2441no—
541D4—2441no—
542D4—2442no—
543D4—2441no—
544D4—2441no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—42no—
548C2×C4—82no—
549C2×C4—611no—
550C2×C4—612no—
551C2×C4—612no—
552C2×C4—612no—
553C2×C4—612no—
554C2×C4—611no—
555C2×C4—611no—
556C2×C4—1222no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1221no—
561C2×C4—4882no—
562C2×C4—2442no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2442no—
567C2×C4—2442no—
568C2×C4—2441no—
569C2×C4—2442no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4—2442no—
574C2×C4C2×C22⋊#692312yes—
575C2×C4C2×C22⋊#692312yes—
576C2×C4C2×C22⋊#692311yescharacteristic
577C2×C4C2×C22⋊#692311yescharacteristic
578C2×C4C23×#692311yescharacteristic
579C8—2442no—
580C8—2441no—
581C8—2441no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—611no—
586C22—611no—
587C22—1221no—
588C22—1221no—
589C22—2441no—
590C22—2441no—
591C22—2441no—
592C22C22×C22⋊#692311yescharacteristic
593C4—21no—
594C4—22no—
595C4—22no—
596C4—21no—
597C4—41no—
598C4—42no—
599C4—612no—
600C4—1221no—
601C4—1222no—
602C4—1222no—
603C4—1221no—
604C4—4882no—
605C4—4881no—
606C4—2442no—
607C4—2442no—
608C4—2441no—
609C4#1684075212yes—
610C2—81no—
611C2—1221no—
612C2—1221no—
613C2—4881no—
614C2#2146242011yescharacteristic
615C2C2×#1684075211yescharacteristic
616C2#2146244911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#554C4
socleC22×C61C22×C22⋊C4
prime radical#16840309C22
cosocleC2×C4×D61C24
hypercenter#554#6923
hypocenterC61#17616
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #554

Complements

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

NH ≅ G/Ntype
1. #21462713C2semidirect
2. #21462339C2semidirect
3. #21461262C2semidirect
4. #21461261C2semidirect
5. #21461251C2semidirect
7. #21461202C2semidirect
9. #21461193C2semidirect
10. #21461190C2semidirect
36. #16840495C4semidirect
37. #16840489C4semidirect
38. #16840485C4semidirect
40. #16840482C4semidirect
41. #16840473C4semidirect
52. C61×#554C4semidirect
326. #554#6923semidirect
361. C61#17616semidirect

Maximal subgroups

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

subgroupindex
#214627132
#214623392
#214612622
#214612612
#214612512
#214612492
#214612022
#214611982
#214611932
#214611902
#214596822
#1761661