Finite Group Databasebeta

← #30611362

Subgroups — #30611362

Order 15616.

918 conjugacy classes of subgroups.

The subgroup lattice has 606 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#21461261C212yesmaximal, complemented
4#21461260C212yesmaximal, complemented
5#21461252C211yesmaximal, characteristic, complemented
6#21461248C211yesmaximal, characteristic
7#21461201C211yesmaximal, characteristic, complemented
8#21461199C211yesmaximal, characteristic
9#21461191C212yesmaximal, complemented
10#21461190C212yesmaximal, complemented
11#21459676C211yesmaximal, 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#16840616C2212yes—
27#16840616C2211yescharacteristic
28#16840616C2211yescharacteristic
29#16840615C2212yes—
30#16840615C2211yescharacteristic
31#16840615C2211yescharacteristic
32#16840613C2212yes—
33#16840613C2212yes—
34#16840613C2212yes—
35#16840613C2212yes—
36#16840498C411yescharacteristic, complemented
37#16840491C411yescharacteristic, complemented
38#16840490C411yescharacteristic, complemented
39#16840482—21no—
40#16840481—21no—
41#16840481C412yescomplemented
42#16840471C412yescomplemented
43#16840449—22no—
44#16840355—22no—
45#16840332—21no—
46#16840316C2211yescharacteristic
47#16840310C2211yescharacteristic
48#16840294C2212yes—
49#16840291C2212yes—
50#16840241—21no—
51C4×Q8×D61C2211yescharacteristic
52C61×#559C411yescharacteristic, 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×#11758224—21no—
76C2×Q8×D61D411yescharacteristic
77#14519777—21no—
78#14519777C2×C411yescharacteristic
79#14519776C2×C411yescharacteristic
80#14519775—21no—
81#14519775C2×C411yescharacteristic
82#14519770—22no—
83#14519770—22no—
84#14519770—22no—
85#14519770—22no—
86#14519769C2×C412yes—
87#14519768—22no—
88#14519768C2×C412yes—
89#14519767—22no—
90#14519767—21no—
91#14519764—21no—
92#14519764—21no—
93#14519764C2×C412yes—
94#14519759—22no—
95#14519759—22no—
96#14519758—22no—
97#14519757C2311yescharacteristic
98#14519756—22no—
99#14519755—21no—
100#14519754—22no—
101#14519754—22no—
102#14519754—22no—
103#14519753C2311yescharacteristic
104#14519748—21no—
105#14519748C2312yes—
106Q8×Dic61C2×C412yes—
107Q8×Dic61D411yescharacteristic
108C4×Dic122—21no—
109C4×Dic122—21no—
110C4×Dic122C2×C412yes—
111C4×Dic122C2×C411yescharacteristic
112C4×Dic122C2×C412yes—
113C4×Dic122C2×C411yescharacteristic
114#14519742—21no—
115#14519742D412yes—
116#14519742D411yescharacteristic
117D4×Dic61—21no—
118D4×Dic61—21no—
119D4×Dic61C2×C412yes—
120D4×Dic61C2×C412yes—
121C4×C22⋊D61—22no—
122#14519737—22no—
123#14519736—22no—
124#14519734—21no—
125#14519733—22no—
126#14519733—22no—
127C2×#11758196—22no—
128C4⋊C4×D61D412yes—
129C4⋊C4×D61D411yescharacteristic
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—
140C22×#6923—21no—
141C22×#6923—22no—
142C22×#6923—21no—
143C2×Q8×C61C22⋊C411yescharacteristic
144C2×Q8×C61C22×C411yescharacteristic
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61C22×C412yes—
148C22×C4×C61—22no—
149C2×C22⋊D61—22no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152Q8×D61—22no—
153Q8×D61—22no—
154#11758224—42no—
155#11758224—42no—
156C2×C4×D61C2×D412yes—
157C2×C4×D61C2×D412yes—
158C2×C4×D61C2×D411yescharacteristic
159C2×C4×D61C2×D411yescharacteristic
160C2×C4×D61C2411yescharacteristic
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122C22⋊C412yes—
164C2×Dic122C22⋊C411yescharacteristic
165C2×Dic122C22×C411yescharacteristic
166C2×#92420—22no—
167C2×#92420—21no—
168C2×#92420—21no—
169C4⋊#6923—42no—
170C4⋊#6923—42no—
171C4×#6923—44no—
172C4×#6923—42no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61C22⋊C411yescharacteristic
175C4⋊C4×C61C22⋊C412yes—
176C4⋊C4×C61C22×C411yescharacteristic
177C22⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C42×C61—21no—
181C42×C61C22×C412yes—
182C42×C61C22×C411yescharacteristic
183C22⋊Dic61—22no—
184C22⋊Dic61—22no—
185C22⋊Dic61—22no—
186C22⋊Dic61—22no—
187C22⋊Dic61—22no—
188#11758198—21no—
189#11758198—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—22no—
192C4⋊Dic61—22no—
193C4⋊Dic61—21no—
194C4⋊Dic61C22⋊C411yescharacteristic
195C4⋊Dic61C22⋊C412yes—
196C4⋊Dic61C22×C411yescharacteristic
197C4⋊Dic61C22×C412yes—
198#11758196—22no—
199#11758196—22no—
200#11758196—22no—
201#11758196—21no—
202#11758196—21no—
203#11758196—22no—
204#11758196—21no—
205#11758196—21no—
206#11758196—21no—
207#11758196C22⋊C412yes—
208#11758196C22⋊C412yes—
209#11758196C22⋊C412yes—
210C4×Dic61—22no—
211C4×Dic61—21no—
212C4×Dic61—22no—
213C4×Dic61C22×C411yescharacteristic
214C4×Dic61C22×C412yes—
215C4×Dic61C2×D412yes—
216C4×Dic61C2×D411yescharacteristic
217C4×Dic61C2×D412yes—
218C4×Dic61C2×D411yescharacteristic
219C23×C61—22no—
220C22×D61C22×D411yescharacteristic
221C2×#6923—22no—
222C2×#6923—41no—
223C2×#6923—41no—
224C2×#6923—42no—
225C2×#6923—41no—
226C2×#6923—41no—
227C2×#6923—42no—
228C2×#6923—41no—
229C2×#6923—42no—
230Q8×C61—22no—
231Q8×C61—41no—
232D4×C61—42no—
233D4×C61—42no—
234D4×C61—81no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
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—42no—
244C22⋊D61—42no—
245C2×Dic61—21no—
246C2×Dic61—22no—
247C2×Dic61—21no—
248C2×Dic61—44no—
249C2×Dic61—82no—
250C2×Dic61C2×C22⋊C412yes—
251C2×Dic61C2×C22⋊C411yescharacteristic
252C2×Dic61C2×C22⋊C412yes—
253C2×Dic61C2×C22⋊C411yescharacteristic
254C2×Dic61C22×D411yescharacteristic
255C4×D61—22no—
256C4×D61—22no—
257C4×D61—42no—
258C4×D61—41no—
259C4×D61—41no—
260C4×D61#13512yes—
261C4×D61#13512yes—
262Dic122—22no—
263Dic122—24no—
264Dic122—42no—
265Dic122—41no—
266Dic122—81no—
267#92420—42no—
268#92420—41no—
269#92420—41no—
270#17616—611nomaximal, Sylow
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—42no—
274C22×C61C22×C22⋊C411yescharacteristic
275C2×D61—21no—
276C2×D61—21no—
277C2×D61C2×#13511yescharacteristic
278C2×D61C2×#13511yescharacteristic
279#6923—42no—
280#6923—81no—
281#6923—82no—
282C4×C61—21no—
283C4×C61—22no—
284C4×C61—22no—
285C4×C61—21no—
286C4×C61—41no—
287C4×C61—41no—
288C4×C61#41612yes—
289Dic61—21no—
290Dic61—22no—
291Dic61—21no—
292Dic61—22no—
293Dic61—41no—
294Dic61—42no—
295Dic61—41no—
296Dic61#41612yes—
297#3507—611no—
298#2982—611no—
299#2356—611no—
300#1521—612no—
301#1521—612no—
302#1518—611no—
303#1517—611no—
304#1503—611no—
305#1502—611no—
306#1500—612no—
307#1500—612no—
308C2×C61—82no—
309C2×C61#288411yescharacteristic
310C2×C61C2×#41611yescharacteristic
311C2×C61#290111yescharacteristic
312D61—21no—
313D61—21no—
314#559—611no—
315#559#692311yescharacteristic, complemented
316#557—611no—
317#557—611no—
318#556—612no—
319#556—1221no—
320#556—1221no—
321#554—612no—
322#552—1222no—
323#539—1222no—
324#536—1221no—
325#533—611no—
326#532—611no—
327#529—612no—
328#529—612no—
329#520—1221no—
330C2×C4×Q8—611no—
331C8×Q8—1221no—
332C8×Q8—1222no—
333C8×Q8—1221no—
334#431—612no—
335#431—611no—
336#431—612no—
337#431—611no—
338#391—1221no—
339#387—1222no—
340#383—612no—
341#382—611no—
342C4×C4⋊C4—611no—
343C4×C4⋊C4—1221no—
344#376—612no—
345#328—611no—
346#328—612no—
347#328—611no—
348#328—611no—
349#328—612no—
350#328—611no—
351#327—612no—
352#327—612no—
353#327—612no—
354#327—612no—
355C61#1761611yescharacteristic, complemented, Sylow
356C2×#41—1221no—
357C22×Q8—611no—
358C4⋊Q8—611no—
359C4⋊Q8—611no—
360C4⋊Q8—611no—
361C4⋊Q8—1221no—
362C4⋊Q8—1221no—
363#127C2×#692311yescharacteristic
364#126—1222no—
365#126—1222no—
366#126—1222no—
367#126—1222no—
368#125—612no—
369#125—612no—
370#125—1221no—
371#125—1222no—
372#125—1222no—
373#124—21no—
374#124—612no—
375#124—1221no—
376#124—1221no—
377#124C2×#692312yes—
378#123—22no—
379#123—1222no—
380#123—1222no—
381#123—1222no—
382C22⋊Q8—611no—
383C22⋊Q8—611no—
384C22⋊Q8—1222no—
385C22⋊Q8—1222no—
386C22⋊Q8—1222no—
387C22⋊Q8—1221no—
388C22⋊Q8—1222no—
389#121—612no—
390#121—1221no—
391C4×Q8—611no—
392C4×Q8—612no—
393C4×Q8—611no—
394C4×Q8—612no—
395C4×Q8—611no—
396C4×Q8—612no—
397C4×Q8—612no—
398C4×Q8—611no—
399C4×Q8—1221no—
400C4×Q8—1221no—
401C4×Q8—1221no—
402C4×Q8C2×#692311yescharacteristic
403C4×Q8C2×#692311yescharacteristic
404C4×D4—21no—
405C4×D4—612no—
406C4×D4—612no—
407C4×D4—1221no—
408C4×D4—1222no—
409C4×D4—1221no—
410C4×D4—1222no—
411C4×D4C2×#692312yes—
412#117—1221no—
413#117—1222no—
414#117—1222no—
415#117—1222no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—612no—
418C2×C4⋊C4—1222no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C42—612no—
422C2×C42—611no—
423C2×C42—1222no—
424C2×C42—1221no—
425C4⋊C8—1222no—
426C4⋊C8—1221no—
427C4⋊C8—1222no—
428C4⋊C8—1221no—
429C22⋊C8—611no—
430C22⋊C8—612no—
431C22⋊C8—611no—
432C4×C8—1222no—
433C4×C8—1222no—
434C4×C8—1221no—
435C4×C8—1221no—
436#95—611no—
437#95—612no—
438#95—611no—
439#95—1222no—
440#95—1222no—
441#95—1221no—
442#95—1222no—
443#41—2442no—
444#41—2442no—
445C2×Q8—612no—
446C2×Q8—611no—
447C2×Q8—611no—
448C2×Q8—1221no—
449C2×Q8—1222no—
450C2×Q8—1222no—
451C2×Q8—1221no—
452C2×Q8C22⋊#692311yescharacteristic
453C2×Q8C22×#692311yescharacteristic
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—1222no—
457C2×D4C22×#692312yes—
458C22×C4—22no—
459C22×C4—611no—
460C22×C4—612no—
461C22×C4—612no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1222no—
469C2×C8—1222no—
470C2×C8—1221no—
471C2×C8—1221no—
472C4⋊C4—21no—
473C4⋊C4—612no—
474C4⋊C4—612no—
475C4⋊C4—612no—
476C4⋊C4—611no—
477C4⋊C4—612no—
478C4⋊C4—611no—
479C4⋊C4—612no—
480C4⋊C4—1221no—
481C4⋊C4—1222no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1222no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C4⋊C4—1222no—
490C4⋊C4—1221no—
491C4⋊C4—1222no—
492C4⋊C4—1221no—
493C4⋊C4—2442no—
494C4⋊C4—2442no—
495C4⋊C4C22⋊#692311yescharacteristic
496C4⋊C4C22⋊#692312yes—
497C4⋊C4C22×#692311yescharacteristic
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—1221no—
502C22⋊C4—1221no—
503C22⋊C4—1222no—
504C22⋊C4—1222no—
505C22⋊C4—1222no—
506C22⋊C4—1222no—
507C22⋊C4—1222no—
508C42—21no—
509C42—612no—
510C42—612no—
511C42—611no—
512C42—611no—
513C42—611no—
514C42—612no—
515C42—1222no—
516C42—1222no—
517C42—1221no—
518C42—2442no—
519C42—2444no—
520C42C22×#692312yes—
521C42C22×#692311yescharacteristic
522C23—22no—
523C23—611no—
524Q8—22no—
525Q8—41no—
526Q8—1224no—
527Q8—1222no—
528Q8—4881no—
529Q8—2441no—
530Q8—2442no—
531D4—42no—
532D4—42no—
533D4—81no—
534D4—2442no—
535D4—2442no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—82no—
539C2×C4—611no—
540C2×C4—612no—
541C2×C4—612no—
542C2×C4—612no—
543C2×C4—611no—
544C2×C4—612no—
545C2×C4—611no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1221no—
550C2×C4—1222no—
551C2×C4—1221no—
552C2×C4—4882no—
553C2×C4—2442no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2444no—
557C2×C4—2441no—
558C2×C4—2442no—
559C2×C4—2442no—
560C2×C4—2441no—
561C2×C4—2442no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4C2×C22⋊#692312yes—
566C2×C4C2×C22⋊#692312yes—
567C2×C4C2×C22⋊#692311yescharacteristic
568C2×C4C2×C22⋊#692311yescharacteristic
569C2×C4C23×#692311yescharacteristic
570C8—2442no—
571C8—2441no—
572C8—2441no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—611no—
577C22—611no—
578C22—1221no—
579C22—1221no—
580C22C22×C22⋊#692311yescharacteristic
581C4—21no—
582C4—22no—
583C4—22no—
584C4—21no—
585C4—41no—
586C4—41no—
587C4—612no—
588C4—1221no—
589C4—1222no—
590C4—1222no—
591C4—1221no—
592C4—4882no—
593C4—4881no—
594C4—2442no—
595C4—2441no—
596C4—2442no—
597C4—2441no—
598C4#1684075212yes—
599C2—82no—
600C2—1221no—
601C2—1221no—
602C2#2146242111yescharacteristic
603C2C2×#1684075211yescharacteristic
604C2#2146244911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#559C4
socleC22×C61C22×C22⋊C4
prime radical#16840310C22
cosocleC2×C4×D61C24
hypercenter#559#6923
hypocenterC61#17616
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #559

Complements

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

NH ≅ G/Ntype
1. #21462713C2semidirect
2. #21462339C2semidirect
3. #21461261C2semidirect
4. #21461260C2semidirect
5. #21461252C2semidirect
7. #21461201C2semidirect
9. #21461191C2semidirect
10. #21461190C2semidirect
36. #16840498C4semidirect
37. #16840491C4semidirect
38. #16840490C4semidirect
41. #16840481C4semidirect
42. #16840471C4semidirect
52. C61×#559C4semidirect
315. #559#6923semidirect
355. C61#17616semidirect

Maximal subgroups

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

subgroupindex
#214627132
#214623392
#214612612
#214612602
#214612522
#214612482
#214612012
#214611992
#214611912
#214611902
#214596762
#1761661