Finite Group Databasebeta

← #30613552

Subgroups — #30613552

Order 15616.

624 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462750C211yesmaximal, characteristic
2#21462720C211yesmaximal, characteristic, complemented
3#21461462C211yesmaximal, characteristic, complemented
4#21461455C211yesmaximal, characteristic, complemented
5#21461449C211yesmaximal, characteristic, complemented
6#21461445C211yesmaximal, characteristic, complemented
7#21461433C211yesmaximal, characteristic, complemented
8#21461429C211yesmaximal, characteristic, complemented
9#21461414C211yesmaximal, characteristic, complemented
10#21461407C211yesmaximal, characteristic, complemented
11#21461384C211yesmaximal, characteristic, complemented
12#21461151C211yesmaximal, characteristic
13#21461148C211yesmaximal, characteristic, complemented
14#21461039C211yesmaximal, characteristic, complemented
15#21454641C211yesmaximal, characteristic
16#16840814—21no—
17#16840814—21no—
18#16840813C2211yescharacteristic
19#16840813C2211yescharacteristic
20#16840811C2211yescharacteristic
21#16840811C2211yescharacteristic
22#16840811C2211yescharacteristic
23#16840811C2211yescharacteristic
24#16840797C2211yescharacteristic
25#16840797C2211yescharacteristic
26#16840795—21no—
27#16840795—21no—
28#16840785—21no—
29#16840784C2211yescharacteristic
30#16840781C2211yescharacteristic
31#16840781C2211yescharacteristic
32#16840780—21no—
33#16840778—21no—
34#16840766C2211yescharacteristic
35#16840765—21no—
36#16840625C2211yescharacteristic
37#16840625C2211yescharacteristic
38#16840623C2211yescharacteristic
39#16840623C2211yescharacteristic
40#16840612C2211yescharacteristic
41#16840612C2211yescharacteristic
42#16840612C2211yescharacteristic
43#16840612C2211yescharacteristic
44Q8⋊#92420C2211yescharacteristic
45Q8⋊#92420C2211yescharacteristic
46D4⋊#92420C2211yescharacteristic
47D4⋊#92420C2211yescharacteristic
48#92420⋊C8C2211yescharacteristic
49#92420⋊C8C2211yescharacteristic
50#16840603C2211yescharacteristic
51C8×#92420C2211yescharacteristic
52C2×#14519806C411yescharacteristic
53C4×#11758224C2211yescharacteristic
54C4×Q8×D61C2211yescharacteristic
55#16839870C411yescharacteristic
56#16839867C411yescharacteristic
57#16839862C411yescharacteristic
58#16839856C2211yescharacteristic
59#16839854C2211yescharacteristic
60#16839853C2211yescharacteristic
61SD16×Dic61C2211yescharacteristic
62#16839803C411yescharacteristic
63#16839769C411yescharacteristic
64#16839760C411yescharacteristic
65#16839714C2211yescharacteristic
66C61×#440C411yescharacteristic
67C2×#11758213—21no—
68C2×#11758217—21no—
69C2×#11758217—21no—
70#14519844—21no—
71#14519844—21no—
72#14519844—21no—
73#14519844—21no—
74#14519844—21no—
75#14519844—21no—
76C4⋊#92420—21no—
77C4⋊#92420—21no—
78C4⋊#92420D411yescharacteristic
79C4⋊#92420D411yescharacteristic
80C4⋊#92420C2311yescharacteristic
81C4⋊#92420C2311yescharacteristic
82C4⋊#92420C2311yescharacteristic
83C4⋊#92420C2311yescharacteristic
84#14519834—21no—
85#14519834—21no—
86#14519834D411yescharacteristic
87#14519834D411yescharacteristic
88C22⋊#92420—21no—
89C22⋊#92420—21no—
90#14519832—21no—
91#14519832—21no—
92#14519832—21no—
93C4×#92420—21no—
94C4×#92420—21no—
95C4×#92420—21no—
96C4×#92420—21no—
97C4×#92420C2311yescharacteristic
98C4×#92420C2311yescharacteristic
99C4×#92420C2311yescharacteristic
100C4×#92420C2311yescharacteristic
101C2×#11758224C2×C411yescharacteristic
102C2×Q8×D61C2×C411yescharacteristic
103#14519806—24no—
104C2×Dic244C2×C411yescharacteristic
105C2×#11758202C2×C411yescharacteristic
106C2×#11758200C2×C411yescharacteristic
107C2×#11758189C2×C411yescharacteristic
108C2×#11758189C2×C411yescharacteristic
109Q8×Dic61C2311yescharacteristic
110C4×Dic122—21no—
111C4×Dic122C2×C411yescharacteristic
112C4×Dic122C2×C411yescharacteristic
113#14519742—21no—
114#14519742C2311yescharacteristic
115#14519742C2311yescharacteristic
116D4×Dic61C2311yescharacteristic
117C4×C22⋊D61—21no—
118#14519737—21no—
119#14519735C2×C411yescharacteristic
120#14519733—21no—
121C4⋊C4×D61—21no—
122C4⋊C4×D61C2×C411yescharacteristic
123C2×C4×Dic61—21no—
124C42×D61—21no—
125C42×D61C2311yescharacteristic
126#14519710C2×C411yescharacteristic
127#14519709C2×C411yescharacteristic
128#14519708C2×C411yescharacteristic
129#14519702C2×C411yescharacteristic
130#14519702C2×C411yescharacteristic
131#14519701C2×C411yescharacteristic
132#14519700C2×C411yescharacteristic
133#14519700C2×C411yescharacteristic
134#14519698C2×C411yescharacteristic
135#14519688C2×C411yescharacteristic
136C8×Dic61C2311yescharacteristic
137C8×Dic61C2311yescharacteristic
138C2×SD16×C61C2×C411yescharacteristic
139C4×Q8×C61C2×C411yescharacteristic
140C4×D4×C61C2×C411yescharacteristic
141#107×C61C2×C411yescharacteristic
142#103×C61C2×C411yescharacteristic
143#102×C61C2×C411yescharacteristic
144#97×C61C2×C411yescharacteristic
145C2×Q8×C61C22×C411yescharacteristic
146C2×D4×C61C22×C411yescharacteristic
147C22×C4×C61—21no—
148C2×C22⋊D61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151Q8×D61—22no—
152Q8×D61—41no—
153#11758224—22no—
154#11758224—41no—
155C2×C4×D61—21no—
156C2×C4×D61—21no—
157C2×C4×D61C22×C411yescharacteristic
158C2×C4×D61C2×D411yescharacteristic
159C2×C4×D61#4111yescharacteristic
160C2×Dic122—21no—
161C2×Dic122C22×C411yescharacteristic
162C2×Dic122C22×C411yescharacteristic
163#11758217—81no—
164#11758217—81no—
165C2×#92420—21no—
166C2×#92420—21no—
167C2×#92420—21no—
168C2×#92420—21no—
169C2×#92420—21no—
170C2×#92420—21no—
171C2×#92420C2×D411yescharacteristic
172C2×#92420C2×D411yescharacteristic
173C2×#92420C2×D411yescharacteristic
174C2×#92420C2×D411yescharacteristic
175#11758213—81no—
176SD16×C61—22no—
177C2×C8×C61C22×C411yescharacteristic
178C2×C8×C61C22×C411yescharacteristic
179C4⋊C4×C61—21no—
180C4⋊C4×C61C22×C411yescharacteristic
181C4⋊C4×C61C22×C411yescharacteristic
182C22⋊C4×C61—21no—
183C42×C61—21no—
184C42×C61C22×C411yescharacteristic
185C22⋊Dic61—21no—
186#11758202—22no—
187#11758200—22no—
188#11758198—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61C22×C411yescharacteristic
191C4⋊Dic61C22×C411yescharacteristic
192#11758196—21no—
193#11758196—21no—
194#11758196—21no—
195#11758196—21no—
196#11758196—21no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61C2×D411yescharacteristic
204C4×Dic61#4111yescharacteristic
205C4×Dic61C2411yescharacteristic
206C2×#92418C22×C411yescharacteristic
207C2×#92418C22×C411yescharacteristic
208Dic244M1614yescomplemented
209#11758189—22no—
210#11758189—41no—
211C23×C61—21no—
212C22×D61C4×D411yescharacteristic
213Q8×C61—21no—
214Q8×C61—21no—
215D4×C61—21no—
216D4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—42no—
221C2×C4×C61C4×D411yescharacteristic
222C2×C4×C61C4×D411yescharacteristic
223C2×C4×C61C23×C411yescharacteristic
224C22⋊D61—42no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—42no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61C4×D411yescharacteristic
233C2×Dic61C22×D411yescharacteristic
234C2×Dic61C2×#4111yescharacteristic
235C4×D61—21no—
236C4×D61—21no—
237C4×D61—21no—
238C4×D61—21no—
239C4×D61—41no—
240C4×D61—42no—
241C4×D61—42no—
242Dic122—21no—
243Dic122—21no—
244Dic122—42no—
245Dic122C2×M1612yes—
246Dic122C2×M1612yes—
247#92420—21no—
248#92420—21no—
249#92420—21no—
250#92420—21no—
251#92420—41no—
252#92420—41no—
253#92420—41no—
254#92420—41no—
255#92420—41no—
256#92420—41no—
257C8×C61—21no—
258C8×C61C2×M1612yes—
259#92418—21no—
260#92418—21no—
261#18053—611nomaximal, Sylow
262C22×C61—41no—
263C22×C61—41no—
264C22×C61—41no—
265C22×C61C2×C4×D411yescharacteristic
266C2×D61—21no—
267C2×D61—21no—
268C2×D61—21no—
269C4×C61—21no—
270C4×C61—21no—
271C4×C61—41no—
272C4×C61—41no—
273C4×C61—41no—
274C4×C61C22×M1611yescharacteristic
275C4×C61#56811yescharacteristic
276Dic61—22no—
277Dic61—22no—
278Dic61—21no—
279Dic61—21no—
280Dic61—41no—
281Dic61—41no—
282Dic61#57711yescharacteristic
283Dic61#57811yescharacteristic
284#2991—611no—
285#2984—611no—
286C4×#137—611no—
287#1592—611no—
288#1590—611no—
289#1588—611no—
290#1587—611no—
291#1584—611no—
292#1583—611no—
293#1579—611no—
294C8×Q16—611no—
295#1569—611no—
296#1488—611no—
297#1487—611no—
298#1453—611no—
299C2×C61—81no—
300C2×C61D4×M1611yescharacteristic
301C2×C61#294711yescharacteristic
302C2×C61#295511yescharacteristic
303D61—41no—
304C2×#137—611no—
305C4×#41—611no—
306C2×C4×Q8—611no—
307#446—611no—
308#446—611no—
309#446—1221no—
310#446—1221no—
311C8×Q8—611no—
312C8×Q8—611no—
313C8×Q8—611no—
314C8×Q8—611no—
315#441—611no—
316#441—611no—
317#440—611no—
318#440#692311yescharacteristic
319C4×Q16—611no—
320C4×Q16—611no—
321C4×SD16—611no—
322C4×SD16—611no—
323#436—611no—
324#436—611no—
325#436—1221no—
326#436—1221no—
327#432—611no—
328#432—1221no—
329#431—611no—
330#431—611no—
331#431—1221no—
332#431—1221no—
333#428—611no—
334#423—611no—
335#423—1221no—
336#419—611no—
337#417—611no—
338C4×M16—611no—
339#335—611no—
340#335—611no—
341#334—611no—
342#334—611no—
343Q8⋊C8—611no—
344Q8⋊C8—611no—
345Q8⋊C8—611no—
346Q8⋊C8—611no—
347Q8⋊C8—611no—
348Q8⋊C8—611no—
349D4⋊C8—611no—
350D4⋊C8—611no—
351#322—611no—
352#322—611no—
353#322—611no—
354C82—611no—
355C61#1805311yescharacteristic, complemented, Sylow
356C2×#41—611no—
357C22×Q8—611no—
358#137—1224no—
359C2×Q16—611no—
360C2×Q16—611no—
361C2×SD16—611no—
362C2×SD16C2×#692311yescharacteristic
363C2×M16—611no—
364C2×M16—611no—
365C2×M16—1221no—
366C2×M16—1221no—
367C2×M16—1221no—
368C4×Q8—611no—
369C4×Q8—611no—
370C4×Q8—611no—
371C4×Q8—611no—
372C4×Q8—611no—
373C4×Q8—1221no—
374C4×Q8—1221no—
375C4×Q8C2×#692311yescharacteristic
376C4×D4—611no—
377C4×D4—1221no—
378C4×D4—1221no—
379C4×D4C2×#692311yescharacteristic
380#117—611no—
381#117—1221no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—1221no—
384C2×C42—611no—
385C2×C42—1221no—
386C2×C42—1221no—
387#107—611no—
388#107C2×#692311yescharacteristic
389#106—611no—
390#106—611no—
391C4⋊C8—611no—
392C4⋊C8—611no—
393C4⋊C8—611no—
394C4⋊C8—611no—
395C4⋊C8—611no—
396C4⋊C8—611no—
397C4⋊C8—1221no—
398C4⋊C8—1221no—
399C4⋊C8—1221no—
400C4⋊C8—1221no—
401C4⋊C8—1221no—
402C4⋊C8—1221no—
403C4⋊C8—1221no—
404C4⋊C8—1221no—
405#103—611no—
406#103—611no—
407#103—611no—
408#103—611no—
409#103—611no—
410#103C2×#692311yescharacteristic
411#102—611no—
412#102C2×#692311yescharacteristic
413C22⋊C8—611no—
414C22⋊C8—611no—
415C22⋊C8—1221no—
416C22⋊C8—1221no—
417C22⋊C8—1221no—
418C22⋊C8—1221no—
419#97—611no—
420#97—1221no—
421#97—1221no—
422#97—1221no—
423#97C2×#692311yescharacteristic
424C4×C8—611no—
425C4×C8—611no—
426C4×C8—611no—
427C4×C8—611no—
428C4×C8—611no—
429C4×C8—611no—
430C4×C8—1221no—
431C4×C8—1221no—
432C4×C8—1221no—
433C4×C8—1221no—
434#41—1222no—
435#41—2441no—
436C2×Q8—611no—
437C2×Q8—611no—
438C2×Q8—1221no—
439C2×Q8—1222no—
440C2×Q8—2441no—
441C2×Q8C22×#692311yescharacteristic
442C2×D4—1221no—
443C2×D4C22×#692311yescharacteristic
444C22×C4—21no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—1221no—
452Q16—614no—
453Q16—1222no—
454SD16—22no—
455SD16—1222no—
456M16—1222no—
457M16—4881no—
458M16—4881no—
459M16—4881no—
460M16—2441no—
461C2×C8—611no—
462C2×C8—611no—
463C2×C8—611no—
464C2×C8—611no—
465C2×C8—611no—
466C2×C8—611no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8C22×#692311yescharacteristic
474C2×C8C22×#692311yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—611no—
477C4⋊C4—611no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4C22×#692311yescharacteristic
485C4⋊C4C22×#692311yescharacteristic
486C22⋊C4—21no—
487C22⋊C4—1221no—
488C22⋊C4—1221no—
489C42—21no—
490C42—611no—
491C42—611no—
492C42—611no—
493C42—1221no—
494C42—1221no—
495C42—1221no—
496C42—1221no—
497C42—1221no—
498C42—1221no—
499C42C22×#692311yescharacteristic
500C23—21no—
501C23—611no—
502Q8—21no—
503Q8—21no—
504Q8—612no—
505Q8—612no—
506Q8—1221no—
507Q8—1221no—
508Q8—2442no—
509D4—21no—
510D4—21no—
511D4—2442no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—42no—
516C2×C4—611no—
517C2×C4—611no—
518C2×C4—611no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1221no—
527C2×C4—2441no—
528C2×C4—2442no—
529C2×C4—2442no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2442no—
533C2×C4D4×#692311yescharacteristic
534C2×C4D4×#692311yescharacteristic
535C2×C4C23×#692311yescharacteristic
536C8—21no—
537C8—1221no—
538C8—1221no—
539C8—1221no—
540C8—1221no—
541C8—1221no—
542C8—1221no—
543C8—2441no—
544C8—2441no—
545C8—2441no—
546C8—2441no—
547C8—2441no—
548C8—2441no—
549C8D61⋊M1612yes—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—1221no—
554C22—1221no—
555C22—1221no—
556C22C2×D4×#692311yescharacteristic
557C4—21no—
558C4—21no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—611no—
563C4—611no—
564C4—1221no—
565C4—1222no—
566C4—1222no—
567C4—1221no—
568C4—2441no—
569C4—2441no—
570C4C2×D61⋊M1611yescharacteristic
571C4#1684084111yescharacteristic
572C2—81no—
573C2—2441no—
574C2#2146256011yescharacteristic
575C2#2146257711yescharacteristic
576C2#2146259111yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#440C4
socleC22×C61C2×C4×D4
prime radicalC2×#11758224C2×C4
cosocleC4×Dic61C24
hypercenter#440#6923
hypocenterC61#18053
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. #21462720C2semidirect
3. #21461462C2semidirect
4. #21461455C2semidirect
5. #21461449C2semidirect
6. #21461445C2semidirect
7. #21461433C2semidirect
8. #21461429C2semidirect
9. #21461414C2semidirect
10. #21461407C2semidirect
11. #21461384C2semidirect
13. #21461148C2semidirect
14. #21461039C2semidirect
208. Dic244M16semidirect
355. C61#18053semidirect

Maximal subgroups

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

subgroupindex
#214627502
#214627202
#214614622
#214614552
#214614492
#214614452
#214614332
#214614292
#214614142
#214614072
#214613842
#214611512
#214611482
#214610392
#214546412
#1805361