Finite Group Databasebeta

← #29663930

Subgroups — #29663930

Order 15616.

670 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#498×D61C211yesmaximal, characteristic, complemented
2#21455633C211yesmaximal, characteristic
3#21455303C211yesmaximal, characteristic
4C8×#11758226C211yesmaximal, characteristic
5#21445203C211yesmaximal, characteristic, complemented
6#21445201C211yesmaximal, characteristic, complemented
7#21445199C211yesmaximal, characteristic, complemented
8#21445197C211yesmaximal, characteristic, complemented
9#21445195C211yesmaximal, characteristic, complemented
10#21445193C211yesmaximal, characteristic, complemented
11#21445191C211yesmaximal, characteristic, complemented
12#21444573C211yesmaximal, characteristic, complemented
13#21444498C211yesmaximal, characteristic, complemented
14#21444489C211yesmaximal, characteristic, complemented
15C61×#1672C211yesmaximal, characteristic, complemented
16C2×#14519809—21no—
17C4⋊Q8×D61C2211yescharacteristic
18C4⋊Q8×D61C2211yescharacteristic
19C4×#11758226C2211yescharacteristic
20#16840160C2211yescharacteristic
21#16840159—21no—
22#16840159—21no—
23#16840158C2211yescharacteristic
24#16840158C2211yescharacteristic
25#16840057C2211yescharacteristic
26#16840053C2211yescharacteristic
27#16840049C2211yescharacteristic
28#16840045—21no—
29#16840041—21no—
30#16839972C2211yescharacteristic
31#16839970—21no—
32#16839968C2211yescharacteristic
33#16839966—21no—
34#16839964C2211yescharacteristic
35Q8×#92418C2211yescharacteristic
36C8×D244—21no—
37C8×D244C2211yescharacteristic
38#16839827—21no—
39#16839827C2211yescharacteristic
40#16839814—21no—
41#16839814C2211yescharacteristic
42#16839803—21no—
43#106×D61—21no—
44#106×D61—21no—
45#16839784C2211yescharacteristic
46#16839784C2211yescharacteristic
47#16839769—21no—
48#16839762—21no—
49C4×C8×D61—21no—
50C4×C8×D61C2211yescharacteristic
51#16839513C2211yescharacteristic
52#16839512C2211yescharacteristic
53#16839512C2211yescharacteristic
54#16839511C2211yescharacteristic
55#16839510C2211yescharacteristic
56#16839510C2211yescharacteristic
57#16839509C2211yescharacteristic
58#16839508C2211yescharacteristic
59#16839507C2211yescharacteristic
60C61×#498C2211yescharacteristic
61C61×#477C2211yescharacteristic
62C61×#462C2211yescharacteristic
63C8×Q8×C61C2211yescharacteristic
64C61×#334C2211yescharacteristic
65C61×#333C2211yescharacteristic
66C61×#332C2211yescharacteristic
67C2×#11758226D411yescharacteristic
68C2×Q8×D61—21no—
69C2×Q8×D61—21no—
70#14519809—44no—
71C2×#11758202—21no—
72C2×#11758190—21no—
73C2×#11758201—21no—
74C2×#11758189—21no—
75C2×#11758189—21no—
76C2×C8×D61—21no—
77C2×C8×D61—21no—
78C2×C8×D61D411yescharacteristic
79C2×C8×D61D411yescharacteristic
80#14519777C2311yescharacteristic
81#14519777C2311yescharacteristic
82#14519776—21no—
83#14519776—21no—
84#14519776—21no—
85#14519776—21no—
86#14519775—21no—
87#14519775—21no—
88Q8×Dic61D411yescharacteristic
89C4×D244—21no—
90C4×D244C2311yescharacteristic
91#14519739—21no—
92#14519739D411yescharacteristic
93#14519735—21no—
94#14519735D411yescharacteristic
95C4⋊C4×D61—21no—
96C4⋊C4×D61—21no—
97C4⋊C4×D61—21no—
98C4⋊C4×D61—21no—
99C42×D61—21no—
100C42×D61C2311yescharacteristic
101#14519710—21no—
102#14519709—21no—
103#14519709—21no—
104#14519709—21no—
105#14519708—21no—
106#14519708—21no—
107#14519708—21no—
108#14519708—21no—
109#14519708—21no—
110C4⋊#92418—21no—
111C4⋊#92418C2311yescharacteristic
112C4⋊#92418C2311yescharacteristic
113C4⋊#92418C2311yescharacteristic
114#14519706—21no—
115#14519706—21no—
116#14519702—21no—
117#14519701—21no—
118#14519701—21no—
119#14519700—21no—
120#14519699—21no—
121#14519697—21no—
122#14519689—21no—
123#14519689—21no—
124#14519689—21no—
125C4×#92418—21no—
126C4×#92418C2311yescharacteristic
127C8×Dic61—21no—
128C8×Dic61D411yescharacteristic
129C8×Dic61D411yescharacteristic
130C2×Q16×C61—21no—
131C4⋊Q8×C61C2311yescharacteristic
132C4⋊Q8×C61C2311yescharacteristic
133C4×Q8×C61C2311yescharacteristic
134#107×C61—21no—
135#106×C61—21no—
136#106×C61—21no—
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140C4⋊C8×C61C2311yescharacteristic
141#103×C61—21no—
142#103×C61—21no—
143C4×C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61C2×D411yescharacteristic
148#11758226—22no—
149#11758226SD1614yes—
150Q8×D61—42no—
151Q8×D61—42no—
152Q8×D61—82no—
153C2×D244—21no—
154C2×D244C2×D411yescharacteristic
155C2×C4×D61—21no—
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—41no—
159C2×C4×D61—41no—
160C2×C4×D61C2×D411yescharacteristic
161C2×C4×D61C2×D411yescharacteristic
162C2×C4×D61#4111yescharacteristic
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—41no—
166C2×Dic122—41no—
167Q16×C61—42no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61C2×D411yescharacteristic
172C2×C8×C61C2×D411yescharacteristic
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61C2×D411yescharacteristic
179C42×C61—21no—
180C42×C61C2411yescharacteristic
181#11758202—42no—
182#11758201—42no—
183#11758198—21no—
184#11758198—21no—
185#11758198—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61C2×D411yescharacteristic
192#11758196—41no—
193#11758196—41no—
194#11758196—41no—
195#11758196—41no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61C2×D411yescharacteristic
199C4×Dic61C2×D411yescharacteristic
200C4×Dic61#4111yescharacteristic
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418C2×D411yescharacteristic
205C2×#92418C2×D411yescharacteristic
206#11758190—42no—
207#11758189—42no—
208#11758189—42no—
209C8×D61—22no—
210C8×D61—22no—
211C8×D61—42no—
212C8×D61—42no—
213C22×D61—21no—
214C22×D61#12111yescharacteristic
215Q8×C61—21no—
216Q8×C61—41no—
217Q8×C61—41no—
218Q8×C61—81no—
219Q8×C61C2×SD1612yes—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×Dic61—21no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61#12111yescharacteristic
231C2×Dic61#12111yescharacteristic
232C2×Dic61#12111yescharacteristic
233D244—24no—
234D244—21no—
235D244C2×SD1612yes—
236C4×D61—21no—
237C4×D61—21no—
238C4×D61—24no—
239C4×D61—22no—
240C4×D61—41no—
241C4×D61—41no—
242C4×D61—42no—
243C4×D61—82no—
244C4×D61—82no—
245C4×D61C2×SD1612yes—
246C4×D61#13512yes—
247Dic122—41no—
248Dic122—41no—
249Dic122—82no—
250Dic122—82no—
251Dic122—81no—
252C8×C61—21no—
253C8×C61—41no—
254C8×C61—41no—
255C8×C61—41no—
256C8×C61#13712yescomplemented
257#92418—21no—
258#92418—41no—
259#92418—41no—
260#92418—41no—
261#92418#13712yescomplemented
262#19270—611nomaximal, Sylow
263C22×C61C2×#12111yescharacteristic
264C2×D61—21no—
265C2×D61—21no—
266C2×D61—22no—
267C2×D61—41no—
268C2×D61—41no—
269C2×D61#46512yes—
270C4×C61—22no—
271C4×C61—21no—
272C4×C61—41no—
273C4×C61—81no—
274C4×C61—81no—
275C4×C61C22×SD1611yescharacteristic
276C4×C61C2×#13511yescharacteristic
277C4×C61C2×#13711yescharacteristic
278C4×C61C2×#13711yescharacteristic
279Dic61—21no—
280Dic61—41no—
281Dic61—81no—
282Dic61—81no—
283Dic61#46512yes—
284Dic61#47112yes—
285C2×#498—611no—
286#3079—611no—
287#3039—611no—
288C8×#41—611no—
289#1672—611no—
290#1672—611no—
291#1672—611no—
292#1672D6111yescharacteristic, complemented
293#1671—611no—
294#1671—611no—
295#1671—611no—
296#1671—611no—
297#1566—611no—
298#1548—611no—
299#1546—611no—
300C2×C61#303911yescharacteristic
301C2×C61C2×#46511yescharacteristic
302C2×C61C2×#47111yescharacteristic
303D61—22no—
304D61—42no—
305C2×#137—1221no—
306C2×C4⋊Q8—611no—
307C2×C4⋊Q8—611no—
308C4×#41—611no—
309#498—611no—
310#498—611no—
311#498—611no—
312#498—1221no—
313#498—1221no—
314#498C2×D6111yescharacteristic
315#477—611no—
316#477C2×D6111yescharacteristic
317#476—611no—
318#476—611no—
319#475—1221no—
320#474—1221no—
321#462—611no—
322#462—1221no—
323#462C2×D6111yescharacteristic
324#461—611no—
325#461—611no—
326#461—1221no—
327C8×Q8—611no—
328C8×Q8C2×D6111yescharacteristic
329C8×D4—611no—
330C8×D4—611no—
331C8×D4—1221no—
332C8×D4—1221no—
333#431—611no—
334#431—1221no—
335#428—1221no—
336C2×#106—1221no—
337C2×#106—1221no—
338#423—611no—
339#423—611no—
340#419—1221no—
341#417—1221no—
342C2×C4×C8—611no—
343C2×C4×C8—1221no—
344#334—611no—
345#334—611no—
346#334—611no—
347#334C2×D6111yescharacteristic
348#333—611no—
349#333—611no—
350#333—611no—
351#333C2×D6111yescharacteristic
352#332—611no—
353#332—611no—
354#332—611no—
355#332C2×D6111yescharacteristic
356C61#1927011yescharacteristic, complemented, Sylow
357C2×#41—611no—
358C22×Q8—1221no—
359C22×Q8—1221no—
360#137—2444no—
361C2×Q16—21no—
362C2×Q16—1221no—
363C2×SD16—1221no—
364C2×SD16—1221no—
365C2×M16—1221no—
366C2×M16—1221no—
367C22×C8—611no—
368C22×C8—611no—
369C22×C8—1221no—
370C22×C8—1221no—
371C4⋊Q8—611no—
372C4⋊Q8—611no—
373C4⋊Q8—1221no—
374C4⋊Q8—1221no—
375C4⋊Q8—1221no—
376C4⋊Q8—1221no—
377C4⋊Q8—1221no—
378C4⋊Q8—1221no—
379C4⋊Q8C22×D6111yescharacteristic
380C4⋊Q8C22×D6111yescharacteristic
381C4×Q8—611no—
382C4×Q8C22×D6111yescharacteristic
383C4×D4—611no—
384C4×D4—611no—
385C4×D4—1221no—
386C4×D4—1221no—
387#117—611no—
388#117—1221no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C42—611no—
394C2×C42—1221no—
395#107—21no—
396#107—1221no—
397#106—21no—
398#106—21no—
399#106—1221no—
400#106—1221no—
401#106—1221no—
402#106—1221no—
403#106—1221no—
404#106—1221no—
405#106—1221no—
406#106—1221no—
407C4⋊C8—21no—
408C4⋊C8—611no—
409C4⋊C8—611no—
410C4⋊C8—611no—
411C4⋊C8—1221no—
412C4⋊C8—1221no—
413C4⋊C8—1221no—
414C4⋊C8C22×D6111yescharacteristic
415C4⋊C8C22×D6111yescharacteristic
416C4⋊C8C22×D6111yescharacteristic
417#103—21no—
418#103—21no—
419#103—1221no—
420#103—1221no—
421#103—1221no—
422#103—1221no—
423#102—1221no—
424#102—1221no—
425C22⋊C8—1221no—
426C22⋊C8—1221no—
427C22⋊C8—1221no—
428C4×C8—21no—
429C4×C8—611no—
430C4×C8—611no—
431C4×C8—611no—
432C4×C8—1221no—
433C4×C8—1221no—
434C4×C8C22×D6111yescharacteristic
435#41—614no—
436#41—1222no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—1221no—
440C2×Q8—1221no—
441C2×Q8—4882no—
442C2×Q8—2441no—
443C2×Q8—2441no—
444C2×Q8—2442no—
445C2×Q8—2442no—
446C2×Q8D4×D6111yescharacteristic
447C2×D4—611no—
448C2×D4—1221no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—2441no—
456C22×C4—2441no—
457Q16—42no—
458Q16—2442no—
459SD16—2442no—
460SD16—2442no—
461M16—2442no—
462M16—2442no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—611no—
467C2×C8—611no—
468C2×C8—1222no—
469C2×C8—1222no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—2442no—
474C2×C8—2442no—
475C2×C8D4×D6111yescharacteristic
476C2×C8D4×D6111yescharacteristic
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—611no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—2441no—
489C4⋊C4—2441no—
490C4⋊C4—2441no—
491C4⋊C4—2441no—
492C4⋊C4D4×D6111yescharacteristic
493C22⋊C4—1221no—
494C22⋊C4—1221no—
495C22⋊C4—1221no—
496C42—21no—
497C42—611no—
498C42—611no—
499C42—611no—
500C42—1221no—
501C42—1221no—
502C42C23×D6111yescharacteristic
503C23—611no—
504C23—1221no—
505Q8—21no—
506Q8—41no—
507Q8—41no—
508Q8—81no—
509Q8—4881no—
510Q8—4882no—
511Q8—4882no—
512Q8—2441no—
513Q8—2441no—
514Q8SD16×D6112yes—
515D4—612no—
516D4—1224no—
517D4—1221no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—611no—
523C2×C4—611no—
524C2×C4—612no—
525C2×C4—612no—
526C2×C4—611no—
527C2×C4—1222no—
528C2×C4—1224no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—4882no—
533C2×C4—4882no—
534C2×C4—2442no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4C2×D4×D6111yescharacteristic
540C2×C4C2×D4×D6111yescharacteristic
541C2×C4#41×D6111yescharacteristic
542C8—21no—
543C8—41no—
544C8—41no—
545C8—41no—
546C8—612no—
547C8—1221no—
548C8—2441no—
549C8—2441no—
550C8—2441no—
551C8#1451980912yescomplemented
552C22—612no—
553C22—1221no—
554C22—1221no—
555C22—1222no—
556C22—2441no—
557C22—2441no—
558C22#121×D6111yescharacteristic
559C4—22no—
560C4—21no—
561C4—41no—
562C4—81no—
563C4—81no—
564C4—612no—
565C4—612no—
566C4—1221no—
567C4—4881no—
568C4—4881no—
569C4—2441no—
570C4C2×SD16×D6111yescharacteristic
571C4#135×D6111yescharacteristic
572C4#137×D6111yescharacteristic
573C4C2×#1451980911yescharacteristic
574C2—1222no—
575C2—2442no—
576C2#2145530311yescharacteristic
577C2#465×D6111yescharacteristic
578C2#471×D6111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1672C2
socleC22×C61C2×#121
prime radicalC2×#11758226D4
hypercenter#1672D61
hypocenterC61#19270
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1672

Complements

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

NH ≅ G/Ntype
1. #498×D61C2semidirect
5. #21445203C2semidirect
6. #21445201C2semidirect
7. #21445199C2semidirect
8. #21445197C2semidirect
9. #21445195C2semidirect
10. #21445193C2semidirect
11. #21445191C2semidirect
12. #21444573C2semidirect
13. #21444498C2semidirect
14. #21444489C2semidirect
15. C61×#1672C2semidirect
256. C8×C61#137semidirect
261. #92418#137semidirect
292. #1672D61semidirect
356. C61#19270semidirect
551. C8#14519809semidirect

Maximal subgroups

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

subgroupindex
#498×D612
#214556332
#214553032
C8×#117582262
#214452032
#214452012
#214451992
#214451972
#214451952
#214451932
#214451912
#214445732
#214444982
#214444892
C61×#16722
#1927061