Finite Group Databasebeta

← #29670111

Subgroups — #29670111

Order 15616.

640 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11758201C211yesmaximal, characteristic, complemented
2#21457686C211yesmaximal, characteristic
3#21457465C211yesmaximal, characteristic
4#21445451C211yesmaximal, characteristic, complemented
5#21445320C211yesmaximal, characteristic, complemented
6#21445249C211yesmaximal, characteristic, complemented
7#21445191C211yesmaximal, characteristic, complemented
8#21445187C211yesmaximal, characteristic, complemented
9#21445082C211yesmaximal, characteristic, complemented
10#21445081C211yesmaximal, characteristic, complemented
11#21445040C211yesmaximal, characteristic, complemented
12#21444969C211yesmaximal, characteristic
13#21444663C211yesmaximal, characteristic, complemented
14#21444629C211yesmaximal, characteristic, complemented
15Q8×SD16×C61C211yesmaximal, characteristic, complemented
16C2×#14519808—21no—
17Q8×D244C2211yescharacteristic
18#16840417C2211yescharacteristic
19#16840158—21no—
20#16840158C2211yescharacteristic
21#16840158C2211yescharacteristic
22#16840141C2211yescharacteristic
23#16840113C2211yescharacteristic
24#16840112C2211yescharacteristic
25#16840094—21no—
26#16840049—21no—
27#16840049C2211yescharacteristic
28#16840047—21no—
29#16840047C2211yescharacteristic
30#16840047C2211yescharacteristic
31#16840018—21no—
32#16839990—21no—
33#16839972C2211yescharacteristic
34#16839967C2211yescharacteristic
35#16839964C2211yescharacteristic
36#16839922—21no—
37#16839918—21no—
38#16839899—21no—
39Q8×#92418C2211yescharacteristic
40C4×#11758201—21no—
41C4×#11758201C2211yescharacteristic
42#16839840C2211yescharacteristic
43C8×D244C2211yescharacteristic
44#103×D61—21no—
45#16839514C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839507C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839484C2211yescharacteristic
56Q82×C61C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic
58C61×#498—21no—
59C61×#498C2211yescharacteristic
60C61×#476—21no—
61C61×#476C2211yescharacteristic
62C61×#475—21no—
63C61×#475C2211yescharacteristic
64C8×Q8×C61C2211yescharacteristic
65C4×SD16×C61—21no—
66C4×SD16×C61C2211yescharacteristic
67C2×#11758222—21no—
68C2×Q8×D61—21no—
69#14519808—44no—
70C2×#11758202—21no—
71C2×#11758202—21no—
72C2×#11758202—21no—
73C2×#11758200—21no—
74C2×#11758201—21no—
75C2×#11758201D411yescharacteristic
76C2×#11758194—21no—
77C2×C8×D61—21no—
78#14519777—21no—
79#14519777C2311yescharacteristic
80#14519777C2311yescharacteristic
81#14519759—41no—
82#14519757—21no—
83#14519757—41no—
84#14519756—21no—
85#14519748—21no—
86C4×D244—21no—
87C4×D244C2311yescharacteristic
88C4⋊C4×D61—21no—
89#14519708—21no—
90#14519708—21no—
91#14519708—21no—
92#14519708—21no—
93#14519708—21no—
94#14519708—41no—
95#14519708D411yescharacteristic
96C4⋊#92418—21no—
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100#14519702—21no—
101#14519702—21no—
102#14519701—21no—
103#14519701—21no—
104#14519701—21no—
105#14519701—41no—
106#14519701D411yescharacteristic
107#14519700—21no—
108#14519700—21no—
109#14519697—21no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112#14519689—21no—
113C4×#92418—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C22×Q8×C61—21no—
117C2×SD16×C61D411yescharacteristic
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61—21no—
121C4⋊Q8×C61—41no—
122C4⋊Q8×C61C2311yescharacteristic
123C4⋊Q8×C61C2311yescharacteristic
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—41no—
126C4×Q8×C61—21no—
127C4×Q8×C61—21no—
128C4×Q8×C61C2311yescharacteristic
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61C2311yescharacteristic
132#106×C61—21no—
133#106×C61—21no—
134#106×C61—21no—
135#106×C61—21no—
136#106×C61D411yescharacteristic
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C4×C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—41no—
148C2×Q8×C61—41no—
149C2×Q8×C61—42no—
150C2×Q8×C61—81no—
151C2×Q8×C61C2×D411yescharacteristic
152C2×Q8×C61C2×D411yescharacteristic
153C2×D4×C61C2×D411yescharacteristic
154C22×C4×C61—21no—
155C22×C4×C61—41no—
156C2×C22⋊D61—41no—
157Q8×D61—42no—
158Q8×D61—81no—
159#11758222—42no—
160#11758222—81no—
161C2×D244C2×D411yescharacteristic
162C2×C4×D61—21no—
163C2×C4×D61—41no—
164C2×C4×D61—41no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—41no—
168SD16×C61—22no—
169C2×C8×C61—21no—
170C2×C8×C61C2×D411yescharacteristic
171C2×C8×C61C2×D411yescharacteristic
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—41no—
179C4⋊C4×C61—41no—
180C4⋊C4×C61—41no—
181C4⋊C4×C61C2×D411yescharacteristic
182C4⋊C4×C61C2×D411yescharacteristic
183C4⋊C4×C61C2×D411yescharacteristic
184C22⋊C4×C61—21no—
185C22⋊C4×C61—41no—
186C42×C61—21no—
187C42×C61—21no—
188C42×C61—41no—
189C42×C61C2411yescharacteristic
190C22⋊Dic61—41no—
191#11758202—42no—
192#11758202—42no—
193#11758202—81no—
194#11758201—42no—
195#11758201SD1614yescomplemented
196#11758200—81no—
197#11758198—21no—
198#11758198—41no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—41no—
204C4⋊Dic61C2×D411yescharacteristic
205#11758196—41no—
206#11758194—42no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418C2×D411yescharacteristic
212C2×#92418C2×D411yescharacteristic
213C8×D61—81no—
214C23×C61—21no—
215C22×D61—21no—
216Q8×C61—21no—
217Q8×C61—21no—
218Q8×C61—41no—
219Q8×C61—41no—
220Q8×C61—82no—
221Q8×C61—81no—
222Q8×C61—161no—
223Q8×C61C2×SD1612yes—
224Q8×C61#13712yescomplemented
225D4×C61—21no—
226D4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—82no—
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61H(5,𝔽2)11yescharacteristic
240C22⋊D61—82no—
241C2×Dic61—21no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244D244—21no—
245D244C2×SD1612yes—
246C4×D61—41no—
247C4×D61—41no—
248C4×D61—82no—
249C4×D61—82no—
250Dic122—41no—
251Dic122—41no—
252Dic122—41no—
253Dic122—41no—
254Dic122—82no—
255C8×C61—21no—
256C8×C61—21no—
257C8×C61—41no—
258#92418—21no—
259#92418—41no—
260#92418—41no—
261#92418—41no—
262#92418—41no—
263#92418C2×SD1612yes—
264#19805—611nomaximal, Sylow
265C22×C61—41no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61D4211yescharacteristic
269C2×D61—22no—
270C2×D61—41no—
271C2×D61—41no—
272C4×C61—22no—
273C4×C61—22no—
274C4×C61—21no—
275C4×C61—41no—
276C4×C61—41no—
277C4×C61—41no—
278C4×C61—81no—
279C4×C61—81no—
280C4×C61C22×SD1611yescharacteristic
281C4×C61C2×#13711yescharacteristic
282C4×C61#57711yescharacteristic
283C4×C61#57711yescharacteristic
284Dic61—41no—
285Dic61—81no—
286Dic61—81no—
287Q8×SD16—611no—
288Q8×SD16D6111yescharacteristic, complemented
289#3298—611no—
290#3276—611no—
291#1716—611no—
292#1691—611no—
293#1680—611no—
294#1671—611no—
295#1670—611no—
296#1651—611no—
297#1651—611no—
298#1645—611no—
299#1635—611no—
300#1583—611no—
301SD16⋊C8—611no—
302C2×C61—81no—
303C2×C61#326611yescharacteristic
304C2×C61D4×SD1611yescharacteristic
305C2×C61#328911yescharacteristic
306D61—42no—
307C2×#137—1221no—
308Q82C2×D6111yescharacteristic
309D4×Q8—611no—
310D4×Q8C2×D6111yescharacteristic
311#547—611no—
312#498—21no—
313#498—611no—
314#498—611no—
315#498—1221no—
316#498C2×D6111yescharacteristic
317#494—611no—
318#487—611no—
319#487—611no—
320#483—1221no—
321#476—21no—
322#476—611no—
323#476—1221no—
324#476C2×D6111yescharacteristic
325#475—21no—
326#475—611no—
327#475—611no—
328#475—1221no—
329#475C2×D6111yescharacteristic
330#470—1221no—
331#465—1221no—
332#462—611no—
333#461—611no—
334#461—611no—
335#452—1221no—
336#451—1221no—
337#448—1221no—
338C8×Q8—611no—
339C8×Q8C2×D6111yescharacteristic
340C4×SD16—21no—
341C4×SD16—611no—
342C4×SD16—1221no—
343C4×SD16C2×D6111yescharacteristic
344#436—611no—
345C8×D4—611no—
346C2×#103—1221no—
347#335—611no—
348#334—611no—
349#333—611no—
350#332—611no—
351#332—611no—
352#332—611no—
353Q8⋊C8—611no—
354Q8⋊C8—611no—
355D4⋊C8—611no—
356D4⋊C8—611no—
357#322—611no—
358C61#1980511yescharacteristic, complemented, Sylow
359C2×#41—1221no—
360C22×Q8—21no—
361C22×Q8—1221no—
362#137—2444no—
363C2×Q16—1221no—
364C2×Q16—1221no—
365C2×Q16—1221no—
366C2×SD16—611no—
367C2×SD16—1221no—
368C2×SD16—1221no—
369C2×SD16C22⋊D6111yescharacteristic
370C2×M16—1221no—
371C22×C8—1221no—
372C4⋊Q8—21no—
373C4⋊Q8—21no—
374C4⋊Q8—21no—
375C4⋊Q8—41no—
376C4⋊Q8—611no—
377C4⋊Q8—611no—
378C4⋊Q8—1221no—
379C4⋊Q8C22×D6111yescharacteristic
380C4⋊Q8C22×D6111yescharacteristic
381#123—2441no—
382C22⋊Q8—21no—
383C22⋊Q8—41no—
384C22⋊Q8—1221no—
385C22⋊Q8—1221no—
386C22⋊Q8—2441no—
387#121—1221no—
388C4×Q8—21no—
389C4×Q8—21no—
390C4×Q8C22×D6111yescharacteristic
391C4×Q8C22×D6111yescharacteristic
392C4×D4—21no—
393C4×D4—611no—
394C4×D4—1221no—
395C4×D4C22×D6111yescharacteristic
396C2×C4⋊C4—1221no—
397#106—21no—
398#106—21no—
399#106—21no—
400#106—21no—
401#106—611no—
402#106—1221no—
403#106—1221no—
404#106—1221no—
405#106—1221no—
406#106—1221no—
407#106—2441no—
408#106C22⋊D6111yescharacteristic
409C4⋊C8—21no—
410C4⋊C8—611no—
411C4⋊C8—611no—
412C4⋊C8—611no—
413C4⋊C8—1221no—
414C4⋊C8C22×D6111yescharacteristic
415#103—21no—
416#103—611no—
417#103—1221no—
418#103—1221no—
419#103—1221no—
420#103—1221no—
421#103—1221no—
422#103—1221no—
423#103—1221no—
424#103—2441no—
425#103C22⋊D6111yescharacteristic
426#102—21no—
427#102—611no—
428#102—1221no—
429#102C22⋊D6111yescharacteristic
430C22⋊C8—1221no—
431C22⋊C8—1221no—
432C4×C8—21no—
433C4×C8—611no—
434C4×C8—611no—
435C4×C8—1221no—
436C4×C8C22×D6111yescharacteristic
437#41—4881no—
438#41—2442no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—41no—
442C2×Q8—41no—
443C2×Q8—42no—
444C2×Q8—81no—
445C2×Q8—1221no—
446C2×Q8—1221no—
447C2×Q8—4881no—
448C2×Q8—2441no—
449C2×Q8—2442no—
450C2×Q8D4×D6111yescharacteristic
451C2×Q8C2×C22⋊D6111yescharacteristic
452C2×D4—611no—
453C2×D4—2441no—
454C2×D4C2×C22⋊D6111yescharacteristic
455C22×C4—21no—
456C22×C4—41no—
457C22×C4—1221no—
458C22×C4—2441no—
459C22×C4—2441no—
460Q16—4881no—
461Q16—2442no—
462Q16—2442no—
463SD16—22no—
464SD16—614no—
465SD16—4881no—
466SD16—2442no—
467M16—2442no—
468C2×C8—21no—
469C2×C8—611no—
470C2×C8—611no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—4881no—
476C2×C8C2×C22⋊D6111yescharacteristic
477C2×C8C2×C22⋊D6111yescharacteristic
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—611no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—2441no—
493C4⋊C4—2441no—
494C4⋊C4D4×D6111yescharacteristic
495C4⋊C4C2×C22⋊D6111yescharacteristic
496C4⋊C4C2×C22⋊D6111yescharacteristic
497C22⋊C4—21no—
498C22⋊C4—41no—
499C22⋊C4—1221no—
500C22⋊C4—2441no—
501C22⋊C4—2441no—
502C42—21no—
503C42—21no—
504C42—41no—
505C42C23×D6111yescharacteristic
506C23—21no—
507C23—1221no—
508Q8—21no—
509Q8—21no—
510Q8—41no—
511Q8—41no—
512Q8—82no—
513Q8—81no—
514Q8—161no—
515Q8—4882no—
516Q8—2441no—
517Q8—2441no—
518Q8—2441no—
519Q8—2441no—
520Q8SD16×D6112yes—
521Q8#1451980812yescomplemented
522D4—21no—
523D4—21no—
524D4—612no—
525D4—1221no—
526D4—4882no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—82no—
537C2×C4—1221no—
538C2×C4—4882no—
539C2×C4—4882no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4C2×D4×D6111yescharacteristic
545C2×C4C22×C22⋊D6111yescharacteristic
546C2×C4#1451982211yescharacteristic
547C8—21no—
548C8—21no—
549C8—41no—
550C8—612no—
551C8—1221no—
552C8—2441no—
553C8—2441no—
554C8—2441no—
555C8—2441no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—1222no—
560C22—2441no—
561C22—2441no—
562C22D4×C22⋊D6111yescharacteristic
563C4—22no—
564C4—22no—
565C4—21no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—4881no—
572C4—4881no—
573C4—2441no—
574C4C2×SD16×D6111yescharacteristic
575C4C2×#1451980811yescharacteristic
576C4#1684056811yescharacteristic
577C4#1684057011yescharacteristic
578C2—81no—
579C2—2442no—
580C2#2145733011yescharacteristic
581C2SD16×C22⋊D6111yescharacteristic
582C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingQ8×SD16×C61C2
socleC22×C61D42
prime radical#16840417C22
hypercenterQ8×SD16D61
hypocenterC61#19805
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ Q8×SD16

Complements

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

NH ≅ G/Ntype
1. Q8×#11758201C2semidirect
4. #21445451C2semidirect
5. #21445320C2semidirect
6. #21445249C2semidirect
7. #21445191C2semidirect
8. #21445187C2semidirect
9. #21445082C2semidirect
10. #21445081C2semidirect
11. #21445040C2semidirect
13. #21444663C2semidirect
14. #21444629C2semidirect
15. Q8×SD16×C61C2semidirect
195. #11758201SD16semidirect
224. Q8×C61#137semidirect
288. Q8×SD16D61semidirect
358. C61#19805semidirect
521. Q8#14519808semidirect

Maximal subgroups

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

subgroupindex
Q8×#117582012
#214576862
#214574652
#214454512
#214453202
#214452492
#214451912
#214451872
#214450822
#214450812
#214450402
#214449692
#214446632
#214446292
Q8×SD16×C612
#1980561