Finite Group Databasebeta

← #29671456

Subgroups — #29671456

Order 15616.

638 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458796C211yesmaximal, characteristic
2Q8×#11758201C211yesmaximal, characteristic, complemented
3#21457486C211yesmaximal, characteristic
4#21445407C211yesmaximal, characteristic, complemented
5#21445315C211yesmaximal, characteristic, complemented
6#21445291C211yesmaximal, characteristic, complemented
7#21445203C211yesmaximal, characteristic, complemented
8#21445169C211yesmaximal, characteristic, complemented
9#21445096C211yesmaximal, characteristic, complemented
10#21445081C211yesmaximal, characteristic, complemented
11#21444944C211yesmaximal, characteristic
12#21444897C211yesmaximal, characteristic, complemented
13#21444709C211yesmaximal, characteristic, complemented
14Q16×#92418C211yesmaximal, characteristic, complemented
15C61×#3387C211yesmaximal, characteristic, complemented
16C2×#14519809—21no—
17#16840434C2211yescharacteristic
18Q8×D244C2211yescharacteristic
19#16840158—21no—
20#16840158C2211yescharacteristic
21#16840158C2211yescharacteristic
22#16840136—21no—
23#16840136C2211yescharacteristic
24#16840136C2211yescharacteristic
25#16840135C2211yescharacteristic
26#16840049—21no—
27#16840049C2211yescharacteristic
28#16840047—21no—
29#16840047C2211yescharacteristic
30#16840009—21no—
31#16839972C2211yescharacteristic
32#16839964—21no—
33#16839964C2211yescharacteristic
34#16839962—21no—
35#16839962C2211yescharacteristic
36#16839962C2211yescharacteristic
37#16839927—21no—
38#16839902—21no—
39Q8×#92418C2211yescharacteristic
40Q8×#92418C2211yescharacteristic
41C4×#11758201—21no—
42C4×#11758201—21no—
43C4×#11758201C2211yescharacteristic
44#16839841C2211yescharacteristic
45#16839515C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53Q8⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55C8×#92418C2211yescharacteristic
56Q82×C61C2211yescharacteristic
57C61×#557C2211yescharacteristic
58C61×#501—21no—
59C61×#498C2211yescharacteristic
60C61×#479—21no—
61C61×#477C2211yescharacteristic
62C61×#475—21no—
63C61×#475C2211yescharacteristic
64C61×#446C2211yescharacteristic
65C61×#441—21no—
66C4×Q16×C61C2211yescharacteristic
67C2×#11758226—21no—
68C2×Q8×D61—21no—
69#14519809—44no—
70C2×#11758202—21no—
71C2×#11758190—21no—
72C2×#11758190—21no—
73C2×#11758201—21no—
74C2×#11758201—21no—
75C2×#11758201—21no—
76C2×#11758201—21no—
77C2×#11758201—21no—
78C2×#11758201—41no—
79C2×#11758201D411yescharacteristic
80C2×#11758189—21no—
81#14519777—21no—
82#14519777C2311yescharacteristic
83#14519774—21no—
84#14519774C2311yescharacteristic
85#14519757—21no—
86#14519757—41no—
87#14519752—21no—
88#14519752—41no—
89C4×D244—21no—
90C4×D244—21no—
91C4×D244C2311yescharacteristic
92#14519708—21no—
93#14519708—21no—
94#14519708—21no—
95#14519708—21no—
96#14519708—41no—
97#14519708D411yescharacteristic
98C4⋊#92418—21no—
99C4⋊#92418—21no—
100C4⋊#92418C2311yescharacteristic
101C4⋊#92418C2311yescharacteristic
102C4⋊#92418C2311yescharacteristic
103#14519701—21no—
104#14519701—21no—
105#14519701D411yescharacteristic
106#14519700—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697—21no—
110#14519697D411yescharacteristic
111#14519689—21no—
112C4×#92418—21no—
113C4×#92418—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C2×Q16×C61D411yescharacteristic
117C4⋊Q8×C61—21no—
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61—41no—
121C4⋊Q8×C61C2311yescharacteristic
122C4⋊Q8×C61C2311yescharacteristic
123#125×C61—21no—
124#125×C61—41no—
125C4×Q8×C61—21no—
126C4×Q8×C61—21no—
127C4×Q8×C61—21no—
128C4×Q8×C61—21no—
129C4×Q8×C61C2311yescharacteristic
130C4×Q8×C61C2311yescharacteristic
131C4×Q8×C61C2311yescharacteristic
132#107×C61—21no—
133#107×C61D411yescharacteristic
134#106×C61—21no—
135#106×C61—21no—
136#106×C61—21no—
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61—21no—
141#103×C61D411yescharacteristic
142#103×C61D411yescharacteristic
143#97×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—41no—
148C2×Q8×C61C2×D411yescharacteristic
149C2×Q8×C61C2×D411yescharacteristic
150C2×Q8×C61C2×D411yescharacteristic
151#11758226—42no—
152#11758226—81no—
153Q8×D61—42no—
154Q8×D61—81no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—41no—
159C2×D244—41no—
160C2×D244C2×D411yescharacteristic
161C2×C4×D61—21no—
162C2×C4×D61—41no—
163C2×C4×D61—41no—
164C2×Dic122—21no—
165C2×Dic122—41no—
166Q16×C61SD1614yescomplemented
167C2×C8×C61—21no—
168C2×C8×C61C2×D411yescharacteristic
169C2×C8×C61C2×D411yescharacteristic
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—41no—
178C4⋊C4×C61—41no—
179C4⋊C4×C61—41no—
180C4⋊C4×C61—41no—
181C4⋊C4×C61—41no—
182C4⋊C4×C61C2×D411yescharacteristic
183C4⋊C4×C61C2×D411yescharacteristic
184C4⋊C4×C61C2×D411yescharacteristic
185C42×C61—21no—
186C42×C61—21no—
187C42×C61—21no—
188C42×C61—41no—
189C42×C61—41no—
190C42×C61C2411yescharacteristic
191#11758202—42no—
192#11758201—22no—
193#11758201—42no—
194#11758201—42no—
195#11758201—81no—
196#11758201—82no—
197#11758201—81no—
198#11758201—81no—
199#11758198—21no—
200#11758198—41no—
201#11758198—41no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—41no—
205C4⋊Dic61C2×D411yescharacteristic
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418C2×D411yescharacteristic
211C2×#92418C2×D411yescharacteristic
212#11758190—42no—
213#11758190—81no—
214#11758189—42no—
215C22×D61—21no—
216C22×D61—41no—
217Q8×C61—21no—
218Q8×C61—21no—
219Q8×C61—21no—
220Q8×C61—21no—
221Q8×C61—41no—
222Q8×C61—41no—
223Q8×C61—81no—
224Q8×C61—161no—
225Q8×C61C2×SD1612yes—
226Q8×C61C2×SD1612yes—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61H(5,𝔽2)11yescharacteristic
238C2×Dic61—21no—
239C2×Dic61—41no—
240D244—21no—
241D244—21no—
242D244—41no—
243D244—41no—
244D244—41no—
245D244—41no—
246D244—82no—
247D244—81no—
248D244—161no—
249C4×D61—41no—
250C4×D61—41no—
251C4×D61—82no—
252C4×D61—82no—
253Dic122—41no—
254Dic122—41no—
255Dic122—82no—
256C8×C61—21no—
257C8×C61—41no—
258C8×C61C2×SD1612yes—
259#92418—22no—
260#92418—21no—
261#92418—41no—
262#92418—41no—
263#92418—41no—
264#92418#13712yescomplemented
265#19897—611nomaximal, Sylow
266C22×C61D4211yescharacteristic
267C2×D61—41no—
268C2×D61—41no—
269C2×D61—41no—
270C2×D61—81no—
271C2×D61—81no—
272C2×D61—81no—
273C4×C61—22no—
274C4×C61—22no—
275C4×C61—21no—
276C4×C61—41no—
277C4×C61—41no—
278C4×C61—41no—
279C4×C61—41no—
280C4×C61—81no—
281C4×C61—81no—
282C4×C61C22×SD1611yescharacteristic
283C4×C61C2×#13711yescharacteristic
284C4×C61#57711yescharacteristic
285C4×C61#57711yescharacteristic
286Dic61—41no—
287Dic61—81no—
288#3392—611no—
289#3387D6111yescharacteristic, complemented
290Q8×SD16—611no—
291#3278—611no—
292#1705—611no—
293#1691—611no—
294#1686—611no—
295#1672—611no—
296#1666—611no—
297#1653—611no—
298#1651—611no—
299#1631—611no—
300#1623—611no—
301#1590—611no—
302C8×Q16—611no—
303C2×C61#327811yescharacteristic
304C2×C61D4×SD1611yescharacteristic
305C2×C61#329111yescharacteristic
306D61—81no—
307D61—161no—
308C2×#137—1221no—
309Q82C2×D6111yescharacteristic
310#557C2×D6111yescharacteristic
311#550—611no—
312D4×Q8—611no—
313#501—21no—
314#498—611no—
315#498—611no—
316#498—1221no—
317#498C2×D6111yescharacteristic
318#492—611no—
319#492—611no—
320#492—611no—
321#492—1221no—
322#479—21no—
323#477C2×D6111yescharacteristic
324#476—611no—
325#476—1221no—
326#475—21no—
327#475—611no—
328#475—1221no—
329#475C2×D6111yescharacteristic
330#468—1221no—
331#462—611no—
332#461—611no—
333#461—1221no—
334#460—611no—
335#460—611no—
336#460—1221no—
337#452—1221no—
338#448—1221no—
339#446C2×D6111yescharacteristic
340C8×Q8—611no—
341C8×Q8—611no—
342#441—21no—
343C4×Q16C2×D6111yescharacteristic
344C4×SD16—611no—
345C4×SD16—1221no—
346C4×SD16—1221no—
347#436—611no—
348#335—611no—
349#334—611no—
350#333—611no—
351#332—611no—
352#332—611no—
353#331—611no—
354Q8⋊C8—611no—
355Q8⋊C8—611no—
356Q8⋊C8—611no—
357D4⋊C8—611no—
358C82—611no—
359C61#1989711yescharacteristic, complemented, Sylow
360C2×#41—1221no—
361C22×Q8—1221no—
362#137—2444no—
363C2×Q16—1221no—
364C2×Q16C22⋊D6111yescharacteristic
365C2×SD16—611no—
366C2×SD16—1221no—
367C2×SD16—1221no—
368C2×SD16—1221no—
369C2×SD16—1221no—
370C2×SD16—1221no—
371C2×SD16—1221no—
372C2×SD16—1221no—
373C2×SD16—2441no—
374C2×M16—1221no—
375C4⋊Q8—21no—
376C4⋊Q8—21no—
377C4⋊Q8—21no—
378C4⋊Q8—41no—
379C4⋊Q8—611no—
380C4⋊Q8—1221no—
381C4⋊Q8C22×D6111yescharacteristic
382C4⋊Q8C22×D6111yescharacteristic
383#127—611no—
384#127—1221no—
385#125—21no—
386#125—41no—
387C22⋊Q8—1221no—
388C22⋊Q8—2441no—
389#121—1221no—
390#121—2441no—
391C4×Q8—21no—
392C4×Q8—21no—
393C4×Q8—21no—
394C4×Q8—21no—
395C4×Q8C22×D6111yescharacteristic
396C4×Q8C22×D6111yescharacteristic
397C4×Q8C22×D6111yescharacteristic
398C4×D4—611no—
399C4×D4—1221no—
400C4×D4—1221no—
401#107—21no—
402#107C22⋊D6111yescharacteristic
403#106—21no—
404#106—21no—
405#106—21no—
406#106—611no—
407#106—1221no—
408#106—1221no—
409#106—1221no—
410#106—1221no—
411#106—2441no—
412C4⋊C8—21no—
413C4⋊C8—611no—
414C4⋊C8—611no—
415C4⋊C8—611no—
416C4⋊C8—1221no—
417C4⋊C8—1221no—
418C4⋊C8C22×D6111yescharacteristic
419#103—21no—
420#103—21no—
421#103—611no—
422#103—1221no—
423#103—1221no—
424#103—1221no—
425#103C22⋊D6111yescharacteristic
426#103C22⋊D6111yescharacteristic
427#102—611no—
428#102—1221no—
429#102—1221no—
430#102—1221no—
431C22⋊C8—1221no—
432#97—21no—
433C4×C8—611no—
434C4×C8—611no—
435C4×C8—1221no—
436C4×C8—1221no—
437C4×C8C22×D6111yescharacteristic
438#41—4881no—
439#41—2442no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—41no—
443C2×Q8—1221no—
444C2×Q8—4881no—
445C2×Q8—2441no—
446C2×Q8—2442no—
447C2×Q8D4×D6111yescharacteristic
448C2×Q8C2×C22⋊D6111yescharacteristic
449C2×Q8C2×C22⋊D6111yescharacteristic
450C2×D4—611no—
451C2×D4—1221no—
452C2×D4—1221no—
453C2×D4—1221no—
454C2×D4—2441no—
455C2×D4—2441no—
456C22×C4—1221no—
457C22×C4—2441no—
458C22×C4—2441no—
459Q16—2442no—
460Q16#1175820114yescomplemented
461SD16—1222no—
462SD16—4881no—
463SD16—4882no—
464SD16—4881no—
465SD16—4881no—
466SD16—4881no—
467SD16—2442no—
468SD16—2442no—
469SD16—2442no—
470M16—2442no—
471C2×C8—21no—
472C2×C8—611no—
473C2×C8—611no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8C2×C22⋊D6111yescharacteristic
479C2×C8C2×C22⋊D6111yescharacteristic
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—611no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—2441no—
496C4⋊C4D4×D6111yescharacteristic
497C4⋊C4C2×C22⋊D6111yescharacteristic
498C4⋊C4C2×C22⋊D6111yescharacteristic
499C22⋊C4—1221no—
500C22⋊C4—2441no—
501C22⋊C4—2441no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—41no—
506C42—41no—
507C42C23×D6111yescharacteristic
508C23—1221no—
509C23—2441no—
510Q8—21no—
511Q8—21no—
512Q8—21no—
513Q8—21no—
514Q8—41no—
515Q8—41no—
516Q8—81no—
517Q8—161no—
518Q8—4882no—
519Q8—2441no—
520Q8—2441no—
521Q8C2×#1175820112yes—
522Q8C2×#1175820112yes—
523D4—1221no—
524D4—1221no—
525D4—9761no—
526D4—4881no—
527D4—4882no—
528D4—2441no—
529D4—2441no—
530D4—2441no—
531D4—2441no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—1221no—
541C2×C4—4882no—
542C2×C4—4882no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4C2×D4×D6111yescharacteristic
547C2×C4C22×C22⋊D6111yescharacteristic
548C2×C4#1451982211yescharacteristic
549C8—21no—
550C8—41no—
551C8—612no—
552C8—1222no—
553C8—1221no—
554C8—2441no—
555C8—2441no—
556C8—2441no—
557C8C2×#1175820112yes—
558C22—4881no—
559C22—4881no—
560C22—4881no—
561C22—2441no—
562C22—2441no—
563C22—2441no—
564C22D4×C22⋊D6111yescharacteristic
565C4—22no—
566C4—22no—
567C4—21no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—81no—
573C4—81no—
574C4—4881no—
575C4—2441no—
576C4C22×#1175820111yescharacteristic
577C4#137×D6111yescharacteristic
578C4#1684057011yescharacteristic
579C4#1684057111yescharacteristic
580C2—9761no—
581C2—4881no—
582C2#2145747911yescharacteristic
583C2D4×#1175820111yescharacteristic
584C2#2145761011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3387C2
socleC22×C61D42
prime radical#16840434C22
hypercenter#3387D61
hypocenterC61#19897
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3387

Complements

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

NH ≅ G/Ntype
2. Q8×#11758201C2semidirect
4. #21445407C2semidirect
5. #21445315C2semidirect
6. #21445291C2semidirect
7. #21445203C2semidirect
8. #21445169C2semidirect
9. #21445096C2semidirect
10. #21445081C2semidirect
12. #21444897C2semidirect
13. #21444709C2semidirect
14. Q16×#92418C2semidirect
15. C61×#3387C2semidirect
166. Q16×C61SD16semidirect
264. #92418#137semidirect
289. #3387D61semidirect
359. C61#19897semidirect
460. Q16#11758201semidirect

Maximal subgroups

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

subgroupindex
#214587962
Q8×#117582012
#214574862
#214454072
#214453152
#214452912
#214452032
#214451692
#214450962
#214450812
#214449442
#214448972
#214447092
Q16×#924182
C61×#33872
#1989761