Finite Group Databasebeta

← #29662873

Subgroups — #29662873

Order 15616.

682 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456273C211yesmaximal, characteristic
2#21455849C211yesmaximal, characteristic
3#475×D61C211yesmaximal, characteristic, complemented
4Q8×#11758189C211yesmaximal, characteristic, complemented
5#21445314C211yesmaximal, characteristic, complemented
6#21445312C211yesmaximal, characteristic, complemented
7#21445290C211yesmaximal, characteristic, complemented
8#21445288C211yesmaximal, characteristic, complemented
9#21445189C211yesmaximal, characteristic, complemented
10#21445187C211yesmaximal, characteristic, complemented
11#21445165C211yesmaximal, characteristic, complemented
12#21444568C211yesmaximal, characteristic, complemented
13#21444493C211yesmaximal, characteristic
14#21444489C211yesmaximal, characteristic, complemented
15C61×#1666C211yesmaximal, characteristic, complemented
16C2×#14519806—21no—
17C2×#14519806—21no—
18#16840310C2211yescharacteristic
19C4⋊Q8×D61C2211yescharacteristic
20C4×Q8×D61C2211yescharacteristic
21#16840153—21no—
22#16840153—21no—
23#16840142C2211yescharacteristic
24#16840141C2211yescharacteristic
25#16840134C2211yescharacteristic
26#16840079—21no—
27#16840048C2211yescharacteristic
28#16840048C2211yescharacteristic
29#16840047C2211yescharacteristic
30#16840047C2211yescharacteristic
31#16840046—21no—
32#16840045—21no—
33#16840044C2211yescharacteristic
34#16839959C2211yescharacteristic
35#16839899—21no—
36#16839892—21no—
37#16839892C2211yescharacteristic
38#16839889—21no—
39#16839889C2211yescharacteristic
40Q8×#92418C2211yescharacteristic
41#16839853—21no—
42SD16×Dic61—21no—
43#16839814C2211yescharacteristic
44#106×D61—21no—
45#16839784—21no—
46#16839784C2211yescharacteristic
47C4⋊C8×D61C2211yescharacteristic
48#103×D61—21no—
49C4×#11758189—21no—
50C4×#11758189C2211yescharacteristic
51#16839515C2211yescharacteristic
52#16839514C2211yescharacteristic
53#16839512C2211yescharacteristic
54#16839509C2211yescharacteristic
55#16839509C2211yescharacteristic
56#16839508C2211yescharacteristic
57#16839507C2211yescharacteristic
58#16839507C2211yescharacteristic
59#16839506C2211yescharacteristic
60C61×#492C2211yescharacteristic
61C61×#475C2211yescharacteristic
62C61×#460C2211yescharacteristic
63C8×Q8×C61C2211yescharacteristic
64C61×#334C2211yescharacteristic
65C61×#332C2211yescharacteristic
66C61×#331C2211yescharacteristic
67C2×#11758224—21no—
68C2×Q8×D61—21no—
69C2×Q8×D61D411yescharacteristic
70#14519806—44no—
71#14519806—44no—
72C2×Dic244—21no—
73C2×Dic244—21no—
74C2×#11758202—21no—
75C2×#11758202—21no—
76C2×#11758200—21no—
77C2×#11758200—21no—
78C2×#11758200—21no—
79C2×#11758189—21no—
80C2×#11758189D411yescharacteristic
81C2×#11758189D411yescharacteristic
82C2×C8×D61—21no—
83#14519777C2311yescharacteristic
84#14519777C2311yescharacteristic
85#14519776—21no—
86#14519776—21no—
87#14519775—21no—
88#14519764—21no—
89#14519748—21no—
90#14519748—21no—
91Q8×Dic61D411yescharacteristic
92C4×Dic122—21no—
93C4×Dic122C2311yescharacteristic
94#14519742—21no—
95#14519742D411yescharacteristic
96D4×Dic61—21no—
97D4×Dic61—21no—
98C4⋊C4×D61—21no—
99C4⋊C4×D61—21no—
100C4⋊C4×D61—21no—
101C4⋊C4×D61D411yescharacteristic
102C42×D61—21no—
103C42×D61C2311yescharacteristic
104#14519709—21no—
105#14519709—21no—
106#14519708—21no—
107#14519708—21no—
108#14519708—21no—
109C4⋊#92418—21no—
110C4⋊#92418C2311yescharacteristic
111C4⋊#92418C2311yescharacteristic
112C4⋊#92418C2311yescharacteristic
113#14519706—21no—
114#14519706—21no—
115#14519706—21no—
116#14519706—21no—
117#14519702—21no—
118#14519702—21no—
119#14519701—21no—
120#14519701—21no—
121#14519700—21no—
122#14519698—21no—
123#14519689—21no—
124#14519687—21no—
125#14519687—21no—
126#14519687D411yescharacteristic
127#14519687D411yescharacteristic
128C4×#92418—21no—
129C4×#92418C2311yescharacteristic
130C8×Dic61—21no—
131C2×SD16×C61—21no—
132C2×SD16×C61—21no—
133C2×SD16×C61—21no—
134C4⋊Q8×C61C2311yescharacteristic
135#127×C61C2311yescharacteristic
136C4×Q8×C61C2311yescharacteristic
137#106×C61—21no—
138C4⋊C8×C61—21no—
139C4⋊C8×C61C2311yescharacteristic
140C4⋊C8×C61C2311yescharacteristic
141C4⋊C8×C61C2311yescharacteristic
142#103×C61—21no—
143#102×C61—21no—
144C4×C8×C61—21no—
145C4×C8×C61C2311yescharacteristic
146C2×Q8×C61—21no—
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×C22⋊D61—41no—
152C22×Dic61—41no—
153Q8×D61—22no—
154Q8×D61—41no—
155Q8×D61—42no—
156Q8×D61—42no—
157#11758224—42no—
158#11758224—42no—
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—41no—
162C2×C4×D61C2×D411yescharacteristic
163C2×C4×D61C2×D411yescharacteristic
164C2×C4×D61#4111yescharacteristic
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—41no—
169C2×Dic122C2×D411yescharacteristic
170SD16×C61—42no—
171SD16×C61—42no—
172SD16×C61—42no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61C2×D411yescharacteristic
177C2×C8×C61C2×D411yescharacteristic
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61C2×D411yescharacteristic
182C42×C61—21no—
183C42×C61C2411yescharacteristic
184C22⋊Dic61—41no—
185C22⋊Dic61—41no—
186#11758202—42no—
187#11758202—42no—
188#11758200—42no—
189#11758200—42no—
190#11758200—42no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61C2×D411yescharacteristic
197#11758196—21no—
198#11758196—21no—
199#11758196—21no—
200#11758196—41no—
201#11758196—41no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61C2×D411yescharacteristic
206C4×Dic61C2×D411yescharacteristic
207C4×Dic61#4111yescharacteristic
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418C2×D411yescharacteristic
212C2×#92418C2×D411yescharacteristic
213Dic244—42no—
214Dic244—42no—
215#11758189—22no—
216#11758189—42no—
217#11758189SD1614yescomplemented
218C8×D61—42no—
219C23×C61—41no—
220C22×D61#12111yescharacteristic
221Q8×C61—21no—
222Q8×C61—41no—
223Q8×C61—41no—
224Q8×C61#13712yescomplemented
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—81no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—41no—
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C22⋊D61—82no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—82no—
242C2×Dic61#12111yescharacteristic
243C2×Dic61#12111yescharacteristic
244C2×Dic61#12111yescharacteristic
245C4×D61—21no—
246C4×D61—21no—
247C4×D61—41no—
248C4×D61—42no—
249C4×D61—81no—
250C4×D61—82no—
251C4×D61C2×SD1612yes—
252C4×D61AGL(1,ℤ/8)12yes—
253Dic122—21no—
254Dic122—21no—
255Dic122—42no—
256Dic122—41no—
257Dic122—41no—
258Dic122—41no—
259Dic122—41no—
260Dic122—82no—
261C8×C61—21no—
262C8×C61—41no—
263C8×C61—41no—
264C8×C61—41no—
265C8×C61C2×SD1612yes—
266#92418—21no—
267#92418—41no—
268#92418—41no—
269#92418—41no—
270#92418C2×SD1612yes—
271#19171—611nomaximal, Sylow
272C22×C61—81no—
273C22×C61—81no—
274C22×C61—81no—
275C22×C61C2×#12111yescharacteristic
276C2×D61—21no—
277C2×D61—21no—
278C2×D61#46012yes—
279C4×C61—22no—
280C4×C61—21no—
281C4×C61—41no—
282C4×C61—81no—
283C4×C61C22×SD1611yescharacteristic
284C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
285C4×C61C2×#13711yescharacteristic
286C4×C61#57511yescharacteristic
287Dic61—21no—
288Dic61—21no—
289Dic61—41no—
290Dic61—41no—
291Dic61—81no—
292Dic61—81no—
293Dic61#46012yes—
294#3152—611no—
295#3103—611no—
296C2×#475—611no—
297Q8×M16—611no—
298#1690—611no—
299#1690—611no—
300#1686—611no—
301#1686—611no—
302#1670—611no—
303#1670—611no—
304#1666—611no—
305#1666D6111yescharacteristic, complemented
306#1565—611no—
307#1547—611no—
308#1546—611no—
309C2×C61—161no—
310C2×C61C2×#46011yescharacteristic
311C2×C61#305611yescharacteristic
312C2×C61#306911yescharacteristic
313D61—22no—
314C2×#137—1221no—
315C2×#137—1221no—
316#532—611no—
317C2×C4⋊Q8—611no—
318C2×C4×Q8—611no—
319#497—1221no—
320#497—1221no—
321#494—611no—
322#494—611no—
323#492—611no—
324#492C2×D6111yescharacteristic
325#481—1221no—
326#475—611no—
327#475—611no—
328#475—611no—
329#475—611no—
330#475—611no—
331#475—1221no—
332#475—1221no—
333#475C2×D6111yescharacteristic
334#460—611no—
335#460C2×D6111yescharacteristic
336#448—1221no—
337#446—611no—
338#446—611no—
339#446—1221no—
340#446—1221no—
341C8×Q8—611no—
342C8×Q8C2×D6111yescharacteristic
343C4×SD16—1221no—
344C4×SD16—1221no—
345#431—611no—
346C2×#106—1221no—
347#423—611no—
348#423—1221no—
349C2×C4⋊C8—611no—
350C2×#103—1221no—
351C4×M16—611no—
352C4×M16—1221no—
353#335—611no—
354#335—611no—
355#334—611no—
356#334C2×D6111yescharacteristic
357#332—611no—
358#332—611no—
359#332—611no—
360#332—611no—
361#332—611no—
362#332C2×D6111yescharacteristic
363#331—611no—
364#331C2×D6111yescharacteristic
365C61#1917111yescharacteristic, complemented, Sylow
366C2×#41—1221no—
367C22×Q8—611no—
368C22×Q8—1221no—
369#137—2444no—
370#137—2444no—
371C2×Q16—1221no—
372C2×Q16—1221no—
373C2×Q16—1221no—
374C2×Q16—1221no—
375C2×SD16—21no—
376C2×SD16—21no—
377C2×SD16—21no—
378C2×SD16—1221no—
379C2×SD16—1221no—
380C2×SD16—1221no—
381C2×M16—611no—
382C2×M16—611no—
383C2×M16—1221no—
384C22×C8—1221no—
385C4⋊Q8—611no—
386C4⋊Q8—611no—
387C4⋊Q8—1221no—
388C4⋊Q8—1221no—
389C4⋊Q8—1221no—
390C4⋊Q8C22×D6111yescharacteristic
391#127C22×D6111yescharacteristic
392#124—1221no—
393#121—1221no—
394#121—1221no—
395C4×Q8—611no—
396C4×Q8—611no—
397C4×Q8—611no—
398C4×Q8—1221no—
399C4×Q8—1221no—
400C4×Q8C22×D6111yescharacteristic
401C4×D4—1221no—
402C4×D4—1221no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—1221no—
407C2×C42—611no—
408C2×C42—1221no—
409#106—21no—
410#106—1221no—
411#106—1221no—
412#106—1221no—
413#106—1221no—
414#106—1221no—
415C4⋊C8—21no—
416C4⋊C8—611no—
417C4⋊C8—611no—
418C4⋊C8—611no—
419C4⋊C8—1221no—
420C4⋊C8—1221no—
421C4⋊C8—1221no—
422C4⋊C8—1221no—
423C4⋊C8—1221no—
424C4⋊C8C22×D6111yescharacteristic
425C4⋊C8C22×D6111yescharacteristic
426C4⋊C8C22×D6111yescharacteristic
427#103—21no—
428#103—1221no—
429#103—1221no—
430#103—1221no—
431#103—1221no—
432#103—1221no—
433#102—21no—
434#102—1221no—
435C22⋊C8—1221no—
436#97—611no—
437#97—611no—
438#97—1221no—
439#97—1221no—
440C4×C8—21no—
441C4×C8—611no—
442C4×C8—1221no—
443C4×C8—1221no—
444C4×C8C22×D6111yescharacteristic
445#41—2442no—
446#41—2442no—
447C2×Q8—21no—
448C2×Q8—611no—
449C2×Q8—1221no—
450C2×Q8—1222no—
451C2×Q8—1221no—
452C2×Q8—1221no—
453C2×Q8—2441no—
454C2×Q8—2442no—
455C2×Q8—2442no—
456C2×Q8—2441no—
457C2×Q8D4×D6111yescharacteristic
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—2441no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—2441no—
468C22×C4—2441no—
469Q16—2442no—
470Q16—2442no—
471Q16—2442no—
472Q16—2442no—
473SD16—42no—
474SD16—42no—
475SD16—42no—
476SD16—2442no—
477SD16—2442no—
478SD16—2442no—
479M16—614no—
480M16—1222no—
481M16—2442no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—611no—
486C2×C8—611no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—2442no—
491C2×C8D4×D6111yescharacteristic
492C2×C8D4×D6111yescharacteristic
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—611no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—2441no—
506C4⋊C4—2441no—
507C4⋊C4D4×D6111yescharacteristic
508C22⋊C4—2441no—
509C22⋊C4—2441no—
510C42—21no—
511C42—611no—
512C42—611no—
513C42—611no—
514C42—1221no—
515C42—1221no—
516C42—1221no—
517C42C23×D6111yescharacteristic
518C23—41no—
519C23—611no—
520Q8—21no—
521Q8—41no—
522Q8—41no—
523Q8—1221no—
524Q8—1221no—
525Q8—4882no—
526Q8—2442no—
527Q8—2441no—
528Q8—2441no—
529Q8—2441no—
530Q8—2441no—
531Q8#1451980612yescomplemented
532D4—41no—
533D4—41no—
534D4—41no—
535D4—41no—
536D4—81no—
537D4—4882no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—41no—
541C2×C4—611no—
542C2×C4—612no—
543C2×C4—612no—
544C2×C4—611no—
545C2×C4—611no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—4881no—
551C2×C4—4882no—
552C2×C4—4882no—
553C2×C4—2441no—
554C2×C4—2442no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4C2×D4×D6111yescharacteristic
558C2×C4C2×D4×D6111yescharacteristic
559C2×C4#41×D6111yescharacteristic
560C8—21no—
561C8—41no—
562C8—41no—
563C8—41no—
564C8—612no—
565C8—1221no—
566C8—2441no—
567C8—2441no—
568C8—2441no—
569C8SD16×D6112yes—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—612no—
574C22—1221no—
575C22—1221no—
576C22#121×D6111yescharacteristic
577C4—22no—
578C4—21no—
579C4—41no—
580C4—81no—
581C4—612no—
582C4—1221no—
583C4—1221no—
584C4—4881no—
585C4—4881no—
586C4—2441no—
587C4—2441no—
588C4C2×SD16×D6111yescharacteristic
589C4AGL(1,ℤ/8)×D6111yescharacteristic
590C4C2×#1451980611yescharacteristic
591C4#1684055911yescharacteristic
592C2—161no—
593C2—1222no—
594C2#460×D6111yescharacteristic
595C2#2145544211yescharacteristic
596C2#2145555111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1666C2
socleC22×C61C2×#121
prime radical#16840310C22
hypercenter#1666D61
hypocenterC61#19171
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1666

Complements

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

NH ≅ G/Ntype
3. #475×D61C2semidirect
4. Q8×#11758189C2semidirect
5. #21445314C2semidirect
6. #21445312C2semidirect
7. #21445290C2semidirect
8. #21445288C2semidirect
9. #21445189C2semidirect
10. #21445187C2semidirect
11. #21445165C2semidirect
12. #21444568C2semidirect
14. #21444489C2semidirect
15. C61×#1666C2semidirect
217. #11758189SD16semidirect
224. Q8×C61#137semidirect
305. #1666D61semidirect
365. C61#19171semidirect
531. Q8#14519806semidirect

Maximal subgroups

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

subgroupindex
#214562732
#214558492
#475×D612
Q8×#117581892
#214453142
#214453122
#214452902
#214452882
#214451892
#214451872
#214451652
#214445682
#214444932
#214444892
C61×#16662
#1917161