Finite Group Databasebeta

← #29670950

Subgroups — #29670950

Order 15616.

596 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458682C211yesmaximal, characteristic, complemented
2#21458447C211yesmaximal, characteristic, complemented
3#21457747C211yesmaximal, characteristic
4#21445493C211yesmaximal, characteristic, complemented
5#21445315C211yesmaximal, characteristic, complemented
6#21445275C211yesmaximal, characteristic, complemented
7#21445218C211yesmaximal, characteristic, complemented
8#21445216C211yesmaximal, characteristic, complemented
9#21445025C211yesmaximal, characteristic, complemented
10#21444970C211yesmaximal, characteristic, complemented
11#21444963C211yesmaximal, characteristic, complemented
12#21444953C211yesmaximal, characteristic, complemented
13#21444648C211yesmaximal, characteristic, complemented
14SD16⋊#92418C211yesmaximal, characteristic, complemented
15C61×#3335C211yesmaximal, characteristic, complemented
16#16840477C2211yescharacteristic
17#16840472C2211yescharacteristic
18#16840167C2211yescharacteristic
19#16840165C2211yescharacteristic
20#16840161—21no—
21#16840141C2211yescharacteristic
22#16840114C2211yescharacteristic
23#16840112C2211yescharacteristic
24#16840102—21no—
25#16840100—21no—
26#16840074—21no—
27#16840062—21no—
28#16840059—21no—
29#16840057C2211yescharacteristic
30#16840052C2211yescharacteristic
31#16840052C2211yescharacteristic
32#16840052C2211yescharacteristic
33#16840049C2211yescharacteristic
34#16840005—21no—
35#16839993—21no—
36#16839964C2211yescharacteristic
37#16839907—21no—
38Q8×#92418C2211yescharacteristic
39C4×#11758199—21no—
40C4×#11758199C2211yescharacteristic
41#16839841C2211yescharacteristic
42D4×#92418C2211yescharacteristic
43#16839801—21no—
44#16839758—21no—
45#16839515C2211yescharacteristic
46#16839514C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839510C2211yescharacteristic
50#16839508C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839483C2211yescharacteristic
56C61×#554C2211yescharacteristic
57C61×#547C2211yescharacteristic
58C61×#498C2211yescharacteristic
59C61×#483—21no—
60C61×#481—21no—
61C61×#476C2211yescharacteristic
62C61×#475C2211yescharacteristic
63C61×#465—21no—
64C4×SD16×C61C2211yescharacteristic
65C8×D4×C61C2211yescharacteristic
66C2×#106×C61—21no—
67C2×#11758202—41no—
68C2×#11758201—21no—
69C2×#11758199D411yescharacteristic
70C2×#11758189—21no—
71C22×#92418—21no—
72#14519777C2311yescharacteristic
73#14519777C2311yescharacteristic
74#14519769—21no—
75#14519763—41no—
76#14519755—41no—
77#14519753—21no—
78#14519751—21no—
79#14519748—21no—
80C4×D244—21no—
81C4×D244C2311yescharacteristic
82#14519736—21no—
83#14519735—21no—
84C4⋊C4×D61—21no—
85#14519711—21no—
86#14519711—21no—
87#14519710—21no—
88#14519710—21no—
89#14519710—21no—
90#14519710D411yescharacteristic
91#14519708—21no—
92#14519708—21no—
93#14519708—21no—
94#14519708—21no—
95C4⋊#92418—21no—
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99#14519701—21no—
100#14519701—21no—
101#14519701—21no—
102#14519701—21no—
103#14519700—41no—
104#14519698—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698D411yescharacteristic
108#14519697—21no—
109#14519697D411yescharacteristic
110C22⋊#92418—21no—
111#14519689—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×#41×C61—21no—
116C2×SD16×C61D411yescharacteristic
117C22×C8×C61—21no—
118C4⋊Q8×C61C2311yescharacteristic
119C4⋊Q8×C61C2311yescharacteristic
120#125×C61—21no—
121#123×C61—41no—
122C22⋊Q8×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—41no—
125#121×C61—21no—
126C4×Q8×C61C2311yescharacteristic
127C4×D4×C61—21no—
128C4×D4×C61C2311yescharacteristic
129C4×D4×C61C2311yescharacteristic
130C2×C4⋊C4×C61—21no—
131C2×C4⋊C4×C61—21no—
132#106×C61—21no—
133#106×C61—21no—
134#106×C61—21no—
135#106×C61—21no—
136#106×C61D411yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61D411yescharacteristic
140#102×C61—21no—
141#102×C61D411yescharacteristic
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144#41×C61—42no—
145#41×C61—81no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61C2×D411yescharacteristic
149C2×D4×C61—41no—
150C2×D4×C61C2×D411yescharacteristic
151C2×D4×C61C2×D411yescharacteristic
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—41no—
155C22×C4×C61—41no—
156C2×C22⋊D61—41no—
157C2×D244C2×D411yescharacteristic
158C2×C4×D61—21no—
159C2×C4×D61—41no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162SD16×C61—22no—
163C2×C8×C61—21no—
164C2×C8×C61—42no—
165C2×C8×C61C2×D411yescharacteristic
166C2×C8×C61C2×D411yescharacteristic
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—41no—
173C4⋊C4×C61—41no—
174C4⋊C4×C61—41no—
175C4⋊C4×C61C2×D411yescharacteristic
176C4⋊C4×C61C2×D411yescharacteristic
177C4⋊C4×C61C2×D411yescharacteristic
178C22⋊C4×C61—21no—
179C22⋊C4×C61—21no—
180C22⋊C4×C61—41no—
181C22⋊C4×C61—41no—
182C42×C61—21no—
183C42×C61C2411yescharacteristic
184C22⋊Dic61—41no—
185#11758202—82no—
186#11758201—81no—
187#11758199—22no—
188#11758198—21no—
189#11758198—41no—
190#11758198—41no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61C2×D411yescharacteristic
196#11758196—41no—
197#11758196—41no—
198C4×Dic61—41no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—81no—
204C2×#92418C2×D411yescharacteristic
205C2×#92418C2×D411yescharacteristic
206#11758189—42no—
207C23×C61—21no—
208C23×C61—21no—
209C22×D61—21no—
210Q8×C61—21no—
211Q8×C61—21no—
212Q8×C61—41no—
213Q8×C61—41no—
214Q8×C61—81no—
215D4×C61—21no—
216D4×C61—21no—
217D4×C61—21no—
218D4×C61—82no—
219D4×C61#13712yes—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—82no—
231C2×C4×C61—161no—
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61H(5,𝔽2)11yescharacteristic
235C22⋊D61—161no—
236C2×Dic61—21no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239D244—21no—
240D244—21no—
241C4×D61—41no—
242C4×D61—41no—
243C4×D61—161no—
244Dic122—81no—
245Dic122—81no—
246C8×C61—21no—
247C8×C61—41no—
248C8×C61#13512yes—
249#92418—21no—
250#92418—21no—
251#92418—41no—
252#92418—41no—
253#92418—41no—
254#92418—41no—
255#19861—611nomaximal, Sylow
256C22×C61—22no—
257C22×C61—41no—
258C22×C61—41no—
259C22×C61—41no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61D4211yescharacteristic
263C2×D61—41no—
264C2×D61—41no—
265C2×D61—41no—
266C4×C61—21no—
267C4×C61—41no—
268C4×C61—41no—
269C4×C61—41no—
270C4×C61—41no—
271C4×C61—81no—
272C4×C61—81no—
273C4×C61C2×#13511yescharacteristic
274C4×C61C2×#13711yescharacteristic
275C4×C61#57711yescharacteristic
276C4×C61#57811yescharacteristic
277Dic61—41no—
278Dic61—81no—
279Dic61—81no—
280#3380—611no—
281#3357—611no—
282#3335D6111yescharacteristic, complemented
283#3303—611no—
284#1723—611no—
285#1691—611no—
286#1684—611no—
287#1675—611no—
288#1675—611no—
289#1643—611no—
290#1635—611no—
291#1634—611no—
292#1632—611no—
293#1581—611no—
294SD16⋊C8—611no—
295C2×C61—42no—
296C2×C61—81no—
297C2×C61#326611yescharacteristic
298C2×C61#328311yescharacteristic
299C2×C61#328811yescharacteristic
300D61—81no—
301#555—611no—
302#554—611no—
303#554C2×D6111yescharacteristic
304#547C2×D6111yescharacteristic
305#500—611no—
306#500—611no—
307#499—1221no—
308#498C2×D6111yescharacteristic
309#494—611no—
310#487—611no—
311#487—611no—
312#485—1221no—
313#485—1221no—
314#483—21no—
315#481—21no—
316#480—1221no—
317#478—1221no—
318#478—1221no—
319#477—611no—
320#476—611no—
321#476—611no—
322#476—611no—
323#476—611no—
324#476C2×D6111yescharacteristic
325#475C2×D6111yescharacteristic
326#468—1221no—
327#465—21no—
328#465—1221no—
329#461—611no—
330#449—1221no—
331C8×Q8—611no—
332C4×SD16C2×D6111yescharacteristic
333C4×D8—611no—
334C4×D8—1221no—
335#436—611no—
336C8×D4—611no—
337C8×D4C2×D6111yescharacteristic
338#428—1221no—
339C2×#106—21no—
340#416—1221no—
341#335—611no—
342#335—611no—
343#333—611no—
344#333—611no—
345#333—611no—
346#332—611no—
347Q8⋊C8—611no—
348D4⋊C8—611no—
349D4⋊C8—611no—
350D4⋊C8—611no—
351#322—611no—
352C61#1986111yescharacteristic, complemented, Sylow
353C2×#41—21no—
354C2×Q16—2441no—
355C2×SD16—1221no—
356C2×SD16C22⋊D6111yescharacteristic
357C2×D8—611no—
358C2×M16—1221no—
359C22×C8—21no—
360C22×C8—1221no—
361C4⋊Q8—611no—
362C4⋊Q8—611no—
363C4⋊Q8C22×D6111yescharacteristic
364C4⋊Q8C22×D6111yescharacteristic
365#125—21no—
366#125—1221no—
367#124—2441no—
368#123—41no—
369C22⋊Q8—21no—
370C22⋊Q8—21no—
371C22⋊Q8—41no—
372C22⋊Q8—1221no—
373C22⋊Q8—2441no—
374#121—21no—
375#121—1221no—
376#121—1221no—
377C4×Q8C22×D6111yescharacteristic
378C4×D4—21no—
379C4×D4—611no—
380C4×D4—1221no—
381C4×D4C22×D6111yescharacteristic
382C4×D4C22×D6111yescharacteristic
383#117—1221no—
384#117—1221no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—1221no—
388#107—611no—
389#107—1221no—
390#107—1221no—
391#107—1221no—
392#107—1221no—
393#107—1221no—
394#106—21no—
395#106—21no—
396#106—21no—
397#106—21no—
398#106—1221no—
399#106—1221no—
400#106—1221no—
401#106—1221no—
402#106C22⋊D6111yescharacteristic
403C4⋊C8—611no—
404C4⋊C8—611no—
405C4⋊C8—611no—
406C4⋊C8—1221no—
407C4⋊C8C22×D6111yescharacteristic
408#103—21no—
409#103—1221no—
410#103—1221no—
411#103—1221no—
412#103—1221no—
413#103—2441no—
414#103C22⋊D6111yescharacteristic
415#102—21no—
416#102—611no—
417#102—611no—
418#102—1221no—
419#102—1221no—
420#102—1221no—
421#102—1221no—
422#102C22⋊D6111yescharacteristic
423C22⋊C8—21no—
424C22⋊C8—1221no—
425C22⋊C8—1221no—
426C4×C8—611no—
427C4×C8—611no—
428C4×C8—1221no—
429C4×C8C22×D6111yescharacteristic
430#41—42no—
431#41—81no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—1221no—
435C2×Q8—1221no—
436C2×Q8C2×C22⋊D6111yescharacteristic
437C2×D4—41no—
438C2×D4—611no—
439C2×D4—2441no—
440C2×D4D4×D6111yescharacteristic
441C2×D4C2×C22⋊D6111yescharacteristic
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—1221no—
447C22×C4—2441no—
448Q16—4882no—
449SD16—22no—
450SD16—4881no—
451D8—1222no—
452M16—2442no—
453C2×C8—21no—
454C2×C8—42no—
455C2×C8—611no—
456C2×C8—611no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—4881no—
462C2×C8C2×C22⋊D6111yescharacteristic
463C2×C8C2×C22⋊D6111yescharacteristic
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—611no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—2441no—
478C4⋊C4—2441no—
479C4⋊C4D4×D6111yescharacteristic
480C4⋊C4C2×C22⋊D6111yescharacteristic
481C4⋊C4C2×C22⋊D6111yescharacteristic
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—1221no—
487C22⋊C4—2441no—
488C22⋊C4—2441no—
489C22⋊C4—2441no—
490C42—21no—
491C42—2441no—
492C42C23×D6111yescharacteristic
493C23—21no—
494C23—21no—
495C23—1221no—
496Q8—21no—
497Q8—21no—
498Q8—41no—
499Q8—41no—
500Q8—81no—
501Q8—4881no—
502Q8—4881no—
503D4—21no—
504D4—21no—
505D4—21no—
506D4—82no—
507D4—1221no—
508D4—1221no—
509D4—9761no—
510D4#1451980612yes—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—82no—
522C2×C4—161no—
523C2×C4—1221no—
524C2×C4—9761no—
525C2×C4—2441no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4C2×D4×D6111yescharacteristic
530C2×C4C22×C22⋊D6111yescharacteristic
531C2×C4#1451982211yescharacteristic
532C8—21no—
533C8—41no—
534C8—1221no—
535C8—1221no—
536C8—2441no—
537C8—2441no—
538C8—2441no—
539C8—2441no—
540C8#1451979712yes—
541C22—22no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—2441no—
548C22—2441no—
549C22—2441no—
550C22D4×C22⋊D6111yescharacteristic
551C4—21no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4—81no—
557C4—81no—
558C4—4881no—
559C4—4881no—
560C4—2441no—
561C4C2×#1451979711yescharacteristic
562C4C2×#1451980611yescharacteristic
563C4#1684056811yescharacteristic
564C4#1684057711yescharacteristic
565C2—42no—
566C2—81no—
567C2—4881no—
568C2#2145732911yescharacteristic
569C2#2145753711yescharacteristic
570C2#2145758111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3335C2
socleC22×C61D42
prime radical#21457747C2
hypercenter#3335D61
hypocenterC61#19861
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3335

Complements

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

NH ≅ G/Ntype
1. #21458682C2semidirect
2. #21458447C2semidirect
4. #21445493C2semidirect
5. #21445315C2semidirect
6. #21445275C2semidirect
7. #21445218C2semidirect
8. #21445216C2semidirect
9. #21445025C2semidirect
10. #21444970C2semidirect
11. #21444963C2semidirect
12. #21444953C2semidirect
13. #21444648C2semidirect
14. SD16⋊#92418C2semidirect
15. C61×#3335C2semidirect
282. #3335D61semidirect
352. C61#19861semidirect

Maximal subgroups

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

subgroupindex
#214586822
#214584472
#214577472
#214454932
#214453152
#214452752
#214452182
#214452162
#214450252
#214449702
#214449632
#214449532
#214446482
SD16⋊#924182
C61×#33352
#1986161