Finite Group Databasebeta

← #29669566

Subgroups — #29669566

Order 15616.

616 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458751C211yesmaximal, characteristic, complemented
2SD16×Dic122C211yesmaximal, characteristic
3D4×#11758202C211yesmaximal, characteristic
4#21445454C211yesmaximal, characteristic, complemented
5#21445311C211yesmaximal, characteristic, complemented
6#21445246C211yesmaximal, characteristic, complemented
7#21445201C211yesmaximal, characteristic, complemented
8#21445192C211yesmaximal, characteristic
9#21445096C211yesmaximal, characteristic, complemented
10#21445080C211yesmaximal, characteristic
11#21445039C211yesmaximal, characteristic, complemented
12#21444984C211yesmaximal, characteristic
13#21444656C211yesmaximal, characteristic
14#21444636C211yesmaximal, characteristic, complemented
15C61×#3359C211yesmaximal, characteristic
16C22×#11758202—21no—
17Q8×Dic122C2211yescharacteristic
18D4×Dic122C2211yescharacteristic
19#16840172—21no—
20#16840160C2211yescharacteristic
21#16840159—21no—
22#16840158C2211yescharacteristic
23#16840158C2211yescharacteristic
24#16840141C2211yescharacteristic
25#16840114C2211yescharacteristic
26#16840066—21no—
27#16840057C2211yescharacteristic
28#16840052C2211yescharacteristic
29#16840050—21no—
30#16840047—21no—
31#16840047C2211yescharacteristic
32#16840045—21no—
33#16840044C2211yescharacteristic
34#16840001—21no—
35#16839972C2211yescharacteristic
36#16839969C2211yescharacteristic
37#16839918—21no—
38#16839917—21no—
39#16839893C2211yescharacteristic
40C8×Dic122C2211yescharacteristic
41#16839867—21no—
42C4×#11758202C2211yescharacteristic
43SD16×Dic61—21no—
44D4×#92418C2211yescharacteristic
45#16839513C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839510C2211yescharacteristic
48#16839508C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839507C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53#16839492C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic
56C61×#557C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic
58C61×#487—21no—
59C61×#487C2211yescharacteristic
60C61×#475—21no—
61C61×#475C2211yescharacteristic
62C61×#461—21no—
63C61×#461C2211yescharacteristic
64C8×Q8×C61C2211yescharacteristic
65C4×SD16×C61—21no—
66C4×SD16×C61C2211yescharacteristic
67C22×Dic122—21no—
68C2×#11758202—21no—
69C2×#11758202—21no—
70C2×#11758202—21no—
71C2×#11758202—21no—
72C2×#11758202—44no—
73C2×#11758202D411yescharacteristic
74C22×#92418—21no—
75#14519777—21no—
76#14519777C2311yescharacteristic
77#14519777C2311yescharacteristic
78#14519776—21no—
79#14519776—21no—
80#14519776—41no—
81#14519756—21no—
82#14519754—41no—
83Q8×Dic61—21no—
84C4×Dic122—21no—
85C4×Dic122C2311yescharacteristic
86D4×Dic61—21no—
87#14519710—21no—
88#14519710D411yescharacteristic
89#14519709—21no—
90#14519709—21no—
91#14519708—21no—
92#14519708—21no—
93#14519708—21no—
94#14519708—21no—
95#14519708—21no—
96#14519708—41no—
97C4⋊#92418—21no—
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100C4⋊#92418C2311yescharacteristic
101#14519706—21no—
102#14519701—21no—
103#14519701—21no—
104#14519701—21no—
105#14519701D411yescharacteristic
106#14519700—21no—
107#14519700—21no—
108#14519700—41no—
109#14519700D411yescharacteristic
110#14519698—21no—
111C22⋊#92418—21no—
112#14519688—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C8×Dic61—21no—
116C22×Q8×C61—21no—
117C2×SD16×C61—21no—
118C2×SD16×C61D411yescharacteristic
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61—21no—
121C4⋊Q8×C61C2311yescharacteristic
122C4⋊Q8×C61C2311yescharacteristic
123#125×C61—21no—
124#125×C61—41no—
125C22⋊Q8×C61—21no—
126C22⋊Q8×C61—41no—
127C4×Q8×C61—21no—
128C4×Q8×C61—21no—
129C4×Q8×C61C2311yescharacteristic
130C4×Q8×C61C2311yescharacteristic
131C4×D4×C61—21no—
132C4×D4×C61C2311yescharacteristic
133#106×C61—21no—
134#106×C61D411yescharacteristic
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137#103×C61—21no—
138#103×C61—21no—
139#103×C61—21no—
140#103×C61—21no—
141#103×C61D411yescharacteristic
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—42no—
149C2×Q8×C61—81no—
150C2×Q8×C61C2×D411yescharacteristic
151C2×Q8×C61C2×D411yescharacteristic
152C2×D4×C61C2×D411yescharacteristic
153C22×C4×C61—21no—
154C22×C4×C61—41no—
155C22×Dic61—41no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—42no—
159C2×Dic122—41no—
160C2×Dic122—41no—
161C2×Dic122—81no—
162C2×Dic122C2×D411yescharacteristic
163SD16×C61—22no—
164SD16×C61—42no—
165C2×C8×C61—21no—
166C2×C8×C61C2×D411yescharacteristic
167C2×C8×C61C2×D411yescharacteristic
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—41no—
175C4⋊C4×C61—41no—
176C4⋊C4×C61—41no—
177C4⋊C4×C61—41no—
178C4⋊C4×C61C2×D411yescharacteristic
179C4⋊C4×C61C2×D411yescharacteristic
180C4⋊C4×C61C2×D411yescharacteristic
181C22⋊C4×C61—21no—
182C22⋊C4×C61—41no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—41no—
186C42×C61C2411yescharacteristic
187C22⋊Dic61—41no—
188#11758202—42no—
189#11758202—42no—
190#11758202—42no—
191#11758202—81no—
192#11758202SD1614yescomplemented
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—41no—
198C4⋊Dic61C2×D411yescharacteristic
199#11758196—21no—
200#11758196—41no—
201#11758196—41no—
202C4×Dic61—21no—
203C4×Dic61—41no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—42no—
209C2×#92418C2×D411yescharacteristic
210C2×#92418C2×D411yescharacteristic
211C23×C61—21no—
212Q8×C61—21no—
213Q8×C61—21no—
214Q8×C61—21no—
215Q8×C61—41no—
216Q8×C61—41no—
217Q8×C61—41no—
218Q8×C61—41no—
219Q8×C61—82no—
220Q8×C61C2×SD1612yes—
221D4×C61—21no—
222D4×C61#13712yes—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—82no—
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61H(5,𝔽2)11yescharacteristic
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—82no—
241Dic122—21no—
242Dic122—41no—
243Dic122—41no—
244Dic122—82no—
245Dic122—81no—
246Dic122—161no—
247Dic122C2×SD1612yes—
248C8×C61—21no—
249C8×C61—21no—
250C8×C61—41no—
251#92418—21no—
252#92418—41no—
253#92418—41no—
254#92418—41no—
255#92418—41no—
256#92418C2×SD1612yes—
257#19769—611nomaximal, Sylow
258C22×C61—22no—
259C22×C61—41no—
260C22×C61—41no—
261C22×C61D4211yescharacteristic
262C4×C61—22no—
263C4×C61—21no—
264C4×C61—41no—
265C4×C61—41no—
266C4×C61—41no—
267C4×C61—41no—
268C4×C61—81no—
269C4×C61—81no—
270C4×C61C22×SD1611yescharacteristic
271C4×C61C2×#13711yescharacteristic
272C4×C61#57711yescharacteristic
273C4×C61#57811yescharacteristic
274Dic61—22no—
275Dic61—41no—
276Dic61—81no—
277Dic61—81no—
278#3387—611no—
279Q8×SD16—611no—
280#3359D6111yescharacteristic
281D4×Q16—611no—
282#1716—611no—
283#1690—611no—
284#1680—611no—
285#1672—611no—
286#1671—611no—
287#1653—611no—
288#1651—611no—
289#1645—611no—
290#1637—611no—
291#1582—611no—
292#1579—611no—
293C2×C61—42no—
294C2×C61#327011yescharacteristic
295C2×C61D4×SD1611yescharacteristic
296C2×C61#328911yescharacteristic
297C22×Q16—1221no—
298Q82—611no—
299#557C2×D6111yescharacteristic
300D4×Q8—611no—
301D4×Q8C2×D6111yescharacteristic
302#501—1221no—
303#498—611no—
304#498—611no—
305#498—611no—
306#498—1221no—
307#494—611no—
308#487—21no—
309#487—611no—
310#487C2×D6111yescharacteristic
311#479—1221no—
312#477—611no—
313#476—611no—
314#476—1221no—
315#475—21no—
316#475—611no—
317#475—611no—
318#475—1221no—
319#475—1221no—
320#475C2×D6111yescharacteristic
321#467—1221no—
322#462—611no—
323#462—611no—
324#461—21no—
325#461C2×D6111yescharacteristic
326#451—1221no—
327#451—1221no—
328#446—611no—
329C8×Q8—611no—
330C8×Q8C2×D6111yescharacteristic
331#441—1221no—
332C4×Q16—611no—
333C4×SD16—21no—
334C4×SD16—1221no—
335C4×SD16C2×D6111yescharacteristic
336C8×D4—611no—
337#334—611no—
338#334—611no—
339#333—611no—
340#332—611no—
341#332—611no—
342#332—611no—
343Q8⋊C8—611no—
344Q8⋊C8—611no—
345Q8⋊C8—611no—
346D4⋊C8—611no—
347#322—611no—
348C61#1976911yescharacteristic, complemented, Sylow
349C22×Q8—21no—
350C22×Q8—1221no—
351C2×Q16—611no—
352C2×Q16—1221no—
353C2×Q16—1221no—
354C2×Q16—1221no—
355C2×Q16—1221no—
356C2×Q16—2444no—
357C2×SD16—21no—
358C2×SD16C22⋊D6111yescharacteristic
359C22×C8—1221no—
360C4⋊Q8—21no—
361C4⋊Q8—21no—
362C4⋊Q8—611no—
363C4⋊Q8—611no—
364C4⋊Q8—1221no—
365C4⋊Q8—1221no—
366C4⋊Q8—1221no—
367C4⋊Q8—2441no—
368C4⋊Q8C22×D6111yescharacteristic
369C4⋊Q8C22×D6111yescharacteristic
370#125—21no—
371#125—41no—
372C22⋊Q8—21no—
373C22⋊Q8—41no—
374C22⋊Q8—1221no—
375C22⋊Q8—2441no—
376C4×Q8—21no—
377C4×Q8—21no—
378C4×Q8—611no—
379C4×Q8—1221no—
380C4×Q8—1221no—
381C4×Q8C22×D6111yescharacteristic
382C4×Q8C22×D6111yescharacteristic
383C4×D4—21no—
384C4×D4—1221no—
385C4×D4C22×D6111yescharacteristic
386#107—611no—
387#107—1221no—
388#106—21no—
389#106—1221no—
390#106—1221no—
391#106—1221no—
392#106—1221no—
393#106—1221no—
394#106—1221no—
395#106—1221no—
396#106—2441no—
397#106C22⋊D6111yescharacteristic
398C4⋊C8—21no—
399C4⋊C8—611no—
400C4⋊C8—611no—
401C4⋊C8—611no—
402C4⋊C8—1221no—
403C4⋊C8—1221no—
404C4⋊C8C22×D6111yescharacteristic
405#103—21no—
406#103—21no—
407#103—21no—
408#103—21no—
409#103—611no—
410#103—611no—
411#103—1221no—
412#103—1221no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#103—2441no—
417#103C22⋊D6111yescharacteristic
418#102—1221no—
419#102C22⋊D6111yescharacteristic
420C22⋊C8—1221no—
421#97—1221no—
422C4×C8—21no—
423C4×C8—611no—
424C4×C8—611no—
425C4×C8—1221no—
426C4×C8C22×D6111yescharacteristic
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—41no—
430C2×Q8—42no—
431C2×Q8—81no—
432C2×Q8—611no—
433C2×Q8—1221no—
434C2×Q8—1221no—
435C2×Q8—4881no—
436C2×Q8—2441no—
437C2×Q8—2441no—
438C2×Q8—2442no—
439C2×Q8D4×D6111yescharacteristic
440C2×Q8C2×C22⋊D6111yescharacteristic
441C2×D4C2×C22⋊D6111yescharacteristic
442C22×C4—21no—
443C22×C4—41no—
444C22×C4—2441no—
445Q16—614no—
446Q16—4881no—
447Q16—2442no—
448Q16—2442no—
449Q16—2442no—
450SD16—22no—
451SD16—42no—
452C2×C8—21no—
453C2×C8—611no—
454C2×C8—611no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—2442no—
460C2×C8C2×C22⋊D6111yescharacteristic
461C2×C8C2×C22⋊D6111yescharacteristic
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—41no—
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—1221no—
478C4⋊C4—2441no—
479C4⋊C4—2441no—
480C4⋊C4—2441no—
481C4⋊C4D4×D6111yescharacteristic
482C4⋊C4C2×C22⋊D6111yescharacteristic
483C4⋊C4C2×C22⋊D6111yescharacteristic
484C22⋊C4—21no—
485C22⋊C4—41no—
486C22⋊C4—2441no—
487C42—21no—
488C42—21no—
489C42—41no—
490C42—1221no—
491C42—2441no—
492C42C23×D6111yescharacteristic
493C23—21no—
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—41no—
498Q8—41no—
499Q8—41no—
500Q8—41no—
501Q8—82no—
502Q8—612no—
503Q8—1221no—
504Q8—9761no—
505Q8—4881no—
506Q8—4882no—
507Q8—2441no—
508Q8—2441no—
509Q8SD16×D6112yes—
510D4—21no—
511D4#1451980712yes—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—82no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—4882no—
525C2×C4—2441no—
526C2×C4—2441no—
527C2×C4C2×D4×D6111yescharacteristic
528C2×C4C22×C22⋊D6111yescharacteristic
529C2×C4#1451982211yescharacteristic
530C8—21no—
531C8—21no—
532C8—41no—
533C8—612no—
534C8—1221no—
535C8—2441no—
536C8—2441no—
537C8—2441no—
538C8—2441no—
539C22—22no—
540C22—41no—
541C22—41no—
542C22D4×C22⋊D6111yescharacteristic
543C4—22no—
544C4—21no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—81no—
550C4—81no—
551C4—1222no—
552C4—4881no—
553C4—4881no—
554C4—2441no—
555C4C2×SD16×D6111yescharacteristic
556C4C2×#1451980711yescharacteristic
557C4#1684056811yescharacteristic
558C4#1684057711yescharacteristic
559C2—42no—
560C2#2145738511yescharacteristic
561C2SD16×C22⋊D6111yescharacteristic
562C2#2145759011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3359C2
socleC22×C61D42
prime radicalC2×D4×C61C2×D4
hypercenter#3359D61
hypocenterC61#19769
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3359

Complements

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

NH ≅ G/Ntype
1. #21458751C2semidirect
4. #21445454C2semidirect
5. #21445311C2semidirect
6. #21445246C2semidirect
7. #21445201C2semidirect
9. #21445096C2semidirect
11. #21445039C2semidirect
14. #21444636C2semidirect
192. #11758202SD16semidirect
348. C61#19769semidirect

Maximal subgroups

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

subgroupindex
#214587512
SD16×Dic1222
D4×#117582022
#214454542
#214453112
#214452462
#214452012
#214451922
#214450962
#214450802
#214450392
#214449842
#214446562
#214446362
C61×#33592
#1976961