Finite Group Databasebeta

← #29669941

Subgroups — #29669941

Order 15616.

582 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458600C211yesmaximal, characteristic, complemented
2#21458502C211yesmaximal, characteristic, complemented
3#21457736C211yesmaximal, characteristic
4#21445463C211yesmaximal, characteristic, complemented
5#21445371C211yesmaximal, characteristic, complemented
6#21445272C211yesmaximal, characteristic, complemented
7#21445242C211yesmaximal, characteristic, complemented
8#21445198C211yesmaximal, characteristic, complemented
9#21445060C211yesmaximal, characteristic, complemented
10#21445054C211yesmaximal, characteristic
11#21445033C211yesmaximal, characteristic, complemented
12#21444989C211yesmaximal, characteristic, complemented
13#21444656C211yesmaximal, characteristic, complemented
14#21444631C211yesmaximal, characteristic, complemented
15C61×#3335C211yesmaximal, characteristic
16#16840491C2211yescharacteristic
17#16840482C2211yescharacteristic
18#16840165C2211yescharacteristic
19#16840132—21no—
20#16840115—21no—
21#16840114C2211yescharacteristic
22#16840113C2211yescharacteristic
23#16840112C2211yescharacteristic
24#16840112C2211yescharacteristic
25#16840100—21no—
26#16840072—21no—
27#16840066—21no—
28#16840054C2211yescharacteristic
29#16840042C2211yescharacteristic
30#16840042C2211yescharacteristic
31#16840001—21no—
32#16839982—21no—
33#16839972C2211yescharacteristic
34#16839969C2211yescharacteristic
35#16839967—21no—
36#16839967C2211yescharacteristic
37#16839924—21no—
38#16839893C2211yescharacteristic
39C8×Dic122C2211yescharacteristic
40#16839862—21no—
41#16839854—21no—
42C4×#11758200C2211yescharacteristic
43D4×#92418C2211yescharacteristic
44#16839758—21no—
45#16839514C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839507C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52#16839492C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic
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—21no—
68C2×#11758202—21no—
69C2×#11758200—21no—
70C2×#11758200D411yescharacteristic
71C22×#92418—21no—
72#14519777—21no—
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519770—41no—
76#14519768—21no—
77#14519768—41no—
78#14519764—21no—
79#14519756—21no—
80Q8×Dic61—21no—
81C4×Dic122—21no—
82C4×Dic122C2311yescharacteristic
83#14519742—21no—
84#14519736—21no—
85#14519710—21no—
86#14519710—21no—
87#14519710—21no—
88#14519710—41no—
89#14519708D411yescharacteristic
90C4⋊#92418—21no—
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94#14519706—21no—
95#14519702—21no—
96#14519702—21no—
97#14519701—21no—
98#14519701—21no—
99#14519701—21no—
100#14519701—41no—
101#14519700—21no—
102#14519700—21no—
103#14519700—21no—
104#14519700—21no—
105#14519700—21no—
106#14519700D411yescharacteristic
107#14519698—21no—
108#14519698—21no—
109#14519698D411yescharacteristic
110C22⋊#92418—21no—
111#14519688—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
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×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122C2×D411yescharacteristic
159SD16×C61—41no—
160C2×C8×C61—21no—
161C2×C8×C61—42no—
162C2×C8×C61C2×D411yescharacteristic
163C2×C8×C61C2×D411yescharacteristic
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—41no—
170C4⋊C4×C61—41no—
171C4⋊C4×C61—41no—
172C4⋊C4×C61C2×D411yescharacteristic
173C4⋊C4×C61C2×D411yescharacteristic
174C4⋊C4×C61C2×D411yescharacteristic
175C22⋊C4×C61—21no—
176C22⋊C4×C61—21no—
177C22⋊C4×C61—41no—
178C22⋊C4×C61—41no—
179C42×C61—21no—
180C42×C61C2411yescharacteristic
181C22⋊Dic61—41no—
182C22⋊Dic61—41no—
183#11758202—42no—
184#11758202—81no—
185#11758200—22no—
186#11758200—42no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—41no—
192C4⋊Dic61—41no—
193C4⋊Dic61C2×D411yescharacteristic
194#11758196—21no—
195#11758196—41no—
196#11758196—41no—
197C4×Dic61—21no—
198C4×Dic61—41no—
199C4×Dic61—41no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—81no—
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207C23×C61—21no—
208C23×C61—21no—
209Q8×C61—21no—
210Q8×C61—21no—
211Q8×C61—41no—
212Q8×C61—41no—
213Q8×C61—81no—
214D4×C61—21no—
215D4×C61—21no—
216D4×C61—21no—
217D4×C61—82no—
218D4×C61#13512yes—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—82no—
230C2×C4×C61—161no—
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61H(5,𝔽2)11yescharacteristic
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238Dic122—21no—
239Dic122—21no—
240Dic122—41no—
241Dic122—41no—
242Dic122—41no—
243Dic122—41no—
244C8×C61—21no—
245C8×C61—21no—
246C8×C61—41no—
247#92418—21no—
248#92418—21no—
249#92418—41no—
250#92418—41no—
251#92418—41no—
252#92418—41no—
253#19794—611nomaximal, Sylow
254C22×C61—22no—
255C22×C61—41no—
256C22×C61—41no—
257C22×C61—41no—
258C22×C61—41no—
259C22×C61—41no—
260C22×C61D4211yescharacteristic
261C4×C61—21no—
262C4×C61—41no—
263C4×C61—41no—
264C4×C61—41no—
265C4×C61—41no—
266C4×C61—81no—
267C4×C61—81no—
268C4×C61C2×#13511yescharacteristic
269C4×C61#57511yescharacteristic
270C4×C61#57711yescharacteristic
271C4×C61#57811yescharacteristic
272Dic61—41no—
273Dic61—41no—
274Dic61—81no—
275Dic61—81no—
276#3372—611no—
277#3364—611no—
278#3335D6111yescharacteristic
279#3302—611no—
280#1718—611no—
281#1700—611no—
282#1683—611no—
283#1679—611no—
284#1672—611no—
285#1648—611no—
286#1647—611no—
287#1644—611no—
288#1638—611no—
289#1582—611no—
290SD16⋊C8—611no—
291C2×C61—42no—
292C2×C61—81no—
293C2×C61#326911yescharacteristic
294C2×C61#328211yescharacteristic
295C2×C61#328911yescharacteristic
296#557—611no—
297#556—611no—
298#554C2×D6111yescharacteristic
299#547C2×D6111yescharacteristic
300#500—611no—
301#498C2×D6111yescharacteristic
302#491—1221no—
303#488—1221no—
304#487—611no—
305#487—611no—
306#487—611no—
307#487—611no—
308#485—1221no—
309#483—21no—
310#481—21no—
311#479—1221no—
312#479—1221no—
313#477—611no—
314#476C2×D6111yescharacteristic
315#475C2×D6111yescharacteristic
316#474—611no—
317#474—611no—
318#467—1221no—
319#465—21no—
320#464—1221no—
321#462—611no—
322#462—611no—
323#461—611no—
324#461—1221no—
325#452—1221no—
326#446—611no—
327C8×Q8—611no—
328#440—1221no—
329C4×Q16—1221no—
330C4×SD16—611no—
331C4×SD16C2×D6111yescharacteristic
332C8×D4—611no—
333C8×D4C2×D6111yescharacteristic
334C2×#106—21no—
335#416—1221no—
336#335—611no—
337#334—611no—
338#333—611no—
339#332—611no—
340#332—611no—
341#332—611no—
342Q8⋊C8—611no—
343Q8⋊C8—611no—
344D4⋊C8—611no—
345D4⋊C8—611no—
346#322—611no—
347C61#1979411yescharacteristic, complemented, Sylow
348C2×#41—21no—
349C2×Q16—1221no—
350C2×Q16—1221no—
351C2×SD16—611no—
352C2×SD16—1221no—
353C2×SD16C22⋊D6111yescharacteristic
354C22×C8—21no—
355C22×C8—1221no—
356C4⋊Q8—611no—
357C4⋊Q8—611no—
358C4⋊Q8—1221no—
359C4⋊Q8C22×D6111yescharacteristic
360C4⋊Q8C22×D6111yescharacteristic
361#126—2441no—
362#125—21no—
363#125—1221no—
364#125—2441no—
365#124—1221no—
366#123—41no—
367C22⋊Q8—21no—
368C22⋊Q8—21no—
369C22⋊Q8—41no—
370C22⋊Q8—1221no—
371#121—21no—
372C4×Q8—611no—
373C4×Q8—1221no—
374C4×Q8—1221no—
375C4×Q8—1221no—
376C4×Q8C22×D6111yescharacteristic
377C4×D4—21no—
378C4×D4C22×D6111yescharacteristic
379C4×D4C22×D6111yescharacteristic
380#117—1221no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383#107—1221no—
384#107—1221no—
385#107—1221no—
386#107—2441no—
387#106—21no—
388#106—21no—
389#106—21no—
390#106—21no—
391#106—611no—
392#106C22⋊D6111yescharacteristic
393C4⋊C8—611no—
394C4⋊C8—611no—
395C4⋊C8—611no—
396C4⋊C8—1221no—
397C4⋊C8—1221no—
398C4⋊C8C22×D6111yescharacteristic
399#103—21no—
400#103—611no—
401#103—1221no—
402#103—1221no—
403#103—1221no—
404#103—1221no—
405#103—1221no—
406#103—1221no—
407#103—1221no—
408#103—1221no—
409#103—1221no—
410#103—1221no—
411#103—2441no—
412#103C22⋊D6111yescharacteristic
413#102—21no—
414#102—611no—
415#102—1221no—
416#102—1221no—
417#102C22⋊D6111yescharacteristic
418C22⋊C8—21no—
419C22⋊C8—1221no—
420#97—1221no—
421C4×C8—611no—
422C4×C8—611no—
423C4×C8—1221no—
424C4×C8C22×D6111yescharacteristic
425#41—42no—
426#41—81no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—611no—
430C2×Q8—1221no—
431C2×Q8—1221no—
432C2×Q8C2×C22⋊D6111yescharacteristic
433C2×D4—41no—
434C2×D4D4×D6111yescharacteristic
435C2×D4C2×C22⋊D6111yescharacteristic
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—41no—
439C22×C4—41no—
440Q16—4881no—
441Q16—2442no—
442SD16—41no—
443SD16—1222no—
444SD16—2442no—
445C2×C8—21no—
446C2×C8—42no—
447C2×C8—611no—
448C2×C8—611no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8—1221no—
452C2×C8—1221no—
453C2×C8—4881no—
454C2×C8C2×C22⋊D6111yescharacteristic
455C2×C8C2×C22⋊D6111yescharacteristic
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—611no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—2441no—
471C4⋊C4—2441no—
472C4⋊C4—2441no—
473C4⋊C4—2441no—
474C4⋊C4D4×D6111yescharacteristic
475C4⋊C4C2×C22⋊D6111yescharacteristic
476C4⋊C4C2×C22⋊D6111yescharacteristic
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—2441no—
482C22⋊C4—2441no—
483C42—21no—
484C42—1221no—
485C42—2441no—
486C42—2441no—
487C42C23×D6111yescharacteristic
488C23—21no—
489C23—21no—
490Q8—21no—
491Q8—21no—
492Q8—41no—
493Q8—41no—
494Q8—81no—
495Q8—1221no—
496Q8—1221no—
497Q8—2441no—
498Q8—2441no—
499Q8—2441no—
500Q8—2441no—
501D4—21no—
502D4—21no—
503D4—21no—
504D4—82no—
505D4#1451979812yes—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—82no—
517C2×C4—161no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—2441no—
521C2×C4—2441no—
522C2×C4C2×D4×D6111yescharacteristic
523C2×C4C22×C22⋊D6111yescharacteristic
524C2×C4#1451982211yescharacteristic
525C8—21no—
526C8—21no—
527C8—41no—
528C8—1221no—
529C8—1221no—
530C8—2441no—
531C8—2441no—
532C8—2441no—
533C8—2441no—
534C22—22no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22D4×C22⋊D6111yescharacteristic
541C4—21no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—81no—
547C4—81no—
548C4—4881no—
549C4—4881no—
550C4—2441no—
551C4—2441no—
552C4C2×#1451979811yescharacteristic
553C4#1684056011yescharacteristic
554C4#1684056811yescharacteristic
555C4#1684057711yescharacteristic
556C2—42no—
557C2—81no—
558C2#2145737011yescharacteristic
559C2#2145752611yescharacteristic
560C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3335C2
socleC22×C61D42
prime radicalC61×#547C22
hypercenter#3335D61
hypocenterC61#19794
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. #21458600C2semidirect
2. #21458502C2semidirect
4. #21445463C2semidirect
5. #21445371C2semidirect
6. #21445272C2semidirect
7. #21445242C2semidirect
8. #21445198C2semidirect
9. #21445060C2semidirect
11. #21445033C2semidirect
12. #21444989C2semidirect
13. #21444656C2semidirect
14. #21444631C2semidirect
347. C61#19794semidirect

Maximal subgroups

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

subgroupindex
#214586002
#214585022
#214577362
#214454632
#214453712
#214452722
#214452422
#214451982
#214450602
#214450542
#214450332
#214449892
#214446562
#214446312
C61×#33352
#1979461