Finite Group Databasebeta

← #29670899

Subgroups — #29670899

Order 15616.

580 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458811C211yesmaximal, characteristic
2#21458504C211yesmaximal, characteristic, complemented
3#21458251C211yesmaximal, characteristic, complemented
4#21445438C211yesmaximal, characteristic, complemented
5#21445337C211yesmaximal, characteristic, complemented
6#21445251C211yesmaximal, characteristic, complemented
7#21445165C211yesmaximal, characteristic, complemented
8#21445157C211yesmaximal, characteristic
9#21445096C211yesmaximal, characteristic, complemented
10#21445087C211yesmaximal, characteristic, complemented
11#21445005C211yesmaximal, characteristic, complemented
12#21444928C211yesmaximal, characteristic
13#21444653C211yesmaximal, characteristic, complemented
14#21444638C211yesmaximal, characteristic, complemented
15C61×#3400C211yesmaximal, characteristic, complemented
16#16840485C2211yescharacteristic
17#16840485C2211yescharacteristic
18#16840167C2211yescharacteristic
19#16840158C2211yescharacteristic
20#16840144—21no—
21#16840134C2211yescharacteristic
22#16840119—21no—
23#16840114C2211yescharacteristic
24#16840108C2211yescharacteristic
25#16840102—21no—
26#16840102—21no—
27#16840066—21no—
28#16840047C2211yescharacteristic
29#16840047C2211yescharacteristic
30#16840039C2211yescharacteristic
31#16840000—21no—
32#16839984—21no—
33#16839981—21no—
34#16839977—21no—
35#16839972C2211yescharacteristic
36#16839972C2211yescharacteristic
37#16839954C2211yescharacteristic
38Q8×#92418C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40C4×#11758202—21no—
41C4×#11758201C2211yescharacteristic
42C4×#11758200—21no—
43C8×D244C2211yescharacteristic
44#16839799—21no—
45#16839513C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839508C2211yescharacteristic
48#16839507C2211yescharacteristic
49#16839507C2211yescharacteristic
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53Q8⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839483C2211yescharacteristic
56C61×#559C2211yescharacteristic
57C61×#557C2211yescharacteristic
58C61×#497—21no—
59C61×#492C2211yescharacteristic
60C61×#464—21no—
61C61×#462C2211yescharacteristic
62C61×#461—21no—
63C61×#460C2211yescharacteristic
64C61×#446C2211yescharacteristic
65C61×#440—21no—
66C4×Q16×C61C2211yescharacteristic
67C2×#11758202—21no—
68C2×#11758202—21no—
69C2×#11758200—21no—
70C2×#11758200—21no—
71C2×#11758201D411yescharacteristic
72C2×#11758199—21no—
73C2×C8×D61—21no—
74#14519777C2311yescharacteristic
75#14519777C2311yescharacteristic
76#14519772—41no—
77#14519772—41no—
78#14519765—21no—
79#14519765—21no—
80#14519753—21no—
81#14519753—21no—
82C4×Dic122—21no—
83C4×Dic122—21no—
84C4×D244C2311yescharacteristic
85#14519735—21no—
86#14519735—21no—
87#14519711—21no—
88#14519711—21no—
89#14519710—21no—
90#14519708—21no—
91#14519708—21no—
92#14519708—41no—
93#14519708D411yescharacteristic
94C4⋊#92418—21no—
95C4⋊#92418—21no—
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99#14519701—21no—
100#14519701—21no—
101#14519701D411yescharacteristic
102#14519700—21no—
103#14519700—21no—
104#14519700—21no—
105#14519700—21no—
106#14519698—21no—
107#14519698—41no—
108#14519697—21no—
109#14519697D411yescharacteristic
110#14519689—21no—
111C4×#92418—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×Q16×C61—21no—
116C2×Q16×C61D411yescharacteristic
117C2×SD16×C61—21no—
118C2×SD16×C61—21no—
119C2×SD16×C61—21no—
120C4⋊Q8×C61—21no—
121C4⋊Q8×C61C2311yescharacteristic
122#127×C61C2311yescharacteristic
123#125×C61—21no—
124#125×C61—41no—
125#124×C61—21no—
126#124×C61—21no—
127#123×C61—41no—
128C4×Q8×C61—21no—
129C4×Q8×C61—21no—
130C4×Q8×C61C2311yescharacteristic
131C4×Q8×C61C2311yescharacteristic
132C4×Q8×C61C2311yescharacteristic
133C4×D4×C61—21no—
134C4×D4×C61—21no—
135#107×C61D411yescharacteristic
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61D411yescharacteristic
140#103×C61D411yescharacteristic
141#102×C61—21no—
142#97×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61C2×D411yescharacteristic
146C2×Q8×C61C2×D411yescharacteristic
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C22×C4×C61—41no—
152C2×D244C2×D411yescharacteristic
153C2×C4×D61—21no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156Q16×C61—22no—
157Q16×C61—81no—
158SD16×C61—42no—
159SD16×C61—81no—
160SD16×C61—81no—
161C2×C8×C61—21no—
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—41no—
176C22⋊C4×C61—41no—
177C22⋊C4×C61—41no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61—21no—
181C42×C61—41no—
182C42×C61C2411yescharacteristic
183#11758202—42no—
184#11758202—81no—
185#11758201—41no—
186#11758200—42no—
187#11758200—81no—
188#11758199—81no—
189#11758198—21no—
190#11758198—41no—
191#11758198—41no—
192#11758198—41no—
193#11758198—41no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61C2×D411yescharacteristic
199#11758196—41no—
200#11758196—41no—
201C4×Dic61—41no—
202C4×Dic61—41no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209C8×D61—42no—
210C23×C61—41no—
211C22×D61—21no—
212Q8×C61—21no—
213Q8×C61—21no—
214Q8×C61—21no—
215Q8×C61—21no—
216Q8×C61—21no—
217Q8×C61—21no—
218Q8×C61—41no—
219Q8×C61—41no—
220D4×C61—41no—
221D4×C61—41no—
222D4×C61—41no—
223D4×C61—41no—
224D4×C61—81no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—41no—
232C2×C4×C61—161no—
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61H(5,𝔽2)11yescharacteristic
236C2×Dic61—21no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239D244—21no—
240D244—21no—
241C4×D61—41no—
242C4×D61—41no—
243Dic122—41no—
244Dic122—41no—
245Dic122—41no—
246Dic122—41no—
247C8×C61—21no—
248C8×C61—41no—
249C8×C61#13512yes—
250#92418—21no—
251#92418—21no—
252#92418—41no—
253#92418—41no—
254#92418—41no—
255#92418—41no—
256#19858—611nomaximal, Sylow
257C22×C61—81no—
258C22×C61—81no—
259C22×C61—81no—
260C22×C61D4211yescharacteristic
261C2×D61—41no—
262C2×D61—41no—
263C2×D61—41no—
264C4×C61—21no—
265C4×C61—41no—
266C4×C61—41no—
267C4×C61—41no—
268C4×C61—41no—
269C4×C61—41no—
270C4×C61—41no—
271C4×C61—81no—
272C4×C61C2×#13511yescharacteristic
273C4×C61#57511yescharacteristic
274C4×C61#57711yescharacteristic
275C4×C61#57711yescharacteristic
276Dic61—41no—
277Dic61—81no—
278Dic61—81no—
279#3400D6111yescharacteristic, complemented
280#3394—611no—
281#3364—611no—
282#3342—611no—
283#1714—611no—
284#1694—611no—
285#1680—611no—
286#1666—611no—
287#1665—611no—
288#1653—611no—
289#1652—611no—
290#1640—611no—
291#1628—611no—
292#1582—611no—
293#1579—611no—
294C2×C61—161no—
295C2×C61#327111yescharacteristic
296C2×C61#328411yescharacteristic
297C2×C61#328911yescharacteristic
298D61—81no—
299#559C2×D6111yescharacteristic
300#557C2×D6111yescharacteristic
301#556—611no—
302#556—611no—
303#500—611no—
304#498—611no—
305#497—21no—
306#495—1221no—
307#492—611no—
308#492C2×D6111yescharacteristic
309#488—1221no—
310#487—611no—
311#486—611no—
312#485—1221no—
313#485—1221no—
314#479—1221no—
315#475—611no—
316#475—611no—
317#474—611no—
318#467—1221no—
319#464—21no—
320#464—1221no—
321#464—1221no—
322#463—1221no—
323#462—611no—
324#462—611no—
325#462C2×D6111yescharacteristic
326#461—21no—
327#460C2×D6111yescharacteristic
328#459—611no—
329#446C2×D6111yescharacteristic
330C8×Q8—611no—
331C8×Q8—611no—
332#440—21no—
333C4×Q16—1221no—
334C4×Q16C2×D6111yescharacteristic
335C4×SD16—611no—
336C4×SD16—1221no—
337C8×D4—611no—
338#427—1221no—
339#334—611no—
340#334—611no—
341#332—611no—
342#332—611no—
343#332—611no—
344#331—611no—
345Q8⋊C8—611no—
346Q8⋊C8—611no—
347Q8⋊C8—611no—
348D4⋊C8—611no—
349#322—611no—
350C61#1985811yescharacteristic, complemented, Sylow
351C2×Q16—21no—
352C2×Q16—1221no—
353C2×Q16—1221no—
354C2×Q16C22⋊D6111yescharacteristic
355C2×SD16—21no—
356C2×SD16—21no—
357C2×SD16—21no—
358C2×SD16—611no—
359C2×SD16—1221no—
360C2×SD16—1221no—
361C2×D8—1221no—
362C22×C8—1221no—
363C4⋊Q8—21no—
364C4⋊Q8—611no—
365C4⋊Q8—611no—
366C4⋊Q8C22×D6111yescharacteristic
367#127C22×D6111yescharacteristic
368#126—2441no—
369#126—2441no—
370#125—21no—
371#125—41no—
372#124—21no—
373#124—21no—
374#124—1221no—
375#124—1221no—
376#123—41no—
377C22⋊Q8—1221no—
378C22⋊Q8—1221no—
379C4×Q8—21no—
380C4×Q8—21no—
381C4×Q8—1221no—
382C4×Q8—1221no—
383C4×Q8C22×D6111yescharacteristic
384C4×Q8C22×D6111yescharacteristic
385C4×Q8C22×D6111yescharacteristic
386C4×D4—21no—
387C4×D4—21no—
388C4×D4—611no—
389#117—1221no—
390#117—1221no—
391#107—1221no—
392#107—1221no—
393#107—1221no—
394#107C22⋊D6111yescharacteristic
395#106—611no—
396#106—1221no—
397#106—1221no—
398#106—2441no—
399C4⋊C8—21no—
400C4⋊C8—611no—
401C4⋊C8—611no—
402C4⋊C8—611no—
403C4⋊C8—1221no—
404C4⋊C8—1221no—
405C4⋊C8C22×D6111yescharacteristic
406#103—21no—
407#103—611no—
408#103—1221no—
409#103—1221no—
410#103—1221no—
411#103—1221no—
412#103—1221no—
413#103—1221no—
414#103C22⋊D6111yescharacteristic
415#103C22⋊D6111yescharacteristic
416#102—21no—
417#102—611no—
418#102—1221no—
419#102—1221no—
420#102—2441no—
421C22⋊C8—1221no—
422#97—21no—
423C4×C8—611no—
424C4×C8—611no—
425C4×C8—1221no—
426C4×C8—1221no—
427C4×C8C22×D6111yescharacteristic
428C2×Q8—21no—
429C2×Q8—1221no—
430C2×Q8—1221no—
431C2×Q8D4×D6111yescharacteristic
432C2×Q8C2×C22⋊D6111yescharacteristic
433C2×Q8C2×C22⋊D6111yescharacteristic
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—611no—
438C22×C4—41no—
439C22×C4—1221no—
440Q16—22no—
441Q16—81no—
442Q16—4881no—
443Q16—2442no—
444SD16—42no—
445SD16—81no—
446SD16—81no—
447SD16—4881no—
448SD16—2442no—
449SD16—2441no—
450D8—4881no—
451C2×C8—21no—
452C2×C8—611no—
453C2×C8—611no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—2442no—
459C2×C8C2×C22⋊D6111yescharacteristic
460C2×C8C2×C22⋊D6111yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—611no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—2441no—
475C4⋊C4—2441no—
476C4⋊C4D4×D6111yescharacteristic
477C4⋊C4C2×C22⋊D6111yescharacteristic
478C4⋊C4C2×C22⋊D6111yescharacteristic
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—1221no—
483C22⋊C4—2441no—
484C22⋊C4—2441no—
485C22⋊C4—2441no—
486C22⋊C4—2441no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—41no—
491C42—2441no—
492C42—2441no—
493C42C23×D6111yescharacteristic
494C23—41no—
495C23—1221no—
496Q8—21no—
497Q8—21no—
498Q8—21no—
499Q8—21no—
500Q8—21no—
501Q8—21no—
502Q8—41no—
503Q8—41no—
504Q8—2441no—
505Q8—2441no—
506Q8—2441no—
507Q8—2441no—
508D4—41no—
509D4—41no—
510D4—41no—
511D4—41no—
512D4—81no—
513D4—1221no—
514D4—1221no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—161no—
523C2×C4—1221no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4C2×D4×D6111yescharacteristic
529C2×C4C22×C22⋊D6111yescharacteristic
530C2×C4#1451982211yescharacteristic
531C8—21no—
532C8—41no—
533C8—1221no—
534C8—1221no—
535C8—2441no—
536C8—2441no—
537C8—2441no—
538C8—2441no—
539C8#1451979712yes—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—2441no—
544C22—2441no—
545C22—2441no—
546C22D4×C22⋊D6111yescharacteristic
547C4—21no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—81no—
555C4—4881no—
556C4—4881no—
557C4—2441no—
558C4C2×#1451979711yescharacteristic
559C4#1684056111yescharacteristic
560C4#1684057111yescharacteristic
561C4#1684057311yescharacteristic
562C2—161no—
563C2—4881no—
564C2#2145739811yescharacteristic
565C2#2145754911yescharacteristic
566C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3400C2
socleC22×C61D42
prime radical#16839954C22
hypercenter#3400D61
hypocenterC61#19858
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3400

Complements

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

NH ≅ G/Ntype
2. #21458504C2semidirect
3. #21458251C2semidirect
4. #21445438C2semidirect
5. #21445337C2semidirect
6. #21445251C2semidirect
7. #21445165C2semidirect
9. #21445096C2semidirect
10. #21445087C2semidirect
11. #21445005C2semidirect
13. #21444653C2semidirect
14. #21444638C2semidirect
15. C61×#3400C2semidirect
279. #3400D61semidirect
350. C61#19858semidirect

Maximal subgroups

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

subgroupindex
#214588112
#214585042
#214582512
#214454382
#214453372
#214452512
#214451652
#214451572
#214450962
#214450872
#214450052
#214449282
#214446532
#214446382
C61×#34002
#1985861