Finite Group Databasebeta

← #29667625

Subgroups — #29667625

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#21457156C211yesmaximal, characteristic
2#21456972C211yesmaximal, characteristic, complemented
3#21456843C211yesmaximal, characteristic
4#21445373C211yesmaximal, characteristic, complemented
5#21445366C211yesmaximal, characteristic, complemented
6#21445337C211yesmaximal, characteristic, complemented
7#21445332C211yesmaximal, characteristic, complemented
8#21445249C211yesmaximal, characteristic, complemented
9#21445242C211yesmaximal, characteristic, complemented
10#21445213C211yesmaximal, characteristic, complemented
11#21445208C211yesmaximal, characteristic, complemented
12#21444604C211yesmaximal, characteristic, complemented
13#21444517C211yesmaximal, characteristic, complemented
14C2×#16839507C211yesmaximal, characteristic, complemented
15C61×#2994C211yesmaximal, characteristic
16#16840319C2211yescharacteristic
17C2×#14519777C2211yescharacteristic
18#16840167C2211yescharacteristic
19#16840167C2211yescharacteristic
20#16840157—21no—
21#16840132—21no—
22#16840113C2211yescharacteristic
23#16840113C2211yescharacteristic
24#16840098—21no—
25#16840098—21no—
26#16840066—21no—
27#16840058—21no—
28#16840058—21no—
29#16840047C2211yescharacteristic
30#16840042C2211yescharacteristic
31#16840001—21no—
32#16839989—21no—
33#16839981—21no—
34#16839972C2211yescharacteristic
35#16839972C2211yescharacteristic
36#16839972C2211yescharacteristic
37#16839967C2211yescharacteristic
38#16839967C2211yescharacteristic
39#16839967C2211yescharacteristic
40#16839917—21no—
41#16839900—21no—
42#16839819C2211yescharacteristic
43C2×C4⋊#92418C2211yescharacteristic
44#16839515C2211yescharacteristic
45#16839515C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839513C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839509C2211yescharacteristic
50#16839507C2211yescharacteristic
51#16839507C2211yescharacteristic
52#16839507C2211yescharacteristic
53#16839507C2211yescharacteristic
54#16839507C2211yescharacteristic
55#16839507C2211yescharacteristic
56C61×#529C2211yescharacteristic
57C61×#446C2211yescharacteristic
58C8×Q8×C61C2211yescharacteristic
59C61×#436C2211yescharacteristic
60C61×#435—21no—
61C8×D4×C61—21no—
62C8×D4×C61C2211yescharacteristic
63C61×#433—21no—
64C61×#431C2211yescharacteristic
65C61×#424—21no—
66C2×C4⋊C8×C61C2211yescharacteristic
67C22×Dic122—21no—
68C2×#11758202—21no—
69C2×#11758202—21no—
70C2×#11758202—21no—
71C2×#11758200—21no—
72C2×#11758200—21no—
73C2×#11758200—21no—
74C22×#92418—21no—
75#14519777—21no—
76#14519777—21no—
77#14519777—21no—
78#14519777C2311yescharacteristic
79#14519777C2311yescharacteristic
80#14519777C2311yescharacteristic
81#14519777C2311yescharacteristic
82#14519769—21no—
83#14519756—21no—
84#14519756—21no—
85C4×Dic122—21no—
86C4×Dic122—21no—
87#14519736—21no—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61—21no—
90#14519711—21no—
91#14519711—21no—
92#14519711—21no—
93#14519711—21no—
94#14519710—21no—
95#14519708—21no—
96C4⋊#92418—21no—
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100C4⋊#92418C2311yescharacteristic
101#14519702—21no—
102#14519702—21no—
103#14519702—21no—
104#14519702—21no—
105#14519701—21no—
106#14519700—21no—
107#14519700—21no—
108#14519700—21no—
109#14519700—21no—
110#14519700—21no—
111#14519700—21no—
112#14519698—21no—
113C22⋊#92418—21no—
114#14519688—21no—
115C2×M16×C61—21no—
116C22×C8×C61—21no—
117C22×C8×C61—21no—
118#126×C61—21no—
119#125×C61D411yescharacteristic
120#124×C61D411yescharacteristic
121#123×C61—21no—
122C22⋊Q8×C61D411yescharacteristic
123#121×C61D411yescharacteristic
124C4×Q8×C61C2311yescharacteristic
125C4×D4×C61—21no—
126C4×D4×C61C2311yescharacteristic
127#117×C61—21no—
128C2×C42×C61C2311yescharacteristic
129C4⋊C8×C61—21no—
130C4⋊C8×C61—21no—
131C4⋊C8×C61—21no—
132C4⋊C8×C61D411yescharacteristic
133C4⋊C8×C61D411yescharacteristic
134C4⋊C8×C61C2311yescharacteristic
135C4⋊C8×C61C2311yescharacteristic
136C22⋊C8×C61—21no—
137C22⋊C8×C61—21no—
138C22⋊C8×C61D411yescharacteristic
139C22⋊C8×C61D411yescharacteristic
140#97×C61—21no—
141C4×C8×C61—21no—
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61C2×D411yescharacteristic
145C2×D4×C61—21no—
146C2×D4×C61C2×D411yescharacteristic
147C22×C4×C61—21no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61#4111yescharacteristic
151C22×Dic61—41no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—41no—
157C2×Dic122—81no—
158C2×Dic122—81no—
159M16×C61—81no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—81no—
163C2×C8×C61—81no—
164C2×C8×C61C2×D411yescharacteristic
165C2×C8×C61C2×D411yescharacteristic
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×C61C2×D411yescharacteristic
173C4⋊C4×C61C2×D411yescharacteristic
174C22⋊C4×C61—21no—
175C22⋊C4×C61—21no—
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C22⋊C4×C61—21no—
179C42×C61—21no—
180C42×C61C2×D411yescharacteristic
181C42×C61C2×D411yescharacteristic
182C42×C61#4111yescharacteristic
183C42×C61C2411yescharacteristic
184C22⋊Dic61—41no—
185#11758202—81no—
186#11758202—81no—
187#11758202—81no—
188#11758200—81no—
189#11758200—81no—
190#11758200—81no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—41no—
200C4⋊Dic61—41no—
201#11758196—41no—
202#11758196—41no—
203C4×Dic61—41no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—81no—
209C23×C61—21no—
210C23×C61#12111yescharacteristic
211Q8×C61—21no—
212Q8×C61—21no—
213D4×C61—21no—
214D4×C61—21no—
215D4×C61—42no—
216C2×C4×C61—22no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
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#12111yescharacteristic
227C2×C4×C61#12111yescharacteristic
228C2×C4×C61#12111yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—82no—
237Dic122—41no—
238Dic122—41no—
239Dic122—41no—
240Dic122—41no—
241Dic122—41no—
242Dic122—41no—
243Dic122—41no—
244Dic122—41no—
245Dic122—82no—
246C8×C61—21no—
247C8×C61—21no—
248C8×C61—21no—
249C8×C61—21no—
250C8×C61—41no—
251C8×C61—41no—
252#92418—41no—
253#92418—41no—
254#92418—41no—
255#92418—41no—
256#19590—611nomaximal, Sylow
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—41no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61#47312yes—
263C22×C61C2×#12111yescharacteristic
264C4×C61—21no—
265C4×C61—21no—
266C4×C61—21no—
267C4×C61—21no—
268C4×C61—41no—
269C4×C61—41no—
270C4×C61—41no—
271C4×C61#57611yescharacteristic
272C4×C61#57711yescharacteristic
273C4×C61#57711yescharacteristic
274C4×C61#57811yescharacteristic
275Dic61—81no—
276Dic61—81no—
277Dic61—81no—
278Dic61—81no—
279#3248—611no—
280#3230—611no—
281#3218—611no—
282#2994D6111yescharacteristic
283#1700—611no—
284#1699—611no—
285#1694—611no—
286#1693—611no—
287#1680—611no—
288#1679—611no—
289#1674—611no—
290#1673—611no—
291#1573—611no—
292#1552—611no—
293C2×#332—611no—
294C2×C61—22no—
295C2×C61—81no—
296C2×C61#304611yescharacteristic
297C2×C61#306211yescharacteristic
298C2×C61C2×#47311yescharacteristic
299#533—611no—
300C2×C4⋊Q8—611no—
301#529C2×D6111yescharacteristic
302#500—611no—
303#500—611no—
304#497—1221no—
305#491—1221no—
306#487—611no—
307#487—611no—
308#484—1221no—
309#484—1221no—
310#479—1221no—
311#477—1221no—
312#477—1221no—
313#475—611no—
314#474—611no—
315#467—1221no—
316#465—1221no—
317#464—1221no—
318#462—611no—
319#462—611no—
320#462—611no—
321#461—611no—
322#461—611no—
323#461—611no—
324#451—1221no—
325#448—1221no—
326#446C2×D6111yescharacteristic
327C8×Q8C2×D6111yescharacteristic
328#436C2×D6111yescharacteristic
329#435—21no—
330C8×D4—21no—
331C8×D4C2×D6111yescharacteristic
332#433—21no—
333#432—611no—
334#431C2×D6111yescharacteristic
335#424—21no—
336C2×C4⋊C8—611no—
337C2×C4⋊C8C2×D6111yescharacteristic
338#335—611no—
339#335—611no—
340#334—611no—
341#334—611no—
342#332—611no—
343#332—611no—
344#332—611no—
345#332—611no—
346#332—611no—
347#332—611no—
348#332—611no—
349#332—611no—
350C61#1959011yescharacteristic, complemented, Sylow
351C22×Q8—1221no—
352C2×Q16—1221no—
353C2×Q16—1221no—
354C2×Q16—1221no—
355C2×SD16—1221no—
356C2×SD16—1221no—
357C2×SD16—1221no—
358C2×M16—21no—
359C22×C8—21no—
360C22×C8—21no—
361C22×C8—1221no—
362C4⋊Q8—611no—
363C4⋊Q8—611no—
364C4⋊Q8—611no—
365C4⋊Q8—611no—
366C4⋊Q8—1221no—
367C4⋊Q8—1221no—
368C4⋊Q8—1221no—
369#126—21no—
370#125—1221no—
371#125D24411yescharacteristic
372#124D24411yescharacteristic
373#123—21no—
374C22⋊Q8—1221no—
375C22⋊Q8—1221no—
376C22⋊Q8D24411yescharacteristic
377#121D24411yescharacteristic
378C4×Q8—1221no—
379C4×Q8—1221no—
380C4×Q8C22×D6111yescharacteristic
381C4×D4—21no—
382C4×D4C22×D6111yescharacteristic
383#117—21no—
384#117—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C42C22×D6111yescharacteristic
388#107—1221no—
389#107—1221no—
390#107—1221no—
391#107—1221no—
392#107—1221no—
393#106—1221no—
394C4⋊C8—21no—
395C4⋊C8—21no—
396C4⋊C8—21no—
397C4⋊C8—611no—
398C4⋊C8—611no—
399C4⋊C8—611no—
400C4⋊C8—611no—
401C4⋊C8—1221no—
402C4⋊C8C22⋊D6111yescharacteristic
403C4⋊C8C22⋊D6111yescharacteristic
404C4⋊C8C22×D6111yescharacteristic
405C4⋊C8C22×D6111yescharacteristic
406#103—1221no—
407#103—1221no—
408#103—1221no—
409#103—1221no—
410#103—1221no—
411#103—1221no—
412#103—1221no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#103—1221no—
417#102—1221no—
418C22⋊C8—21no—
419C22⋊C8—21no—
420C22⋊C8—1221no—
421C22⋊C8C22⋊D6111yescharacteristic
422C22⋊C8C22⋊D6111yescharacteristic
423#97—21no—
424#97—1221no—
425C4×C8—21no—
426C4×C8C22×D6111yescharacteristic
427C4×C8C22×D6111yescharacteristic
428C2×Q8—1221no—
429C2×Q8—1221no—
430C2×Q8—1221no—
431C2×Q8—1221no—
432C2×Q8—4881no—
433C2×Q8—4881no—
434C2×Q8—2441no—
435C2×Q8C2×D24411yescharacteristic
436C2×D4—21no—
437C2×D4C2×D24411yescharacteristic
438C22×C4—21no—
439C22×C4—2441no—
440C22×C4C2×D24411yescharacteristic
441C22×C4#1175822211yescharacteristic
442C22×C4C2×C22⋊D6111yescharacteristic
443Q16—4881no—
444Q16—4881no—
445Q16—4881no—
446SD16—4881no—
447SD16—4881no—
448SD16—4881no—
449M16—81no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—81no—
453C2×C8—81no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—4881no—
459C2×C8C2×C22⋊D6111yescharacteristic
460C2×C8C2×C22⋊D6111yescharacteristic
461C2×C8C2×C22⋊D6111yescharacteristic
462C2×C8C2×C22⋊D6111yescharacteristic
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—2441no—
476C4⋊C4—2441no—
477C4⋊C4—2441no—
478C4⋊C4—2441no—
479C4⋊C4C2×D24411yescharacteristic
480C4⋊C4C2×D24411yescharacteristic
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—2441no—
487C42—21no—
488C42—2441no—
489C42C2×D24411yescharacteristic
490C42#1175822211yescharacteristic
491C42C2×C22⋊D6111yescharacteristic
492C42C23×D6111yescharacteristic
493C23—21no—
494C23#1451975111yescharacteristic
495Q8—21no—
496Q8—21no—
497Q8—4882no—
498Q8—2441no—
499Q8—2441no—
500Q8—2441no—
501Q8—2441no—
502Q8—2441no—
503Q8—2441no—
504Q8—2441no—
505Q8—2441no—
506D4—21no—
507D4—21no—
508D4—42no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—4882no—
520C2×C4—2441no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4#1451975111yescharacteristic
525C2×C4#1451975111yescharacteristic
526C2×C4#1451975111yescharacteristic
527C2×C4C22×C22⋊D6111yescharacteristic
528C2×C4C22×D24411yescharacteristic
529C2×C4C2×#1175822211yescharacteristic
530C8—21no—
531C8—21no—
532C8—21no—
533C8—21no—
534C8—41no—
535C8—41no—
536C8—2441no—
537C8—2441no—
538C8—2441no—
539C8—2441no—
540C22—21no—
541C22—21no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22#1684003712yes—
546C22C2×#1451975111yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—4881no—
555C4—4881no—
556C4—4881no—
557C4—4881no—
558C4#1684056711yescharacteristic
559C4#1684057311yescharacteristic
560C4#1684057411yescharacteristic
561C4#1684057711yescharacteristic
562C2—22no—
563C2—81no—
564C2#2145536911yescharacteristic
565C2#2145549911yescharacteristic
566C2C2×#1684003711yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2994C2
socleC22×C61C2×#121
prime radical#121×C61D4
hypercenter#2994D61
hypocenterC61#19590
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2994

Complements

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

NH ≅ G/Ntype
2. #21456972C2semidirect
4. #21445373C2semidirect
5. #21445366C2semidirect
6. #21445337C2semidirect
7. #21445332C2semidirect
8. #21445249C2semidirect
9. #21445242C2semidirect
10. #21445213C2semidirect
11. #21445208C2semidirect
12. #21444604C2semidirect
13. #21444517C2semidirect
14. C2×#16839507C2semidirect
350. C61#19590semidirect

Maximal subgroups

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

subgroupindex
#214571562
#214569722
#214568432
#214453732
#214453662
#214453372
#214453322
#214452492
#214452422
#214452132
#214452082
#214446042
#214445172
C2×#168395072
C61×#29942
#1959061