Finite Group Databasebeta

← #29650050

Subgroups — #29650050

Order 15616.

624 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455888C211yesmaximal, characteristic
2C8×#11758226C211yesmaximal, characteristic
3#21454750C211yesmaximal, characteristic, complemented
4#21444669C211yesmaximal, characteristic, complemented
5#21444657C211yesmaximal, characteristic, complemented
6#21444653C211yesmaximal, characteristic, complemented
7#21444638C211yesmaximal, characteristic, complemented
8#21444635C211yesmaximal, characteristic, complemented
9#21444634C211yesmaximal, characteristic, complemented
10#21444630C211yesmaximal, characteristic, complemented
11#21444609C211yesmaximal, characteristic, complemented
12#21444257C211yesmaximal, characteristic
13#21444238C211yesmaximal, characteristic, complemented
14#21444105C211yesmaximal, characteristic, complemented
15C61×#1584C211yesmaximal, characteristic, complemented
16#16840316C2211yescharacteristic
17C4×#11758226C2211yescharacteristic
18#16840105C411yescharacteristic
19#16840101C411yescharacteristic
20#16840045C411yescharacteristic
21#16840041C411yescharacteristic
22#16839970C411yescharacteristic
23#16839966C411yescharacteristic
24#16839927C411yescharacteristic
25#16839923C411yescharacteristic
26#16839892C2211yescharacteristic
27#16839890—21no—
28Q8×#92418C2211yescharacteristic
29Q8×#92418C2211yescharacteristic
30#16839886—21no—
31#16839886C2211yescharacteristic
32C4×#11758202C2211yescharacteristic
33C4×#11758190C2211yescharacteristic
34C4×#11758201C2211yescharacteristic
35C8×D244—21no—
36C8×D244C2211yescharacteristic
37#16839827—21no—
38#16839827C2211yescharacteristic
39#16839814—21no—
40#16839814C2211yescharacteristic
41#16839813C2211yescharacteristic
42#16839787—21no—
43#16839784C2211yescharacteristic
44#16839719—21no—
45C4×#11758189C2211yescharacteristic
46C4×C8×D61—21no—
47C4×C8×D61C2211yescharacteristic
48#16839514C2211yescharacteristic
49#16839513C2211yescharacteristic
50#16839512C2211yescharacteristic
51#16839494C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53Q8⋊#92418C2211yescharacteristic
54#16839492C2211yescharacteristic
55#16839491C2211yescharacteristic
56#16839489C2211yescharacteristic
57#16839484C2211yescharacteristic
58#16839483C2211yescharacteristic
59#16839483C2211yescharacteristic
60C61×#446C2211yescharacteristic
61C8×Q8×C61C2211yescharacteristic
62C4×Q16×C61C2211yescharacteristic
63C61×#335C2211yescharacteristic
64C61×Q8⋊C8C2211yescharacteristic
65C61×Q8⋊C8C2211yescharacteristic
66C61×#322C2211yescharacteristic
67C2×#11758226C2×C411yescharacteristic
68C2×#11758202C2×C411yescharacteristic
69C2×#11758190C2×C411yescharacteristic
70C2×#11758201C2×C411yescharacteristic
71C2×#11758189—21no—
72C2×#11758189—21no—
73C2×C8×D61—21no—
74C2×C8×D61—21no—
75C2×C8×D61D411yescharacteristic
76C2×C8×D61D411yescharacteristic
77#14519776C2×C411yescharacteristic
78#14519775C2×C411yescharacteristic
79#14519767—21no—
80#14519757C2×C411yescharacteristic
81#14519755—21no—
82#14519753C2×C411yescharacteristic
83Q8×Dic61C2×C411yescharacteristic
84C4×Dic122—21no—
85C4×Dic122C2311yescharacteristic
86C4×D244—21no—
87C4×D244C2311yescharacteristic
88#14519739—21no—
89#14519739C2×C411yescharacteristic
90#14519735—21no—
91#14519735C2×C411yescharacteristic
92#14519734—21no—
93C42×D61—21no—
94C42×D61C2311yescharacteristic
95#14519710C2×C411yescharacteristic
96#14519709C2×C411yescharacteristic
97#14519708C2×C411yescharacteristic
98C4⋊#92418—21no—
99C4⋊#92418—21no—
100C4⋊#92418C2311yescharacteristic
101C4⋊#92418C2311yescharacteristic
102#14519706—21no—
103#14519706—21no—
104#14519706—21no—
105#14519706C2×C411yescharacteristic
106#14519706C2×C411yescharacteristic
107#14519702C2×C411yescharacteristic
108#14519701C2×C411yescharacteristic
109#14519701C2×C411yescharacteristic
110#14519700C2×C411yescharacteristic
111#14519699C2×C411yescharacteristic
112#14519697C2×C411yescharacteristic
113#14519689—21no—
114#14519689—21no—
115#14519689—21no—
116#14519689C2×C411yescharacteristic
117#14519689C2×C411yescharacteristic
118#14519687—21no—
119#14519687—21no—
120C4×#92418—21no—
121C4×#92418—21no—
122C4×#92418C2311yescharacteristic
123C4×#92418C2311yescharacteristic
124C4×#92418C2311yescharacteristic
125C8×Dic61—21no—
126C8×Dic61—21no—
127C8×Dic61D411yescharacteristic
128C8×Dic61D411yescharacteristic
129C2×Q16×C61C2×C411yescharacteristic
130C4×Q8×C61C2311yescharacteristic
131C4×Q8×C61C2311yescharacteristic
132#107×C61C2×C411yescharacteristic
133C4⋊C8×C61—21no—
134C4⋊C8×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137#103×C61C2×C411yescharacteristic
138#103×C61C2×C411yescharacteristic
139#97×C61—21no—
140C4×C8×C61—21no—
141C4×C8×C61C2311yescharacteristic
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61C22×C411yescharacteristic
145C2×Q8×C61C22×C411yescharacteristic
146#11758226—22no—
147#11758226—22no—
148C2×D244—21no—
149C2×D244C22×C411yescharacteristic
150C2×C4×D61—21no—
151C2×C4×D61—21no—
152C2×C4×D61—21no—
153C2×C4×D61C22×C411yescharacteristic
154C2×C4×D61C2×D411yescharacteristic
155C2×C4×D61#4111yescharacteristic
156C2×Dic122—21no—
157C2×Dic122C22×C411yescharacteristic
158Q16×C61—22no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61C22×C411yescharacteristic
163C2×C8×C61C22×C411yescharacteristic
164C2×C8×C61C2×D411yescharacteristic
165C2×C8×C61C2×D411yescharacteristic
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61C22×C411yescharacteristic
169C4⋊C4×C61C22×C411yescharacteristic
170C42×C61—21no—
171C42×C61—21no—
172C42×C61C2411yescharacteristic
173#11758202—22no—
174#11758201—22no—
175#11758198—21no—
176#11758198—21no—
177#11758198—21no—
178#11758198—21no—
179#11758198—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61C22×C411yescharacteristic
183C4⋊Dic61C22×C411yescharacteristic
184#11758196—21no—
185#11758196—21no—
186#11758196—21no—
187#11758196—21no—
188#11758196—21no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61C22×C411yescharacteristic
193C4×Dic61C2×D411yescharacteristic
194C4×Dic61#4111yescharacteristic
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418C22×C411yescharacteristic
199C2×#92418C22×C411yescharacteristic
200C2×#92418C2×D411yescharacteristic
201C2×#92418C2×D411yescharacteristic
202#11758190—22no—
203#11758189—42no—
204#11758189—42no—
205C8×D61—22no—
206C8×D61—41no—
207C8×D61—42no—
208C8×D61—42no—
209C22×D61—21no—
210C22×D61C4×D411yescharacteristic
211Q8×C61—21no—
212Q8×C61—21no—
213Q8×C61—21no—
214Q8×C61#13112yes—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61C23×C411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61C4×D411yescharacteristic
226C2×Dic61C4×D411yescharacteristic
227C2×Dic61C4×D411yescharacteristic
228D244—24no—
229D244—21no—
230D244#13112yes—
231C4×D61—24no—
232C4×D61—21no—
233C4×D61—21no—
234C4×D61—21no—
235C4×D61—21no—
236C4×D61—22no—
237C4×D61—41no—
238C4×D61—41no—
239C4×D61—42no—
240Dic122—21no—
241Dic122—21no—
242Dic122—42no—
243C8×C61—21no—
244C8×C61—21no—
245C8×C61—21no—
246C8×C61—41no—
247C8×C61—41no—
248C8×C61—41no—
249C8×C61#13712yes—
250#92418—21no—
251#92418—21no—
252#92418—21no—
253#92418—21no—
254#92418—41no—
255#92418—41no—
256#92418—41no—
257#17916—611nomaximal, Sylow
258C22×C61C2×C4×D411yescharacteristic
259C2×D61—21no—
260C2×D61—22no—
261C2×D61—21no—
262C2×D61—21no—
263C2×D61—41no—
264C2×D61—41no—
265C4×C61—22no—
266C4×C61—21no—
267C4×C61—41no—
268C4×C61—41no—
269C4×C61—41no—
270C4×C61C2×#13111yescharacteristic
271C4×C61C2×#13111yescharacteristic
272C4×C61C2×#13711yescharacteristic
273C4×C61#57511yescharacteristic
274Dic61—21no—
275Dic61—21no—
276Dic61—41no—
277Dic61—41no—
278Dic61—41no—
279Dic61#44412yes—
280#3107—611no—
281C8×#41—611no—
282#2960—611no—
283#1584—611no—
284#1584D6111yescharacteristic, complemented
285#1582—611no—
286#1582—611no—
287#1579—611no—
288#1579—611no—
289SD16⋊C8—611no—
290SD16⋊C8—611no—
291#1574—611no—
292#1489—611no—
293#1485—611no—
294#1451—611no—
295C2×C61#292911yescharacteristic
296C2×C61C4×#13711yescharacteristic
297C2×C61C2×#44411yescharacteristic
298D61—41no—
299D61—42no—
300#533—611no—
301C4×#41—611no—
302#485—611no—
303#485—611no—
304#475—611no—
305#474—611no—
306#462—611no—
307#461—611no—
308#452—611no—
309#452—611no—
310#446—611no—
311#446—1221no—
312#446C2×D6111yescharacteristic
313C8×Q8—611no—
314C8×Q8—611no—
315C8×Q8—611no—
316C8×Q8—1221no—
317C8×Q8C2×D6111yescharacteristic
318C4×Q16—611no—
319C4×Q16C2×D6111yescharacteristic
320C4×SD16—611no—
321C4×SD16—611no—
322C8×D4—611no—
323C8×D4—611no—
324C8×D4—1221no—
325C8×D4—1221no—
326#431—611no—
327#431—611no—
328#431—1221no—
329#424—1221no—
330#423—611no—
331#405—1221no—
332C4×M16—611no—
333C2×C4×C8—611no—
334C2×C4×C8—1221no—
335#335—611no—
336#335C2×D6111yescharacteristic
337#334—611no—
338#334—611no—
339Q8⋊C8—611no—
340Q8⋊C8—611no—
341Q8⋊C8—611no—
342Q8⋊C8—611no—
343Q8⋊C8C2×D6111yescharacteristic
344Q8⋊C8C2×D6111yescharacteristic
345D4⋊C8—611no—
346D4⋊C8—611no—
347#322—611no—
348#322—611no—
349#322—611no—
350#322C2×D6111yescharacteristic
351C61#1791611yescharacteristic, complemented, Sylow
352C2×#41—611no—
353C2×Q16—611no—
354C2×Q16C4×D6111yescharacteristic
355C2×SD16—611no—
356C2×SD16—611no—
357C2×M16—1221no—
358C2×M16—1221no—
359C22×C8—611no—
360C22×C8—611no—
361C22×C8—1221no—
362C22×C8—1221no—
363C4⋊Q8—611no—
364C4⋊Q8—611no—
365#125—1221no—
366C22⋊Q8—611no—
367C22⋊Q8—611no—
368C22⋊Q8—1221no—
369C4×Q8—611no—
370C4×Q8—611no—
371C4×Q8—1221no—
372C4×Q8C22×D6111yescharacteristic
373C4×Q8C22×D6111yescharacteristic
374C4×D4—611no—
375C4×D4—611no—
376C4×D4—1221no—
377C4×D4—1221no—
378#117—611no—
379#117—1221no—
380#117—1221no—
381C2×C42—611no—
382C2×C42—1221no—
383#107—611no—
384#107C4×D6111yescharacteristic
385#106—611no—
386#106—611no—
387C4⋊C8—21no—
388C4⋊C8—21no—
389C4⋊C8—611no—
390C4⋊C8—611no—
391C4⋊C8—611no—
392C4⋊C8—611no—
393C4⋊C8—1221no—
394C4⋊C8—1221no—
395C4⋊C8—1221no—
396C4⋊C8—1221no—
397C4⋊C8—1221no—
398C4⋊C8C22×D6111yescharacteristic
399C4⋊C8C22×D6111yescharacteristic
400#103—611no—
401#103—611no—
402#103—611no—
403#103—611no—
404#103C4×D6111yescharacteristic
405#103C4×D6111yescharacteristic
406#102—611no—
407#102—611no—
408C22⋊C8—611no—
409C22⋊C8—611no—
410C22⋊C8—1221no—
411C22⋊C8—1221no—
412C22⋊C8—1221no—
413#97—21no—
414#97—1221no—
415#97—1221no—
416C4×C8—21no—
417C4×C8—611no—
418C4×C8—611no—
419C4×C8—611no—
420C4×C8—611no—
421C4×C8—611no—
422C4×C8—1221no—
423C4×C8—1221no—
424C4×C8—1221no—
425C4×C8—1221no—
426C4×C8C22×D6111yescharacteristic
427C4×C8C22×D6111yescharacteristic
428C4×C8C22×D6111yescharacteristic
429#41—1222no—
430#41—1222no—
431C2×Q8—611no—
432C2×Q8—1221no—
433C2×Q8C2×C4×D6111yescharacteristic
434C2×Q8C2×C4×D6111yescharacteristic
435C2×D4—611no—
436C2×D4—1221no—
437C22×C4—611no—
438C22×C4—611no—
439C22×C4—611no—
440C22×C4—1221no—
441C22×C4—1221no—
442C22×C4—1221no—
443Q16—22no—
444Q16—1222no—
445SD16—1222no—
446SD16—1222no—
447M16—2442no—
448M16—2442no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—611no—
453C2×C8—611no—
454C2×C8—611no—
455C2×C8—611no—
456C2×C8—1222no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—2441no—
461C2×C8—2442no—
462C2×C8—2442no—
463C2×C8C2×C4×D6111yescharacteristic
464C2×C8C2×C4×D6111yescharacteristic
465C2×C8D4×D6111yescharacteristic
466C2×C8D4×D6111yescharacteristic
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—611no—
470C4⋊C4—611no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4C2×C4×D6111yescharacteristic
479C4⋊C4C2×C4×D6111yescharacteristic
480C22⋊C4—1221no—
481C22⋊C4—1221no—
482C22⋊C4—1221no—
483C22⋊C4—1221no—
484C22⋊C4—1221no—
485C42—21no—
486C42—21no—
487C42—611no—
488C42—611no—
489C42—611no—
490C42—1221no—
491C42—1221no—
492C42—1221no—
493C42C23×D6111yescharacteristic
494C23—611no—
495C23—1221no—
496Q8—21no—
497Q8—21no—
498Q8—21no—
499Q8—1221no—
500Q8—1221no—
501Q8—2442no—
502Q8#1451978512yes—
503D4—612no—
504D4—1224no—
505D4—1221no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—611no—
511C2×C4—611no—
512C2×C4—611no—
513C2×C4—1222no—
514C2×C4—1224no—
515C2×C4—1221no—
516C2×C4—1221no—
517C2×C4—1221no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—2442no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4C22×C4×D6111yescharacteristic
526C2×C4C2×D4×D6111yescharacteristic
527C2×C4#41×D6111yescharacteristic
528C8—21no—
529C8—21no—
530C8—21no—
531C8—41no—
532C8—41no—
533C8—41no—
534C8—1221no—
535C8—1221no—
536C8—1221no—
537C8—1221no—
538C8—2441no—
539C8—2441no—
540C8—2441no—
541C8#1451980912yes—
542C22—1221no—
543C22—1221no—
544C22—1222no—
545C22—1221no—
546C22—2441no—
547C22—2441no—
548C22C4×D4×D6111yescharacteristic
549C4—21no—
550C4—22no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—612no—
555C4—1221no—
556C4—1221no—
557C4—2441no—
558C4—2441no—
559C4—2441no—
560C4#131×D6111yescharacteristic
561C4C2×#1451978511yescharacteristic
562C4C2×#1451980911yescharacteristic
563C4#1684055911yescharacteristic
564C2—2441no—
565C2—2442no—
566C2#2145451011yescharacteristic
567C2C4×#1451980911yescharacteristic
568C2#444×D6111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1584C2
socleC22×C61C2×C4×D4
prime radicalC2×#11758226C2×C4
cosocleC42×C61C24
hypercenter#1584D61
hypocenterC61#17916
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1584

Complements

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

NH ≅ G/Ntype
3. #21454750C2semidirect
4. #21444669C2semidirect
5. #21444657C2semidirect
6. #21444653C2semidirect
7. #21444638C2semidirect
8. #21444635C2semidirect
9. #21444634C2semidirect
10. #21444630C2semidirect
11. #21444609C2semidirect
13. #21444238C2semidirect
14. #21444105C2semidirect
15. C61×#1584C2semidirect
284. #1584D61semidirect
351. C61#17916semidirect

Maximal subgroups

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

subgroupindex
#214558882
C8×#117582262
#214547502
#214446692
#214446572
#214446532
#214446382
#214446352
#214446342
#214446302
#214446092
#214442572
#214442382
#214441052
C61×#15842
#1791661