Finite Group Databasebeta

← #29650549

Subgroups — #29650549

Order 15616.

630 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456385C211yesmaximal, characteristic
2#21454797C211yesmaximal, characteristic, complemented
3#21454793C211yesmaximal, characteristic, complemented
4#21444714C211yesmaximal, characteristic, complemented
5#21444711C211yesmaximal, characteristic, complemented
6#21444702C211yesmaximal, characteristic, complemented
7#21444623C211yesmaximal, characteristic, complemented
8SD16×#92418C211yesmaximal, characteristic, complemented
9#21444618C211yesmaximal, characteristic, complemented
10#21444611C211yesmaximal, characteristic, complemented
11#21444579C211yesmaximal, characteristic, complemented
12#21444278C211yesmaximal, characteristic, complemented
13#21444276C211yesmaximal, characteristic, complemented
14C82×D61C211yesmaximal, characteristic, complemented
15C61×#1590C211yesmaximal, characteristic, complemented
16#16840299C2211yescharacteristic
17#16840294C2211yescharacteristic
18#16840162C411yescharacteristic
19#16840143C411yescharacteristic
20#16840118C411yescharacteristic
21#16840115C411yescharacteristic
22#16840002C411yescharacteristic
23#16839997C411yescharacteristic
24#16839992C411yescharacteristic
25#16839988C411yescharacteristic
26#16839892C2211yescharacteristic
27Q8×#92418C2211yescharacteristic
28#16839886C2211yescharacteristic
29C4×#11758202C2211yescharacteristic
30C4×#11758190C2211yescharacteristic
31C4×#11758199C2211yescharacteristic
32#16839841C2211yescharacteristic
33#16839835—21no—
34#16839831—21no—
35D4×#92418C2211yescharacteristic
36#16839827—21no—
37#16839827C2211yescharacteristic
38#16839825—21no—
39#16839822—21no—
40#16839820—21no—
41#16839818C2211yescharacteristic
42#16839818C2211yescharacteristic
43#16839720—21no—
44#16839719—21no—
45C4×C8×D61C2211yescharacteristic
46C4×C8×D61C2211yescharacteristic
47C4×C8×D61C2211yescharacteristic
48#16839514C2211yescharacteristic
49#16839514C2211yescharacteristic
50#16839513C2211yescharacteristic
51#16839494C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53#16839492C2211yescharacteristic
54#16839491C2211yescharacteristic
55D4⋊#92418C2211yescharacteristic
56#16839489C2211yescharacteristic
57C8×#92418C2211yescharacteristic
58C8×#92418C2211yescharacteristic
59C8×#92418C2211yescharacteristic
60C61×#446C2211yescharacteristic
61C4×SD16×C61C2211yescharacteristic
62C61×#436C2211yescharacteristic
63C61×#334C2211yescharacteristic
64C61×Q8⋊C8C2211yescharacteristic
65C61×D4⋊C8C2211yescharacteristic
66C82×C61C2211yescharacteristic
67C2×#11758202C2×C411yescharacteristic
68C2×#11758190C2×C411yescharacteristic
69C2×#11758199C2×C411yescharacteristic
70C2×#11758189—21no—
71C22×#92418—21no—
72C2×C8×D61—21no—
73C2×C8×D61—21no—
74C2×C8×D61C2×C411yescharacteristic
75C2×C8×D61C2×C411yescharacteristic
76C2×C8×D61D411yescharacteristic
77C2×C8×D61D411yescharacteristic
78#14519772—21no—
79#14519770—21no—
80#14519768C2×C411yescharacteristic
81#14519768C2×C411yescharacteristic
82#14519766C2×C411yescharacteristic
83#14519764C2×C411yescharacteristic
84#14519762—21no—
85#14519759—21no—
86#14519757C2×C411yescharacteristic
87#14519753C2×C411yescharacteristic
88#14519752C2×C411yescharacteristic
89#14519748C2×C411yescharacteristic
90C4×Dic122C2311yescharacteristic
91C4×D244—21no—
92C4×D244C2311yescharacteristic
93C4×C22⋊D61—21no—
94#14519736—21no—
95#14519734—21no—
96C42×D61C2311yescharacteristic
97#14519710C2×C411yescharacteristic
98#14519710C2×C411yescharacteristic
99#14519709C2×C411yescharacteristic
100C4⋊#92418—21no—
101C4⋊#92418C2311yescharacteristic
102C4⋊#92418C2311yescharacteristic
103#14519706—21no—
104#14519706—21no—
105#14519702C2×C411yescharacteristic
106#14519701C2×C411yescharacteristic
107#14519700C2×C411yescharacteristic
108#14519699C2×C411yescharacteristic
109#14519698C2×C411yescharacteristic
110#14519697C2×C411yescharacteristic
111C22⋊#92418—21no—
112#14519689—21no—
113#14519689—21no—
114#14519689—21no—
115#14519689—21no—
116#14519687—21no—
117#14519687—21no—
118#14519687—21no—
119C4×#92418—21no—
120C4×#92418C2311yescharacteristic
121C4×#92418C2311yescharacteristic
122C4×#92418C2311yescharacteristic
123C8×Dic61—21no—
124C8×Dic61—21no—
125C8×Dic61C2×C411yescharacteristic
126C8×Dic61C2×C411yescharacteristic
127C8×Dic61D411yescharacteristic
128C8×Dic61D411yescharacteristic
129C2×SD16×C61C2×C411yescharacteristic
130C2×M16×C61—21no—
131C4×Q8×C61C2311yescharacteristic
132C4×D4×C61C2311yescharacteristic
133#106×C61C2×C411yescharacteristic
134C4⋊C8×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137#103×C61C2×C411yescharacteristic
138#102×C61C2×C411yescharacteristic
139C22⋊C8×C61—21no—
140#97×C61—21no—
141C4×C8×C61C2311yescharacteristic
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61C22×C411yescharacteristic
145C2×D4×C61C22×C411yescharacteristic
146C22×C4×C61—21no—
147C2×C22⋊D61—21no—
148C2×D244—21no—
149C2×D244C22×C411yescharacteristic
150C2×C4×D61—21no—
151C2×C4×D61C22×C411yescharacteristic
152C2×C4×D61C2×D411yescharacteristic
153C2×C4×D61#4111yescharacteristic
154C2×Dic122C22×C411yescharacteristic
155SD16×C61—22no—
156M16×C61—42no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61C22×C411yescharacteristic
161C2×C8×C61C22×C411yescharacteristic
162C2×C8×C61C2×D411yescharacteristic
163C2×C8×C61C2×D411yescharacteristic
164C4⋊C4×C61—21no—
165C4⋊C4×C61C22×C411yescharacteristic
166C4⋊C4×C61C22×C411yescharacteristic
167C22⋊C4×C61—21no—
168C42×C61—21no—
169C42×C61C2411yescharacteristic
170C22⋊Dic61—21no—
171C22⋊Dic61—21no—
172C22⋊Dic61—21no—
173#11758202—22no—
174#11758199—22no—
175#11758198—21no—
176#11758198—21no—
177#11758198—21no—
178#11758198—21no—
179#11758198—21no—
180#11758198—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61C22×C411yescharacteristic
185C4⋊Dic61C22×C411yescharacteristic
186#11758196—21no—
187#11758196—21no—
188#11758196—21no—
189#11758196—21no—
190C4×Dic61—21no—
191C4×Dic61C22×C411yescharacteristic
192C4×Dic61C2×D411yescharacteristic
193C4×Dic61#4111yescharacteristic
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—42no—
198C2×#92418C22×C411yescharacteristic
199C2×#92418C22×C411yescharacteristic
200C2×#92418C2×D411yescharacteristic
201C2×#92418C2×D411yescharacteristic
202#11758190—22no—
203#11758189—42no—
204C8×D61—24no—
205C8×D61—22no—
206C8×D61—22no—
207C8×D61—22no—
208C8×D61—22no—
209C8×D61—42no—
210C23×C61—21no—
211C22×D61—21no—
212C22×D61C4×D411yescharacteristic
213Q8×C61—21no—
214Q8×C61—21no—
215D4×C61—21no—
216D4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61C23×C411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C2×#4111yescharacteristic
225C22⋊D61—42no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61C4×D411yescharacteristic
230C2×Dic61C4×D411yescharacteristic
231C2×Dic61C4×D411yescharacteristic
232D244—21no—
233D244—21no—
234D244—42no—
235C4×D61—22no—
236C4×D61—21no—
237C4×D61—21no—
238C4×D61—21no—
239C4×D61—21no—
240C4×D61—41no—
241C4×D61—41no—
242Dic122—21no—
243Dic122—21no—
244C8×C61—22no—
245C8×C61—21no—
246C8×C61—21no—
247C8×C61—41no—
248C8×C61—41no—
249C8×C61#13112yes—
250C8×C61#13512yes—
251#92418—22no—
252#92418—21no—
253#92418—21no—
254#92418—41no—
255#92418—41no—
256#92418#13112yes—
257#92418#13512yes—
258#17959—611nomaximal, Sylow
259C22×C61—41no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61C2×C4×D411yescharacteristic
263C2×D61—21no—
264C2×D61—21no—
265C2×D61—41no—
266C2×D61—41no—
267C2×D61—41no—
268C2×D61#44312yes—
269C4×C61—21no—
270C4×C61—41no—
271C4×C61—41no—
272C4×C61—41no—
273C4×C61C2×#13111yescharacteristic
274C4×C61C2×#13111yescharacteristic
275C4×C61C2×#13511yescharacteristic
276C4×C61C2×#13511yescharacteristic
277Dic61—21no—
278Dic61—21no—
279Dic61—21no—
280Dic61—41no—
281Dic61—41no—
282Dic61—41no—
283#3168—611no—
284#2966—611no—
285#2966—611no—
286#1591—611no—
287#1590—611no—
288#1590D6111yescharacteristic, complemented
289#1589—611no—
290C8×Q16—611no—
291C8×SD16—611no—
292C8×SD16—611no—
293C8×D8—611no—
294#1568—611no—
295#1494—611no—
296#1494—611no—
297C2×C82—611no—
298C2×C61—81no—
299C2×C61#293111yescharacteristic
300C2×C61C4×#13511yescharacteristic
301C2×C61C2×#44311yescharacteristic
302D61—22no—
303D61—81no—
304#529—611no—
305#529—611no—
306#499—611no—
307#495—611no—
308#488—611no—
309#488—611no—
310#467—611no—
311#466—611no—
312#465—611no—
313#465—611no—
314#446—611no—
315#446C2×D6111yescharacteristic
316C8×Q8—611no—
317C8×Q8—611no—
318C4×Q16—611no—
319C4×SD16—611no—
320C4×SD16C2×D6111yescharacteristic
321C4×D8—611no—
322#436—611no—
323#436C2×D6111yescharacteristic
324#435—1221no—
325#435—1221no—
326C8×D4—611no—
327C8×D4—611no—
328C8×D4—1221no—
329C8×D4—1221no—
330#433—1221no—
331#433—1221no—
332#432—611no—
333#432—611no—
334#405—1221no—
335#405—1221no—
336C2×C4×C8—611no—
337C2×C4×C8—611no—
338C2×C4×C8—611no—
339#335—611no—
340#335—611no—
341#334—611no—
342#334C2×D6111yescharacteristic
343Q8⋊C8—611no—
344Q8⋊C8—611no—
345Q8⋊C8—611no—
346Q8⋊C8C2×D6111yescharacteristic
347D4⋊C8—611no—
348D4⋊C8—611no—
349D4⋊C8—611no—
350D4⋊C8C2×D6111yescharacteristic
351C82—611no—
352C82—611no—
353C82—611no—
354C82C2×D6111yescharacteristic
355C61#1795911yescharacteristic, complemented, Sylow
356C2×Q16—611no—
357C2×SD16—611no—
358C2×SD16C4×D6111yescharacteristic
359C2×D8—611no—
360C2×M16—21no—
361C2×M16—1221no—
362C22×C8—611no—
363C22×C8—611no—
364C22×C8—611no—
365C22×C8—611no—
366C22×C8—1221no—
367C22×C8—1221no—
368C22×C8—1221no—
369#126—1221no—
370#126—1221no—
371#125—611no—
372#125—611no—
373#124—611no—
374#124—611no—
375#123—1221no—
376#123—1221no—
377C22⋊Q8—611no—
378C22⋊Q8—611no—
379#121—611no—
380#121—611no—
381C4×Q8—611no—
382C4×Q8C22×D6111yescharacteristic
383C4×D4—611no—
384C4×D4—1221no—
385C4×D4—1221no—
386C4×D4C22×D6111yescharacteristic
387#117—1221no—
388#117—1221no—
389C2×C42—611no—
390#107—611no—
391#107—611no—
392#106—611no—
393#106C4×D6111yescharacteristic
394C4⋊C8—21no—
395C4⋊C8—611no—
396C4⋊C8—611no—
397C4⋊C8—1221no—
398C4⋊C8—1221no—
399C4⋊C8—1221no—
400C4⋊C8C22×D6111yescharacteristic
401C4⋊C8C22×D6111yescharacteristic
402#103—611no—
403#103—611no—
404#103—611no—
405#103C4×D6111yescharacteristic
406#102—611no—
407#102—611no—
408#102—611no—
409#102C4×D6111yescharacteristic
410C22⋊C8—21no—
411C22⋊C8—1221no—
412C22⋊C8—1221no—
413C22⋊C8—1221no—
414C22⋊C8—1221no—
415C22⋊C8—1221no—
416#97—21no—
417#97—1221no—
418#97—1221no—
419#97—1221no—
420C4×C8—611no—
421C4×C8—611no—
422C4×C8—611no—
423C4×C8—611no—
424C4×C8—611no—
425C4×C8—611no—
426C4×C8—611no—
427C4×C8—1221no—
428C4×C8—1221no—
429C4×C8—1221no—
430C4×C8C22×D6111yescharacteristic
431C4×C8C22×D6111yescharacteristic
432C4×C8C22×D6111yescharacteristic
433C2×Q8—611no—
434C2×Q8C2×C4×D6111yescharacteristic
435C2×D4—611no—
436C2×D4—1221no—
437C2×D4—1221no—
438C2×D4C2×C4×D6111yescharacteristic
439C22×C4—21no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—1221no—
444Q16—1222no—
445SD16—22no—
446SD16—1222no—
447D8—1222no—
448M16—42no—
449M16—2442no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—611no—
454C2×C8—611no—
455C2×C8—611no—
456C2×C8—611no—
457C2×C8—1222no—
458C2×C8—1222no—
459C2×C8—1224no—
460C2×C8—1221no—
461C2×C8—1222no—
462C2×C8—1222no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—2442no—
466C2×C8—2442no—
467C2×C8C2×C4×D6111yescharacteristic
468C2×C8C2×C4×D6111yescharacteristic
469C2×C8D4×D6111yescharacteristic
470C2×C8D4×D6111yescharacteristic
471C4⋊C4—21no—
472C4⋊C4—611no—
473C4⋊C4—611no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4C2×C4×D6111yescharacteristic
482C4⋊C4C2×C4×D6111yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—1221no—
485C22⋊C4—1221no—
486C22⋊C4—1221no—
487C22⋊C4—1221no—
488C22⋊C4—1221no—
489C22⋊C4—1221no—
490C22⋊C4—1221no—
491C22⋊C4—1221no—
492C22⋊C4—1221no—
493C42—21no—
494C42—611no—
495C42—611no—
496C42—611no—
497C42—1221no—
498C42C23×D6111yescharacteristic
499C23—21no—
500C23—611no—
501C23—1221no—
502Q8—21no—
503Q8—21no—
504Q8—1221no—
505Q8—1221no—
506D4—21no—
507D4—21no—
508D4—1221no—
509D4—1221no—
510D4—2442no—
511D4—2442no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—611no—
518C2×C4—611no—
519C2×C4—611no—
520C2×C4—1222no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1221no—
527C2×C4—1221no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4C22×C4×D6111yescharacteristic
531C2×C4C2×D4×D6111yescharacteristic
532C2×C4#41×D6111yescharacteristic
533C8—22no—
534C8—21no—
535C8—21no—
536C8—41no—
537C8—41no—
538C8—612no—
539C8—612no—
540C8—1222no—
541C8—1221no—
542C8—1221no—
543C8—2441no—
544C8—2441no—
545C8#1451978512yes—
546C8#1451979812yes—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—612no—
551C22—1221no—
552C22—1221no—
553C22—2441no—
554C22—2441no—
555C22—2441no—
556C22C4×D4×D6111yescharacteristic
557C4—21no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—1221no—
562C4—1221no—
563C4—1221no—
564C4—2441no—
565C4—2441no—
566C4—2441no—
567C4#131×D6111yescharacteristic
568C4C2×#1451978511yescharacteristic
569C4#135×D6111yescharacteristic
570C4C2×#1451979811yescharacteristic
571C2—81no—
572C2—1222no—
573C2—4881no—
574C2#2145452811yescharacteristic
575C2C4×#1451979811yescharacteristic
576C2#443×D6111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1590C2
socleC22×C61C2×C4×D4
prime radical#16839997C4
cosocleC42×C61C24
hypercenter#1590D61
hypocenterC61#17959
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1590

Complements

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

NH ≅ G/Ntype
2. #21454797C2semidirect
3. #21454793C2semidirect
4. #21444714C2semidirect
5. #21444711C2semidirect
6. #21444702C2semidirect
7. #21444623C2semidirect
8. SD16×#92418C2semidirect
9. #21444618C2semidirect
10. #21444611C2semidirect
11. #21444579C2semidirect
12. #21444278C2semidirect
13. #21444276C2semidirect
14. C82×D61C2semidirect
15. C61×#1590C2semidirect
288. #1590D61semidirect
355. C61#17959semidirect

Maximal subgroups

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

subgroupindex
#214563852
#214547972
#214547932
#214447142
#214447112
#214447022
#214446232
SD16×#924182
#214446182
#214446112
#214445792
#214442782
#214442762
C82×D612
C61×#15902
#1795961