Finite Group Databasebeta

← #29648013

Subgroups — #29648013

Order 15616.

960 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459834C211yesmaximal, characteristic
2#21454837C211yesmaximal, characteristic, complemented
3#21454244C211yesmaximal, characteristic, complemented
4#21444433C211yesmaximal, characteristic, complemented
5#21444432C212yesmaximal, complemented
6#21444426C211yesmaximal, characteristic, complemented
7#21444423C212yesmaximal, complemented
8#21444413C211yesmaximal, characteristic, complemented
9#21444412C212yesmaximal, complemented
10#21444404C212yesmaximal, complemented
11C61×#1524C211yesmaximal, characteristic, complemented
12C2×#14519783—21no—
13Q8×Dic122C411yescharacteristic
14#16840485C411yescharacteristic
15#16840483C412yes—
16#16840482—21no—
17#16840479—22no—
18#16840477C411yescharacteristic
19#16840476C412yes—
20D4×Dic122C411yescharacteristic
21#16840421—21no—
22#16840383—21no—
23#16840380—22no—
24#16840319C2211yescharacteristic
25#16840318C2212yes—
26#16840291C2212yes—
27#16840239—21no—
28C4×#11758222C2211yescharacteristic
29C8×Dic122—21no—
30#16839841—21no—
31#16839838—22no—
32#16839819C2211yescharacteristic
33#16839817C2212yes—
34#16839719—22no—
35#16839717—21no—
36C4×#11758194C2211yescharacteristic
37#16839714C2212yes—
38#16839714C2212yes—
39#97×D61C2212yes—
40#16839504C2212yes—
41#16839503C2212yes—
42#16839503C2212yes—
43#16839503C2212yes—
44#16839503C2212yes—
45#16839502C2212yes—
46C61×#529C2211yescharacteristic
47C4×M16×C61C2211yescharacteristic
48C2×#97×C61C2211yescharacteristic
49C61×#330C2211yescharacteristic
50C61×#330C2211yescharacteristic
51C61×#329C2211yescharacteristic
52C61×#329C2211yescharacteristic
53C2×#11758222D411yescharacteristic
54C22×Dic122—21no—
55#14519783—28no—
56C2×#11758194—21no—
57C2×#11758189—22no—
58C2×#11758189—22no—
59C2×C8×D61—22no—
60#14519777—21no—
61#14519777C2×C411yescharacteristic
62#14519777C2×C411yescharacteristic
63#14519776—22no—
64#14519776C2×C412yes—
65#14519776C2×C412yes—
66#14519772—22no—
67#14519772—22no—
68#14519770—22no—
69#14519770—22no—
70#14519769C2312yes—
71#14519765—21no—
72#14519764—22no—
73#14519763—22no—
74#14519763—22no—
75#14519763—22no—
76#14519759—22no—
77#14519756—21no—
78#14519756—21no—
79#14519756—21no—
80#14519755—22no—
81#14519755—22no—
82#14519754—22no—
83#14519753C2×C412yes—
84#14519751—21no—
85#14519749—22no—
86#14519748C2×C412yes—
87Q8×Dic61C2×C412yes—
88C4×Dic122—21no—
89C4×Dic122—21no—
90C4×Dic122—21no—
91C4×Dic122—21no—
92C4×Dic122C2×C411yescharacteristic
93C4×Dic122D411yescharacteristic
94#14519742—22no—
95#14519742—22no—
96#14519742—22no—
97C4×D244C2×C411yescharacteristic
98C4×D244D411yescharacteristic
99D4×Dic61C2×C412yes—
100C4×C22⋊D61C2×C412yes—
101C4×C22⋊D61D412yes—
102#14519737—22no—
103#14519736—21no—
104#14519736C2×C412yes—
105#14519736D411yescharacteristic
106#14519735C2×C412yes—
107#14519735C2×C412yes—
108#14519734D412yes—
109#14519733—22no—
110#14519733—22no—
111C42×D61C2312yes—
112C4⋊#92418—21no—
113#14519706—22no—
114C22⋊#92418—21no—
115#14519689—22no—
116#14519688C2312yes—
117#14519688C2312yes—
118#14519687—22no—
119C4×#92418—21no—
120C8×Dic61—22no—
121C8×Dic61—22no—
122C2×M16×C61—21no—
123C22×C8×C61—21no—
124#126×C61—21no—
125#125×C61C2311yescharacteristic
126#124×C61C2311yescharacteristic
127#123×C61—21no—
128C22⋊Q8×C61C2×C411yescharacteristic
129#121×C61C2×C411yescharacteristic
130C4×Q8×C61C2×C411yescharacteristic
131C4×D4×C61—21no—
132C4×D4×C61C2×C411yescharacteristic
133#117×C61—21no—
134C2×C42×C61C2311yescharacteristic
135#97×C61—21no—
136#97×C61C2311yescharacteristic
137#97×C61C2311yescharacteristic
138#97×C61C2311yescharacteristic
139#97×C61C2311yescharacteristic
140C4×C8×C61—21no—
141C2×Q8×C61C22×C411yescharacteristic
142C2×D4×C61—21no—
143C2×D4×C61C22×C411yescharacteristic
144C22×C4×C61—21no—
145C22×C4×C61C22×C411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C2×C22⋊D61—22no—
149C2×C22⋊D61C22⋊C412yes—
150C22×Dic61—22no—
151#11758222—24no—
152C2×D244C22⋊C411yescharacteristic
153C2×C4×D61C22×C412yes—
154C2×C4×D61C2×D412yes—
155C2×C4×D61C2×D412yes—
156C2×Dic122—21no—
157C2×Dic122—22no—
158C2×Dic122—21no—
159C2×Dic122—22no—
160C2×Dic122—44no—
161C2×Dic122C22⋊C411yescharacteristic
162M16×C61—42no—
163C2×C8×C61—24no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61C22×C411yescharacteristic
173C4⋊C4×C61C22×C411yescharacteristic
174C22⋊C4×C61—21no—
175C22⋊C4×C61—21no—
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C22⋊C4×C61—21no—
179C42×C61—21no—
180C42×C61C22×C411yescharacteristic
181C42×C61C2×D411yescharacteristic
182C42×C61C2×D411yescharacteristic
183C42×C61C2411yescharacteristic
184C22⋊Dic61—22no—
185C22⋊Dic61—22no—
186C22⋊Dic61—22no—
187C22⋊Dic61C22⋊C412yes—
188#11758198—22no—
189#11758198—22no—
190#11758198—22no—
191#11758198C22⋊C412yes—
192#11758198C22⋊C412yes—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—22no—
196C4⋊Dic61C22⋊C411yescharacteristic
197C4⋊Dic61C22⋊C411yescharacteristic
198C4⋊Dic61C22×C412yes—
199C4⋊Dic61C22×C412yes—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196C22⋊C412yes—
206#11758196C22⋊C412yes—
207C4×Dic61—22no—
208C4×Dic61—22no—
209C4×Dic61—22no—
210C4×Dic61—22no—
211C4×Dic61C22×C412yes—
212C4×Dic61C2×D412yes—
213C4×Dic61C2×D412yes—
214#11758194—24no—
215C2×#92418—22no—
216C2×#92418—22no—
217#11758189—28no—
218#11758189—44no—
219C8×D61—28no—
220C23×C61—21no—
221C23×C61C2×C22⋊C411yescharacteristic
222C22×D61C2×C22⋊C412yes—
223Q8×C61—41no—
224D4×C61—41no—
225D4×C61—81no—
226C2×C4×C61—22no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—42no—
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C2×C4×C61C2×C22⋊C411yescharacteristic
235C2×C4×C61C2×C22⋊C411yescharacteristic
236C2×C4×C61C23×C411yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C22⋊D61—24no—
240C22⋊D61—82no—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—44no—
244C2×Dic61C2×C22⋊C412yes—
245C2×Dic61C2×C22⋊C412yes—
246C2×Dic61C2×C22⋊C412yes—
247D244—22no—
248C4×D61—24no—
249C4×D61—42no—
250C4×D61—42no—
251Dic122—42no—
252Dic122—44no—
253Dic122—82no—
254Dic122—81no—
255Dic122#10414yes—
256C8×C61—22no—
257C8×C61—22no—
258C8×C61—22no—
259C8×C61—41no—
260#92418—24no—
261#92418—42no—
262#17654—611nomaximal, Sylow
263C22×C61—21no—
264C22×C61—21no—
265C22×C61—21no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61C22×C22⋊C411yescharacteristic
270C2×D61—22no—
271C2×D61—22no—
272C2×D61—22no—
273C4×C61—21no—
274C4×C61—21no—
275C4×C61—21no—
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—41no—
279C4×C61—41no—
280C4×C61—41no—
281C4×C61C2×#10412yes—
282Dic61—22no—
283Dic61—22no—
284Dic61—42no—
285Dic61—42no—
286Dic61C2×#10414yes—
287#3518—611no—
288#2971—611no—
289#2875—611no—
290#1531—611no—
291#1531—612no—
292#1529—612no—
293#1529—611no—
294#1526—611no—
295#1526—612no—
296#1524—612no—
297#1524D6111yescharacteristic, complemented
298C2×C61—41no—
299C2×C61—81no—
300C2×C61#288911yescharacteristic
301C2×C61C22×#10411yescharacteristic
302C2×C61#290111yescharacteristic
303D61—42no—
304C2×#131—1221no—
305Q82—611no—
306#556—612no—
307#556—611no—
308#556—1221no—
309#556—1222no—
310#555—612no—
311#555—611no—
312D4×Q8—611no—
313#548—1221no—
314#542—1221no—
315#542—1222no—
316#533—612no—
317#533—611no—
318#529—612no—
319#529C2×D6111yescharacteristic
320#519—1221no—
321C4×#41—611no—
322C8×Q8—1221no—
323#436—1221no—
324#436—1222no—
325#432—611no—
326#432—612no—
327#405—1222no—
328#405—1221no—
329C4×M16—611no—
330C4×M16—612no—
331C4×M16—612no—
332C4×M16C2×D6111yescharacteristic
333C2×#97—612no—
334C2×#97C2×D6111yescharacteristic
335#330—612no—
336#330—612no—
337#330—612no—
338#330—612no—
339#330—612no—
340#330C2×D6111yescharacteristic
341#330C2×D6111yescharacteristic
342#329—612no—
343#329C2×D6111yescharacteristic
344#329C2×D6111yescharacteristic
345C61#1765411yescharacteristic, complemented, Sylow
346C2×#41—611no—
347C22×Q8—1221no—
348#131—1228no—
349C2×M16—21no—
350C2×M16—1222no—
351C2×M16—1222no—
352C2×M16—1221no—
353C22×C8—21no—
354C22×C8—1222no—
355C4⋊Q8—612no—
356C4⋊Q8—612no—
357C4⋊Q8—611no—
358C4⋊Q8—611no—
359C4⋊Q8—1221no—
360C4⋊Q8—1222no—
361#126—21no—
362#126—1222no—
363#126—1222no—
364#126—1222no—
365#126—1222no—
366#125—612no—
367#125C22×D6111yescharacteristic
368#124—1221no—
369#124—1222no—
370#124—1222no—
371#124—1222no—
372#124—1222no—
373#124C22×D6111yescharacteristic
374#123—21no—
375#123—1222no—
376C22⋊Q8—612no—
377C22⋊Q8—1222no—
378C22⋊Q8—1222no—
379C22⋊Q8—1222no—
380C22⋊Q8—1221no—
381C22⋊Q8—1221no—
382C22⋊Q8—1221no—
383C22⋊Q8C4×D6111yescharacteristic
384#121—612no—
385#121—1221no—
386#121—1222no—
387#121C4×D6111yescharacteristic
388C4×Q8—611no—
389C4×Q8—611no—
390C4×Q8—612no—
391C4×Q8—1221no—
392C4×Q8—1221no—
393C4×Q8—1222no—
394C4×Q8—1222no—
395C4×Q8—1222no—
396C4×Q8—1221no—
397C4×Q8—1221no—
398C4×Q8C4×D6111yescharacteristic
399C4×D4—21no—
400C4×D4—612no—
401C4×D4—612no—
402C4×D4—612no—
403C4×D4—611no—
404C4×D4—611no—
405C4×D4—1222no—
406C4×D4C4×D6111yescharacteristic
407#117—21no—
408#117—611no—
409#117—612no—
410#117—612no—
411#117—612no—
412#117—612no—
413#117—1222no—
414#117—1222no—
415#117—1221no—
416C2×C42—612no—
417C2×C42C22×D6111yescharacteristic
418C4⋊C8—1222no—
419C4⋊C8—1221no—
420C22⋊C8—1222no—
421C22⋊C8—1221no—
422#97—21no—
423#97—612no—
424#97—612no—
425#97—1222no—
426#97C22×D6111yescharacteristic
427#97C22×D6111yescharacteristic
428#97C22×D6111yescharacteristic
429#97C22×D6111yescharacteristic
430C4×C8—21no—
431C4×C8—1222no—
432C4×C8—1222no—
433C4×C8—1221no—
434#41—1224no—
435C2×Q8—611no—
436C2×Q8—1222no—
437C2×Q8—1222no—
438C2×Q8—1221no—
439C2×Q8—1221no—
440C2×Q8—2444no—
441C2×Q8C2×C4×D6111yescharacteristic
442C2×D4—21no—
443C2×D4—612no—
444C2×D4—611no—
445C2×D4—1222no—
446C2×D4C2×C4×D6111yescharacteristic
447C22×C4—21no—
448C22×C4—612no—
449C22×C4—612no—
450C22×C4—612no—
451C22×C4—1222no—
452C22×C4C2×C4×D6111yescharacteristic
453C22×C4D4×D6111yescharacteristic
454C22×C4D4×D6111yescharacteristic
455M16—42no—
456M16—1228no—
457M16—1224no—
458M16—2444no—
459C2×C8—24no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—1228no—
465C2×C8—1222no—
466C2×C8—1222no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—612no—
472C4⋊C4—612no—
473C4⋊C4—611no—
474C4⋊C4—611no—
475C4⋊C4—612no—
476C4⋊C4—612no—
477C4⋊C4—1222no—
478C4⋊C4—1222no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4C2×C4×D6111yescharacteristic
486C4⋊C4C2×C4×D6111yescharacteristic
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—21no—
491C22⋊C4—21no—
492C22⋊C4—612no—
493C22⋊C4—612no—
494C22⋊C4—612no—
495C22⋊C4—1222no—
496C22⋊C4—1222no—
497C22⋊C4—1222no—
498C22⋊C4—1222no—
499C22⋊C4—1222no—
500C22⋊C4—1222no—
501C42—21no—
502C42—612no—
503C42—612no—
504C42—612no—
505C42—1222no—
506C42—1222no—
507C42—1222no—
508C42—1222no—
509C42C2×C4×D6111yescharacteristic
510C42D4×D6111yescharacteristic
511C42D4×D6111yescharacteristic
512C42C23×D6111yescharacteristic
513C23—21no—
514C23—612no—
515C23C22⋊C4×D6111yescharacteristic
516Q8—41no—
517Q8—614no—
518Q8—4881no—
519Q8—4882no—
520Q8—2444no—
521Q8—2442no—
522D4—41no—
523D4—81no—
524D4—1224no—
525D4—1222no—
526D4—4882no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—612no—
535C2×C4—612no—
536C2×C4—612no—
537C2×C4—1224no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—2442no—
541C2×C4—2442no—
542C2×C4—2444no—
543C2×C4C22⋊C4×D6111yescharacteristic
544C2×C4C22⋊C4×D6111yescharacteristic
545C2×C4C22⋊C4×D6111yescharacteristic
546C2×C4C22×C4×D6111yescharacteristic
547C2×C4C2×D4×D6111yescharacteristic
548C2×C4C2×D4×D6111yescharacteristic
549C8—22no—
550C8—22no—
551C8—22no—
552C8—41no—
553C8—1224no—
554C8—2442no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—1222no—
562C22—1222no—
563C22—1222no—
564C22C2×C22⋊C4×D6111yescharacteristic
565C4—21no—
566C4—21no—
567C4—21no—
568C4—21no—
569C4—21no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—614no—
574C4—1222no—
575C4—1222no—
576C4—2442no—
577C4—2442no—
578C4#104×D6112yes—
579C2—41no—
580C2—81no—
581C2—2442no—
582C2#2145430011yescharacteristic
583C2C2×#104×D6111yescharacteristic
584C2#2145436311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1524C2
socleC22×C61C22×C22⋊C4
prime radical#21459834C2
cosocleC42×C61C24
hypercenter#1524D61
hypocenterC61#17654
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1524

Complements

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

NH ≅ G/Ntype
2. #21454837C2semidirect
3. #21454244C2semidirect
4. #21444433C2semidirect
5. #21444432C2semidirect
6. #21444426C2semidirect
7. #21444423C2semidirect
8. #21444413C2semidirect
9. #21444412C2semidirect
10. #21444404C2semidirect
11. C61×#1524C2semidirect
297. #1524D61semidirect
345. C61#17654semidirect

Maximal subgroups

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

subgroupindex
#214598342
#214548372
#214542442
#214444332
#214444322
#214444262
#214444232
#214444132
#214444122
#214444042
C61×#15242
#1765461