Finite Group Databasebeta

← #29648004

Subgroups — #29648004

Order 15616.

964 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#21459797C211yesmaximal, characteristic
2#21454833C211yesmaximal, characteristic, complemented
3#21444437C212yesmaximal, complemented
4#21444433C211yesmaximal, characteristic, complemented
5#21444424C211yesmaximal, characteristic, complemented
6#21444420C212yesmaximal, complemented
7#21444413C211yesmaximal, characteristic, complemented
8#21444408C212yesmaximal, complemented
9#21444405C211yesmaximal, characteristic, complemented
10C2×#16839503C212yesmaximal, complemented
11C61×#2875C211yesmaximal, characteristic, complemented
12#16840490C411yescharacteristic
13#16840472C411yescharacteristic
14#16840469C411yescharacteristic
15#16840458C412yes—
16#16840452C412yes—
17#16840451—21no—
18#16840445—22no—
19#16840416C411yescharacteristic
20#16840411—21no—
21#16840370—21no—
22#16840354—22no—
23#16840319C2211yescharacteristic
24#16840296C2211yescharacteristic
25#16840296C2212yes—
26C2×#14519769C2212yes—
27#16840241—21no—
28#16839893—21no—
29#16839834—22no—
30D4×#92418—21no—
31#16839819C2212yes—
32#16839819C2211yescharacteristic
33C4×#11758194C2211yescharacteristic
34C2×#14519688C2212yes—
35#16839503C2211yescharacteristic
36#16839503C2212yes—
37#16839503C2212yes—
38#16839503C2212yes—
39#16839503C2211yescharacteristic
40#16839503C2212yes—
41#16839503C2212yes—
42#16839501C2211yescharacteristic
43#16839501C2211yescharacteristic
44#16839501C2212yes—
45C2×#131×C61—21no—
46C4×#41×C61C2211yescharacteristic
47C61×#405—21no—
48C61×#405—22no—
49C4×M16×C61C2212yes—
50C4×M16×C61C2211yescharacteristic
51C2×#97×C61C2212yes—
52C2×#97×C61C2211yescharacteristic
53C2×#11758224—21no—
54C2×#11758194—21no—
55C22×#92418—22no—
56#14519777C2×C411yescharacteristic
57#14519777C2×C411yescharacteristic
58#14519775—21no—
59#14519772—22no—
60#14519771—22no—
61#14519771—21no—
62#14519770—22no—
63#14519769—22no—
64#14519769—22no—
65#14519769C2×C412yes—
66#14519769C2×C412yes—
67#14519769C2311yescharacteristic
68#14519769C2312yes—
69#14519768—21no—
70#14519767—22no—
71#14519767—22no—
72#14519765C2311yescharacteristic
73#14519761—22no—
74#14519760—21no—
75#14519760—22no—
76#14519759—22no—
77#14519758—22no—
78#14519757—21no—
79#14519756—21no—
80#14519756—21no—
81#14519756C2×C411yescharacteristic
82#14519756C2×C412yes—
83#14519755—22no—
84#14519754—22no—
85#14519753—21no—
86#14519751C2×C411yescharacteristic
87#14519751C2×C412yes—
88#14519749—22no—
89C4×Dic122—21no—
90C4×Dic122—21no—
91C4×Dic122C2×C412yes—
92C4×Dic122C2×C411yescharacteristic
93#14519742—21no—
94C4×D244C2×C412yes—
95C4×D244C2×C411yescharacteristic
96D4×Dic61—21no—
97C4×C22⋊D61—21no—
98C4×C22⋊D61—22no—
99#14519737—22no—
100#14519736C2×C412yes—
101#14519734—22no—
102#14519734—21no—
103#14519733—22no—
104C2×C4⋊Dic61C2×C412yes—
105C2×C4⋊Dic61C2×C412yes—
106C2×#11758196—22no—
107C2×#11758196—22no—
108C4⋊C4×D61—21no—
109C4⋊#92418—21no—
110C4⋊#92418—22no—
111C22⋊#92418—22no—
112C22⋊#92418—21no—
113#14519688—22no—
114#14519688C2311yescharacteristic
115#14519688C2312yes—
116#14519688C2311yescharacteristic
117C4×#92418—21no—
118C2×#41×C61D411yescharacteristic
119#131×C61—28no—
120C2×M16×C61—22no—
121C2×M16×C61—21no—
122C22×C8×C61—22no—
123C22×C8×C61—21no—
124C4×Q8×C61C2×C411yescharacteristic
125C4×Q8×C61D411yescharacteristic
126C4×D4×C61C2×C412yes—
127C4×D4×C61C2×C411yescharacteristic
128C4×D4×C61D411yescharacteristic
129C4×D4×C61D412yes—
130#117×C61D411yescharacteristic
131#117×C61D412yes—
132C2×C42×C61C2312yes—
133C2×C42×C61C2311yescharacteristic
134#97×C61—22no—
135#97×C61—21no—
136#97×C61C2311yescharacteristic
137#97×C61C2311yescharacteristic
138#97×C61C2312yes—
139C4×C8×C61—22no—
140C4×C8×C61—21no—
141#41×C61—24no—
142C2×Q8×C61C22⋊C411yescharacteristic
143C2×D4×C61C22⋊C412yes—
144C2×D4×C61C22⋊C411yescharacteristic
145C22×C4×C61C22×C412yes—
146C22×C4×C61C22×C411yescharacteristic
147C22×C4×C61C2×D412yes—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D412yes—
150C22×C4×C61C2×D411yescharacteristic
151C2×C22⋊D61—22no—
152C2×C22⋊D61—21no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155#11758224—44no—
156C2×D244C22×C411yescharacteristic
157C2×C4×D61—21no—
158C2×C4×D61—22no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122C22×C411yescharacteristic
162M16×C61—28no—
163M16×C61—24no—
164C2×C8×C61—28no—
165C2×C8×C61—24no—
166C2×C8×C61—21no—
167C2×C8×C61—22no—
168C2×C8×C61—21no—
169C4⋊C4×C61C22⋊C411yescharacteristic
170C4⋊C4×C61C22⋊C412yes—
171C4⋊C4×C61C22⋊C411yescharacteristic
172C4⋊C4×C61C22⋊C412yes—
173C22⋊C4×C61C22⋊C412yes—
174C22⋊C4×C61C22⋊C412yes—
175C22⋊C4×C61C22⋊C412yes—
176C42×C61C22×C412yes—
177C42×C61C22×C411yescharacteristic
178C42×C61C2×D411yescharacteristic
179C42×C61C2×D412yes—
180C42×C61C2×D412yes—
181C42×C61C2×D411yescharacteristic
182C42×C61C2411yescharacteristic
183C22⋊Dic61—22no—
184C22⋊Dic61—22no—
185C22⋊Dic61—21no—
186#11758198—21no—
187#11758198—22no—
188#11758198—22no—
189#11758198—21no—
190#11758198—21no—
191#11758198—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—22no—
194C4⋊Dic61—22no—
195C4⋊Dic61—21no—
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—21no—
205#11758196—22no—
206#11758196—21no—
207#11758196—22no—
208#11758196—21no—
209#11758196—21no—
210#11758196—22no—
211#11758196—22no—
212C4×Dic61—22no—
213C4×Dic61—22no—
214C4×Dic61—21no—
215#11758194—42no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—22no—
219C2×#92418—44no—
220C23×C61C2×C22⋊C411yescharacteristic
221C23×C61C2×C22⋊C412yes—
222C22×D61—21no—
223Q8×C61—22no—
224D4×C61—24no—
225D4×C61#10414yes—
226C2×C4×C61—24no—
227C2×C4×C61—22no—
228C2×C4×C61—42no—
229C2×C4×C61—41no—
230C2×C4×C61—42no—
231C2×C4×C61—41no—
232C2×C4×C61C2×C22⋊C411yescharacteristic
233C2×C4×C61C2×C22⋊C412yes—
234C2×C4×C61C2×C22⋊C412yes—
235C2×C4×C61C2×C22⋊C412yes—
236C2×C4×C61C2×C22⋊C411yescharacteristic
237C2×C4×C61C2×C22⋊C411yescharacteristic
238C2×C4×C61C23×C411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C22⋊D61—44no—
242C22⋊D61—81no—
243C2×Dic61—21no—
244C2×Dic61—22no—
245C2×Dic61—21no—
246C2×Dic61—22no—
247C2×Dic61—21no—
248C2×Dic61—44no—
249C2×Dic61—82no—
250D244—41no—
251C4×D61—42no—
252C4×D61—82no—
253Dic122—42no—
254Dic122—41no—
255Dic122—81no—
256C8×C61—22no—
257C8×C61—24no—
258C8×C61—22no—
259#92418—22no—
260#92418—42no—
261#92418—41no—
262#17653—611nomaximal, Sylow
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—21no—
266C22×C61—21no—
267C22×C61—21no—
268C22×C61C2×#10414yes—
269C22×C61C22×C22⋊C411yescharacteristic
270C2×D61—41no—
271C2×D61—41no—
272C2×D61—41no—
273C4×C61—21no—
274C4×C61—21no—
275C4×C61—22no—
276C4×C61—21no—
277C4×C61—22no—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—22no—
281C4×C61C2×#10412yes—
282Dic61—41no—
283Dic61—42no—
284Dic61—41no—
285Dic61—42no—
286Dic61—41no—
287#3515—611no—
288#2970—611no—
289#2875D6111yescharacteristic, complemented
290#1532—612no—
291#1531—611no—
292#1529—611no—
293#1528—612no—
294#1526—611no—
295#1524—612no—
296#1524—611no—
297C2×#330—612no—
298C2×C61—24no—
299C2×C61—41no—
300C2×C61#288911yescharacteristic
301C2×C61C22×#10411yescharacteristic
302C2×C61#290111yescharacteristic
303D61—81no—
304C2×#131—21no—
305#557—611no—
306#554—611no—
307#554—611no—
308#553—612no—
309#552—612no—
310#552—1221no—
311#552—1222no—
312#547—611no—
313#547—1221no—
314#541—1221no—
315#539—1222no—
316#533—611no—
317#529—611no—
318#529—612no—
319C2×#125—612no—
320#520—1221no—
321C4×#41C2×D6111yescharacteristic
322#446—1221no—
323#435—1222no—
324C8×D4—1221no—
325#432—612no—
326#432—611no—
327#405—21no—
328#405—22no—
329C4×M16—611no—
330C4×M16C2×D6112yes—
331C4×M16C2×D6111yescharacteristic
332C2×#97—612no—
333C2×#97C2×D6112yes—
334C2×#97C2×D6111yescharacteristic
335#330—611no—
336#330—612no—
337#330—612no—
338#330—612no—
339#330—611no—
340#330—612no—
341#330—612no—
342#329—611no—
343#329—612no—
344#329—611no—
345C61#1765311yescharacteristic, complemented, Sylow
346C2×#41—1221no—
347C2×#41C22⋊D6111yescharacteristic
348#131—28no—
349C2×M16—22no—
350C2×M16—21no—
351C2×M16—1221no—
352C22×C8—22no—
353C22×C8—21no—
354C22×C8—1222no—
355C4⋊Q8—611no—
356C4⋊Q8—611no—
357C4⋊Q8—1221no—
358#126—1222no—
359#126—1222no—
360#126—1222no—
361#126—1221no—
362#125—612no—
363#125—612no—
364#125—611no—
365#125—612no—
366#125—1222no—
367#125—1222no—
368#125—1221no—
369#125—1222no—
370#125—1222no—
371#124—611no—
372#123—1222no—
373#123—1221no—
374#123—1222no—
375#123—1222no—
376#123—1222no—
377C22⋊Q8—611no—
378C22⋊Q8—612no—
379C22⋊Q8—1221no—
380C22⋊Q8—1221no—
381C22⋊Q8—1221no—
382C22⋊Q8—1222no—
383C22⋊Q8—1221no—
384C22⋊Q8—1222no—
385#121—612no—
386#121—611no—
387#121—1222no—
388C4×Q8—612no—
389C4×Q8—611no—
390C4×Q8—1221no—
391C4×Q8—1221no—
392C4×Q8—1221no—
393C4×Q8C4×D6111yescharacteristic
394C4×Q8D24411yescharacteristic
395C4×D4—611no—
396C4×D4—612no—
397C4×D4—1221no—
398C4×D4—1222no—
399C4×D4—1221no—
400C4×D4—1222no—
401C4×D4C4×D6111yescharacteristic
402C4×D4C4×D6112yes—
403C4×D4D24411yescharacteristic
404C4×D4D24412yes—
405#117—612no—
406#117—1222no—
407#117—1221no—
408#117—1222no—
409#117C22⋊D6111yescharacteristic
410#117C22⋊D6112yes—
411C2×C4⋊C4—612no—
412C2×C4⋊C4—612no—
413C2×C4⋊C4—1222no—
414C2×C4⋊C4—1222no—
415C2×C4⋊C4—1221no—
416C2×C42C22×D6112yes—
417C2×C42C22×D6111yescharacteristic
418C4⋊C8—1221no—
419C4⋊C8—1222no—
420C22⋊C8—1222no—
421C22⋊C8—1221no—
422#97—22no—
423#97—21no—
424#97—611no—
425#97—612no—
426#97—611no—
427#97—1222no—
428#97C22×D6111yescharacteristic
429#97C22×D6111yescharacteristic
430#97C22×D6112yes—
431C4×C8—22no—
432C4×C8—21no—
433C4×C8—1221no—
434#41—24no—
435#41—2444no—
436C2×Q8—611no—
437C2×Q8—1221no—
438C2×Q8—1221no—
439C2×Q8#1175819811yescharacteristic
440C2×D4—611no—
441C2×D4—1221no—
442C2×D4—1222no—
443C2×D4#1175819812yes—
444C2×D4#1175819811yescharacteristic
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1221no—
449C22×C4C2×C4×D6112yes—
450C22×C4C2×C4×D6111yescharacteristic
451C22×C4C2×D24412yes—
452C22×C4C2×D24411yescharacteristic
453C22×C4C2×C22⋊D6112yes—
454C22×C4C2×C22⋊D6111yescharacteristic
455M16—28no—
456M16—24no—
457M16—2442no—
458C2×C8—28no—
459C2×C8—24no—
460C2×C8—21no—
461C2×C8—22no—
462C2×C8—21no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1222no—
466C2×C8—2444no—
467C4⋊C4—611no—
468C4⋊C4—611no—
469C4⋊C4—612no—
470C4⋊C4—612no—
471C4⋊C4—1221no—
472C4⋊C4—1222no—
473C4⋊C4—1221no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—1222no—
477C4⋊C4—1222no—
478C4⋊C4—1222no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1222no—
484C4⋊C4—1221no—
485C4⋊C4—1222no—
486C4⋊C4—1221no—
487C4⋊C4#1175819811yescharacteristic
488C4⋊C4#1175819811yescharacteristic
489C4⋊C4#1175819812yes—
490C4⋊C4#1175819812yes—
491C22⋊C4—1222no—
492C22⋊C4—1222no—
493C22⋊C4—1221no—
494C22⋊C4—1221no—
495C22⋊C4—1221no—
496C22⋊C4—1221no—
497C22⋊C4—1222no—
498C22⋊C4—1222no—
499C22⋊C4—1221no—
500C22⋊C4#1175819812yes—
501C22⋊C4#1175819812yes—
502C22⋊C4#1175819812yes—
503C42—1222no—
504C42—1222no—
505C42—1221no—
506C42C2×C4×D6112yes—
507C42C2×C4×D6111yescharacteristic
508C42C2×D24411yescharacteristic
509C42C2×D24412yes—
510C42C2×C22⋊D6112yes—
511C42C2×C22⋊D6111yescharacteristic
512C42C23×D6111yescharacteristic
513C23—1221no—
514C23C2×#1175819811yescharacteristic
515C23C2×#1175819812yes—
516Q8—22no—
517Q8—4881no—
518Q8—2441no—
519Q8—2442no—
520D4—24no—
521D4—4881no—
522D4—2441no—
523D4—2444no—
524D4#1451970514yes—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—1221no—
532C2×C4—1222no—
533C2×C4—1222no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—4882no—
537C2×C4—4882no—
538C2×C4—2444no—
539C2×C4—2442no—
540C2×C4C2×#1175819811yescharacteristic
541C2×C4C2×#1175819812yes—
542C2×C4C2×#1175819812yes—
543C2×C4C2×#1175819811yescharacteristic
544C2×C4C2×#1175819812yes—
545C2×C4C2×#1175819811yescharacteristic
546C2×C4C22×C4×D6111yescharacteristic
547C2×C4C22×C22⋊D6111yescharacteristic
548C2×C4C22×D24411yescharacteristic
549C8—22no—
550C8—24no—
551C8—22no—
552C8—1222no—
553C8—2442no—
554C8—2441no—
555C22—22no—
556C22—22no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—2441no—
561C22—2441no—
562C22—2441no—
563C22C2×#1451970514yes—
564C22C22×#1175819811yescharacteristic
565C4—21no—
566C4—22no—
567C4—21no—
568C4—22no—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—22no—
573C4—2441no—
574C4—2442no—
575C4—2441no—
576C4—2442no—
577C4—2441no—
578C4C2×#1451970512yes—
579C2—24no—
580C2—41no—
581C2—4881no—
582C2#2145430111yescharacteristic
583C2C22×#1451970511yescharacteristic
584C2#2145436411yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#2875C2
socleC22×C61C22×C22⋊C4
prime radical#21459797C2
cosocleC42×C61C24
hypercenter#2875D61
hypocenterC61#17653
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×#41 ◁ #2875

Complements

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

NH ≅ G/Ntype
2. #21454833C2semidirect
3. #21444437C2semidirect
4. #21444433C2semidirect
5. #21444424C2semidirect
6. #21444420C2semidirect
7. #21444413C2semidirect
8. #21444408C2semidirect
9. #21444405C2semidirect
10. C2×#16839503C2semidirect
11. C61×#2875C2semidirect
289. #2875D61semidirect
345. C61#17653semidirect

Maximal subgroups

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

subgroupindex
#214597972
#214548332
#214444372
#214444332
#214444242
#214444202
#214444132
#214444082
#214444052
C2×#168395032
C61×#28752
#1765361