Finite Group Databasebeta

← #30610214

Subgroups — #30610214

Order 15616.

1048 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462683C211yesmaximal, characteristic, complemented
2#21462672C211yesmaximal, characteristic, complemented
3#21461102C212yesmaximal, complemented
4#21461095C212yesmaximal, complemented
5#21461089C212yesmaximal, complemented
6#21461084C212yesmaximal, complemented
7#21461074C212yesmaximal, complemented
8#21461065C212yesmaximal, complemented
9C2×#16840344C211yesmaximal, characteristic
10#16840795—24no—
11#16840791—24no—
12#16840783C2212yescomplemented
13#16840779C2212yescomplemented
14#16840733C2212yes—
15#16840733C2212yes—
16#16840732C2212yescomplemented
17C2×C22⋊#92420C2212yescomplemented
18#16840608C2214yescomplemented
19#16840606C2214yes—
20#16840605C2214yes—
21#16840605C2214yescomplemented
22#16840344—24no—
23#16840344—24no—
24C2×#14519772C2211yescharacteristic
25C2×#14519770C411yescharacteristic
26C2×#14519764C2211yescharacteristic
27C2×#14519763C411yescharacteristic
28C2×#14519760C411yescharacteristic
29C2×#14519758C2211yescharacteristic
30C2×#14519755C411yescharacteristic
31C2×#14519749C2211yescharacteristic
32C2×#14519748C411yescharacteristic
33C2×#14519745C411yescharacteristic
34C2×#14519745C411yescharacteristic
35C2×C4×C22⋊D61C2211yescharacteristic
36C2×#14519737C2211yescharacteristic
37C2×C22⋊C4×D61C2211yescharacteristic
38C2×#121×C61C411yescharacteristic
39C2×#11758213—22no—
40C2×#11758213—22no—
41C2×#11758217—22no—
42C22×#92420—22no—
43#14519844—22no—
44C4⋊#92420—22no—
45#14519834—22no—
46#14519834—22no—
47C22⋊#92420—22no—
48C22⋊#92420—22no—
49C22⋊#92420C2314yes—
50C22⋊#92420C2314yes—
51#14519832—22no—
52C4×#92420—22no—
53C24×D61C2×C411yescharacteristic
54C22×Dic122C2×C411yescharacteristic
55C22×C22⋊D61C2×C411yescharacteristic
56C22×C22⋊D61C2×C411yescharacteristic
57C22×C22⋊D61C2×C411yescharacteristic
58C22×C22⋊D61C2×C411yescharacteristic
59C23×Dic61C2311yescharacteristic
60C22×C4×D61C2×C411yescharacteristic
61C22×C4×D61C2311yescharacteristic
62#14519772—22no—
63#14519772—22no—
64#14519770—24no—
65#14519764—22no—
66#14519764—22no—
67#14519763—24no—
68#14519760—24no—
69#14519758—22no—
70#14519758—22no—
71#14519755—24no—
72#14519749—22no—
73#14519749—22no—
74#14519748—24no—
75#14519745—24no—
76#14519745—24no—
77C4×C22⋊D61—22no—
78C4×C22⋊D61—22no—
79#14519737D414yes—
80#14519737D414yes—
81C2×C4⋊Dic61C2×C411yescharacteristic
82C2×#11758196C2×C411yescharacteristic
83C2×#11758196C2×C411yescharacteristic
84C2×#11758196C2×C411yescharacteristic
85C2×#11758196C2311yescharacteristic
86C2×C22⋊Dic61C2×C411yescharacteristic
87C2×C22⋊Dic61C2×C411yescharacteristic
88C2×C22⋊Dic61C2×C411yescharacteristic
89C2×C22⋊Dic61C2×C411yescharacteristic
90C2×C22⋊Dic61C2311yescharacteristic
91C2×C22⋊Dic61C2311yescharacteristic
92C2×#11758198C2×C411yescharacteristic
93C2×#11758198C2×C411yescharacteristic
94C2×#11758198C2×C411yescharacteristic
95C2×#11758198C2×C411yescharacteristic
96C2×#11758198C2×C411yescharacteristic
97C2×#11758198C2311yescharacteristic
98C22⋊C4×D61—22no—
99C22⋊C4×D61—22no—
100C2×C4×Dic61C2×C411yescharacteristic
101C2×C4×Dic61C2311yescharacteristic
102C22×D4×C61C2×C411yescharacteristic
103C22×D4×C61C2×C411yescharacteristic
104C22×D4×C61C2×C411yescharacteristic
105C23×C4×C61C2×C411yescharacteristic
106#121×C61—24no—
107C2×C4⋊C4×C61C2×C411yescharacteristic
108C2×C22⋊C4×C61C2×C411yescharacteristic
109C2×C22⋊C4×C61C2×C411yescharacteristic
110C24×C61C22×C411yescharacteristic
111C24×C61C22×C411yescharacteristic
112C24×C61C22×C411yescharacteristic
113C23×D61—22no—
114C23×D61—42no—
115C23×D61—81no—
116C23×D61—81no—
117C23×D61C22×C411yescharacteristic
118C23×D61C22×C411yescharacteristic
119C2×D4×C61—22no—
120C2×D4×C61—22no—
121C2×D4×C61—22no—
122C2×D4×C61—81no—
123C2×D4×C61—81no—
124C2×D4×C61—81no—
125C22×C4×C61—22no—
126C22×C4×C61—81no—
127C22×C4×C61C22×C411yescharacteristic
128C22×C4×C61C22×C411yescharacteristic
129C22×C4×C61C22×C411yescharacteristic
130C22×C4×C61C22×C411yescharacteristic
131C2×C22⋊D61—22no—
132C2×C22⋊D61—22no—
133C2×C22⋊D61—22no—
134C2×C22⋊D61—42no—
135C2×C22⋊D61—42no—
136C2×C22⋊D61—81no—
137C2×C22⋊D61—81no—
138C2×C22⋊D61C22⋊C414yes—
139C22×Dic61—81no—
140C22×Dic61C22×C411yescharacteristic
141C22×Dic61C22×C411yescharacteristic
142C22×Dic61C22×C411yescharacteristic
143C22×Dic61C22×C411yescharacteristic
144C22×Dic61C22×C411yescharacteristic
145C22×Dic61C2×D412yes—
146C22×Dic61C2×D412yes—
147C22×Dic61C2411yescharacteristic
148C2×C4×D61—22no—
149C2×C4×D61—81no—
150C2×C4×D61—81no—
151C2×C4×D61C2×D412yes—
152C2×C4×D61C2×D412yes—
153C2×Dic122—22no—
154C2×Dic122—81no—
155#11758217—82no—
156C2×#92420—24no—
157C2×#92420—24no—
158C2×#92420—82no—
159#11758213—82no—
160#11758213—82no—
161C4⋊C4×C61—22no—
162C22⋊C4×C61—22no—
163C22⋊C4×C61C22⋊C414yes—
164C22⋊Dic61—22no—
165C22⋊Dic61—22no—
166C22⋊Dic61—22no—
167C22⋊Dic61—22no—
168C22⋊Dic61—21no—
169C22⋊Dic61—21no—
170C22⋊Dic61—21no—
171C22⋊Dic61—21no—
172#11758198—22no—
173#11758198—22no—
174#11758198—22no—
175#11758198—22no—
176#11758198—21no—
177#11758198—21no—
178#11758198C22⋊C414yes—
179C4⋊Dic61—22no—
180#11758196—22no—
181#11758196—22no—
182#11758196—21no—
183#11758196—21no—
184#11758196C22⋊C414yes—
185C4×Dic61—22no—
186C4×Dic61C2×D412yes—
187C4×Dic61C2×D412yes—
188C23×C61—22no—
189C23×C61—21no—
190C23×C61—21no—
191C23×C61—22no—
192C23×C61—22no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61—41no—
199C23×C61C2×C22⋊C412yes—
200C23×C61C23×C411yescharacteristic
201C22×D61—22no—
202C22×D61—21no—
203C22×D61—22no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61—44no—
207C22×D61—41no—
208C22×D61—41no—
209C22×D61—82no—
210C22×D61—82no—
211C22×D61—88no—
212C22×D61C2×C22⋊C412yes—
213D4×C61—82no—
214D4×C61—82no—
215D4×C61—82no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—82no—
223C2×C4×C61C2×C22⋊C412yes—
224C2×C4×C61C2×C22⋊C412yes—
225C22⋊D61—44no—
226C22⋊D61—44no—
227C22⋊D61—82no—
228C22⋊D61—82no—
229C2×Dic61—22no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—22no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61C2×C22⋊C412yes—
241C2×Dic61C2×C22⋊C412yes—
242C2×Dic61C22×D411yescharacteristic
243C2×Dic61C22×D411yescharacteristic
244C4×D61—42no—
245C4×D61—42no—
246C4×D61—82no—
247Dic122—82no—
248#92420—44no—
249#92420—44no—
250#17405—611nomaximal, Sylow
251C22×C61—21no—
252C22×C61—21no—
253C22×C61—22no—
254C22×C61—22no—
255C22×C61—22no—
256C22×C61—41no—
257C22×C61—41no—
258C22×C61—41no—
259C22×C61—41no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61—82no—
263C22×C61—82no—
264C22×C61—82no—
265C22×C61C22×C22⋊C411yescharacteristic
266C22×C61#56812yes—
267C2×D61—22no—
268C2×D61—22no—
269C2×D61—22no—
270C2×D61—44no—
271C2×D61—41no—
272C2×D61—41no—
273C2×D61—41no—
274C2×D61—88no—
275C2×D61—82no—
276C4×C61—22no—
277C4×C61—22no—
278C4×C61—41no—
279C4×C61—41no—
280Dic61—22no—
281Dic61—21no—
282Dic61—21no—
283Dic61—22no—
284Dic61—41no—
285Dic61—41no—
286Dic61—41no—
287C2×#538—611no—
288#2975—611no—
289#2973—611no—
290#1473—612no—
291#1471—612no—
292#1469—612no—
293#1468—612no—
294#1465—612no—
295#1462—612no—
296C2×C61—22no—
297C2×C61—42no—
298C2×C61—81no—
299C2×C61—81no—
300C2×C61#287911yescharacteristic
301C2×C61#288311yescharacteristic
302C2×C61#288811yescharacteristic
303D61—42no—
304D61—81no—
305#538—1224no—
306#538—1224no—
307C2×#126—611no—
308C2×#126—611no—
309C2×#124—611no—
310C2×#124—611no—
311C2×#123—611no—
312C2×#123—611no—
313C2×C22⋊Q8—611no—
314C2×#121—611no—
315C2×#121—611no—
316C2×#121#692311yescharacteristic
317C2×#120—611no—
318C2×#120—611no—
319C2×C4×D4—611no—
320C2×C4×D4—611no—
321C22×C22⋊C4—611no—
322#436—1224no—
323#435—1224no—
324#432—612no—
325#431—612no—
326#407—612no—
327#407—612no—
328#407—612no—
329C2×C22⋊C8—612no—
330#324—614no—
331#323—614no—
332#323—614no—
333#323—614no—
334C61#1740511yescharacteristic, complemented, Sylow
335C25—611no—
336C22×Q8—611no—
337C22×D4—611no—
338C22×D4—611no—
339C22×D4—611no—
340C22×D4—611no—
341C22×D4C2×#692311yescharacteristic
342C22×D4C2×#692311yescharacteristic
343C22×D4C2×#692311yescharacteristic
344C23×C4—611no—
345C23×C4—611no—
346C23×C4—611no—
347C23×C4C2×#692311yescharacteristic
348C2×M16—1222no—
349C2×M16—1222no—
350C2×M16—1222no—
351C22×C8—1222no—
352#126—1222no—
353#126—1222no—
354#126—1224no—
355#124—1222no—
356#124—1222no—
357#124—1224no—
358#123—1224no—
359#123—1222no—
360#123—1222no—
361C22⋊Q8—1224no—
362#121—24no—
363#121—1222no—
364#121—1222no—
365#121—1224no—
366#120—1224no—
367#120—1224no—
368C4×D4—614no—
369C4×D4—614no—
370C4×D4—1222no—
371C4×D4—1222no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4C2×#692311yescharacteristic
378C2×C22⋊C4—611no—
379C2×C22⋊C4—611no—
380C2×C22⋊C4—611no—
381C2×C22⋊C4—611no—
382C2×C22⋊C4—611no—
383C2×C22⋊C4—611no—
384C2×C22⋊C4—611no—
385C2×C22⋊C4—611no—
386C2×C22⋊C4—611no—
387C2×C22⋊C4—611no—
388C2×C22⋊C4—611no—
389C2×C22⋊C4—611no—
390C2×C22⋊C4—1222no—
391C2×C22⋊C4—1222no—
392C2×C22⋊C4C2×#692311yescharacteristic
393C2×C22⋊C4C2×#692311yescharacteristic
394C2×C42—611no—
395C2×C42—611no—
396C4⋊C8—1222no—
397C4⋊C8—1222no—
398C22⋊C8—614no—
399C22⋊C8—614no—
400C22⋊C8—1222no—
401C22⋊C8—1222no—
402C22⋊C8—1222no—
403C22⋊C8—1222no—
404#97—1222no—
405C4×C8—1222no—
406C24—611no—
407C24—611no—
408C24—1222no—
409C24—4881no—
410C24—4881no—
411C24—2442no—
412C24C22×#692311yescharacteristic
413C24C22×#692311yescharacteristic
414C24C22×#692311yescharacteristic
415C2×Q8—1222no—
416C2×Q8—4881no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—81no—
421C2×D4—81no—
422C2×D4—81no—
423C2×D4—614no—
424C2×D4—1222no—
425C2×D4—1222no—
426C2×D4—1222no—
427C2×D4—4881no—
428C2×D4—4881no—
429C2×D4—2442no—
430C2×D4—2442no—
431C22×C4—22no—
432C22×C4—81no—
433C22×C4—611no—
434C22×C4—612no—
435C22×C4—612no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—612no—
439C22×C4—612no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—1222no—
444C22×C4—4881no—
445C22×C4—4881no—
446C22×C4—4881no—
447C22×C4C22×#692311yescharacteristic
448C22×C4C22×#692311yescharacteristic
449C22×C4C22×#692311yescharacteristic
450C22×C4C22×#692311yescharacteristic
451M16—4882no—
452M16—4882no—
453M16—4882no—
454C2×C8—1224no—
455C2×C8—1224no—
456C2×C8—4882no—
457C4⋊C4—22no—
458C4⋊C4—614no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—1222no—
462C4⋊C4—1222no—
463C4⋊C4—1222no—
464C22⋊C4—22no—
465C22⋊C4—614no—
466C22⋊C4—1221no—
467C22⋊C4—1221no—
468C22⋊C4—1222no—
469C22⋊C4—1222no—
470C22⋊C4—1221no—
471C22⋊C4—1221no—
472C22⋊C4—1221no—
473C22⋊C4—1221no—
474C22⋊C4—1222no—
475C22⋊C4—1222no—
476C22⋊C4—1222no—
477C22⋊C4—1222no—
478C22⋊C4—1222no—
479C22⋊C4—1222no—
480C22⋊C4C22⋊#692314yes—
481C42—612no—
482C42—612no—
483C42—1222no—
484C23—22no—
485C23—21no—
486C23—21no—
487C23—22no—
488C23—22no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—612no—
496C23—1222no—
497C23—1222no—
498C23—1221no—
499C23—4882no—
500C23—4882no—
501C23—4888no—
502C23—2441no—
503C23—2441no—
504C23—2441no—
505C23—2441no—
506C23—2444no—
507C23C2×C22⋊#692312yes—
508C23C23×#692311yescharacteristic
509Q8—4882no—
510D4—82no—
511D4—82no—
512D4—82no—
513D4—4882no—
514D4—4882no—
515D4—2444no—
516D4—2444no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—82no—
524C2×C4—612no—
525C2×C4—612no—
526C2×C4—611no—
527C2×C4—611no—
528C2×C4—1222no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1222no—
533C2×C4—4882no—
534C2×C4—2442no—
535C2×C4—2442no—
536C2×C4—2442no—
537C2×C4—2442no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4C2×C22⋊#692312yes—
543C2×C4C2×C22⋊#692312yes—
544C8—2444no—
545C8—2444no—
546C22—21no—
547C22—21no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—1222no—
561C22—1222no—
562C22—1222no—
563C22—4882no—
564C22—4888no—
565C22—2444no—
566C22—2441no—
567C22—2441no—
568C22—2441no—
569C22C22×C22⋊#692311yescharacteristic
570C22#1684084112yes—
571C4—22no—
572C4—22no—
573C4—41no—
574C4—41no—
575C4—1222no—
576C4—1221no—
577C4—1221no—
578C4—1222no—
579C4—2441no—
580C4—2441no—
581C4—2441no—
582C2—22no—
583C2—42no—
584C2—81no—
585C2—81no—
586C2—4881no—
587C2—2442no—
588C2#2146240411yescharacteristic
589C2#2146241811yescharacteristic
590C2#2146242911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×#121×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840344C2
cosocleC22×Dic61C24
hypercenterC2×#121#6923
hypocenterC61#17405
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #21462683C2semidirect
2. #21462672C2semidirect
3. #21461102C2semidirect
4. #21461095C2semidirect
5. #21461089C2semidirect
6. #21461084C2semidirect
7. #21461074C2semidirect
8. #21461065C2semidirect
12. #16840783C22semidirect
13. #16840779C22semidirect
16. #16840732C22semidirect
17. C2×C22⋊#92420C22semidirect
18. #16840608C22semidirect
21. #16840605C22semidirect
334. C61#17405semidirect

Maximal subgroups

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

subgroupindex
#214626832
#214626722
#214611022
#214610952
#214610892
#214610842
#214610742
#214610652
C2×#168403442
#1740561