Finite Group Databasebeta

← #29653883

Subgroups — #29653883

Order 15616.

972 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840359C211yesmaximal, characteristic
2#21456591C211yesmaximal, characteristic, complemented
3#21456572C211yesmaximal, characteristic
4#21444848C214yesmaximal, complemented
5#21444780C214yesmaximal, complemented
6#21444390C211yesmaximal, characteristic, complemented
7#21444386C211yesmaximal, characteristic
8C2×#16839488C211yesmaximal, characteristic, complemented
9C61×#2627C211yesmaximal, characteristic, complemented
10#16840359—22no—
11#16840359—22no—
12#16840359—22no—
13#16840359C2214yes—
14C2×#14519772—21no—
15C2×#14519767—21no—
16C2×#14519762—21no—
17C2×#14519760—21no—
18C2×#14519756C2211yescharacteristic
19C2×#14519755—21no—
20C2×#14519751C2211yescharacteristic
21C2×C4×C22⋊D61—21no—
22#16840097C2212yescomplemented
23#16840095—21no—
24#16840090—21no—
25#16840080C2212yes—
26#16840001—22no—
27#16839993—22no—
28#16839918C2212yescomplemented
29#16839917—21no—
30#16839912—21no—
31#16839901C2212yes—
32C2×C22⋊#92418C2211yescharacteristic
33C2×C22⋊#92418C2211yescharacteristic
34#16839498C2214yescomplemented
35#16839495C2214yes—
36#16839488C2214yescomplemented
37C2×C22⋊Q8×C61C2211yescharacteristic
38C22×C4⋊C4×C61C2211yescharacteristic
39C61×#400C2212yescomplemented
40C61×#398—22no—
41C61×#395C2212yescomplemented
42C61×#385—21no—
43C61×#381—21no—
44C61×#380C2211yescharacteristic
45C22×Dic122D411yescharacteristic
46C22×D244D411yescharacteristic
47C22×C22⋊D61—21no—
48C22×C4×D61—21no—
49C2×#11758202—22no—
50C2×#11758201—22no—
51C22×#92418—21no—
52C22×#92418—21no—
53#14519772—24no—
54#14519772—24no—
55#14519767—24no—
56#14519767—24no—
57#14519762—24no—
58#14519762—24no—
59#14519760—24no—
60#14519760—24no—
61#14519756—21no—
62#14519756—21no—
63#14519756D412yes—
64#14519756C2312yes—
65#14519755—24no—
66#14519755—24no—
67#14519751—21no—
68#14519751—21no—
69#14519751D412yes—
70#14519751C2312yes—
71C4×C22⋊D61—24no—
72C4×C22⋊D61—24no—
73C2×C4⋊Dic61D411yescharacteristic
74C2×C4⋊Dic61D411yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×C22⋊Dic61—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×C4×Dic61—21no—
85#14519710—22no—
86#14519708—22no—
87#14519701—22no—
88#14519701—22no—
89#14519700—22no—
90#14519697—22no—
91C22⋊#92418—21no—
92C22⋊#92418—21no—
93C22⋊#92418C2312yes—
94C22⋊#92418C2312yes—
95C22×Q8×C61C2311yescharacteristic
96C23×C4×C61D411yescharacteristic
97C23×C4×C61D411yescharacteristic
98C23×C4×C61C2311yescharacteristic
99C22⋊Q8×C61—21no—
100C22⋊Q8×C61—21no—
101C22⋊Q8×C61—21no—
102C22⋊Q8×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—22no—
110C2×C4⋊C4×C61D412yes—
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2312yes—
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61—21no—
115#95×C61—22no—
116#95×C61—22no—
117#95×C61—22no—
118#95×C61C2312yes—
119C24×C61C2×D411yescharacteristic
120C23×D61—21no—
121C2×Q8×C61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61—81no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—22no—
132C22×C4×C61—21no—
133C22×C4×C61—81no—
134C22×C4×C61—81no—
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2×D412yes—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2411yescharacteristic
143C2×C22⋊D61—22no—
144C2×C22⋊D61—22no—
145C2×C22⋊D61—42no—
146C2×C22⋊D61—42no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C2×D244—21no—
151C2×D244—81no—
152C2×D244C2×D412yes—
153C2×C4×D61—24no—
154C2×C4×D61—82no—
155C2×Dic122—21no—
156C2×Dic122—81no—
157C2×Dic122C2×D412yes—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—41no—
165C4⋊C4×C61—41no—
166C4⋊C4×C61—81no—
167C4⋊C4×C61—82no—
168C22⋊C4×C61—41no—
169C22⋊C4×C61—41no—
170C22⋊C4×C61—42no—
171C22⋊Dic61—22no—
172C22⋊Dic61—22no—
173#11758202—82no—
174#11758201—82no—
175#11758198—24no—
176#11758198—24no—
177#11758198—24no—
178#11758198—22no—
179#11758198—22no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61C2×D412yes—
183C4⋊Dic61C2×D412yes—
184#11758196—24no—
185#11758196—24no—
186#11758196—24no—
187#11758196—22no—
188#11758196—22no—
189C4×Dic61—24no—
190C2×#92418—22no—
191C2×#92418—22no—
192C2×#92418—81no—
193C2×#92418—81no—
194C23×C61—41no—
195C23×C61—41no—
196C23×C61#12011yescharacteristic
197C23×C61#12011yescharacteristic
198C23×C61#13512yes—
199C23×C61#13712yes—
200C23×C61C22×D411yescharacteristic
201C22×D61—22no—
202C22×D61—42no—
203C22×D61—42no—
204C22×D61—42no—
205Q8×C61—42no—
206Q8×C61—42no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—42no—
228C2×C4×C61—81no—
229C2×C4×C61#12011yescharacteristic
230C2×C4×C61#12011yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C22⋊D61—42no—
234C22⋊D61—42no—
235C22⋊D61—44no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242D244—42no—
243D244—81no—
244C4×D61—84no—
245Dic122—42no—
246Dic122—81no—
247#92418—42no—
248#92418—42no—
249#18289—611nomaximal, Sylow
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—21no—
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—21no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61#45212yes—
259C22×C61#45212yes—
260C22×C61C2×#12011yescharacteristic
261C22×C61C2×#13511yescharacteristic
262C22×C61C2×#13711yescharacteristic
263C2×D61—42no—
264C2×D61—42no—
265C2×D61—42no—
266C2×D61—84no—
267C4×C61—21no—
268C4×C61—21no—
269C4×C61—21no—
270C4×C61—21no—
271C4×C61—42no—
272C4×C61—42no—
273C4×C61—82no—
274C4×C61—81no—
275C4×C61—81no—
276Dic61—42no—
277Dic61—42no—
278Dic61—42no—
279C2×#539—611no—
280#3189—611no—
281#3187—611no—
282#2627D6111yescharacteristic, complemented
283#1615—614no—
284#1603—614no—
285#1520—611no—
286#1519—611no—
287C2×#324—611no—
288C2×C61—21no—
289C2×C61—22no—
290C2×C61—21no—
291C2×C61—22no—
292C2×C61C2×#45211yescharacteristic
293C2×C61#302011yescharacteristic
294C2×C61#302611yescharacteristic
295D61—82no—
296#539—614no—
297#539—1222no—
298#539—1222no—
299#539—1222no—
300C2×#126—1221no—
301C2×#125—1221no—
302C2×#123—1221no—
303C2×#123—1221no—
304C2×C22⋊Q8—611no—
305C2×C22⋊Q8—1221no—
306C2×C22⋊Q8C2×D6111yescharacteristic
307C2×#121—611no—
308C2×C4×D4—1221no—
309C22×C4⋊C4C2×D6111yescharacteristic
310#484—612no—
311#484—1221no—
312#483—1221no—
313#481—612no—
314#467—1222no—
315#465—1222no—
316#451—612no—
317#451—1221no—
318#450—1221no—
319#448—612no—
320C2×C22⋊C8—611no—
321C2×C22⋊C8—611no—
322#400C2×D6112yescomplemented
323#398—22no—
324#395C2×D6112yescomplemented
325#385—21no—
326#381—21no—
327#380C2×D6111yescharacteristic
328#328—614no—
329#327—614no—
330#324—614no—
331C61#1828911yescharacteristic, complemented, Sylow
332C22×Q8—611no—
333C22×Q8C22×D6111yescharacteristic
334C22×D4—611no—
335C22×D4—1221no—
336C23×C4—1221no—
337C23×C4C22⋊D6111yescharacteristic
338C23×C4C22⋊D6111yescharacteristic
339C23×C4C22×D6111yescharacteristic
340C2×Q16—1222no—
341C2×SD16—1222no—
342C22×C8—1221no—
343C22×C8—1221no—
344#126—1224no—
345#126—1224no—
346#125—1224no—
347#125—1224no—
348#123—1224no—
349#123—1224no—
350#123—1224no—
351#123—1224no—
352C22⋊Q8—21no—
353C22⋊Q8—21no—
354C22⋊Q8—21no—
355C22⋊Q8—21no—
356C22⋊Q8—612no—
357C22⋊Q8—612no—
358C22⋊Q8—1224no—
359C22⋊Q8—1224no—
360C22⋊Q8—1221no—
361C22⋊Q8—1221no—
362#121—612no—
363#121—612no—
364#121—1221no—
365#121—1221no—
366C4×D4—1224no—
367C4×D4—1224no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4C22⋊D6112yes—
382C2×C4⋊C4C22×D6111yescharacteristic
383C2×C4⋊C4C22×D6112yes—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—1221no—
387C2×C22⋊C4—1221no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C22⋊C4—1221no—
391C2×C42—1221no—
392#107—1222no—
393#106—1222no—
394#103—1222no—
395#103—1222no—
396#103—1222no—
397#102—1222no—
398C22⋊C8—612no—
399C22⋊C8—612no—
400C22⋊C8—1221no—
401C22⋊C8—1221no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95C22×D6112yes—
406C24—1221no—
407C24C2×C22⋊D6111yescharacteristic
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—81no—
411C2×Q8—612no—
412C2×Q8—1221no—
413C2×Q8—4881no—
414C2×D4—612no—
415C2×D4—1222no—
416C2×D4—1222no—
417C2×D4—1221no—
418C2×D4—4881no—
419C2×D4—2442no—
420C2×D4—2442no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—81no—
431C22×C4—81no—
432C22×C4—1224no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—4882no—
437C22×C4D4×D6111yescharacteristic
438C22×C4D4×D6111yescharacteristic
439C22×C4D4×D6111yescharacteristic
440C22×C4D4×D6111yescharacteristic
441C22×C4C2×C22⋊D6112yes—
442C22×C4C2×C22⋊D6112yes—
443C22×C4C2×C22⋊D6111yescharacteristic
444C22×C4C23×D6111yescharacteristic
445Q16—4882no—
446SD16—4882no—
447C2×C8—1222no—
448C2×C8—1222no—
449C2×C8—4881no—
450C2×C8—4881no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—81no—
460C4⋊C4—82no—
461C4⋊C4—612no—
462C4⋊C4—612no—
463C4⋊C4—1224no—
464C4⋊C4—1224no—
465C4⋊C4—1224no—
466C4⋊C4—1222no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1222no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C22⋊C4—42no—
473C22⋊C4—1224no—
474C22⋊C4—1224no—
475C22⋊C4—1224no—
476C22⋊C4—1222no—
477C22⋊C4—1222no—
478C22⋊C4—1222no—
479C22⋊C4—1222no—
480C42—1224no—
481C23—41no—
482C23—41no—
483C23—1222no—
484C23—2442no—
485C23—2442no—
486C23—2442no—
487C23#1451974511yescharacteristic
488C23#1451974511yescharacteristic
489C23#1451979712yes—
490C23#1451980812yes—
491C23C22×C22⋊D6111yescharacteristic
492Q8—42no—
493Q8—42no—
494Q8—4881no—
495Q8—2442no—
496D4—4881no—
497D4—2442no—
498D4—2442no—
499D4—2442no—
500D4—2444no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—81no—
523C2×C4—1222no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—4884no—
527C2×C4—2442no—
528C2×C4—2442no—
529C2×C4—2442no—
530C2×C4#1451974511yescharacteristic
531C2×C4#1451974511yescharacteristic
532C2×C4C2×D4×D6111yescharacteristic
533C2×C4C2×D4×D6111yescharacteristic
534C8—2442no—
535C8—2442no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—42no—
543C22—42no—
544C22—4884no—
545C22—2442no—
546C22—2442no—
547C22—2442no—
548C22#1683992212yes—
549C22#1683992212yes—
550C22C2×#1451974511yescharacteristic
551C22C2×#1451979711yescharacteristic
552C22C2×#1451980811yescharacteristic
553C4—21no—
554C4—21no—
555C4—21no—
556C4—21no—
557C4—42no—
558C4—42no—
559C4—82no—
560C4—81no—
561C4—81no—
562C4—2442no—
563C4—2442no—
564C4—2442no—
565C2—21no—
566C2—22no—
567C2—21no—
568C2—22no—
569C2—4882no—
570C2C2×#1683992211yescharacteristic
571C2#2145518211yescharacteristic
572C2#2145521511yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2627C2
socleC22×C61C2×#120
prime radicalC2×#14519751C22
hypercenter#2627D61
hypocenterC61#18289
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2627

Complements

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

NH ≅ G/Ntype
2. #21456591C2semidirect
4. #21444848C2semidirect
5. #21444780C2semidirect
6. #21444390C2semidirect
8. C2×#16839488C2semidirect
9. C61×#2627C2semidirect
22. #16840097C22semidirect
28. #16839918C22semidirect
34. #16839498C22semidirect
36. #16839488C22semidirect
39. C61×#400C22semidirect
41. C61×#395C22semidirect
282. #2627D61semidirect
322. #400C2×D61semidirect
324. #395C2×D61semidirect
331. C61#18289semidirect

Maximal subgroups

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

subgroupindex
C2×#168403592
#214565912
#214565722
#214448482
#214447802
#214443902
#214443862
C2×#168394882
C61×#26272
#1828961