Finite Group Databasebeta

← #29677067

Subgroups — #29677067

Order 15616.

700 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458835C211yesmaximal, characteristic, complemented
2#21458052C211yesmaximal, characteristic, complemented
3#21457419C211yesmaximal, characteristic, complemented
4#21445410C211yesmaximal, characteristic, complemented
5#21445380C211yesmaximal, characteristic, complemented
6#21445377C211yesmaximal, characteristic, complemented
7#21445293C211yesmaximal, characteristic, complemented
8#21445282C211yesmaximal, characteristic, complemented
9#21445074C211yesmaximal, characteristic, complemented
10#21445054C211yesmaximal, characteristic, complemented
11#21444884C211yesmaximal, characteristic, complemented
12#21444872C211yesmaximal, characteristic, complemented
13#21444710C211yesmaximal, characteristic, complemented
14#21444702C211yesmaximal, characteristic, complemented
15C61×#3332C211yesmaximal, characteristic, complemented
16C2×#14519802—21no—
17#16840485C2211yescharacteristic
18D4×D244C2211yescharacteristic
19#16840165C2211yescharacteristic, complemented
20#16840149—21no—
21#16840138C2211yescharacteristic
22#16840136C2211yescharacteristic
23#16840134C2211yescharacteristic, complemented
24#16840109C2211yescharacteristic
25#16840088—21no—
26#16840042C2211yescharacteristic
27#16840011—21no—
28#16840010—21no—
29#16839977—21no—
30#16839967C2211yescharacteristic, complemented
31#16839964C2211yescharacteristic
32#16839962C2211yescharacteristic
33#16839959—21no—
34#16839957C2211yescharacteristic
35#16839954C2211yescharacteristic
36#16839912—21no—
37#16839911—21no—
38#16839907—21no—
39#16839893C2211yescharacteristic, complemented
40#16839862—21no—
41C4×#11758199C2211yescharacteristic
42#16839841C2211yescharacteristic
43#16839840C2211yescharacteristic
44#16839765—21no—
45#16839514C2211yescharacteristic, complemented
46#16839513C2211yescharacteristic, complemented
47#16839508C2211yescharacteristic, complemented
48#16839507C2211yescharacteristic, complemented
49#16839506C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic, complemented
54#16839489C2211yescharacteristic
55C8×#92418C2211yescharacteristic, complemented
56C61×#550C2211yescharacteristic, complemented
57C61×#547C2211yescharacteristic
58C61×#486C2211yescharacteristic, complemented
59C61×#481—21no—
60C61×#475C2211yescharacteristic, complemented
61C61×#469—21no—
62C61×#459C2211yescharacteristic
63C61×#449—21no—
64C4×SD16×C61C2211yescharacteristic
65C61×#436C2211yescharacteristic, complemented
66C61×#417—21no—
67C22×D244—21no—
68C2×D4×D61—21no—
69#14519802—44no—
70C2×#11758200—21no—
71C2×#11758200—21no—
72C2×#11758200—21no—
73C2×#11758201—21no—
74C2×#11758201—21no—
75C2×#11758201—21no—
76C2×#11758201—21no—
77C2×#11758199—21no—
78C2×#11758199—21no—
79C2×#11758199—41no—
80C2×#11758199D411yescharacteristic
81C2×#11758194—21no—
82C2×#11758189—21no—
83#14519777C2311yescharacteristic
84#14519774C2311yescharacteristic
85#14519772—41no—
86#14519765—21no—
87#14519753—21no—
88#14519752—21no—
89#14519751—21no—
90#14519747—41no—
91C4×Dic122—21no—
92C4×D244C2311yescharacteristic
93#14519735—21no—
94#14519710—21no—
95#14519710—41no—
96#14519710D411yescharacteristic
97C4⋊#92418—21no—
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100C4⋊#92418C2311yescharacteristic
101#14519701—21no—
102#14519700—21no—
103#14519699—21no—
104#14519699—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698D411yescharacteristic
108#14519697—21no—
109#14519697—21no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112#14519689—21no—
113#14519688—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C2×#41×C61—21no—
117C2×#41×C61—21no—
118C2×SD16×C61D411yescharacteristic
119C2×D8×C61—21no—
120C2×M16×C61—21no—
121C4⋊Q8×C61C2311yescharacteristic
122#127×C61—21no—
123#127×C61C2311yescharacteristic
124#123×C61—41no—
125C22⋊Q8×C61—21no—
126#121×C61—21no—
127#121×C61—21no—
128#121×C61—41no—
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61C2311yescharacteristic
132C4×D4×C61C2311yescharacteristic
133C2×C4⋊C4×C61—21no—
134#106×C61—21no—
135#106×C61D411yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137#103×C61—21no—
138#103×C61D411yescharacteristic
139#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—41no—
146#41×C61—42no—
147#41×C61—42no—
148#41×C61—81no—
149#41×C61—81no—
150C2×Q8×C61—21no—
151C2×Q8×C61C2×D411yescharacteristic
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×D4×C61—41no—
156C2×D4×C61—41no—
157C2×D4×C61—41no—
158C2×D4×C61C2×D411yescharacteristic
159C2×D4×C61C2×D411yescharacteristic
160C22×C4×C61—21no—
161C22×C4×C61—21no—
162C22×C4×C61—41no—
163C22×C4×C61—41no—
164C2×C22⋊D61—41no—
165D4×D61—42no—
166D4×D61—81no—
167C2×D244—21no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244—42no—
171C2×D244—41no—
172C2×D244—81no—
173C2×D244C2×D411yescharacteristic
174C2×C4×D61—21no—
175C2×Dic122—21no—
176SD16×C61—41no—
177D8×C61—81no—
178M16×C61—81no—
179C2×C8×C61—21no—
180C2×C8×C61C2×D411yescharacteristic
181C2×C8×C61C2×D411yescharacteristic
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—41no—
186C4⋊C4×C61C2×D411yescharacteristic
187C4⋊C4×C61C2×D411yescharacteristic
188C4⋊C4×C61C2×D411yescharacteristic
189C22⋊C4×C61—21no—
190C22⋊C4×C61—21no—
191C22⋊C4×C61—41no—
192C22⋊C4×C61—41no—
193C42×C61—21no—
194C42×C61C2411yescharacteristic
195#11758201—42no—
196#11758201—42no—
197#11758201—81no—
198#11758201—81no—
199#11758200—42no—
200#11758200—81no—
201#11758200—81no—
202#11758199—22no—
203#11758199—42no—
204#11758199—82no—
205#11758199—81no—
206#11758198—21no—
207#11758198—41no—
208#11758198—41no—
209#11758198—41no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61C2×D411yescharacteristic
213#11758196—41no—
214C4×Dic61—41no—
215#11758194—42no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418C2×D411yescharacteristic
221C2×#92418C2×D411yescharacteristic
222#11758189—81no—
223C23×C61—21no—
224C23×C61—21no—
225C23×C61—41no—
226C22×D61—21no—
227C22×D61—41no—
228C22×D61—82no—
229C22×D61—82no—
230C22×D61—161no—
231Q8×C61—21no—
232Q8×C61—21no—
233Q8×C61—41no—
234Q8×C61—41no—
235D4×C61—21no—
236D4×C61—21no—
237D4×C61—21no—
238D4×C61—21no—
239D4×C61—41no—
240D4×C61—41no—
241D4×C61—41no—
242D4×C61—41no—
243D4×C61—81no—
244D4×C61—82no—
245D4×C61—82no—
246D4×C61—161no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—82no—
257C2×C4×C61—82no—
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61H(5,𝔽2)11yescharacteristic
261C22⋊D61—82no—
262C2×Dic61—21no—
263C2×Dic61—41no—
264D244—21no—
265D244—21no—
266D244—41no—
267D244—41no—
268D244—41no—
269D244—41no—
270D244—81no—
271D244—82no—
272C4×D61—41no—
273C4×D61—41no—
274Dic122—41no—
275Dic122—41no—
276C8×C61—21no—
277C8×C61—21no—
278C8×C61—41no—
279#92418—22no—
280#92418—21no—
281#92418—41no—
282#92418—41no—
283#92418—41no—
284#92418AGL(1,ℤ/8)12yes—
285#20303—611nomaximal, Sylow
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61—81no—
293C22×C61—81no—
294C22×C61—81no—
295C22×C61D4211yescharacteristic
296C2×D61—41no—
297C2×D61—41no—
298C2×D61—41no—
299C2×D61—81no—
300C2×D61—81no—
301C2×D61—81no—
302C2×D61—164no—
303C4×C61—21no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61—41no—
307C4×C61—41no—
308C4×C61—81no—
309C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
310C4×C61#57511yescharacteristic
311C4×C61#57611yescharacteristic
312C4×C61#57611yescharacteristic
313Dic61—41no—
314Dic61—81no—
315#3397—611no—
316#3332D6111yescharacteristic, complemented
317#3324—611no—
318#3272—611no—
319#1706—611no—
320#1701—611no—
321#1701—611no—
322#1686—611no—
323#1685—611no—
324#1650—611no—
325#1647—611no—
326#1621—611no—
327#1619—611no—
328#1590—611no—
329#1589—611no—
330C2×C61—81no—
331C2×C61—81no—
332C2×C61—161no—
333C2×C61#327311yescharacteristic
334C2×C61#327411yescharacteristic
335C2×C61#329111yescharacteristic
336D61—81no—
337D61—161no—
338C2×AGL(1,ℤ/8)—1221no—
339#556—611no—
340#550C2×D6111yescharacteristic, complemented
341#547C2×D6111yescharacteristic
342D42—611no—
343#500—611no—
344#496—1221no—
345#493—611no—
346#492—611no—
347#492—611no—
348#486—611no—
349#486C2×D6111yescharacteristic, complemented
350#482—1221no—
351#481—21no—
352#475C2×D6111yescharacteristic, complemented
353#474—611no—
354#469—21no—
355#468—1221no—
356#468—1221no—
357#463—1221no—
358#461—611no—
359#461—611no—
360#460—611no—
361#460—1221no—
362#459—611no—
363#459—611no—
364#459C2×D6111yescharacteristic
365#450—1221no—
366#450—1221no—
367#449—21no—
368#449—1221no—
369#446—611no—
370#440—1221no—
371C4×SD16C2×D6111yescharacteristic
372C4×D8—611no—
373#436—611no—
374#436—611no—
375#436C2×D6111yescharacteristic, complemented
376#418—1221no—
377#417—21no—
378#335—611no—
379#334—611no—
380#332—611no—
381#332—611no—
382#331—611no—
383#331—611no—
384Q8⋊C8—611no—
385D4⋊C8—611no—
386D4⋊C8—611no—
387D4⋊C8—611no—
388C82—611no—
389C61#2030311yescharacteristic, complemented, Sylow
390C2×#41—21no—
391C2×#41—21no—
392C22×D4—1221no—
393C22×D4—1221no—
394AGL(1,ℤ/8)—2444no—
395C2×SD16—1221no—
396C2×SD16—1221no—
397C2×SD16—1221no—
398C2×SD16—1221no—
399C2×SD16—1221no—
400C2×SD16—1221no—
401C2×SD16—1221no—
402C2×SD16C22⋊D6111yescharacteristic
403C2×D8—21no—
404C2×D8—611no—
405C2×D8—1221no—
406C2×D8—1221no—
407C2×D8—2441no—
408C2×M16—21no—
409C2×M16—1221no—
410C2×M16—1221no—
411C4⋊Q8—611no—
412C4⋊Q8C22×D6111yescharacteristic
413#127—21no—
414#127—611no—
415#127C22×D6111yescharacteristic
416#126—2441no—
417#124—1221no—
418#123—41no—
419C22⋊Q8—21no—
420C22⋊Q8—1221no—
421#121—21no—
422#121—21no—
423#121—41no—
424#121—1221no—
425#121—1221no—
426#120—2441no—
427C4×Q8—1221no—
428C4×Q8C22×D6111yescharacteristic
429C4×D4—21no—
430C4×D4—611no—
431C4×D4C22×D6111yescharacteristic
432C4×D4C22×D6111yescharacteristic
433#117—1221no—
434C2×C4⋊C4—21no—
435#107—611no—
436#107—1221no—
437#107—2441no—
438#106—21no—
439#106C22⋊D6111yescharacteristic
440C4⋊C8—611no—
441C4⋊C8—611no—
442C4⋊C8—611no—
443C4⋊C8—1221no—
444C4⋊C8C22×D6111yescharacteristic
445#103—21no—
446#103—1221no—
447#103—1221no—
448#103C22⋊D6111yescharacteristic
449#102—21no—
450#102—21no—
451#102—21no—
452#102—611no—
453#102—611no—
454#102—1221no—
455#102—1221no—
456#102—1221no—
457#102—1221no—
458#102—1221no—
459#102—1221no—
460#102C22⋊D6111yescharacteristic
461C22⋊C8—21no—
462C22⋊C8—1221no—
463C22⋊C8—1221no—
464#97—1221no—
465C4×C8—611no—
466C4×C8—611no—
467C4×C8C22×D6111yescharacteristic
468C24—2441no—
469#41—42no—
470#41—42no—
471#41—81no—
472#41—81no—
473C2×Q8—21no—
474C2×Q8—1221no—
475C2×Q8C2×C22⋊D6111yescharacteristic
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—41no—
480C2×D4—41no—
481C2×D4—41no—
482C2×D4—611no—
483C2×D4—1221no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—4881no—
487C2×D4—4881no—
488C2×D4—2441no—
489C2×D4—2442no—
490C2×D4—2441no—
491C2×D4—2442no—
492C2×D4D4×D6111yescharacteristic
493C2×D4C2×C22⋊D6111yescharacteristic
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—41no—
497C22×C4—41no—
498C22×C4—1221no—
499SD16—41no—
500SD16—4881no—
501SD16—4881no—
502SD16—4881no—
503SD16—4881no—
504SD16—2442no—
505SD16—2442no—
506SD16—2442no—
507D8—81no—
508D8—1222no—
509D8—4882no—
510D8—4881no—
511D8—2442no—
512M16—81no—
513M16—4881no—
514M16—2442no—
515C2×C8—21no—
516C2×C8—611no—
517C2×C8—611no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8C2×C22⋊D6111yescharacteristic
523C2×C8C2×C22⋊D6111yescharacteristic
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—41no—
528C4⋊C4—611no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—2441no—
532C4⋊C4D4×D6111yescharacteristic
533C4⋊C4C2×C22⋊D6111yescharacteristic
534C4⋊C4C2×C22⋊D6111yescharacteristic
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—41no—
538C22⋊C4—41no—
539C22⋊C4—1221no—
540C22⋊C4—2441no—
541C22⋊C4—2441no—
542C22⋊C4—2441no—
543C42—21no—
544C42—2441no—
545C42C23×D6111yescharacteristic
546C23—21no—
547C23—21no—
548C23—41no—
549C23—1221no—
550C23—9761no—
551C23—4882no—
552C23—4882no—
553C23—2441no—
554Q8—21no—
555Q8—21no—
556Q8—41no—
557Q8—41no—
558Q8—2441no—
559Q8—2441no—
560D4—21no—
561D4—21no—
562D4—21no—
563D4—21no—
564D4—41no—
565D4—41no—
566D4—41no—
567D4—41no—
568D4—81no—
569D4—82no—
570D4—82no—
571D4—161no—
572D4—1221no—
573D4—1221no—
574D4—4881no—
575D4—4882no—
576D4—4882no—
577D4—2441no—
578D4—2441no—
579D4—2441no—
580D4—2441no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—1221no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4C2×D4×D6111yescharacteristic
597C2×C4C22×C22⋊D6111yescharacteristic
598C2×C4#1451982211yescharacteristic
599C8—21no—
600C8—21no—
601C8—41no—
602C8—612no—
603C8—1222no—
604C8—1221no—
605C8—2441no—
606C8—2441no—
607C8—2441no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—9764no—
618C22—4881no—
619C22—4881no—
620C22—4881no—
621C22—2441no—
622C22—2441no—
623C22—2441no—
624C22D4×C22⋊D6111yescharacteristic
625C4—21no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—81no—
631C4—4881no—
632C4—2441no—
633C4AGL(1,ℤ/8)×D6111yescharacteristic
634C4#1684056011yescharacteristic
635C4#1684056511yescharacteristic
636C4#1684056611yescharacteristic
637C2—81no—
638C2—81no—
639C2—161no—
640C2—9761no—
641C2—4881no—
642C2#2145742611yescharacteristic
643C2#2145743311yescharacteristic
644C2#2145761011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3332C2
socleC22×C61D42
hypercenter#3332D61
hypocenterC61#20303
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3332

Complements

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

NH ≅ G/Ntype
1. #21458835C2semidirect
2. #21458052C2semidirect
3. #21457419C2semidirect
4. #21445410C2semidirect
5. #21445380C2semidirect
6. #21445377C2semidirect
7. #21445293C2semidirect
8. #21445282C2semidirect
9. #21445074C2semidirect
10. #21445054C2semidirect
11. #21444884C2semidirect
12. #21444872C2semidirect
13. #21444710C2semidirect
14. #21444702C2semidirect
15. C61×#3332C2semidirect
19. #16840165C22semidirect
23. #16840134C22semidirect
30. #16839967C22semidirect
39. #16839893C22semidirect
45. #16839514C22semidirect
46. #16839513C22semidirect
47. #16839508C22semidirect
48. #16839507C22semidirect
49. #16839506C22semidirect
51. Q8⋊#92418C22semidirect
53. D4⋊#92418C22semidirect
55. C8×#92418C22semidirect
56. C61×#550C22semidirect
58. C61×#486C22semidirect
60. C61×#475C22semidirect
65. C61×#436C22semidirect
316. #3332D61semidirect
340. #550C2×D61semidirect
349. #486C2×D61semidirect
352. #475C2×D61semidirect
375. #436C2×D61semidirect
389. C61#20303semidirect

Maximal subgroups

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

subgroupindex
#214588352
#214580522
#214574192
#214454102
#214453802
#214453772
#214452932
#214452822
#214450742
#214450542
#214448842
#214448722
#214447102
#214447022
C61×#33322
#2030361