Finite Group Databasebeta

← #29662857

Subgroups — #29662857

Order 15616.

728 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456209C211yesmaximal, characteristic, complemented
2#21455818C211yesmaximal, characteristic
3#461×D61C211yesmaximal, characteristic, complemented
4D4×#11758189C211yesmaximal, characteristic, complemented
5#21445379C211yesmaximal, characteristic, complemented
6#21445377C211yesmaximal, characteristic, complemented
7#21445369C211yesmaximal, characteristic, complemented
8#21445367C211yesmaximal, characteristic, complemented
9#21445254C211yesmaximal, characteristic, complemented
10#21445244C211yesmaximal, characteristic, complemented
11#21445242C211yesmaximal, characteristic, complemented
12#21444570C211yesmaximal, characteristic, complemented
13#21444502C211yesmaximal, characteristic, complemented
14#21444493C211yesmaximal, characteristic, complemented
15C61×#1681C211yesmaximal, characteristic, complemented
16C2×SD16×D61—21no—
17C22×#11758189—21no—
18#16840310C2211yescharacteristic
19C4⋊Q8×D61C2211yescharacteristic
20C4×D4×D61C2211yescharacteristic
21#16840120—21no—
22#16840120—21no—
23#16840114C2211yescharacteristic, complemented
24#16840113C2211yescharacteristic, complemented
25#16840108C2211yescharacteristic, complemented
26#16840074—21no—
27#16840043C2211yescharacteristic
28#16840042C2211yescharacteristic
29#16839968C2211yescharacteristic
30#16839967C2211yescharacteristic
31#16839966—21no—
32#16839965—21no—
33#16839964C2211yescharacteristic
34#16839954C2211yescharacteristic
35#16839894—21no—
36D8×Dic61—21no—
37#16839842—21no—
38#16839841C2211yescharacteristic
39#16839839C2211yescharacteristic
40#16839833—21no—
41#16839830—21no—
42D4×#92418C2211yescharacteristic
43#16839814C2211yescharacteristic
44#16839784C2211yescharacteristic
45C4⋊C8×D61C2211yescharacteristic
46#16839760—21no—
47#16839760—21no—
48#103×D61—21no—
49#16839724—21no—
50C4×#11758189C2211yescharacteristic
51#16839514C2211yescharacteristic, complemented
52#16839513C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839509C2211yescharacteristic, complemented
55#16839509C2211yescharacteristic, complemented
56#16839508C2211yescharacteristic, complemented
57#16839507C2211yescharacteristic, complemented
58#16839507C2211yescharacteristic, complemented
59#16839506C2211yescharacteristic, complemented
60C61×#486C2211yescharacteristic, complemented
61C61×#461C2211yescharacteristic
62C61×#459C2211yescharacteristic
63C8×D4×C61C2211yescharacteristic
64C61×#335C2211yescharacteristic, complemented
65C61×#332C2211yescharacteristic, complemented
66C61×#331C2211yescharacteristic, complemented
67C2×#11758224—21no—
68C2×Q8×D61—21no—
69C2×D4×D61D411yescharacteristic
70C22×C4×D61—21no—
71C2×#11758190—21no—
72C2×#11758200—21no—
73C2×#11758201—21no—
74SD16×D61—44no—
75C2×#11758199—21no—
76C2×#11758189—21no—
77C2×#11758189—21no—
78C2×#11758189—21no—
79C2×#11758189—44no—
80C2×#11758189D411yescharacteristic
81C2×#11758189D411yescharacteristic
82C22×#92418—21no—
83C2×C8×D61—21no—
84#14519777C2311yescharacteristic
85#14519777C2311yescharacteristic
86#14519776—21no—
87#14519776—21no—
88#14519775—21no—
89#14519764—21no—
90#14519748—21no—
91#14519748—21no—
92C4×D244C2311yescharacteristic
93D4×Dic61—21no—
94D4×Dic61—21no—
95D4×Dic61D411yescharacteristic
96#14519739D411yescharacteristic
97C4×C22⋊D61—21no—
98#14519737—21no—
99C4⋊C4×D61—21no—
100C4⋊C4×D61—21no—
101C4⋊C4×D61D411yescharacteristic
102C22⋊C4×D61—21no—
103C42×D61C2311yescharacteristic
104#14519711—21no—
105#14519710—21no—
106C4⋊#92418C2311yescharacteristic
107C4⋊#92418C2311yescharacteristic
108C4⋊#92418C2311yescharacteristic
109#14519706—21no—
110#14519706—21no—
111#14519702—21no—
112#14519702—21no—
113#14519702—21no—
114#14519701—21no—
115#14519700—21no—
116#14519700—21no—
117#14519700—21no—
118#14519699—21no—
119#14519698—21no—
120#14519698—21no—
121#14519698—21no—
122#14519698—21no—
123C22⋊#92418—21no—
124#14519689—21no—
125#14519689—21no—
126#14519689—21no—
127#14519687D411yescharacteristic
128#14519687D411yescharacteristic
129C4×#92418C2311yescharacteristic
130C8×Dic61—21no—
131C2×SD16×C61—21no—
132C2×D8×C61—21no—
133C22×C8×C61—21no—
134C4⋊Q8×C61C2311yescharacteristic
135#127×C61C2311yescharacteristic
136C4×D4×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140#103×C61—21no—
141#102×C61—21no—
142#102×C61—21no—
143#102×C61—21no—
144C22⋊C8×C61—21no—
145C4×C8×C61C2311yescharacteristic
146C23×D61—21no—
147C2×Q8×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61C2×D411yescharacteristic
152C22×C4×C61—21no—
153C2×C22⋊D61—21no—
154C2×C22⋊D61—41no—
155C22×Dic61—21no—
156C22×Dic61—41no—
157Q8×D61—42no—
158Q8×D61—42no—
159#11758224—42no—
160#11758224—42no—
161D4×D61—22no—
162D4×D61—41no—
163C2×D244C2×D411yescharacteristic
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—42no—
168C2×C4×D61—42no—
169C2×C4×D61—41no—
170C2×C4×D61C2×D411yescharacteristic
171C2×C4×D61C2×D411yescharacteristic
172C2×C4×D61#4111yescharacteristic
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—41no—
176SD16×C61—42no—
177D8×C61—42no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—42no—
182C2×C8×C61C2×D411yescharacteristic
183C2×C8×C61C2×D411yescharacteristic
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61C2×D411yescharacteristic
187C22⋊C4×C61—21no—
188C42×C61C2411yescharacteristic
189C22⋊Dic61—21no—
190C22⋊Dic61—41no—
191C22⋊Dic61—41no—
192#11758201—42no—
193#11758200—42no—
194#11758199—42no—
195#11758198—21no—
196#11758198—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61C2×D411yescharacteristic
202#11758196—21no—
203#11758196—41no—
204#11758196—41no—
205C4×Dic61C2×D411yescharacteristic
206C4×Dic61C2×D411yescharacteristic
207C4×Dic61#4111yescharacteristic
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—42no—
212C2×#92418C2×D411yescharacteristic
213C2×#92418C2×D411yescharacteristic
214#11758190—42no—
215#11758189—22no—
216#11758189—22no—
217#11758189—42no—
218#11758189—42no—
219#11758189—42no—
220C8×D61—42no—
221C23×C61—21no—
222C23×C61—41no—
223C22×D61—21no—
224C22×D61—42no—
225C22×D61—42no—
226C22×D61—81no—
227C22×D61#12111yescharacteristic
228Q8×C61—41no—
229Q8×C61—41no—
230D4×C61—21no—
231D4×C61—41no—
232D4×C61—41no—
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—81no—
236D4×C61AGL(1,ℤ/8)12yes—
237C2×C4×C61—21no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C22⋊D61—42no—
245C22⋊D61—82no—
246C2×Dic61—21no—
247C2×Dic61—42no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—82no—
251C2×Dic61#12111yescharacteristic
252C2×Dic61#12111yescharacteristic
253C2×Dic61#12111yescharacteristic
254D244—21no—
255D244—21no—
256C4×D61—21no—
257C4×D61—21no—
258C4×D61—41no—
259C4×D61—42no—
260C4×D61—41no—
261C4×D61—41no—
262C4×D61—42no—
263C4×D61—82no—
264C4×D61#13512yes—
265C4×D61AGL(1,ℤ/8)12yes—
266Dic122—41no—
267Dic122—41no—
268Dic122—41no—
269Dic122—41no—
270Dic122—82no—
271C8×C61—21no—
272C8×C61—21no—
273C8×C61—41no—
274C8×C61—41no—
275C8×C61—41no—
276#92418—21no—
277#92418—21no—
278#92418—41no—
279#92418—41no—
280#92418—41no—
281#19169—611nomaximal, Sylow
282C22×C61—22no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—81no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61C2×#12111yescharacteristic
289C2×D61—21no—
290C2×D61—21no—
291C2×D61—41no—
292C2×D61—41no—
293C2×D61—41no—
294C2×D61—84no—
295C2×D61#46312yes—
296C4×C61—21no—
297C4×C61—41no—
298C4×C61—81no—
299C4×C61C2×#13511yescharacteristic
300C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
301C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
302C4×C61#57511yescharacteristic
303Dic61—21no—
304Dic61—21no—
305Dic61—41no—
306Dic61—81no—
307Dic61—81no—
308Dic61#46312yes—
309#3141—611no—
310#3100—611no—
311C2×#461—611no—
312D4×M16—611no—
313#1701—611no—
314#1701—611no—
315#1699—611no—
316#1699—611no—
317#1681—611no—
318#1681D6111yescharacteristic, complemented
319#1679—611no—
320#1679—611no—
321#1565—611no—
322#1549—611no—
323#1547—611no—
324C2×C61—42no—
325C2×C61—161no—
326C2×C61C2×#46311yescharacteristic
327C2×C61#305511yescharacteristic
328C2×C61#306911yescharacteristic
329D61—22no—
330D61—81no—
331C22×SD16—1221no—
332C22×M16—1221no—
333#532—611no—
334C2×C4⋊Q8—611no—
335C2×C4×D4—611no—
336#489—1221no—
337#489—1221no—
338#487—611no—
339#487—611no—
340#486—611no—
341#486C2×D6111yescharacteristic, complemented
342#480—1221no—
343#474—611no—
344#474—611no—
345#461—611no—
346#461—611no—
347#461—611no—
348#461—1221no—
349#461—1221no—
350#461C2×D6111yescharacteristic
351#459—611no—
352#459C2×D6111yescharacteristic
353#447—1221no—
354C4×D8—1221no—
355C4×D8—1221no—
356#436—611no—
357#436—611no—
358#435—1221no—
359#435—1221no—
360C8×D4—611no—
361C8×D4C2×D6111yescharacteristic
362#431—611no—
363#423—611no—
364C2×C4⋊C8—611no—
365#417—1221no—
366#417—1221no—
367C2×#103—1221no—
368#407—1221no—
369C4×M16—611no—
370#335—611no—
371#335C2×D6111yescharacteristic, complemented
372#334—611no—
373#334—611no—
374#332—611no—
375#332—611no—
376#332—611no—
377#332—611no—
378#332—611no—
379#332C2×D6111yescharacteristic, complemented
380#331—611no—
381#331C2×D6111yescharacteristic, complemented
382C61#1916911yescharacteristic, complemented, Sylow
383C2×#41—1221no—
384C22×Q8—1221no—
385C22×D4—611no—
386C23×C4—1221no—
387C2×SD16—21no—
388C2×SD16—1221no—
389C2×SD16—1221no—
390C2×SD16—1221no—
391C2×SD16—2444no—
392C2×D8—21no—
393C2×D8—1221no—
394C2×M16—611no—
395C2×M16—611no—
396C2×M16—1221no—
397C2×M16—1221no—
398C2×M16—1221no—
399C2×M16—2444no—
400C22×C8—21no—
401C22×C8—1221no—
402C22×C8—1221no—
403C4⋊Q8—611no—
404C4⋊Q8—611no—
405C4⋊Q8—1221no—
406C4⋊Q8—1221no—
407C4⋊Q8—1221no—
408C4⋊Q8C22×D6111yescharacteristic
409#127C22×D6111yescharacteristic
410#124—1221no—
411#121—1221no—
412#121—1221no—
413C4×D4—611no—
414C4×D4—611no—
415C4×D4—611no—
416C4×D4—1221no—
417C4×D4—1221no—
418C4×D4—1221no—
419C4×D4—1221no—
420C4×D4C22×D6111yescharacteristic
421C2×C4⋊C4—611no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C42—611no—
426#107—1221no—
427#107—1221no—
428C4⋊C8—611no—
429C4⋊C8—611no—
430C4⋊C8—611no—
431C4⋊C8—1221no—
432C4⋊C8—1221no—
433C4⋊C8C22×D6111yescharacteristic
434C4⋊C8C22×D6111yescharacteristic
435C4⋊C8C22×D6111yescharacteristic
436#103—21no—
437#103—1221no—
438#103—1221no—
439#103—1221no—
440#103—1221no—
441#103—1221no—
442#103—1221no—
443#103—1221no—
444#102—21no—
445#102—21no—
446#102—21no—
447#102—1221no—
448#102—1221no—
449#102—1221no—
450#102—1221no—
451#102—1221no—
452C22⋊C8—21no—
453C22⋊C8—1221no—
454C22⋊C8—1221no—
455C22⋊C8—1221no—
456C22⋊C8—1221no—
457#97—611no—
458#97—611no—
459C4×C8—611no—
460C4×C8—1221no—
461C4×C8C22×D6111yescharacteristic
462C24—1221no—
463#41—2442no—
464#41—2442no—
465C2×Q8—21no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—2441no—
469C2×Q8—2442no—
470C2×Q8—2442no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—611no—
475C2×D4—1221no—
476C2×D4—1222no—
477C2×D4—2441no—
478C2×D4—2441no—
479C2×D4D4×D6111yescharacteristic
480C22×C4—21no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—2442no—
489C22×C4—2442no—
490C22×C4—2441no—
491C22×C4—2441no—
492SD16—42no—
493SD16—2442no—
494SD16—2442no—
495SD16—2442no—
496D8—42no—
497D8—2442no—
498M16—1222no—
499M16—1222no—
500M16—2442no—
501M16—2442no—
502M16—2442no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—42no—
507C2×C8—611no—
508C2×C8—611no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—2442no—
513C2×C8—2442no—
514C2×C8D4×D6111yescharacteristic
515C2×C8D4×D6111yescharacteristic
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—611no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—2441no—
525C4⋊C4—2441no—
526C4⋊C4D4×D6111yescharacteristic
527C22⋊C4—21no—
528C22⋊C4—1221no—
529C22⋊C4—1221no—
530C22⋊C4—1221no—
531C22⋊C4—2441no—
532C22⋊C4—2441no—
533C42—611no—
534C42—611no—
535C42—611no—
536C42C23×D6111yescharacteristic
537C23—21no—
538C23—41no—
539C23—611no—
540C23—1221no—
541C23—4881no—
542C23—2442no—
543C23—2442no—
544Q8—41no—
545Q8—41no—
546Q8—4882no—
547Q8—2441no—
548Q8—2441no—
549Q8—2441no—
550Q8—2441no—
551D4—21no—
552D4—41no—
553D4—41no—
554D4—41no—
555D4—41no—
556D4—81no—
557D4—1221no—
558D4—1221no—
559D4—4882no—
560D4—2442no—
561D4#1451980112yes—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—611no—
567C2×C4—612no—
568C2×C4—612no—
569C2×C4—611no—
570C2×C4—611no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—4882no—
575C2×C4—4882no—
576C2×C4—2441no—
577C2×C4—2442no—
578C2×C4—2442no—
579C2×C4—2442no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4C2×D4×D6111yescharacteristic
585C2×C4C2×D4×D6111yescharacteristic
586C2×C4#41×D6111yescharacteristic
587C8—21no—
588C8—21no—
589C8—41no—
590C8—41no—
591C8—41no—
592C8—1221no—
593C8—1221no—
594C8—2441no—
595C8—2441no—
596C8—2441no—
597C22—22no—
598C22—41no—
599C22—41no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—612no—
604C22—1221no—
605C22—1221no—
606C22—4884no—
607C22—2441no—
608C22—2441no—
609C22—2441no—
610C22#121×D6111yescharacteristic
611C4—21no—
612C4—41no—
613C4—81no—
614C4—612no—
615C4—1221no—
616C4—1221no—
617C4—4881no—
618C4—4881no—
619C4—2441no—
620C4#135×D6111yescharacteristic
621C4AGL(1,ℤ/8)×D6111yescharacteristic
622C4C2×#1451980111yescharacteristic
623C4#1684055911yescharacteristic
624C2—42no—
625C2—161no—
626C2—1222no—
627C2—4881no—
628C2#463×D6111yescharacteristic
629C2#2145543511yescharacteristic
630C2#2145555111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1681C2
socleC22×C61C2×#121
prime radical#21455818C2
hypercenter#1681D61
hypocenterC61#19169
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1681

Complements

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

NH ≅ G/Ntype
1. #21456209C2semidirect
3. #461×D61C2semidirect
4. D4×#11758189C2semidirect
5. #21445379C2semidirect
6. #21445377C2semidirect
7. #21445369C2semidirect
8. #21445367C2semidirect
9. #21445254C2semidirect
10. #21445244C2semidirect
11. #21445242C2semidirect
12. #21444570C2semidirect
13. #21444502C2semidirect
14. #21444493C2semidirect
15. C61×#1681C2semidirect
23. #16840114C22semidirect
24. #16840113C22semidirect
25. #16840108C22semidirect
51. #16839514C22semidirect
52. #16839513C22semidirect
53. #16839512C22semidirect
54. #16839509C22semidirect
55. #16839509C22semidirect
56. #16839508C22semidirect
57. #16839507C22semidirect
58. #16839507C22semidirect
59. #16839506C22semidirect
60. C61×#486C22semidirect
64. C61×#335C22semidirect
65. C61×#332C22semidirect
66. C61×#331C22semidirect
318. #1681D61semidirect
341. #486C2×D61semidirect
371. #335C2×D61semidirect
379. #332C2×D61semidirect
381. #331C2×D61semidirect
382. C61#19169semidirect

Maximal subgroups

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

subgroupindex
#214562092
#214558182
#461×D612
D4×#117581892
#214453792
#214453772
#214453692
#214453672
#214452542
#214452442
#214452422
#214445702
#214445022
#214444932
C61×#16812
#1916961