Finite Group Databasebeta

← #29670128

Subgroups — #29670128

Order 15616.

676 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458685C211yesmaximal, characteristic, complemented
2#21458176C211yesmaximal, characteristic, complemented
3#21457640C211yesmaximal, characteristic, complemented
4#21457419C211yesmaximal, characteristic
5#21445443C211yesmaximal, characteristic, complemented
6#21445340C211yesmaximal, characteristic, complemented
7#21445230C211yesmaximal, characteristic, complemented
8#21445193C211yesmaximal, characteristic, complemented
9#21445170C211yesmaximal, characteristic, complemented
10#21445023C211yesmaximal, characteristic, complemented
11#21444937C211yesmaximal, characteristic, complemented
12#21444893C211yesmaximal, characteristic, complemented
13#21444665C211yesmaximal, characteristic, complemented
14#21444640C211yesmaximal, characteristic, complemented
15C61×#1635C211yesmaximal, characteristic, complemented
16C2×#14519802—21no—
17#16840473C2211yescharacteristic
18#16840469C2211yescharacteristic, complemented
19D4×D244C2211yescharacteristic
20#16840165C2211yescharacteristic, complemented
21#16840162—21no—
22#16840158C2211yescharacteristic, complemented
23#16840136C2211yescharacteristic
24#16840135C2211yescharacteristic
25#16840110C2211yescharacteristic
26#16840092—21no—
27#16840082—21no—
28#16840078—21no—
29#16840064—21no—
30#16840061—21no—
31#16840053C2211yescharacteristic
32#16840052C2211yescharacteristic
33#16840044C2211yescharacteristic, complemented
34#16840010—21no—
35#16839992—21no—
36#16839989—21no—
37#16839964C2211yescharacteristic
38#16839963C2211yescharacteristic
39#16839957C2211yescharacteristic
40#16839914—21no—
41#16839912—21no—
42#16839907—21no—
43#16839886C2211yescharacteristic, complemented
44C4×#11758200C2211yescharacteristic
45C4×#11758199C2211yescharacteristic
46#16839842—21no—
47#16839840C2211yescharacteristic
48D4×#92418C2211yescharacteristic, complemented
49#16839827C2211yescharacteristic
50C2×#14519708—21no—
51#102×D61—21no—
52#16839512C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839511C2211yescharacteristic, complemented
55#16839510C2211yescharacteristic, complemented
56#16839505C2211yescharacteristic
57#16839494C2211yescharacteristic, complemented
58#16839491C2211yescharacteristic
59#16839484C2211yescharacteristic, complemented
60C61×#547C2211yescharacteristic
61C61×#487C2211yescharacteristic, complemented
62C61×#476C2211yescharacteristic
63C61×#476C2211yescharacteristic, complemented
64C61×#332C2211yescharacteristic, complemented
65C61×D4⋊C8C2211yescharacteristic
66C61×D4⋊C8C2211yescharacteristic, complemented
67C22×D244—21no—
68C2×D4×D61—21no—
69#14519802—44no—
70C2×#11758190—21no—
71C2×#11758190—41no—
72C2×#11758200D411yescharacteristic
73C2×#11758201—21no—
74C2×#11758201—21no—
75C2×#11758201—21no—
76C2×#11758199—21no—
77C2×#11758199D411yescharacteristic
78C2×#11758194—21no—
79C22×#92418—21no—
80C2×C8×D61—21no—
81#14519777C2311yescharacteristic
82#14519774C2311yescharacteristic
83#14519768—21no—
84#14519768—21no—
85#14519757—21no—
86#14519756—21no—
87#14519754—41no—
88#14519753—41no—
89#14519752—21no—
90#14519751—21no—
91#14519747—41no—
92C4×Dic122C2311yescharacteristic
93C4×D244C2311yescharacteristic
94D4×Dic61—21no—
95#14519739—21no—
96C2×C4⋊Dic61—21no—
97C4⋊C4×D61—21no—
98#14519710—21no—
99#14519710—21no—
100#14519710D411yescharacteristic
101#14519709—21no—
102#14519708—21no—
103#14519708—21no—
104#14519708—21no—
105#14519708—21no—
106#14519708D411yescharacteristic
107C4⋊#92418C2311yescharacteristic
108C4⋊#92418C2311yescharacteristic
109#14519706—21no—
110#14519701—21no—
111#14519700—21no—
112#14519700D411yescharacteristic
113#14519699—21no—
114#14519699—21no—
115#14519698—21no—
116#14519698D411yescharacteristic
117#14519698D411yescharacteristic
118#14519697—21no—
119#14519697—21no—
120#14519697—21no—
121#14519697D411yescharacteristic
122C22⋊#92418—21no—
123C22⋊#92418—21no—
124#14519689—21no—
125C4×#92418C2311yescharacteristic
126C4×#92418C2311yescharacteristic
127C8×Dic61—21no—
128C2×#41×C61—21no—
129C4⋊Q8×C61C2311yescharacteristic
130C4⋊Q8×C61C2311yescharacteristic
131#123×C61—41no—
132C22⋊Q8×C61—21no—
133#121×C61—21no—
134C4×D4×C61C2311yescharacteristic
135C4×D4×C61C2311yescharacteristic
136C2×C4⋊C4×C61—21no—
137#106×C61—21no—
138#106×C61—21no—
139C4⋊C8×C61C2311yescharacteristic
140C4⋊C8×C61C2311yescharacteristic
141#103×C61—41no—
142#102×C61—21no—
143#102×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—41no—
146#41×C61—42no—
147#41×C61—81no—
148C2×Q8×C61—21no—
149C2×Q8×C61—21no—
150C2×D4×C61—41no—
151C2×D4×C61C2×D411yescharacteristic
152C2×D4×C61C2×D411yescharacteristic
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—41no—
156C2×C22⋊D61—41no—
157C22×Dic61—41no—
158D4×D61—42no—
159D4×D61—81no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—42no—
164C2×D244—41no—
165C2×D244—81no—
166C2×D244C2×D411yescharacteristic
167C2×C4×D61—21no—
168C2×C4×D61—41no—
169C2×Dic122—21no—
170C2×Dic122C2×D411yescharacteristic
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—41no—
179C4⋊C4×C61C2×D411yescharacteristic
180C4⋊C4×C61C2×D411yescharacteristic
181C22⋊C4×C61—21no—
182C22⋊C4×C61—21no—
183C22⋊C4×C61—41no—
184C42×C61C2411yescharacteristic
185C22⋊Dic61—41no—
186#11758201—42no—
187#11758201—42no—
188#11758201—81no—
189#11758200—41no—
190#11758199—42no—
191#11758199SD1614yescomplemented
192#11758198—21no—
193#11758198—41no—
194#11758198—41no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—41no—
198C4⋊Dic61—41no—
199C4⋊Dic61C2×D411yescharacteristic
200C4⋊Dic61C2×D411yescharacteristic
201#11758196—21no—
202#11758196—41no—
203#11758196—41no—
204C4×Dic61—21no—
205#11758194—42no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—42no—
209C2×#92418C2×D411yescharacteristic
210C2×#92418C2×D411yescharacteristic
211C2×#92418C2×D411yescharacteristic
212C2×#92418C2×D411yescharacteristic
213#11758190—81no—
214#11758190—82no—
215C8×D61—81no—
216C23×C61—21no—
217C23×C61—21no—
218C22×D61—21no—
219C22×D61—41no—
220C22×D61—82no—
221C22×D61—82no—
222C22×D61—161no—
223Q8×C61—41no—
224Q8×C61—41no—
225Q8×C61—81no—
226D4×C61—21no—
227D4×C61—21no—
228D4×C61—21no—
229D4×C61—82no—
230D4×C61C2×SD1612yes—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—82no—
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61H(5,𝔽2)11yescharacteristic
243C22⋊D61—82no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—41no—
248C2×Dic61—161no—
249D244—21no—
250D244—41no—
251D244—41no—
252D244—41no—
253D244—41no—
254D244—81no—
255D244—82no—
256D244C2×SD1612yes—
257C4×D61—41no—
258C4×D61—41no—
259C4×D61—161no—
260Dic122—21no—
261Dic122—21no—
262Dic122—81no—
263C8×C61—41no—
264C8×C61—41no—
265C8×C61—41no—
266#92418—21no—
267#92418—21no—
268#92418—21no—
269#92418—41no—
270#92418—41no—
271#92418C2×SD1612yes—
272#19806—611nomaximal, Sylow
273C22×C61—22no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61D4211yescharacteristic
280C2×D61—22no—
281C2×D61—41no—
282C2×D61—41no—
283C2×D61—81no—
284C2×D61—81no—
285C2×D61—81no—
286C2×D61—164no—
287C4×C61—21no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61—81no—
291C4×C61—81no—
292C4×C61C22×SD1611yescharacteristic
293C4×C61#57511yescharacteristic
294C4×C61#57711yescharacteristic
295C4×C61#57711yescharacteristic
296Dic61—41no—
297Dic61—41no—
298Dic61—41no—
299Dic61—81no—
300#3380—611no—
301#3335—611no—
302#3295—611no—
303#3272—611no—
304#1714—611no—
305#1695—611no—
306#1677—611no—
307#1671—611no—
308#1666—611no—
309#1643—611no—
310#1635D6111yescharacteristic, complemented
311#1629—611no—
312#1622—611no—
313#1583—611no—
314#1580—611no—
315C2×C61—42no—
316C2×C61—81no—
317C2×C61#326711yescharacteristic
318C2×C61D4×SD1611yescharacteristic
319C2×C61#328811yescharacteristic
320D61—42no—
321D61—161no—
322C2×AGL(1,ℤ/8)—1221no—
323#554—611no—
324#554—611no—
325#547C2×D6111yescharacteristic
326D42—611no—
327#500—611no—
328#499—1221no—
329#498—611no—
330#492—611no—
331#492—611no—
332#487C2×D6111yescharacteristic, complemented
333#486—611no—
334#483—1221no—
335#481—1221no—
336#481—1221no—
337#478—1221no—
338#478—1221no—
339#476—611no—
340#476—611no—
341#476C2×D6111yescharacteristic
342#476C2×D6111yescharacteristic, complemented
343#475—611no—
344#468—1221no—
345#465—1221no—
346#465—1221no—
347#461—611no—
348#460—611no—
349#459—611no—
350#450—1221no—
351#450—1221no—
352#449—1221no—
353C8×Q8—611no—
354C4×SD16—611no—
355C4×D8—611no—
356C4×D8—1221no—
357#436—611no—
358C8×D4—611no—
359C8×D4—611no—
360C2×#106—1221no—
361C2×#102—1221no—
362#334—611no—
363#334—611no—
364#333—611no—
365#333—611no—
366#332C2×D6111yescharacteristic, complemented
367#331—611no—
368Q8⋊C8—611no—
369D4⋊C8—611no—
370D4⋊C8C2×D6111yescharacteristic
371D4⋊C8C2×D6111yescharacteristic, complemented
372#322—611no—
373C61#1980611yescharacteristic, complemented, Sylow
374C2×#41—21no—
375C22×D4—1221no—
376C22×D4—1221no—
377AGL(1,ℤ/8)—2444no—
378C2×SD16—611no—
379C2×SD16—1221no—
380C2×SD16—1221no—
381C2×SD16—1221no—
382C2×SD16—1221no—
383C2×SD16—2441no—
384C2×D8—611no—
385C2×D8—1221no—
386C2×M16—1221no—
387C22×C8—1221no—
388C22×C8—1221no—
389C4⋊Q8—611no—
390C4⋊Q8C22×D6111yescharacteristic
391C4⋊Q8C22×D6111yescharacteristic
392#127—611no—
393#125—1221no—
394#125—1221no—
395#123—41no—
396C22⋊Q8—21no—
397C22⋊Q8—1221no—
398C22⋊Q8—1221no—
399C22⋊Q8—2441no—
400C22⋊Q8—2441no—
401#121—21no—
402#121—1221no—
403#121—1221no—
404#120—2441no—
405C4×Q8—611no—
406C4×D4—611no—
407C4×D4—1221no—
408C4×D4—1221no—
409C4×D4C22×D6111yescharacteristic
410C4×D4C22×D6111yescharacteristic
411C2×C4⋊C4—21no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414#107—611no—
415#107—1221no—
416#107—1221no—
417#106—21no—
418#106—21no—
419#106—611no—
420#106—1221no—
421#106—1221no—
422#106—1221no—
423#106—1221no—
424#106—1221no—
425C4⋊C8—611no—
426C4⋊C8—611no—
427C4⋊C8—1221no—
428C4⋊C8C22×D6111yescharacteristic
429C4⋊C8C22×D6111yescharacteristic
430#103—41no—
431#103—611no—
432#103—1221no—
433#103—1221no—
434#102—21no—
435#102—21no—
436#102—611no—
437#102—611no—
438#102—611no—
439#102—1221no—
440#102—1221no—
441#102—1221no—
442#102—1221no—
443#102—1221no—
444#102—1221no—
445C22⋊C8—1221no—
446C22⋊C8—1221no—
447C22⋊C8—1221no—
448C4×C8—611no—
449C4×C8—611no—
450C4×C8—1221no—
451C4×C8C22×D6111yescharacteristic
452C24—2441no—
453#41—42no—
454#41—81no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—611no—
458C2×Q8—1221no—
459C2×D4—41no—
460C2×D4—611no—
461C2×D4—1221no—
462C2×D4—1221no—
463C2×D4—1221no—
464C2×D4—4881no—
465C2×D4—4881no—
466C2×D4—2441no—
467C2×D4—2442no—
468C2×D4—2441no—
469C2×D4—2442no—
470C2×D4D4×D6111yescharacteristic
471C2×D4D4×D6111yescharacteristic
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—41no—
475C22×C4—1221no—
476C22×C4—2441no—
477C22×C4—2441no—
478SD16—4882no—
479SD16—4881no—
480SD16—4881no—
481SD16—2442no—
482SD16—2441no—
483SD16—2442no—
484D8—614no—
485D8—2442no—
486M16—2442no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—611no—
491C2×C8—611no—
492C2×C8—611no—
493C2×C8—611no—
494C2×C8—1221no—
495C2×C8—1221no—
496C2×C8—4881no—
497C2×C8—2442no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—41no—
503C4⋊C4—611no—
504C4⋊C4—611no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—2441no—
509C4⋊C4—2441no—
510C4⋊C4—2441no—
511C4⋊C4—2441no—
512C4⋊C4D4×D6111yescharacteristic
513C4⋊C4D4×D6111yescharacteristic
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—41no—
517C22⋊C4—1221no—
518C22⋊C4—2441no—
519C22⋊C4—2441no—
520C22⋊C4—2441no—
521C42—1221no—
522C42C23×D6111yescharacteristic
523C23—21no—
524C23—21no—
525C23—1221no—
526C23—9761no—
527C23—4882no—
528C23—4882no—
529C23—2441no—
530Q8—41no—
531Q8—41no—
532Q8—81no—
533Q8—1221no—
534Q8—1221no—
535Q8—4881no—
536D4—21no—
537D4—21no—
538D4—21no—
539D4—82no—
540D4—612no—
541D4—1221no—
542D4—4881no—
543D4—4882no—
544D4—4882no—
545D4—2441no—
546D4—2441no—
547D4—2441no—
548D4—2441no—
549D4SD16×D6112yes—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—82no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—9761no—
563C2×C4—9761no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4C2×D4×D6111yescharacteristic
568C2×C4C2×D4×D6111yescharacteristic
569C2×C4#1451982211yescharacteristic
570C8—41no—
571C8—41no—
572C8—41no—
573C8—612no—
574C8—1221no—
575C8—1221no—
576C8—1221no—
577C8—2441no—
578C8—2441no—
579C22—22no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—1222no—
586C22—9764no—
587C22—4881no—
588C22—4881no—
589C22—4881no—
590C22—2441no—
591C22—2441no—
592C22#1684039411yescharacteristic
593C4—21no—
594C4—41no—
595C4—41no—
596C4—81no—
597C4—81no—
598C4—4881no—
599C4—2441no—
600C4—2441no—
601C4—2441no—
602C4C2×SD16×D6111yescharacteristic
603C4#1684056111yescharacteristic
604C4#1684056911yescharacteristic
605C4#1684057211yescharacteristic
606C2—42no—
607C2—81no—
608C2—9761no—
609C2—2442no—
610C2#2145734711yescharacteristic
611C2#2145751311yescharacteristic
612C2#2145758711yescharacteristic
subgroupG/H
centerC22#16840394
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1635C2
socleC22×C61D42
prime radical#21457419C2
hypercenter#1635D61
hypocenterC61#19806
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1635

Complements

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

NH ≅ G/Ntype
1. #21458685C2semidirect
2. #21458176C2semidirect
3. #21457640C2semidirect
5. #21445443C2semidirect
6. #21445340C2semidirect
7. #21445230C2semidirect
8. #21445193C2semidirect
9. #21445170C2semidirect
10. #21445023C2semidirect
11. #21444937C2semidirect
12. #21444893C2semidirect
13. #21444665C2semidirect
14. #21444640C2semidirect
15. C61×#1635C2semidirect
18. #16840469C22semidirect
20. #16840165C22semidirect
22. #16840158C22semidirect
33. #16840044C22semidirect
43. #16839886C22semidirect
48. D4×#92418C22semidirect
52. #16839512C22semidirect
53. #16839512C22semidirect
54. #16839511C22semidirect
55. #16839510C22semidirect
57. #16839494C22semidirect
59. #16839484C22semidirect
61. C61×#487C22semidirect
63. C61×#476C22semidirect
64. C61×#332C22semidirect
66. C61×D4⋊C8C22semidirect
191. #11758199SD16semidirect
310. #1635D61semidirect
332. #487C2×D61semidirect
342. #476C2×D61semidirect
366. #332C2×D61semidirect
371. D4⋊C8C2×D61semidirect
373. C61#19806semidirect

Maximal subgroups

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

subgroupindex
#214586852
#214581762
#214576402
#214574192
#214454432
#214453402
#214452302
#214451932
#214451702
#214450232
#214449372
#214448932
#214446652
#214446402
C61×#16352
#1980661