Finite Group Databasebeta

← #29663047

Subgroups — #29663047

Order 15616.

668 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456557C211yesmaximal, characteristic, complemented
2#21455879C211yesmaximal, characteristic, complemented
3#21455274C211yesmaximal, characteristic, complemented
4#21445380C211yesmaximal, characteristic, complemented
5#21445352C211yesmaximal, characteristic, complemented
6#21445318C211yesmaximal, characteristic, complemented
7#21445298C211yesmaximal, characteristic, complemented
8#21445263C211yesmaximal, characteristic, complemented
9#21445235C211yesmaximal, characteristic, complemented
10#21445201C211yesmaximal, characteristic, complemented
11#21445179C211yesmaximal, characteristic, complemented
12#21444571C211yesmaximal, characteristic, complemented
13#21444525C211yesmaximal, characteristic, complemented
14#21444515C211yesmaximal, characteristic, complemented
15C61×#2971C211yesmaximal, characteristic, complemented
16C2×#14519797—21no—
17#16840312C2211yescharacteristic
18#16840312C2211yescharacteristic
19#16840160C2211yescharacteristic
20#16840139C2211yescharacteristic
21#16840109C2211yescharacteristic
22#16840109C2211yescharacteristic
23#16840086—21no—
24#16840057C2211yescharacteristic
25#16840052C2211yescharacteristic
26#16840015—21no—
27#16840015—21no—
28#16839984—21no—
29#16839977—21no—
30#16839972C2211yescharacteristic
31#16839967C2211yescharacteristic
32#16839962C2211yescharacteristic
33#16839962C2211yescharacteristic
34#16839957C2211yescharacteristic
35#16839957C2211yescharacteristic
36#16839907—21no—
37#16839875—21no—
38#16839875—21no—
39C4×#11758201—21no—
40C4×#11758199—21no—
41#16839815C2211yescharacteristic
42C2×C4⋊#92418C2211yescharacteristic
43#16839756—21no—
44#16839515C2211yescharacteristic
45#16839515C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839513C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839510C2211yescharacteristic
50#16839507C2211yescharacteristic
51#16839507C2211yescharacteristic
52#16839505C2211yescharacteristic
53#16839505C2211yescharacteristic
54#16839505C2211yescharacteristic
55#16839505C2211yescharacteristic
56C4×#41×C61C2211yescharacteristic
57C8×Q8×C61C2211yescharacteristic
58C8×Q8×C61C2211yescharacteristic
59C61×#436—21no—
60C61×#436—21no—
61C61×#436C2211yescharacteristic
62C61×#436C2211yescharacteristic
63C61×#432—21no—
64C61×#432C2211yescharacteristic
65C4×M16×C61—21no—
66C4×M16×C61C2211yescharacteristic
67C2×#11758222—21no—
68C2×#11758222—21no—
69#14519797—44no—
70C2×#11758202—21no—
71C2×#11758200—21no—
72C2×#11758201—21no—
73C2×#11758201—21no—
74C2×D488—21no—
75C2×D488—21no—
76C2×#11758199—21no—
77C2×#11758199—21no—
78C22×#92418—21no—
79#14519777C2311yescharacteristic
80#14519777C2311yescharacteristic
81#14519774C2311yescharacteristic
82#14519774C2311yescharacteristic
83#14519765—21no—
84#14519765—21no—
85#14519751—21no—
86#14519751—21no—
87#14519751—21no—
88#14519751—21no—
89C4×D244—21no—
90C4×D244—21no—
91C4×D244—21no—
92C4×D244—21no—
93#14519710—21no—
94#14519709—21no—
95#14519709—21no—
96#14519708—21no—
97C4⋊#92418—21no—
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100C4⋊#92418C2311yescharacteristic
101C4⋊#92418C2311yescharacteristic
102#14519701—21no—
103#14519700—21no—
104#14519700—21no—
105#14519699—21no—
106#14519699—21no—
107#14519699—21no—
108#14519699—21no—
109#14519698—21no—
110#14519697—21no—
111#14519697—21no—
112#14519697—21no—
113#14519697—21no—
114C22⋊#92418—21no—
115C4×#92418—21no—
116C2×#41×C61D411yescharacteristic
117C2×M16×C61—21no—
118C2×M16×C61—21no—
119C2×M16×C61D411yescharacteristic
120C2×M16×C61D411yescharacteristic
121C4×Q8×C61D411yescharacteristic
122C4×Q8×C61C2311yescharacteristic
123C4×D4×C61—21no—
124C4×D4×C61—21no—
125C4×D4×C61D411yescharacteristic
126C4×D4×C61C2311yescharacteristic
127#117×C61—21no—
128#117×C61D411yescharacteristic
129C2×C42×C61—21no—
130C2×C42×C61C2311yescharacteristic
131C4⋊C8×C61—21no—
132C4⋊C8×C61—21no—
133C4⋊C8×C61C2311yescharacteristic
134C4⋊C8×C61C2311yescharacteristic
135C22⋊C8×C61—21no—
136C22⋊C8×C61—21no—
137C22⋊C8×C61—21no—
138#97×C61—21no—
139#97×C61D411yescharacteristic
140#97×C61D411yescharacteristic
141C4×C8×C61—21no—
142C4×C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C4×C8×C61C2311yescharacteristic
145#41×C61—22no—
146#41×C61—41no—
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61—21no—
149C2×D4×C61C2×D411yescharacteristic
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61#4111yescharacteristic
156C2×C22⋊D61—41no—
157C2×C22⋊D61—41no—
158#11758222—42no—
159#11758222—42no—
160#11758222—82no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244—21no—
167C2×C4×D61—41no—
168C2×C4×D61—41no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171M16×C61—22no—
172M16×C61—41no—
173M16×C61—42no—
174M16×C61—42no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61C2×D411yescharacteristic
178C2×C8×C61C2×D411yescharacteristic
179C2×C8×C61C2×D411yescharacteristic
180C2×C8×C61C2×D411yescharacteristic
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61C2×D411yescharacteristic
184C4⋊C4×C61C2×D411yescharacteristic
185C22⋊C4×C61—21no—
186C22⋊C4×C61—21no—
187C22⋊C4×C61—21no—
188C42×C61—21no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61C2×D411yescharacteristic
192C42×C61C2×D411yescharacteristic
193C42×C61#4111yescharacteristic
194C42×C61C2411yescharacteristic
195#11758202—42no—
196#11758201—42no—
197#11758201—42no—
198#11758200—42no—
199#11758199—42no—
200#11758199—42no—
201#11758198—41no—
202#11758198—41no—
203#11758198—41no—
204#11758198—41no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—42no—
214D488—42no—
215D488—42no—
216C23×C61—21no—
217C23×C61#12111yescharacteristic
218C22×D61—41no—
219C22×D61—41no—
220Q8×C61—21no—
221Q8×C61AGL(1,ℤ/8)12yes—
222D4×C61—21no—
223D4×C61—21no—
224D4×C61—42no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—41no—
233C2×C4×C61—42no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—81no—
237C2×C4×C61#12111yescharacteristic
238C2×C4×C61#12111yescharacteristic
239C2×C4×C61#12111yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C22⋊D61—82no—
244C22⋊D61—82no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247D244—41no—
248D244—41no—
249D244—41no—
250D244—41no—
251D244—41no—
252D244—41no—
253D244—41no—
254D244—41no—
255D244—81no—
256D244—81no—
257C4×D61—82no—
258C4×D61—82no—
259Dic122—41no—
260Dic122—41no—
261Dic122—81no—
262C8×C61—21no—
263C8×C61—21no—
264C8×C61—21no—
265C8×C61—41no—
266C8×C61—41no—
267C8×C61#13512yescomplemented
268#92418—41no—
269#92418—41no—
270#92418—41no—
271#92418—41no—
272#19184—611nomaximal, Sylow
273C22×C61—21no—
274C22×C61—21no—
275C22×C61—21no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61C2×#12111yescharacteristic
280C2×D61—81no—
281C2×D61—81no—
282C2×D61—81no—
283C2×D61—81no—
284C2×D61—81no—
285C2×D61—81no—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—22no—
289C4×C61—21no—
290C4×C61—21no—
291C4×C61—41no—
292C4×C61—41no—
293C4×C61C2×#13511yescharacteristic
294C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
295C4×C61#57511yescharacteristic
296C4×C61#57511yescharacteristic
297Dic61—81no—
298Dic61—81no—
299#3184—611no—
300#3106—611no—
301#3035—611no—
302#2971D6111yescharacteristic, complemented
303#1701—611no—
304#1697—611no—
305#1691—611no—
306#1687—611no—
307#1682—611no—
308#1678—611no—
309#1672—611no—
310#1668—611no—
311#1565—611no—
312#1553—611no—
313#1551—611no—
314C2×C61—41no—
315C2×C61—81no—
316C2×C61#303511yescharacteristic
317C2×C61#305511yescharacteristic
318C2×C61#306811yescharacteristic
319D61—161no—
320D61—161no—
321C2×#135—1221no—
322#532—611no—
323#532—611no—
324C4×#41C2×D6111yescharacteristic
325#498—611no—
326#493—611no—
327#486—611no—
328#486—611no—
329#482—1221no—
330#477—611no—
331#476—611no—
332#469—1221no—
333#469—1221no—
334#464—1221no—
335#463—1221no—
336#462—611no—
337#461—611no—
338#460—611no—
339#460—611no—
340#459—611no—
341#459—611no—
342#449—1221no—
343C8×Q8C2×D6111yescharacteristic
344C8×Q8C2×D6111yescharacteristic
345#442—1221no—
346#442—1221no—
347C4×SD16—1221no—
348C4×D8—1221no—
349#436—21no—
350#436—21no—
351#436C2×D6111yescharacteristic
352#436C2×D6111yescharacteristic
353#432—21no—
354#432C2×D6111yescharacteristic
355#431—611no—
356C2×C4⋊C8—611no—
357#416—1221no—
358C4×M16—21no—
359C4×M16C2×D6111yescharacteristic
360#335—611no—
361#335—611no—
362#334—611no—
363#334—611no—
364#333—611no—
365#333—611no—
366#332—611no—
367#332—611no—
368#331—611no—
369#331—611no—
370#331—611no—
371#331—611no—
372C61#1918411yescharacteristic, complemented, Sylow
373C2×#41—1221no—
374C2×#41—1221no—
375C2×#41D24411yescharacteristic
376#135—2444no—
377C2×Q16—1221no—
378C2×SD16—1221no—
379C2×SD16—1221no—
380C2×SD16—1221no—
381C2×D8—1221no—
382C2×D8—1221no—
383C2×D8—1221no—
384C2×D8—1221no—
385C2×M16—21no—
386C2×M16—21no—
387C2×M16C22⋊D6111yescharacteristic
388C2×M16C22⋊D6111yescharacteristic
389C22×C8—1221no—
390C4⋊Q8—611no—
391C4⋊Q8—611no—
392#127—611no—
393#127—611no—
394#124—1221no—
395#124—1221no—
396#121—1221no—
397#121—1221no—
398#121—1221no—
399#121—1221no—
400C4×Q8D24411yescharacteristic
401C4×Q8C22×D6111yescharacteristic
402C4×D4—21no—
403C4×D4—21no—
404C4×D4—1221no—
405C4×D4—1221no—
406C4×D4—1221no—
407C4×D4—1221no—
408C4×D4D24411yescharacteristic
409C4×D4C22×D6111yescharacteristic
410#117—21no—
411#117D24411yescharacteristic
412C2×C42—21no—
413C2×C42C22×D6111yescharacteristic
414#107—1221no—
415#106—1221no—
416#106—1221no—
417#106—1221no—
418C4⋊C8—21no—
419C4⋊C8—21no—
420C4⋊C8—611no—
421C4⋊C8—611no—
422C4⋊C8—611no—
423C4⋊C8—611no—
424C4⋊C8—1221no—
425C4⋊C8C22×D6111yescharacteristic
426C4⋊C8C22×D6111yescharacteristic
427#103—1221no—
428#103—1221no—
429#103—1221no—
430#102—1221no—
431#102—1221no—
432#102—1221no—
433#102—1221no—
434#102—1221no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#102—1221no—
439C22⋊C8—21no—
440C22⋊C8—21no—
441C22⋊C8—21no—
442C22⋊C8—1221no—
443#97—21no—
444#97C22⋊D6111yescharacteristic
445#97C22⋊D6111yescharacteristic
446C4×C8—21no—
447C4×C8—21no—
448C4×C8—1221no—
449C4×C8C22×D6111yescharacteristic
450C4×C8C22×D6111yescharacteristic
451#41—22no—
452#41—41no—
453#41—4882no—
454#41—2442no—
455#41—2442no—
456C2×Q8—1221no—
457C2×Q8—1221no—
458C2×Q8C2×D24411yescharacteristic
459C2×D4—21no—
460C2×D4—1221no—
461C2×D4—1221no—
462C2×D4—1221no—
463C2×D4—1221no—
464C2×D4—1221no—
465C2×D4—1221no—
466C2×D4—2441no—
467C2×D4—2441no—
468C2×D4C2×D24411yescharacteristic
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—2441no—
473C22×C4—2441no—
474C22×C4C2×D24411yescharacteristic
475C22×C4#1175822211yescharacteristic
476C22×C4C2×C22⋊D6111yescharacteristic
477Q16—2442no—
478SD16—2442no—
479SD16—2442no—
480SD16—2442no—
481D8—2442no—
482D8—2442no—
483D8—2442no—
484D8—2442no—
485M16—22no—
486M16—41no—
487M16—42no—
488M16—42no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—2442no—
496C2×C8C2×C22⋊D6111yescharacteristic
497C2×C8C2×C22⋊D6111yescharacteristic
498C2×C8C2×C22⋊D6111yescharacteristic
499C2×C8C2×C22⋊D6111yescharacteristic
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4C2×D24411yescharacteristic
507C4⋊C4C2×D24411yescharacteristic
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—2441no—
512C22⋊C4—2441no—
513C22⋊C4—2441no—
514C22⋊C4—2441no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42C2×D24411yescharacteristic
519C42#1175822211yescharacteristic
520C42C2×C22⋊D6111yescharacteristic
521C42C23×D6111yescharacteristic
522C23—21no—
523C23—2441no—
524C23—2441no—
525C23#1451975111yescharacteristic
526Q8—21no—
527Q8—4881no—
528Q8—2441no—
529Q8—2441no—
530Q8#1451980512yes—
531D4—21no—
532D4—21no—
533D4—42no—
534D4—4881no—
535D4—4881no—
536D4—4882no—
537D4—4882no—
538D4—2441no—
539D4—2441no—
540D4—2441no—
541D4—2441no—
542D4—2441no—
543D4—2441no—
544D4—2441no—
545D4—2441no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—81no—
558C2×C4—4882no—
559C2×C4—4882no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4#1451975111yescharacteristic
563C2×C4#1451975111yescharacteristic
564C2×C4#1451975111yescharacteristic
565C2×C4C22×C22⋊D6111yescharacteristic
566C2×C4C22×D24411yescharacteristic
567C2×C4C2×#1175822211yescharacteristic
568C8—21no—
569C8—21no—
570C8—21no—
571C8—41no—
572C8—41no—
573C8—2441no—
574C8—2441no—
575C8—2441no—
576C8—2441no—
577C8#1451979712yescomplemented
578C22—21no—
579C22—21no—
580C22—21no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—4881no—
585C22—4881no—
586C22—4881no—
587C22—4881no—
588C22—4881no—
589C22—4881no—
590C22C2×#1451975111yescharacteristic
591C4—21no—
592C4—22no—
593C4—21no—
594C4—21no—
595C4—21no—
596C4—41no—
597C4—41no—
598C4—4881no—
599C4—4881no—
600C4C2×#1451979711yescharacteristic
601C4C2×#1451980511yescharacteristic
602C4#1684056011yescharacteristic
603C4#1684056211yescharacteristic
604C2—41no—
605C2—81no—
606C2—9761no—
607C2—9761no—
608C2#2145527411yescharacteristic
609C2#2145543811yescharacteristic
610C2#2145554711yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2971C2
socleC22×C61C2×#121
hypercenter#2971D61
hypocenterC61#19184
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2971

Complements

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

NH ≅ G/Ntype
1. #21456557C2semidirect
2. #21455879C2semidirect
3. #21455274C2semidirect
4. #21445380C2semidirect
5. #21445352C2semidirect
6. #21445318C2semidirect
7. #21445298C2semidirect
8. #21445263C2semidirect
9. #21445235C2semidirect
10. #21445201C2semidirect
11. #21445179C2semidirect
12. #21444571C2semidirect
13. #21444525C2semidirect
14. #21444515C2semidirect
15. C61×#2971C2semidirect
267. C8×C61#135semidirect
302. #2971D61semidirect
372. C61#19184semidirect
577. C8#14519797semidirect

Maximal subgroups

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

subgroupindex
#214565572
#214558792
#214552742
#214453802
#214453522
#214453182
#214452982
#214452632
#214452352
#214452012
#214451792
#214445712
#214445252
#214445152
C61×#29712
#1918461