Finite Group Databasebeta

← #29698531

Subgroups — #29698531

Order 15616.

812 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#21454734C211yesmaximal, characteristic
2#21454732C211yesmaximal, characteristic
3C2×#16839876C211yesmaximal, characteristic
4#21454429C211yesmaximal, characteristic, complemented
5#21454428C211yesmaximal, characteristic, complemented
6#21446608C211yesmaximal, characteristic, complemented
7#21446606C211yesmaximal, characteristic, complemented
8#21446061C211yesmaximal, characteristic, complemented
9#21446060C211yesmaximal, characteristic, complemented
10#21446058C211yesmaximal, characteristic, complemented
11#21445990C211yesmaximal, characteristic, complemented
12#21445872C211yesmaximal, characteristic, complemented
13#21445871C211yesmaximal, characteristic, complemented
14#21445697C211yesmaximal, characteristic, complemented
15C61×#1861C211yesmaximal, characteristic, complemented
16C2×#14519797C2211yescharacteristic
17#16840529C2211yescharacteristic
18#16840528C2211yescharacteristic
19#16840528C2211yescharacteristic
20C2×#14519785C2211yescharacteristic
21C2×#14519784C2211yescharacteristic
22#16839882—22no—
23#16839881—22no—
24#16839877—22no—
25#16839876—22no—
26#16839876C412yescomplemented
27#16839876C412yescomplemented
28#16839876C412yescomplemented
29#16839876C412yescomplemented
30#16839811C2211yescharacteristic
31#16839810—22no—
32#16839810C2211yescharacteristic
33#16839810C2211yescharacteristic
34#16839809—22no—
35C2×#14519712—22no—
36C2×#14519712C2211yescharacteristic
37C2×#14519712C2211yescharacteristic
38C2×#14519712C2211yescharacteristic
39C2×#14519712C2211yescharacteristic
40C2×#14519712C2211yescharacteristic
41C2×#14519712C2211yescharacteristic
42#108×D61—22no—
43#16839798C2211yescharacteristic
44#16839786C2211yescharacteristic
45#16839780C2211yescharacteristic
46#16839779C2211yescharacteristic
47#16839778C2211yescharacteristic
48C2×#14519705C2211yescharacteristic
49#16839720C2211yescharacteristic
50#16839718C2211yescharacteristic
51C2×C4×#92418C2211yescharacteristic
52#16839534C2211yescharacteristic
53#16839533C2211yescharacteristic
54#16839532C2211yescharacteristic
55#16839522C2211yescharacteristic
56C61×#568C2211yescharacteristic
57C61×#430C2211yescharacteristic
58C61×#423C2211yescharacteristic
59C61×#421C2211yescharacteristic
60C2×#104×C61C2211yescharacteristic
61C61×#344C2211yescharacteristic
62C61×#339C2211yescharacteristic
63C2×#11758222C2311yescharacteristic
64#14519797C2×C412yes—
65#14519797C2×C412yes—
66#14519797D412yes—
67#14519797D412yes—
68C2×#11758202Q811yescharacteristic
69C2×#11758200Q811yescharacteristic
70C2×#11758201Q811yescharacteristic
71C2×#11758199Q811yescharacteristic
72#14519785—21no—
73#14519785—21no—
74#14519785—22no—
75#14519785—41no—
76#14519785C2×C412yes—
77#14519785C2×C412yes—
78#14519784—21no—
79#14519784—21no—
80#14519784—22no—
81#14519784—41no—
82#14519784C2×C412yes—
83#14519784C2×C412yes—
84#14519783—22no—
85#14519783—41no—
86C2×#11758194—21no—
87C2×#11758194—22no—
88C2×#11758194C2311yescharacteristic
89C2×#11758194C2311yescharacteristic
90C2×#11758194C2311yescharacteristic
91C2×#11758194C2311yescharacteristic
92C2×#11758189—21no—
93M16×D61—22no—
94M16×D61—22no—
95C22×#92418—21no—
96C22×#92418C2311yescharacteristic
97C22×#92418C2311yescharacteristic
98C2×C8×D61—21no—
99#14519736C2311yescharacteristic
100#14519713—22no—
101#14519712—22no—
102#14519712—22no—
103#14519712—21no—
104#14519712—21no—
105#14519712—21no—
106#14519712—21no—
107#14519712—21no—
108#14519712—21no—
109#14519712—21no—
110#14519712—21no—
111#14519712C2×C412yes—
112#14519712C2×C412yes—
113#14519712D412yes—
114#14519712D412yes—
115#14519710—21no—
116#14519708—21no—
117C4⋊#92418—41no—
118#14519705—22no—
119#14519705C2×C412yes—
120#14519705C2×C412yes—
121#14519704—22no—
122#14519703—22no—
123#14519688—21no—
124#14519687—21no—
125C4×#92418—21no—
126C4×#92418C2×C411yescharacteristic
127C4×#92418C2×C411yescharacteristic
128C4×#92418C2×C411yescharacteristic
129C4×#92418C2×C411yescharacteristic
130C8×Dic61—21no—
131C2×#41×C61C2311yescharacteristic
132#131×C61—22no—
133#131×C61—41no—
134C2×M16×C61—22no—
135C2×M16×C61C2311yescharacteristic
136C2×M16×C61C2311yescharacteristic
137C2×M16×C61C2311yescharacteristic
138C2×M16×C61C2311yescharacteristic
139#117×C61C2311yescharacteristic
140C2×C42×C61C2311yescharacteristic
141#108×C61—22no—
142C4⋊C8×C61—41no—
143#104×C61—22no—
144#104×C61C2×C412yes—
145#104×C61C2×C412yes—
146#41×C61—21no—
147#41×C61—21no—
148#41×C61C22×C412yes—
149#41×C61C2×D412yes—
150C2×Q8×C61C2×Q811yescharacteristic
151C2×D4×C61—21no—
152C2×D4×C61C2×Q811yescharacteristic
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61C2411yescharacteristic
156C2×C22⋊D61—21no—
157#11758222—21no—
158#11758222—21no—
159#11758222C22×C412yes—
160#11758222C2×D412yes—
161C2×D244C2×Q811yescharacteristic
162C2×C4×D61—21no—
163C2×Dic122C2×Q811yescharacteristic
164M16×C61—21no—
165M16×C61—21no—
166M16×C61—21no—
167M16×C61—21no—
168M16×C61—21no—
169M16×C61—21no—
170M16×C61—42no—
171M16×C61C22×C412yes—
172M16×C61C2×D412yes—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—42no—
178C4⋊C4×C61—21no—
179C22⋊C4×C61—21no—
180C42×C61—21no—
181C42×C61—21no—
182C42×C61C22×C411yescharacteristic
183C42×C61C22×C411yescharacteristic
184C22⋊Dic61—21no—
185#11758202C4⋊C412yes—
186#11758202C4⋊C412yes—
187#11758201C4⋊C412yes—
188#11758201C4⋊C412yes—
189#11758200C4⋊C412yes—
190#11758200C4⋊C412yes—
191#11758199C4⋊C412yes—
192#11758199C4⋊C412yes—
193C4⋊Dic61—21no—
194C4×Dic61—21no—
195#11758194—21no—
196#11758194—21no—
197#11758194—22no—
198#11758194—22no—
199#11758194—21no—
200#11758194—21no—
201#11758194—21no—
202#11758194—21no—
203#11758194—42no—
204#11758194C22×C412yes—
205#11758194C2×D412yes—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—22no—
209C2×#92418—22no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—42no—
213C2×#92418C22×C411yescharacteristic
214C2×#92418C22×C411yescharacteristic
215C2×#92418C22×C411yescharacteristic
216C2×#92418C22×C411yescharacteristic
217C2×#92418C2×D411yescharacteristic
218C2×#92418C2×D411yescharacteristic
219C2×#92418C2×D411yescharacteristic
220C2×#92418C2×D411yescharacteristic
221C2×#92418C2×Q811yescharacteristic
222C2×#92418C2×Q811yescharacteristic
223C2×#92418#4111yescharacteristic
224C2×#92418#4111yescharacteristic
225#11758189—22no—
226#11758189—22no—
227#11758189—42no—
228#11758189—42no—
229C8×D61—22no—
230C8×D61—22no—
231C8×D61—42no—
232C8×D61—42no—
233C23×C61—21no—
234C23×C612−1+411yescharacteristic
235C22×D61—21no—
236Q8×C61—21no—
237Q8×C61C2×C4⋊C412yes—
238D4×C61—22no—
239D4×C61—22no—
240D4×C61—21no—
241D4×C61C2×C4⋊C412yes—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—21no—
247C2×C4×C61—42no—
248C2×C4×C61—41no—
249C2×C4×C61C23×C411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×Q811yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61H(5,𝔽2)11yescharacteristic
255C22⋊D61—22no—
256C22⋊D61—22no—
257C2×Dic61—21no—
258C2×Dic61—41no—
259D244—21no—
260D244C2×C4⋊C412yes—
261C4×D61—22no—
262C4×D61—22no—
263Dic122—21no—
264Dic122C2×C4⋊C412yes—
265C8×C61—22no—
266C8×C61—22no—
267C8×C61—41no—
268C8×C61—41no—
269#92418—22no—
270#92418—22no—
271#92418—41no—
272#92418—41no—
273#92418C2×C4⋊C411yescharacteristic
274#92418C2×C4⋊C411yescharacteristic
275#92418C2×C4⋊C411yescharacteristic
276#92418C2×C4⋊C411yescharacteristic
277#92418C4×D411yescharacteristic
278#92418C4×D411yescharacteristic
279#92418C4×D411yescharacteristic
280#92418C4×D411yescharacteristic
281#22694—611nomaximal, Sylow
282C22×C61—22no—
283C22×C61—41no—
284C22×C61—81no—
285C22×C61#52011yescharacteristic
286C22×C61#54711yescharacteristic
287C22×C61D4×Q811yescharacteristic
288C2×D61—22no—
289C2×D61—81no—
290C4×C61—22no—
291C4×C61—22no—
292C4×C61—41no—
293C4×C61—81no—
294C4×C61C22×C4⋊C411yescharacteristic
295C4×C61C2×C4×D411yescharacteristic
296C4×C61D4211yescharacteristic
297C4×C61#55411yescharacteristic
298Dic61—22no—
299Dic61—81no—
300#2957—611no—
301#2957—611no—
302C2×#443—611no—
303#2916—611no—
304#2916—611no—
305#1953—611no—
306#1953—611no—
307#1861—611no—
308#1861—611no—
309#1861—611no—
310#1861D6111yescharacteristic, complemented
311#1848—611no—
312#1814—611no—
313#1814—611no—
314#1772—611no—
315C2×C61—21no—
316C2×C61—21no—
317C2×C61—21no—
318C2×C61—42no—
319C2×C61D4×C4⋊C411yescharacteristic
320D61—42no—
321C2×#135—611no—
322#568—611no—
323#568—611no—
324#568—611no—
325#568C2×D6111yescharacteristic
326C2×#131—611no—
327C2×#131—611no—
328#444—1222no—
329#444—1222no—
330#443—612no—
331#443—612no—
332#443—612no—
333#443—612no—
334#443—1222no—
335#443—1222no—
336#430—611no—
337#430—611no—
338#430—611no—
339#430—1222no—
340#430—1222no—
341#430C2×D6111yescharacteristic
342C2×#108—611no—
343C2×#108—611no—
344C2×#108—611no—
345C2×#108—611no—
346C2×#108—611no—
347C2×#108—611no—
348C2×#108—1222no—
349C2×#108—1222no—
350#427—611no—
351#423—611no—
352#423C2×D6111yescharacteristic
353#421—611no—
354#421—611no—
355#421—611no—
356#421C2×D6111yescharacteristic
357C2×#104—611no—
358C2×#104C2×D6111yescharacteristic
359#405—611no—
360#405—611no—
361C2×C4×C8—611no—
362#344—611no—
363#344—611no—
364#344—611no—
365#344C2×D6111yescharacteristic
366#339—611no—
367#339C2×D6111yescharacteristic
368C61#2269411yescharacteristic, complemented, Sylow
369C2×#41—611no—
370C2×#41C22×D6111yescharacteristic
371#135—612no—
372#135—612no—
373#135—612no—
374#135—612no—
375C2×Q16—611no—
376C2×SD16—611no—
377C2×SD16—611no—
378C2×D8—611no—
379#131—22no—
380#131—41no—
381#131—612no—
382#131—612no—
383#131—612no—
384#131—612no—
385#131—1222no—
386#131—1221no—
387#131—1221no—
388#131—1221no—
389#131—1221no—
390#131—1222no—
391#131—1222no—
392#131—2441no—
393#131—2441no—
394#131—2441no—
395C2×M16—22no—
396C2×M16—611no—
397C2×M16—611no—
398C2×M16—611no—
399C2×M16—611no—
400C2×M16—1221no—
401C2×M16—1221no—
402C2×M16—1222no—
403C2×M16—1222no—
404C2×M16—1222no—
405C2×M16C22×D6111yescharacteristic
406C2×M16C22×D6111yescharacteristic
407C2×M16C22×D6111yescharacteristic
408C2×M16C22×D6111yescharacteristic
409C22×C8—611no—
410C22×C8—611no—
411C22×C8—1221no—
412C22×C8—1221no—
413#117—611no—
414#117C22×D6111yescharacteristic
415C2×C42C22×D6111yescharacteristic
416#108—22no—
417#108—612no—
418#108—612no—
419#108—612no—
420#108—612no—
421#108—1222no—
422#108—1221no—
423#108—1221no—
424#108—1221no—
425#108—1221no—
426#108—1221no—
427#108—1221no—
428#108—1221no—
429#108—1221no—
430#108—1222no—
431#108—1222no—
432#107—1221no—
433#106—1221no—
434C4⋊C8—41no—
435C4⋊C8—2441no—
436#104—22no—
437#104—612no—
438#104—612no—
439#104—1222no—
440#104—1222no—
441#104—1222no—
442#104C4×D6112yes—
443#104C4×D6112yes—
444#97—1221no—
445#97—1221no—
446C4×C8—611no—
447C4×C8—611no—
448C4×C8—611no—
449C4×C8—611no—
450C4×C8—1221no—
451C4×C8—1221no—
452#41—21no—
453#41—21no—
454#41—612no—
455#41—612no—
456#41—1221no—
457#41—1221no—
458#41C2×C4×D6112yes—
459#41D4×D6112yes—
460C2×Q8—611no—
461C2×Q8Q8×D6111yescharacteristic
462C2×D4—21no—
463C2×D4—611no—
464C2×D4—1221no—
465C2×D4Q8×D6111yescharacteristic
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—1221no—
469C22×C4C23×D6111yescharacteristic
470Q16—612no—
471Q16—612no—
472SD16—612no—
473SD16—612no—
474SD16—612no—
475SD16—612no—
476D8—612no—
477D8—612no—
478M16—21no—
479M16—21no—
480M16—21no—
481M16—21no—
482M16—21no—
483M16—21no—
484M16—42no—
485M16—612no—
486M16—612no—
487M16—1222no—
488M16—1222no—
489M16—1222no—
490M16—1222no—
491M16—1221no—
492M16—1221no—
493M16—1221no—
494M16—1221no—
495M16—1221no—
496M16—1221no—
497M16—2442no—
498M16—2442no—
499M16—2442no—
500M16C2×C4×D6112yes—
501M16D4×D6112yes—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—42no—
507C2×C8—611no—
508C2×C8—611no—
509C2×C8—611no—
510C2×C8—611no—
511C2×C8—611no—
512C2×C8—611no—
513C2×C8—611no—
514C2×C8—611no—
515C2×C8—611no—
516C2×C8—611no—
517C2×C8—611no—
518C2×C8—611no—
519C2×C8—1222no—
520C2×C8—1222no—
521C2×C8—1222no—
522C2×C8—1222no—
523C2×C8—1221no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—2442no—
528C2×C8—2442no—
529C2×C8—2442no—
530C4⋊C4—21no—
531C4⋊C4—1221no—
532C22⋊C4—21no—
533C22⋊C4—1221no—
534C42—21no—
535C42—21no—
536C42—1221no—
537C42C2×C4×D6111yescharacteristic
538C42C2×C4×D6111yescharacteristic
539C23—21no—
540C23—1221no—
541C23#1451982411yescharacteristic
542Q8—21no—
543Q8—612no—
544Q8—1221no—
545Q8C4⋊C4×D6112yes—
546D4—22no—
547D4—22no—
548D4—21no—
549D4—612no—
550D4—1222no—
551D4—1222no—
552D4—1221no—
553D4C4⋊C4×D6112yes—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—1221no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—2441no—
565C2×C4C22×C4×D6111yescharacteristic
566C2×C4C2×D4×D6111yescharacteristic
567C2×C4C2×D4×D6111yescharacteristic
568C2×C4C2×Q8×D6111yescharacteristic
569C2×C4#41×D6111yescharacteristic
570C2×C4#1451982211yescharacteristic
571C8—22no—
572C8—22no—
573C8—41no—
574C8—41no—
575C8—611no—
576C8—611no—
577C8—611no—
578C8—611no—
579C8—611no—
580C8—611no—
581C8—611no—
582C8—611no—
583C8—1222no—
584C8—1222no—
585C8—2441no—
586C8—2441no—
587C22—22no—
588C22—41no—
589C22—81no—
590C22—1222no—
591C22—4881no—
592C22#1684024111yescharacteristic
593C22#1684041011yescharacteristic
594C22#1684042511yescharacteristic
595C4—22no—
596C4—22no—
597C4—41no—
598C4—81no—
599C4—1222no—
600C4—4881no—
601C4C2×C4⋊C4×D6111yescharacteristic
602C4C4×D4×D6111yescharacteristic
603C4#1684039411yescharacteristic
604C4#1684046611yescharacteristic
605C2—21no—
606C2—21no—
607C2—21no—
608C2—42no—
609C2—2442no—
610C2#2144871711yescharacteristic
subgroupG/H
centerC4#16840394
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1861C2
socleC2×C61D4×C4⋊C4
prime radicalC2×#14519797C22
cosocleC22×C4×C61C24
hypercenter#1861D61
hypocenterC61#22694
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1861

Complements

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

NH ≅ G/Ntype
4. #21454429C2semidirect
5. #21454428C2semidirect
6. #21446608C2semidirect
7. #21446606C2semidirect
8. #21446061C2semidirect
9. #21446060C2semidirect
10. #21446058C2semidirect
11. #21445990C2semidirect
12. #21445872C2semidirect
13. #21445871C2semidirect
14. #21445697C2semidirect
15. C61×#1861C2semidirect
26. #16839876C4semidirect
27. #16839876C4semidirect
28. #16839876C4semidirect
29. #16839876C4semidirect
310. #1861D61semidirect
368. C61#22694semidirect

Maximal subgroups

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

subgroupindex
#214547342
#214547322
C2×#168398762
#214544292
#214544282
#214466082
#214466062
#214460612
#214460602
#214460582
#214459902
#214458722
#214458712
#214456972
C61×#18612
#2269461