Finite Group Databasebeta

← #29743503

Subgroups — #29743503

Order 15616.

706 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455669C211yesmaximal, characteristic
2#21455661C211yesmaximal, characteristic, complemented
3#21454662C211yesmaximal, characteristic
4#21454413C211yesmaximal, characteristic
5#21447367C211yesmaximal, characteristic, complemented
6#21447332C211yesmaximal, characteristic, complemented
7#21447182C211yesmaximal, characteristic, complemented
8#21447168C211yesmaximal, characteristic, complemented
9#21446601C211yesmaximal, characteristic, complemented
10#21446416C211yesmaximal, characteristic, complemented
11#21446331C211yesmaximal, characteristic
12#21446191C211yesmaximal, characteristic
13#21446095C211yesmaximal, characteristic
14#21446053C211yesmaximal, characteristic
15C61×#1857C211yesmaximal, characteristic, complemented
16C2×#14519808C2211yescharacteristic
17C2×#14519785C2211yescharacteristic
18#16840390C2211yescharacteristic
19#16840243C2211yescharacteristic
20#16840242C2211yescharacteristic
21#16840072—21no—
22#16840069—21no—
23#16840069—21no—
24#16840061—21no—
25#16840057—21no—
26#16840056—21no—
27#16840047—21no—
28#16840041—21no—
29#16839867—21no—
30#16839860—21no—
31C4×#11758202—21no—
32C4×#11758201—21no—
33C2×#14519712C2211yescharacteristic
34#16839805C2211yescharacteristic
35#16839805C2211yescharacteristic
36#16839804C2211yescharacteristic
37#16839803—21no—
38#16839801—21no—
39#16839799—21no—
40#16839798C2211yescharacteristic
41C2×#14519710C2211yescharacteristic
42C2×#14519708C2211yescharacteristic
43#106×D61—21no—
44#16839790C2211yescharacteristic
45#16839789C2211yescharacteristic
46C2×#14519705C2211yescharacteristic
47#16839771C2211yescharacteristic
48#16839764C2211yescharacteristic
49#16839756C2211yescharacteristic
50C2×#14519701C2211yescharacteristic
51C2×#14519701C2211yescharacteristic
52#16839718C2211yescharacteristic
53#16839628C2211yescharacteristic
54#16839531C2211yescharacteristic
55#16839530C2211yescharacteristic
56#16839530C2211yescharacteristic
57#16839526C2211yescharacteristic
58#16839522C2211yescharacteristic
59#16839517C2211yescharacteristic
60C61×#519C2211yescharacteristic
61C61×#428C2211yescharacteristic
62C61×#428C2211yescharacteristic
63C2×#101×C61C2211yescharacteristic
64C61×#391C2211yescharacteristic
65C61×#339C2211yescharacteristic
66C61×#339C2211yescharacteristic
67C2×#11758222C2311yescharacteristic
68#14519808—22no—
69#14519808Q814yescomplemented
70C2×#11758202—21no—
71C2×#11758201—21no—
72#14519785—22no—
73#14519785—22no—
74C2×#11758194C2311yescharacteristic
75C2×#11758194C2311yescharacteristic
76C2×#11758189—21no—
77C22×#92418C2311yescharacteristic
78C22×#92418C2311yescharacteristic
79C2×C8×D61—21no—
80#14519777—21no—
81#14519776—21no—
82#14519769—21no—
83#14519768—21no—
84#14519754—41no—
85C4×Dic122—21no—
86#14519742—21no—
87C4×D244—21no—
88#14519739—21no—
89C4×C22⋊D61—41no—
90#14519737—41no—
91#14519736C2311yescharacteristic
92#14519735—21no—
93#14519735—21no—
94C2×C4⋊Dic61C2311yescharacteristic
95C2×C4⋊Dic61C2311yescharacteristic
96C4⋊C4×D61—21no—
97C4⋊C4×D61—21no—
98#14519712—22no—
99#14519711—21no—
100#14519710—21no—
101#14519710—21no—
102#14519710—21no—
103#14519710—21no—
104#14519710D411yescharacteristic
105#14519710D411yescharacteristic
106#14519709—21no—
107#14519708—21no—
108#14519708—21no—
109#14519708—21no—
110#14519708—21no—
111#14519708D411yescharacteristic
112#14519708D411yescharacteristic
113C4⋊#92418—21no—
114C4⋊#92418—21no—
115#14519706—21no—
116#14519706—21no—
117#14519705—22no—
118#14519702—21no—
119#14519701—21no—
120#14519701—21no—
121#14519701—21no—
122#14519701—21no—
123#14519701—21no—
124#14519701—21no—
125#14519700D411yescharacteristic
126#14519700D411yescharacteristic
127#14519699—21no—
128#14519697D411yescharacteristic
129#14519697D411yescharacteristic
130#14519688—21no—
131C4×#92418—21no—
132#14519684—41no—
133C22×Q8×C61C2311yescharacteristic
134C2×M16×C61C2311yescharacteristic
135C2×M16×C61C2311yescharacteristic
136C4×Q8×C61—21no—
137C4×Q8×C61—21no—
138C4×Q8×C61—21no—
139C4×Q8×C61—21no—
140#117×C61—41no—
141#117×C61C2311yescharacteristic
142#117×C61C2311yescharacteristic
143C2×C4⋊C4×C61C2311yescharacteristic
144C2×C42×C61C2311yescharacteristic
145#107×C61—21no—
146#107×C61—21no—
147#106×C61—21no—
148#106×C61—21no—
149#101×C61—22no—
150#95×C61—41no—
151C2×Q8×C61—21no—
152C2×Q8×C61—21no—
153C2×Q8×C61—21no—
154C2×Q8×C61—21no—
155C2×Q8×C61—21no—
156C2×Q8×C61C2×Q812yes—
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61C2411yescharacteristic
161C2×C22⋊D61—21no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164#11758222—21no—
165#11758222—21no—
166#11758222C2×Q812yes—
167#11758222#4112yes—
168C2×D244C2×D411yescharacteristic
169C2×C4×D61—21no—
170C2×C4×D61—41no—
171C2×C4×D61—41no—
172C2×Dic122—41no—
173C2×Dic122C2×D411yescharacteristic
174M16×C61—21no—
175M16×C61—21no—
176M16×C61—21no—
177M16×C61#4112yes—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—41no—
183C4⋊C4×C61—41no—
184C4⋊C4×C61C2×D411yescharacteristic
185C4⋊C4×C61C2×D411yescharacteristic
186C4⋊C4×C61C2×D411yescharacteristic
187C4⋊C4×C61C2×D411yescharacteristic
188C22⋊C4×C61—21no—
189C22⋊C4×C61—21no—
190C42×C61—21no—
191C42×C61—21no—
192C42×C61—21no—
193C42×C61—21no—
194C42×C61—41no—
195C42×C61—41no—
196C22⋊Dic61—21no—
197#11758202—24no—
198#11758201—24no—
199#11758198—41no—
200#11758198—41no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—41no—
206C4⋊Dic61C2×D411yescharacteristic
207C4⋊Dic61C2×D411yescharacteristic
208#11758196—41no—
209#11758196—41no—
210#11758196—41no—
211#11758196—41no—
212C4×Dic61—21no—
213C4×Dic61—41no—
214C4×Dic61—41no—
215#11758194—21no—
216#11758194—21no—
217#11758194—21no—
218#11758194C2×Q812yes—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418C2×D411yescharacteristic
226C2×#92418C2×D411yescharacteristic
227C2×#92418C2×D411yescharacteristic
228C2×#92418C2×D411yescharacteristic
229#11758189—24no—
230C8×D61—24no—
231C23×C61H(5,𝔽2)11yescharacteristic
232C22×D61—21no—
233Q8×C61—22no—
234Q8×C61—22no—
235Q8×C61—41no—
236Q8×C61—41no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—81no—
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×Q811yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C612−1+411yescharacteristic
255C22⋊D61—22no—
256C22⋊D61—22no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—81no—
263C2×Dic61—81no—
264D244—21no—
265D244C22⋊Q812yes—
266C4×D61—22no—
267C4×D61—22no—
268C4×D61—161no—
269C4×D61—161no—
270Dic122—21no—
271Dic122—161no—
272Dic122C22⋊Q812yes—
273C8×C61—22no—
274C8×C61—41no—
275#92418—22no—
276#92418—21no—
277#92418—21no—
278#92418—21no—
279#92418—21no—
280#92418—41no—
281#26404—611nomaximal, Sylow
282C22×C61—21no—
283C22×C61—21no—
284C22×C61#53911yescharacteristic
285C22×C61D4211yescharacteristic
286C22×C61#55411yescharacteristic
287C2×D61—22no—
288C2×D61—41no—
289C2×D61—41no—
290C4×C61—42no—
291C4×C61—41no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61—81no—
296C4×C61—81no—
297C4×C61C2×C22⋊Q811yescharacteristic
298C4×C61#52511yescharacteristic
299C4×C61#54711yescharacteristic
300C4×C61D4×Q811yescharacteristic
301Dic61—22no—
302Dic61—41no—
303Dic61—41no—
304Dic61—81no—
305Dic61—81no—
306#3083—611no—
307#3082—611no—
308#2947—611no—
309#2912—611no—
310#2097—611no—
311#2090—611no—
312#2064—611no—
313#2061—611no—
314#1952—611no—
315#1922—611no—
316#1908—611no—
317#1882—611no—
318#1866—611no—
319#1860—611no—
320#1857D6111yescharacteristic, complemented
321C2×C61—21no—
322C2×C61—21no—
323C2×C61#244911yescharacteristic
324C2×C61#328911yescharacteristic
325C2×C61#329011yescharacteristic
326D61—42no—
327C2×#137—611no—
328C2×#131—611no—
329#544—611no—
330#520—611no—
331#520—611no—
332#519C2×D6111yescharacteristic
333#479—1221no—
334#479—1221no—
335#479—1221no—
336#478—1221no—
337#477—1221no—
338#477—1221no—
339#475—1221no—
340#474—1221no—
341#441—1221no—
342#440—1221no—
343C4×Q16—1221no—
344C4×SD16—1221no—
345C2×#108—611no—
346#428—611no—
347#428—611no—
348#428—611no—
349#428—1221no—
350#428—1221no—
351#428C2×D6111yescharacteristic
352#428C2×D6111yescharacteristic
353#427—611no—
354#427—1221no—
355C2×#107—611no—
356C2×#106—611no—
357C2×#106—1221no—
358#424—611no—
359#424—611no—
360C2×#104—611no—
361#419—611no—
362#417—611no—
363#416—611no—
364C2×#103—611no—
365C2×#103—611no—
366C2×#101C2×D6111yescharacteristic
367#405—611no—
368#391C2×D6111yescharacteristic
369#382—611no—
370#343—611no—
371#343—611no—
372#343—611no—
373#341—611no—
374#339—611no—
375#339C2×D6111yescharacteristic
376#339C2×D6111yescharacteristic
377#336—611no—
378C61#2640411yescharacteristic, complemented, Sylow
379C2×#41—611no—
380C22×Q8C22×D6111yescharacteristic
381#137—614no—
382#137—1222no—
383C2×Q16—1221no—
384C2×SD16—1221no—
385#131—1222no—
386#131—1222no—
387C2×M16—611no—
388C2×M16—611no—
389C2×M16—1221no—
390C2×M16C22×D6111yescharacteristic
391C2×M16C22×D6111yescharacteristic
392C22×C8—611no—
393C22×C8—611no—
394C22×C8—1221no—
395C4⋊Q8—1221no—
396C4⋊Q8—1221no—
397#125—1221no—
398#125—1221no—
399C22⋊Q8—2441no—
400C4×Q8—21no—
401C4×Q8—21no—
402C4×Q8—21no—
403C4×Q8—21no—
404C4×Q8—1221no—
405C4×Q8—1221no—
406C4×D4—1221no—
407C4×D4—1221no—
408C4×D4—2441no—
409C4×D4—2441no—
410#117—41no—
411#117—611no—
412#117—1221no—
413#117—1221no—
414#117C22×D6111yescharacteristic
415#117C22×D6111yescharacteristic
416C2×C4⋊C4—611no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4C22×D6111yescharacteristic
421C2×C42C22×D6111yescharacteristic
422#108—1222no—
423#107—21no—
424#107—21no—
425#107—611no—
426#107—611no—
427#107—1221no—
428#107—1221no—
429#107—1221no—
430#107—1221no—
431#107—1221no—
432#106—21no—
433#106—21no—
434#106—611no—
435#106—611no—
436#106—1221no—
437#106—1221no—
438#106—1221no—
439#106—1221no—
440#106—1221no—
441C4⋊C8—1221no—
442C4⋊C8—1221no—
443C4⋊C8—1221no—
444C4⋊C8—1221no—
445#104—1222no—
446#103—611no—
447#103—611no—
448#103—1221no—
449#103—1221no—
450#103—1221no—
451#103—1221no—
452#103—1221no—
453#103—1221no—
454#103—1221no—
455#102—611no—
456#102—611no—
457#102—1221no—
458#101—22no—
459#97—1221no—
460C4×C8—1221no—
461#95—41no—
462#95—2441no—
463#41—612no—
464#41—612no—
465#41—1221no—
466#41—1221no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—611no—
473C2×Q8—2441no—
474C2×Q8Q8×D6112yes—
475C2×D4—611no—
476C2×D4—1221no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—2441no—
484C22×C4—2441no—
485C22×C4C23×D6111yescharacteristic
486Q16—1224no—
487SD16—1224no—
488M16—21no—
489M16—21no—
490M16—21no—
491M16—612no—
492M16—1224no—
493M16—1221no—
494M16—1221no—
495M16—1221no—
496M16#1175822212yes—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—611no—
500C2×C8—611no—
501C2×C8—611no—
502C2×C8—611no—
503C2×C8—1224no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—611no—
515C4⋊C4—611no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—2441no—
521C4⋊C4—2441no—
522C4⋊C4—2441no—
523C4⋊C4—2441no—
524C4⋊C4—2441no—
525C4⋊C4D4×D6111yescharacteristic
526C4⋊C4D4×D6111yescharacteristic
527C4⋊C4D4×D6111yescharacteristic
528C4⋊C4D4×D6111yescharacteristic
529C22⋊C4—21no—
530C22⋊C4—21no—
531C22⋊C4—1221no—
532C22⋊C4—2441no—
533C22⋊C4—2441no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—41no—
539C42—41no—
540C42—1221no—
541C42—2441no—
542C42—2441no—
543C23—1221no—
544C23#1451982211yescharacteristic
545Q8—22no—
546Q8—22no—
547Q8—41no—
548Q8—41no—
549Q8—612no—
550Q8—1221no—
551Q8—9761no—
552D4—612no—
553D4—1222no—
554D4—1222no—
555D4—1221no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—81no—
568C2×C4—1221no—
569C2×C4—1222no—
570C2×C4—1222no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—9761no—
574C2×C4—9761no—
575C2×C4—4881no—
576C2×C4—4881no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4C2×D4×D6111yescharacteristic
580C2×C4C2×D4×D6111yescharacteristic
581C2×C4C2×Q8×D6111yescharacteristic
582C2×C4#41×D6111yescharacteristic
583C2×C4C2×#1175822211yescharacteristic
584C2×C4#1451982411yescharacteristic
585C8—22no—
586C8—41no—
587C8—1222no—
588C8—1221no—
589C8—1221no—
590C8—1221no—
591C8—1221no—
592C8—2441no—
593C22—21no—
594C22—21no—
595C22—1222no—
596C22—2441no—
597C22—2441no—
598C22#1684035511yescharacteristic
599C22#1684039411yescharacteristic
600C22#1684046711yescharacteristic
601C4—42no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—81no—
607C4—81no—
608C4—1222no—
609C4—4881no—
610C4—4881no—
611C4—2441no—
612C4—2441no—
613C4C22⋊Q8×D6111yescharacteristic
614C4#1684027111yescharacteristic
615C4#1684041711yescharacteristic
616C4#1684042511yescharacteristic
617C2—21no—
618C2—21no—
619C2—2442no—
620C2#2144962711yescharacteristic
621C2#2145758911yescharacteristic
622C2#2145759811yescharacteristic
subgroupG/H
centerC22#16840394
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1857C2
socleC22×C61D42
prime radicalC2×#11758222C23
hypercenter#1857D61
hypocenterC61#26404
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1857

Complements

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

NH ≅ G/Ntype
2. #21455661C2semidirect
5. #21447367C2semidirect
6. #21447332C2semidirect
7. #21447182C2semidirect
8. #21447168C2semidirect
9. #21446601C2semidirect
10. #21446416C2semidirect
15. C61×#1857C2semidirect
69. #14519808Q8semidirect
320. #1857D61semidirect
378. C61#26404semidirect

Maximal subgroups

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

subgroupindex
#214556692
#214556612
#214546622
#214544132
#214473672
#214473322
#214471822
#214471682
#214466012
#214464162
#214463312
#214461912
#214460952
#214460532
C61×#18572
#2640461