Finite Group Databasebeta

← #29743448

Subgroups — #29743448

Order 15616.

740 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455670C211yesmaximal, characteristic, complemented
2#21455652C211yesmaximal, characteristic, complemented
3#21454652C211yesmaximal, characteristic
4#21454417C211yesmaximal, characteristic, complemented
5#21447347C211yesmaximal, characteristic, complemented
6#21447328C211yesmaximal, characteristic, complemented
7#21447179C211yesmaximal, characteristic, complemented
8#21447163C211yesmaximal, characteristic, complemented
9#21446603C211yesmaximal, characteristic, complemented
10#21446421C211yesmaximal, characteristic, complemented
11#21446329C211yesmaximal, characteristic
12#21446092C211yesmaximal, characteristic, complemented
13#21446047C211yesmaximal, characteristic, complemented
14#21446028C211yesmaximal, characteristic, complemented
15C61×#1883C211yesmaximal, characteristic
16C2×#14519784C2211yescharacteristic
17#16840390C2211yescharacteristic
18#16840245C2211yescharacteristic
19#16840237C2211yescharacteristic
20#16840069—21no—
21#16840064—21no—
22#16840062—21no—
23#16840060—21no—
24#16840057—21no—
25#16840050—21no—
26#16840042—21no—
27#16840040—21no—
28#16839874—21no—
29#16839863—21no—
30SD16×Dic61—21no—
31D8×Dic61—21no—
32C2×#14519712C2211yescharacteristic
33#16839805—21no—
34#16839805C2211yescharacteristic
35#16839804C2211yescharacteristic
36#16839804C2211yescharacteristic
37#16839802—21no—
38#16839802C2211yescharacteristic
39#16839802C2211yescharacteristic
40#16839798—21no—
41#16839798C2211yescharacteristic
42C2×#14519710—21no—
43C2×#14519710C2211yescharacteristic
44C2×#14519710C2211yescharacteristic
45#16839790C2211yescharacteristic
46#16839789C2211yescharacteristic
47#16839763C2211yescharacteristic
48#16839758C2211yescharacteristic
49C2×#14519698C2211yescharacteristic
50C2×#14519698C2211yescharacteristic
51C2×#14519694C2211yescharacteristic
52#16839720C2211yescharacteristic
53#16839632C2211yescharacteristic
54#16839530C2211yescharacteristic
55#16839530C2211yescharacteristic
56#16839526C2211yescharacteristic
57#16839522C2211yescharacteristic
58#16839522C2211yescharacteristic
59#16839517C2211yescharacteristic
60C2×AGL(1,ℤ/8)×C61C2211yescharacteristic
61C61×#520C2211yescharacteristic
62C2×#104×C61C2211yescharacteristic
63C61×#418C2211yescharacteristic
64C61×#390C2211yescharacteristic
65C61×#343C2211yescharacteristic
66C61×#339C2211yescharacteristic
67#14519784—22no—
68#14519784—22no—
69C2×#11758194—21no—
70C2×#11758194C2311yescharacteristic
71C2×#11758194C2311yescharacteristic
72C22×#92418—21no—
73C22×#92418C2311yescharacteristic
74C22×#92418C2311yescharacteristic
75#14519777—21no—
76#14519776—21no—
77#14519769—21no—
78#14519768—21no—
79#14519754—41no—
80Q8×Dic61—21no—
81D4×Dic61—21no—
82D4×Dic61—21no—
83D4×Dic61—21no—
84D4×Dic61—21no—
85D4×Dic61—21no—
86D4×Dic61—41no—
87#14519736—21no—
88#14519736C2311yescharacteristic
89#14519736C2311yescharacteristic
90C2×C4⋊Dic61—21no—
91C2×C4⋊Dic61—41no—
92C2×C4⋊Dic61C2311yescharacteristic
93C2×C4⋊Dic61C2311yescharacteristic
94C2×C22⋊Dic61—41no—
95#14519712—22no—
96#14519711—21no—
97#14519711—21no—
98#14519710—21no—
99#14519710—21no—
100#14519710—21no—
101#14519710—21no—
102#14519710—21no—
103#14519710—21no—
104#14519710—21no—
105#14519710D411yescharacteristic
106#14519710D411yescharacteristic
107#14519709—21no—
108#14519709—21no—
109#14519708—21no—
110#14519708—21no—
111#14519708—21no—
112#14519708D411yescharacteristic
113#14519708D411yescharacteristic
114C4⋊#92418—21no—
115C4⋊#92418—21no—
116#14519706—21no—
117#14519706—21no—
118#14519701—21no—
119#14519701D411yescharacteristic
120#14519701D411yescharacteristic
121#14519698—21no—
122#14519698—21no—
123#14519698—21no—
124#14519698—21no—
125#14519698—21no—
126#14519698D411yescharacteristic
127#14519698D411yescharacteristic
128#14519694—22no—
129#14519687—21no—
130C8×Dic61—21no—
131C2×#41×C61C2311yescharacteristic
132C22×D4×C61C2311yescharacteristic
133AGL(1,ℤ/8)×C61—22no—
134AGL(1,ℤ/8)×C61Q814yescomplemented
135C2×SD16×C61—21no—
136C2×D8×C61—21no—
137C2×M16×C61C2311yescharacteristic
138C2×M16×C61C2311yescharacteristic
139C4×Q8×C61—21no—
140C4×D4×C61—21no—
141C4×D4×C61—41no—
142#117×C61—21no—
143#117×C61C2311yescharacteristic
144C2×C4⋊C4×C61—21no—
145C2×C4⋊C4×C61C2311yescharacteristic
146C2×C22⋊C4×C61—41no—
147C2×C42×C61C2311yescharacteristic
148#104×C61—22no—
149#102×C61—21no—
150#102×C61—21no—
151C24×C61—21no—
152#41×C61—21no—
153#41×C61—21no—
154#41×C61C2×Q812yes—
155#41×C61#4112yes—
156C2×Q8×C61C2×D411yescharacteristic
157C2×D4×C61—21no—
158C2×D4×C61—21no—
159C2×D4×C61—21no—
160C2×D4×C61—21no—
161C2×D4×C61—21no—
162C2×D4×C61—21no—
163C2×D4×C61C2×D411yescharacteristic
164C2×D4×C61C2×Q812yes—
165C22×C4×C61—21no—
166C22×C4×C61—21no—
167C22×C4×C61—21no—
168C22×C4×C61—41no—
169C22×C4×C61C2411yescharacteristic
170C22×Dic61—21no—
171C22×Dic61—21no—
172C22×Dic61—41no—
173C22×Dic61—41no—
174C22×Dic61—41no—
175C2×Dic122—41no—
176SD16×C61—24no—
177D8×C61—24no—
178M16×C61—21no—
179M16×C61—21no—
180M16×C61—21no—
181M16×C61C2×Q812yes—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—41no—
187C4⋊C4×C61C2×D411yescharacteristic
188C4⋊C4×C61C2×D411yescharacteristic
189C22⋊C4×C61—21no—
190C22⋊C4×C61—41no—
191C22⋊C4×C61—82no—
192C42×C61—21no—
193C42×C61—21no—
194C42×C61—21no—
195C42×C61—41no—
196C22⋊Dic61—21no—
197C22⋊Dic61—21no—
198C22⋊Dic61—41no—
199C22⋊Dic61—41no—
200C22⋊Dic61—41no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—41no—
206C4⋊Dic61—41no—
207C4⋊Dic61—82no—
208C4⋊Dic61C2×D411yescharacteristic
209C4⋊Dic61C2×D411yescharacteristic
210C4⋊Dic61C2×D411yescharacteristic
211C4⋊Dic61C2×D411yescharacteristic
212#11758196—41no—
213#11758196—41no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C4×Dic61—41no—
217#11758194—24no—
218#11758194—21no—
219#11758194—21no—
220#11758194—21no—
221#11758194#4112yes—
222C2×#92418—24no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418C2×D411yescharacteristic
230C2×#92418C2×D411yescharacteristic
231C2×#92418C2×D411yescharacteristic
232C2×#92418C2×D411yescharacteristic
233C23×C61—21no—
234C23×C61—22no—
235C23×C61—41no—
236C23×C61—41no—
237C23×C61—81no—
238C23×C61—81no—
239C23×C61H(5,𝔽2)11yescharacteristic
240Q8×C61—21no—
241Q8×C61C22⋊Q812yes—
242D4×C61—22no—
243D4×C61—22no—
244D4×C61—22no—
245D4×C61—21no—
246D4×C61—22no—
247D4×C61—41no—
248D4×C61—41no—
249D4×C61C22⋊Q812yes—
250C2×C4×C61—21no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—22no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—81no—
260C2×C4×C61—161no—
261C2×C4×C61C22×D411yescharacteristic
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61C22×Q811yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C612−1+411yescharacteristic
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—81no—
274C2×Dic61—81no—
275C2×Dic61—161no—
276C2×Dic61—161no—
277C2×Dic61—161no—
278Dic122—161no—
279C8×C61—22no—
280C8×C61—41no—
281#92418—21no—
282#92418—22no—
283#92418—21no—
284#92418—21no—
285#92418—21no—
286#92418—41no—
287#26400—611nomaximal, Sylow
288C22×C61—21no—
289C22×C61—21no—
290C22×C61—22no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61—42no—
294C22×C61—82no—
295C22×C61—81no—
296C22×C61—81no—
297C22×C61—161no—
298C22×C61#53911yescharacteristic
299C22×C61D4211yescharacteristic
300C22×C61#55411yescharacteristic
301C4×C61—22no—
302C4×C61—42no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61—81no—
306C4×C61C2×C22⋊Q811yescharacteristic
307C4×C61#52511yescharacteristic
308C4×C61#54711yescharacteristic
309C4×C61D4×Q811yescharacteristic
310Dic61—41no—
311Dic61—41no—
312Dic61—41no—
313Dic61—41no—
314Dic61—81no—
315Dic61—81no—
316#3083—611no—
317#3081—611no—
318#2946—611no—
319#2913—611no—
320#2093—611no—
321#2089—611no—
322#2063—611no—
323#2060—611no—
324#1952—611no—
325#1922—611no—
326#1907—611no—
327#1883D6111yescharacteristic
328#1866—611no—
329#1859—611no—
330#1856—611no—
331C2×C61—21no—
332C2×C61—21no—
333C2×C61—42no—
334C2×C61—82no—
335C2×C61#244911yescharacteristic
336C2×C61#328911yescharacteristic
337C2×C61#329211yescharacteristic
338C2×AGL(1,ℤ/8)C2×D6111yescharacteristic
339C2×#131—611no—
340#544—611no—
341#520—611no—
342#520C2×D6111yescharacteristic
343#518—611no—
344#479—1221no—
345#478—1221no—
346#478—1221no—
347#478—1221no—
348#477—1221no—
349#476—1221no—
350#474—1221no—
351#474—1221no—
352#442—1221no—
353#440—1221no—
354C4×SD16—1221no—
355C4×D8—1221no—
356C2×#108—611no—
357#428—611no—
358#428—611no—
359#428—611no—
360#428—611no—
361#428—611no—
362#428—1221no—
363#428—1221no—
364#427—611no—
365#427—1221no—
366C2×#107—611no—
367C2×#107—611no—
368C2×#107—1221no—
369#424—611no—
370#424—611no—
371C2×#104C2×D6111yescharacteristic
372#418C2×D6111yescharacteristic
373#417—611no—
374#416—611no—
375C2×#102—611no—
376C2×#102—611no—
377C2×#100—611no—
378#405—611no—
379#390C2×D6111yescharacteristic
380#384—611no—
381#343—611no—
382#343—611no—
383#343C2×D6111yescharacteristic
384#341—611no—
385#339—611no—
386#339—611no—
387#339C2×D6111yescharacteristic
388#336—611no—
389C61#2640011yescharacteristic, complemented, Sylow
390C2×#41C22×D6111yescharacteristic
391C22×D4C22×D6111yescharacteristic
392AGL(1,ℤ/8)—22no—
393AGL(1,ℤ/8)Dic12214yescomplemented
394C2×SD16—21no—
395C2×D8—21no—
396#131—1222no—
397#131—1222no—
398C2×M16—611no—
399C2×M16—611no—
400C2×M16—1221no—
401C2×M16C22×D6111yescharacteristic
402C2×M16C22×D6111yescharacteristic
403C22×C8—611no—
404C22×C8—611no—
405C22×C8—1221no—
406C4⋊Q8—1221no—
407C4⋊Q8—1221no—
408#125—1221no—
409#125—1221no—
410C22⋊Q8—2441no—
411C4×Q8—21no—
412C4×Q8—1221no—
413C4×D4—21no—
414C4×D4—41no—
415C4×D4—1221no—
416C4×D4—1221no—
417C4×D4—1221no—
418C4×D4—1221no—
419C4×D4—1221no—
420C4×D4—2441no—
421#117—21no—
422#117—611no—
423#117—611no—
424#117—1221no—
425#117C22×D6111yescharacteristic
426C2×C4⋊C4—21no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—2441no—
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C22⋊C4—41no—
433C2×C22⋊C4—2441no—
434C2×C42C22×D6111yescharacteristic
435#108—1222no—
436#107—611no—
437#107—611no—
438#107—1221no—
439#107—1221no—
440#107—1221no—
441#107—1221no—
442#107—1221no—
443#107—1221no—
444#107—1221no—
445#107—1221no—
446#107—1221no—
447#106—611no—
448#106—611no—
449#106—1221no—
450#106—1221no—
451#106—1221no—
452#106—1221no—
453#106—1221no—
454C4⋊C8—1221no—
455C4⋊C8—1221no—
456C4⋊C8—1221no—
457C4⋊C8—1221no—
458#104—22no—
459#103—611no—
460#103—611no—
461#103—1221no—
462#102—21no—
463#102—21no—
464#102—611no—
465#102—611no—
466#102—1221no—
467#102—1221no—
468#102—1221no—
469#102—1221no—
470#102—1221no—
471#100—1222no—
472#97—1221no—
473C4×C8—1221no—
474C24—21no—
475#41—21no—
476#41—21no—
477#41C2×Dic12212yes—
478#41#1175822412yes—
479C2×Q8—2441no—
480C2×Q8D4×D6111yescharacteristic
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4C2×Dic12212yes—
488C2×D4D4×D6111yescharacteristic
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—41no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—2441no—
496C22×C4—2441no—
497C22×C4—2441no—
498C22×C4C23×D6111yescharacteristic
499SD16—24no—
500D8—24no—
501M16—21no—
502M16—21no—
503M16—21no—
504M16—612no—
505M16—1224no—
506M16—1221no—
507M16—1221no—
508M16—1221no—
509M16C2×Dic12212yes—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—611no—
513C2×C8—611no—
514C2×C8—611no—
515C2×C8—611no—
516C2×C8—1224no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—41no—
526C4⋊C4—611no—
527C4⋊C4—611no—
528C4⋊C4—611no—
529C4⋊C4—611no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—4882no—
535C4⋊C4—2441no—
536C4⋊C4—2441no—
537C4⋊C4—2441no—
538C4⋊C4—2441no—
539C4⋊C4D4×D6111yescharacteristic
540C4⋊C4D4×D6111yescharacteristic
541C22⋊C4—21no—
542C22⋊C4—41no—
543C22⋊C4—82no—
544C22⋊C4—1221no—
545C22⋊C4—1221no—
546C22⋊C4—2441no—
547C22⋊C4—2441no—
548C22⋊C4—2441no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—41no—
553C42—1221no—
554C42—1221no—
555C42—2441no—
556C23—21no—
557C23—22no—
558C23—41no—
559C23—41no—
560C23—81no—
561C23—81no—
562C23#1451982211yescharacteristic
563Q8—21no—
564Q8—9761no—
565Q8#1451975412yes—
566D4—22no—
567D4—22no—
568D4—22no—
569D4—21no—
570D4—22no—
571D4—41no—
572D4—41no—
573D4#1451975412yes—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—81no—
584C2×C4—161no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—9761no—
590C2×C4—9761no—
591C2×C4—9761no—
592C2×C4—4881no—
593C2×C4—4881no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4C2×D4×D6111yescharacteristic
597C2×C4C2×D4×D6111yescharacteristic
598C2×C4C22×Dic12211yescharacteristic
599C2×C4#41×D6111yescharacteristic
600C2×C4C2×#1175822411yescharacteristic
601C2×C4#1451982411yescharacteristic
602C8—22no—
603C8—41no—
604C8—1221no—
605C8—1221no—
606C8—1222no—
607C8—1221no—
608C8—1221no—
609C8—2441no—
610C22—21no—
611C22—21no—
612C22—22no—
613C22—41no—
614C22—41no—
615C22—42no—
616C22—82no—
617C22—81no—
618C22—81no—
619C22—161no—
620C22#1684035711yescharacteristic
621C22#1684039411yescharacteristic
622C22#1684047111yescharacteristic
623C4—22no—
624C4—42no—
625C4—41no—
626C4—41no—
627C4—81no—
628C4—4881no—
629C4—4881no—
630C4—2441no—
631C4—2441no—
632C4—2441no—
633C4—2441no—
634C4C2×#1451975411yescharacteristic
635C4#1684027211yescharacteristic
636C4#1684041011yescharacteristic
637C4D4×Dic12211yescharacteristic
638C2—21no—
639C2—21no—
640C2—42no—
641C2—82no—
642C2#2144963211yescharacteristic
643C2#2145759611yescharacteristic
644C2#2145761511yescharacteristic
subgroupG/H
centerC22#16840394
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1883C2
socleC22×C61D42
prime radicalC2×AGL(1,ℤ/8)×C61C22
hypercenter#1883D61
hypocenterC61#26400
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1883

Complements

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

NH ≅ G/Ntype
1. #21455670C2semidirect
2. #21455652C2semidirect
4. #21454417C2semidirect
5. #21447347C2semidirect
6. #21447328C2semidirect
7. #21447179C2semidirect
8. #21447163C2semidirect
9. #21446603C2semidirect
10. #21446421C2semidirect
12. #21446092C2semidirect
13. #21446047C2semidirect
14. #21446028C2semidirect
134. AGL(1,ℤ/8)×C61Q8semidirect
389. C61#26400semidirect
393. AGL(1,ℤ/8)Dic122semidirect

Maximal subgroups

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

subgroupindex
#214556702
#214556522
#214546522
#214544172
#214473472
#214473282
#214471792
#214471632
#214466032
#214464212
#214463292
#214460922
#214460472
#214460282
C61×#18832
#2640061