Finite Group Databasebeta

← #29743513

Subgroups — #29743513

Order 15616.

732 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455679C211yesmaximal, characteristic, complemented
2#21455650C211yesmaximal, characteristic, complemented
3#21454417C211yesmaximal, characteristic, complemented
4#21447342C211yesmaximal, characteristic, complemented
5#21447327C211yesmaximal, characteristic, complemented
6#21447177C211yesmaximal, characteristic, complemented
7#21447167C211yesmaximal, characteristic, complemented
8#21446599C211yesmaximal, characteristic, complemented
9#21446418C211yesmaximal, characteristic, complemented
10#21446329C211yesmaximal, characteristic
11#21446197C211yesmaximal, characteristic
12#21446096C211yesmaximal, characteristic, complemented
13#21446052C211yesmaximal, characteristic, complemented
14#21446029C211yesmaximal, characteristic, complemented
15C61×#2946C211yesmaximal, characteristic
16C2×#14519784C2211yescharacteristic
17#16840390C2211yescharacteristic
18#16840245C2211yescharacteristic
19#16840067—21no—
20#16840064—21no—
21#16840062—21no—
22#16840062—21no—
23#16840052—21no—
24#16840050—21no—
25#16840045—21no—
26#16840042—21no—
27C2×#14519713C2211yescharacteristic
28#16839805—21no—
29#16839805C2211yescharacteristic
30#16839805C2211yescharacteristic
31#16839805C2211yescharacteristic
32#16839802—21no—
33#16839802C2211yescharacteristic
34#16839798—21no—
35#16839798C2211yescharacteristic
36C2×#14519710—21no—
37C2×#14519710C2211yescharacteristic
38C2×#14519709C2211yescharacteristic
39#16839789C2211yescharacteristic
40#16839788C2211yescharacteristic
41C2×#14519704C2211yescharacteristic
42#16839767C2211yescharacteristic
43#16839763C2211yescharacteristic
44C2×#14519698C2211yescharacteristic
45C2×#14519694C2211yescharacteristic
46#16839665C2211yescharacteristic
47#16839627C2211yescharacteristic
48#16839531C2211yescharacteristic
49#16839530C2211yescharacteristic
50#16839530C2211yescharacteristic
51#16839526C2211yescharacteristic
52#16839523C2211yescharacteristic
53#16839521C2211yescharacteristic
54#16839521C2211yescharacteristic
55#16839516C2211yescharacteristic
56C2×AGL(1,ℤ/8)×C61C2211yescharacteristic
57C61×#520C2211yescharacteristic
58C61×#518C2211yescharacteristic
59C61×#442—21no—
60C61×#440—21no—
61C4×SD16×C61—21no—
62C4×D8×C61—21no—
63C61×#428C2211yescharacteristic
64C61×#416C2211yescharacteristic
65C2×#102×C61C2211yescharacteristic
66C61×#405C2211yescharacteristic
67#14519784—22no—
68#14519784—22no—
69C2×#11758194—21no—
70C2×#11758194C2311yescharacteristic
71C2×#11758194C2311yescharacteristic
72C2×#11758194C2311yescharacteristic
73C22×#92418—21no—
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—41no—
83#14519736—21no—
84#14519736C2311yescharacteristic
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×C22⋊Dic61—41no—
89C2×C4×Dic61C2311yescharacteristic
90#14519713—22no—
91#14519710—21no—
92#14519710—21no—
93#14519710—21no—
94#14519710—21no—
95#14519710—21no—
96#14519710—21no—
97#14519710D411yescharacteristic
98#14519710D411yescharacteristic
99#14519709—21no—
100#14519709—21no—
101#14519708—21no—
102#14519708—21no—
103#14519708—21no—
104#14519708—21no—
105#14519708D411yescharacteristic
106#14519708D411yescharacteristic
107C4⋊#92418—21no—
108C4⋊#92418—21no—
109#14519706—21no—
110#14519706—21no—
111#14519704—22no—
112#14519701—21no—
113#14519698—21no—
114#14519698—21no—
115#14519698—21no—
116#14519698—21no—
117#14519698—21no—
118#14519694—22no—
119#14519684—41no—
120C2×#41×C61C2311yescharacteristic
121C22×D4×C61C2311yescharacteristic
122AGL(1,ℤ/8)×C61—22no—
123AGL(1,ℤ/8)×C61Q814yescomplemented
124C2×SD16×C61—21no—
125C2×D8×C61—21no—
126C2×M16×C61C2311yescharacteristic
127C22×C8×C61C2311yescharacteristic
128C4×Q8×C61—21no—
129C4×D4×C61—21no—
130C4×D4×C61—21no—
131C4×D4×C61—21no—
132C4×D4×C61—21no—
133C4×D4×C61—21no—
134C4×D4×C61—41no—
135#117×C61—21no—
136#117×C61C2311yescharacteristic
137#117×C61C2311yescharacteristic
138C2×C4⋊C4×C61—21no—
139C2×C4⋊C4×C61C2311yescharacteristic
140C2×C22⋊C4×C61—41no—
141#107×C61—21no—
142#106×C61—21no—
143#103×C61D411yescharacteristic
144#103×C61D411yescharacteristic
145#102×C61—21no—
146#102×C61—21no—
147#102×C61D411yescharacteristic
148#102×C61D411yescharacteristic
149#97×C61—21no—
150C4×C8×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—41no—
168C22×C4×C61—41no—
169C22×C4×C61—41no—
170C22×C4×C61C2411yescharacteristic
171C22×Dic61—21no—
172C22×Dic61—21no—
173C22×Dic61—21no—
174C22×Dic61—41no—
175C2×Dic122—41no—
176SD16×C61—24no—
177D8×C61—24no—
178M16×C61—21no—
179M16×C61C2×Q812yes—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61C2×D411yescharacteristic
184C2×C8×C61C2×D411yescharacteristic
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—41no—
188C4⋊C4×C61C2×D411yescharacteristic
189C4⋊C4×C61C2×D411yescharacteristic
190C4⋊C4×C61C2×D411yescharacteristic
191C4⋊C4×C61C2×D411yescharacteristic
192C22⋊C4×C61—21no—
193C22⋊C4×C61—21no—
194C22⋊C4×C61—41no—
195C22⋊C4×C61—41no—
196C22⋊C4×C61—41no—
197C42×C61—21no—
198C42×C61—21no—
199C42×C61—41no—
200C22⋊Dic61—21no—
201C22⋊Dic61—41no—
202C22⋊Dic61—82no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—41no—
208C4⋊Dic61—41no—
209C4⋊Dic61C2×D411yescharacteristic
210C4⋊Dic61C2×D411yescharacteristic
211#11758196—41no—
212#11758196—41no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C4×Dic61—41no—
217#11758194—24no—
218#11758194—21no—
219#11758194—21no—
220#11758194—21no—
221#11758194—21no—
222#11758194—21no—
223#11758194#4112yes—
224C2×#92418—24no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418C2×D411yescharacteristic
231C2×#92418C2×D411yescharacteristic
232C23×C61—21no—
233C23×C61—22no—
234C23×C61—41no—
235C23×C61—41no—
236C23×C61—81no—
237C23×C61—81no—
238C23×C61H(5,𝔽2)11yescharacteristic
239Q8×C61—21no—
240Q8×C61C22⋊Q812yes—
241D4×C61—22no—
242D4×C61—22no—
243D4×C61—22no—
244D4×C61—21no—
245D4×C61—22no—
246D4×C61—41no—
247D4×C61—41no—
248D4×C61C22⋊Q812yes—
249C2×C4×C61—21no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—41no—
257C2×C4×C61—81no—
258C2×C4×C61—161no—
259C2×C4×C61—161no—
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—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—81no—
275C2×Dic61—81no—
276C2×Dic61—161no—
277Dic122—161no—
278C8×C61—22no—
279C8×C61—21no—
280C8×C61—21no—
281#92418—21no—
282#92418—22no—
283#92418—21no—
284#92418—41no—
285#92418—41no—
286#26405—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—22no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61—42no—
293C22×C61—82no—
294C22×C61—81no—
295C22×C61—81no—
296C22×C61—161no—
297C22×C61#53911yescharacteristic
298C22×C61D4211yescharacteristic
299C22×C61#55411yescharacteristic
300C4×C61—22no—
301C4×C61—41no—
302C4×C61—41no—
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—81no—
313Dic61—81no—
314Dic61—81no—
315#3084—611no—
316#3081—611no—
317#2946D6111yescharacteristic
318#2913—611no—
319#2092—611no—
320#2089—611no—
321#2063—611no—
322#2061—611no—
323#1951—611no—
324#1922—611no—
325#1907—611no—
326#1883—611no—
327#1866—611no—
328#1860—611no—
329#1856—611no—
330C2×C61—21no—
331C2×C61—21no—
332C2×C61—42no—
333C2×C61—82no—
334C2×C61#244911yescharacteristic
335C2×C61#328811yescharacteristic
336C2×C61#329011yescharacteristic
337C2×AGL(1,ℤ/8)C2×D6111yescharacteristic
338C2×#131—611no—
339#544—611no—
340#520—611no—
341#520C2×D6111yescharacteristic
342#518C2×D6111yescharacteristic
343#479—1221no—
344#478—1221no—
345#478—1221no—
346#478—1221no—
347#476—1221no—
348#476—1221no—
349#475—1221no—
350#474—1221no—
351#442—21no—
352#440—21no—
353C4×SD16—21no—
354C4×D8—21no—
355C2×#108—611no—
356#428—611no—
357#428—611no—
358#428—611no—
359#428—611no—
360#428—1221no—
361#428—1221no—
362#428C2×D6111yescharacteristic
363#427—611no—
364#427—1221no—
365C2×#107—611no—
366C2×#107—1221no—
367C2×#106—611no—
368#424—611no—
369#424—611no—
370C2×#104—611no—
371#418—611no—
372#417—611no—
373#416C2×D6111yescharacteristic
374C2×#102—611no—
375C2×#102C2×D6111yescharacteristic
376C2×#100—611no—
377#405C2×D6111yescharacteristic
378#390—611no—
379#382—611no—
380#343—611no—
381#343—611no—
382#343—611no—
383#341—611no—
384#339—611no—
385#339—611no—
386#339—611no—
387#336—611no—
388C61#2640511yescharacteristic, complemented, Sylow
389C2×#41C22×D6111yescharacteristic
390C22×D4C22×D6111yescharacteristic
391AGL(1,ℤ/8)—22no—
392AGL(1,ℤ/8)Dic12214yescomplemented
393C2×SD16—21no—
394C2×D8—21no—
395#131—1222no—
396#131—1222no—
397C2×M16—611no—
398C2×M16—611no—
399C2×M16—611no—
400C2×M16—1221no—
401C2×M16C22×D6111yescharacteristic
402C22×C8—611no—
403C22×C8—1221no—
404C22×C8C22×D6111yescharacteristic
405C4⋊Q8—1221no—
406C4⋊Q8—1221no—
407#125—1221no—
408#125—1221no—
409C22⋊Q8—2441no—
410C4×Q8—21no—
411C4×Q8—1221no—
412C4×D4—21no—
413C4×D4—21no—
414C4×D4—21no—
415C4×D4—21no—
416C4×D4—21no—
417C4×D4—41no—
418C4×D4—1221no—
419C4×D4—2441no—
420#117—21no—
421#117—611no—
422#117—1221no—
423#117C22×D6111yescharacteristic
424#117C22×D6111yescharacteristic
425C2×C4⋊C4—21no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4C22×D6111yescharacteristic
430C2×C22⋊C4—41no—
431C2×C22⋊C4—2441no—
432C2×C42—611no—
433#108—1222no—
434#107—21no—
435#107—611no—
436#107—611no—
437#107—1221no—
438#107—1221no—
439#107—1221no—
440#107—1221no—
441#107—1221no—
442#107—1221no—
443#106—21no—
444#106—611no—
445#106—611no—
446#106—1221no—
447#106—1221no—
448#106—1221no—
449#106—1221no—
450#106—1221no—
451#106—1221no—
452C4⋊C8—1221no—
453C4⋊C8—1221no—
454C4⋊C8—1221no—
455C4⋊C8—1221no—
456#104—1222no—
457#103—1221no—
458#103C22⋊D6111yescharacteristic
459#103C22⋊D6111yescharacteristic
460#102—21no—
461#102—21no—
462#102—1221no—
463#102—1221no—
464#102—1221no—
465#102—1221no—
466#102—1221no—
467#102C22⋊D6111yescharacteristic
468#102C22⋊D6111yescharacteristic
469#100—1222no—
470#97—21no—
471C4×C8—21no—
472#95—2441no—
473C24—21no—
474#41—21no—
475#41—21no—
476#41C2×Dic12212yes—
477#41#1175822212yes—
478C2×Q8—2441no—
479C2×Q8C2×C22⋊D6111yescharacteristic
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4C2×Dic12212yes—
487C2×D4C2×C22⋊D6111yescharacteristic
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—41no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—2441no—
497C22×C4C23×D6111yescharacteristic
498SD16—24no—
499D8—24no—
500M16—21no—
501M16—612no—
502M16—1224no—
503M16—1221no—
504M16—1221no—
505M16—1221no—
506M16—1221no—
507M16—1221no—
508M16C2×Dic12212yes—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—611no—
513C2×C8—611no—
514C2×C8—1224no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8C2×C22⋊D6111yescharacteristic
521C2×C8C2×C22⋊D6111yescharacteristic
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—41no—
525C4⋊C4—611no—
526C4⋊C4—611no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—2441no—
532C4⋊C4—2441no—
533C4⋊C4—2441no—
534C4⋊C4—2441no—
535C4⋊C4D4×D6111yescharacteristic
536C4⋊C4D4×D6111yescharacteristic
537C4⋊C4C2×C22⋊D6111yescharacteristic
538C4⋊C4C2×C22⋊D6111yescharacteristic
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—41no—
542C22⋊C4—41no—
543C22⋊C4—41no—
544C22⋊C4—1221no—
545C22⋊C4—4882no—
546C22⋊C4—2441no—
547C42—21no—
548C42—21no—
549C42—41no—
550C42—1221no—
551C42—1221no—
552C42—1221no—
553C42—2441no—
554C23—21no—
555C23—22no—
556C23—41no—
557C23—41no—
558C23—81no—
559C23—81no—
560C23#1451982211yescharacteristic
561Q8—21no—
562Q8—9761no—
563Q8#1451975612yes—
564D4—22no—
565D4—22no—
566D4—22no—
567D4—21no—
568D4—22no—
569D4—41no—
570D4—41no—
571D4#1451975612yes—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—41no—
580C2×C4—81no—
581C2×C4—161no—
582C2×C4—161no—
583C2×C4—161no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—9761no—
587C2×C4—4881no—
588C2×C4—4881no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4C2×D4×D6111yescharacteristic
595C2×C4C22×C22⋊D6111yescharacteristic
596C2×C4C22×Dic12211yescharacteristic
597C2×C4#41×D6111yescharacteristic
598C2×C4C2×#1175822211yescharacteristic
599C2×C4#1451982411yescharacteristic
600C8—22no—
601C8—21no—
602C8—21no—
603C8—1221no—
604C8—1221no—
605C8—1222no—
606C8—2441no—
607C8—2441no—
608C22—21no—
609C22—21no—
610C22—22no—
611C22—41no—
612C22—41no—
613C22—42no—
614C22—82no—
615C22—81no—
616C22—81no—
617C22—161no—
618C22#1684035411yescharacteristic
619C22D4×C22⋊D6111yescharacteristic
620C22#1684047111yescharacteristic
621C4—22no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—81no—
627C4—4881no—
628C4—4881no—
629C4—4881no—
630C4—2441no—
631C4—2441no—
632C4C2×#1451975611yescharacteristic
633C4#1684027111yescharacteristic
634C4#1684041111yescharacteristic
635C4D4×Dic12211yescharacteristic
636C2—21no—
637C2—21no—
638C2—42no—
639C2—82no—
640C2#2144962211yescharacteristic
641C2#2145758311yescharacteristic
642C2#2145759911yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2946C2
socleC22×C61D42
prime radicalC2×AGL(1,ℤ/8)×C61C22
hypercenter#2946D61
hypocenterC61#26405
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2946

Complements

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

NH ≅ G/Ntype
1. #21455679C2semidirect
2. #21455650C2semidirect
3. #21454417C2semidirect
4. #21447342C2semidirect
5. #21447327C2semidirect
6. #21447177C2semidirect
7. #21447167C2semidirect
8. #21446599C2semidirect
9. #21446418C2semidirect
12. #21446096C2semidirect
13. #21446052C2semidirect
14. #21446029C2semidirect
123. AGL(1,ℤ/8)×C61Q8semidirect
388. C61#26405semidirect
392. 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
#214556792
#214556502
#214544172
#214473422
#214473272
#214471772
#214471672
#214465992
#214464182
#214463292
#214461972
#214460962
#214460522
#214460292
C61×#29462
#2640561