Finite Group Databasebeta

← #29663894

Subgroups — #29663894

Order 15616.

786 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#492×D61C211yesmaximal, characteristic, complemented
2#21455627C211yesmaximal, characteristic, complemented
3#21455294C211yesmaximal, characteristic
4C8×#11758224C211yesmaximal, characteristic, complemented
5#21445170C211yesmaximal, characteristic, complemented
6#21445168C211yesmaximal, characteristic, complemented
7#21445166C211yesmaximal, characteristic, complemented
8#21445164C211yesmaximal, characteristic, complemented
9#21445162C211yesmaximal, characteristic, complemented
10#21445160C211yesmaximal, characteristic, complemented
11#21445158C211yesmaximal, characteristic, complemented
12#21444573C211yesmaximal, characteristic, complemented
13#21444502C211yesmaximal, characteristic, complemented
14#21444485C211yesmaximal, characteristic, complemented
15C61×#1665C211yesmaximal, characteristic, complemented
16C2×#14519801—21no—
17C2×SD16×D61—21no—
18C2×SD16×D61—21no—
19C4⋊Q8×D61C2211yescharacteristic
20#127×D61C2211yescharacteristic
21C4×#11758224C2211yescharacteristic
22#16840136C2211yescharacteristic
23#16840135C2211yescharacteristic
24#16840134C2211yescharacteristic
25#16840133—21no—
26#16840133—21no—
27#16840056—21no—
28#16840050—21no—
29#16840048C2211yescharacteristic, complemented
30#16840044C2211yescharacteristic, complemented
31#16840042C2211yescharacteristic, complemented
32#16839963C2211yescharacteristic
33#16839961—21no—
34#16839959C2211yescharacteristic
35#16839957C2211yescharacteristic
36#16839955—21no—
37C8×Dic122C2211yescharacteristic, complemented
38#16839886C2211yescharacteristic, complemented
39D4×#92418C2211yescharacteristic, complemented
40C8×C22⋊D61—21no—
41#16839825—21no—
42#16839814C2211yescharacteristic
43#16839812—21no—
44#16839803—21no—
45#16839784C2211yescharacteristic
46#16839784C2211yescharacteristic
47#16839765—21no—
48#16839760—21no—
49C2×C8×Dic61—21no—
50C4×C8×D61C2211yescharacteristic
51#16839513C2211yescharacteristic, complemented
52#16839512C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839509C2211yescharacteristic, complemented
55#16839508C2211yescharacteristic, complemented
56#16839507C2211yescharacteristic, complemented
57#16839506C2211yescharacteristic
58#16839505C2211yescharacteristic
59#16839505C2211yescharacteristic
60C61×#492C2211yescharacteristic
61C61×#474C2211yescharacteristic, complemented
62C61×#459C2211yescharacteristic
63C8×D4×C61C2211yescharacteristic, complemented
64C61×#334C2211yescharacteristic, complemented
65C61×#332C2211yescharacteristic, complemented
66C61×#331C2211yescharacteristic
67C2×#11758224D411yescharacteristic
68C2×Q8×D61—21no—
69C2×D4×D61—21no—
70C2×D4×D61—21no—
71C2×D4×D61—21no—
72#14519801—44no—
73C2×#11758190—21no—
74C2×#11758190—21no—
75C2×#11758190—21no—
76C2×#11758200—21no—
77C2×#11758200—21no—
78C2×#11758200—21no—
79C2×#11758201—21no—
80C2×#11758201—21no—
81SD16×D61—44no—
82SD16×D61—44no—
83C2×#11758199—21no—
84C2×#11758189—21no—
85C2×#11758189—21no—
86C22×#92418—21no—
87C2×C8×D61D411yescharacteristic
88C2×C8×D61D411yescharacteristic
89#14519777C2311yescharacteristic
90#14519776—21no—
91#14519776—21no—
92#14519775—21no—
93#14519774C2311yescharacteristic
94#14519773—21no—
95#14519773—21no—
96#14519773—21no—
97C4×Dic122C2311yescharacteristic
98#14519742D411yescharacteristic
99D4×Dic61D411yescharacteristic
100C4×C22⋊D61—21no—
101#14519737—21no—
102#14519735D411yescharacteristic
103#14519733—21no—
104C4⋊C4×D61—21no—
105C4⋊C4×D61—21no—
106C2×C4×Dic61—21no—
107C42×D61C2311yescharacteristic
108#14519710—21no—
109#14519709—21no—
110#14519708—21no—
111C4⋊#92418C2311yescharacteristic
112C4⋊#92418C2311yescharacteristic
113C4⋊#92418C2311yescharacteristic
114#14519706—21no—
115#14519706—21no—
116#14519706—21no—
117#14519706—21no—
118#14519702—21no—
119#14519700—21no—
120#14519699—21no—
121#14519698—21no—
122#14519698—21no—
123#14519697—21no—
124C22⋊#92418—21no—
125#14519689—21no—
126C4×#92418C2311yescharacteristic
127C8×Dic61—21no—
128C8×Dic61—21no—
129C8×Dic61—21no—
130C8×Dic61D411yescharacteristic
131C8×Dic61D411yescharacteristic
132C2×SD16×C61—21no—
133C2×SD16×C61—21no—
134C2×D8×C61—21no—
135C22×C8×C61—21no—
136C4⋊Q8×C61C2311yescharacteristic
137#127×C61C2311yescharacteristic
138C4×D4×C61C2311yescharacteristic
139#107×C61—21no—
140C4⋊C8×C61C2311yescharacteristic
141C4⋊C8×C61C2311yescharacteristic
142C4⋊C8×C61C2311yescharacteristic
143#102×C61—21no—
144#102×C61—21no—
145C22⋊C8×C61—21no—
146C4×C8×C61C2311yescharacteristic
147C23×D61—41no—
148C2×Q8×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61C2×D411yescharacteristic
153C22×C4×C61—21no—
154C2×C22⋊D61—21no—
155C2×C22⋊D61—41no—
156C2×C22⋊D61—41no—
157C2×C22⋊D61—41no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160Q8×D61—42no—
161Q8×D61—42no—
162#11758224—22no—
163#11758224SD1614yescomplemented
164D4×D61—42no—
165D4×D61—42no—
166D4×D61—42no—
167D4×D61—42no—
168D4×D61—82no—
169C2×D244—21no—
170C2×D244—21no—
171C2×D244—21no—
172C2×C4×D61—41no—
173C2×C4×D61C2×D411yescharacteristic
174C2×C4×D61C2×D411yescharacteristic
175C2×C4×D61#4111yescharacteristic
176C2×Dic122—21no—
177C2×Dic122—41no—
178C2×Dic122C2×D411yescharacteristic
179SD16×C61—42no—
180SD16×C61—42no—
181D8×C61—42no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—42no—
186C2×C8×C61C2×D411yescharacteristic
187C2×C8×C61C2×D411yescharacteristic
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61C2×D411yescharacteristic
191C22⋊C4×C61—21no—
192C42×C61C2411yescharacteristic
193C22⋊Dic61—21no—
194#11758201—42no—
195#11758201—42no—
196#11758200—42no—
197#11758200—42no—
198#11758200—42no—
199#11758199—42no—
200#11758198—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61C2×D411yescharacteristic
204#11758196—21no—
205#11758196—21no—
206#11758196—41no—
207#11758196—41no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61C2×D411yescharacteristic
212C4×Dic61C2×D411yescharacteristic
213C4×Dic61#4111yescharacteristic
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—42no—
218C2×#92418C2×D411yescharacteristic
219C2×#92418C2×D411yescharacteristic
220#11758190—42no—
221#11758190—42no—
222#11758190—42no—
223#11758189—42no—
224#11758189—42no—
225C8×D61—22no—
226C8×D61—22no—
227C23×C61—21no—
228C23×C61—41no—
229C22×D61—41no—
230C22×D61—82no—
231C22×D61—82no—
232C22×D61—82no—
233C22×D61#12111yescharacteristic
234Q8×C61—41no—
235Q8×C61—41no—
236D4×C61—21no—
237D4×C61—41no—
238D4×C61—41no—
239D4×C61—41no—
240D4×C61—41no—
241D4×C61—81no—
242D4×C61C2×SD1612yescomplemented
243C2×C4×C61—21no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C22⋊D61—24no—
251C22⋊D61—82no—
252C22⋊D61—82no—
253C22⋊D61—82no—
254C2×Dic61—21no—
255C2×Dic61—24no—
256C2×Dic61—21no—
257C2×Dic61—42no—
258C2×Dic61—41no—
259C2×Dic61#12111yescharacteristic
260C2×Dic61#12111yescharacteristic
261C2×Dic61#12111yescharacteristic
262D244—41no—
263D244—41no—
264D244—41no—
265D244—41no—
266D244—81no—
267C4×D61—21no—
268C4×D61—21no—
269C4×D61—22no—
270C4×D61—82no—
271C4×D61C2×SD1612yes—
272C4×D61#13512yes—
273Dic122—21no—
274Dic122—41no—
275Dic122—41no—
276Dic122—82no—
277Dic122C2×SD1612yescomplemented
278C8×C61—21no—
279C8×C61—41no—
280C8×C61—41no—
281C8×C61—41no—
282C8×C61AGL(1,ℤ/8)12yescomplemented
283#92418—21no—
284#92418—41no—
285#92418—41no—
286#92418—41no—
287#92418AGL(1,ℤ/8)12yescomplemented
288#19266—611nomaximal, Sylow
289C22×C61—22no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61—81no—
293C22×C61—81no—
294C22×C61—81no—
295C22×C61C2×#12111yescharacteristic
296C2×D61—21no—
297C2×D61—21no—
298C2×D61—81no—
299C2×D61—81no—
300C2×D61—81no—
301C2×D61—164no—
302C2×D61#46512yes—
303C4×C61—21no—
304C4×C61—41no—
305C4×C61—81no—
306C4×C61C22×SD1611yescharacteristic
307C4×C61C2×#13511yescharacteristic
308C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
309C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
310Dic61—21no—
311Dic61—22no—
312Dic61—41no—
313Dic61—81no—
314Dic61#46512yes—
315Dic61#47112yes—
316C2×#492—611no—
317#3079—611no—
318#3038—611no—
319C8×#41—611no—
320#1666—611no—
321#1666—611no—
322#1666—611no—
323#1666—611no—
324#1665—611no—
325#1665—611no—
326#1665—611no—
327#1665D6111yescharacteristic, complemented
328#1566—611no—
329#1549—611no—
330#1545—611no—
331C2×C61—42no—
332C2×C61—161no—
333C2×C61#303811yescharacteristic
334C2×C61C2×#46511yescharacteristic
335C2×C61C2×#47111yescharacteristic
336D61—22no—
337D61—161no—
338C2×AGL(1,ℤ/8)—1221no—
339C22×SD16—1221no—
340C22×SD16—1221no—
341C2×C4⋊Q8—611no—
342C2×#127—611no—
343C4×#41—611no—
344#492—611no—
345#492—611no—
346#492—611no—
347#492—1221no—
348#492—1221no—
349#492C2×D6111yescharacteristic
350#477—1221no—
351#476—1221no—
352#475—611no—
353#475—611no—
354#474—611no—
355#474C2×D6111yescharacteristic, complemented
356#460—611no—
357#460—611no—
358#460—1221no—
359#459—611no—
360#459—1221no—
361#459C2×D6111yescharacteristic
362C8×Q8—611no—
363C8×Q8—611no—
364C8×D4—611no—
365C8×D4—1221no—
366C8×D4—1221no—
367C8×D4C2×D6111yescharacteristic, complemented
368#431—611no—
369#431—1221no—
370#428—1221no—
371#423—611no—
372#423—611no—
373#418—1221no—
374#417—1221no—
375C2×C4×C8—611no—
376C2×C4×C8—1221no—
377#334—611no—
378#334—611no—
379#334—611no—
380#334C2×D6111yescharacteristic, complemented
381#332—611no—
382#332—611no—
383#332—611no—
384#332C2×D6111yescharacteristic, complemented
385#331—611no—
386#331—611no—
387#331—611no—
388#331C2×D6111yescharacteristic
389C61#1926611yescharacteristic, complemented, Sylow
390C2×#41—611no—
391C22×Q8—1221no—
392C22×D4—1221no—
393C22×D4—1221no—
394C22×D4—1221no—
395AGL(1,ℤ/8)—2444no—
396C2×SD16—21no—
397C2×SD16—21no—
398C2×SD16—1221no—
399C2×SD16—1221no—
400C2×SD16—1221no—
401C2×SD16—1221no—
402C2×SD16—1221no—
403C2×SD16—1221no—
404C2×SD16—1221no—
405C2×SD16—1221no—
406C2×SD16—2444no—
407C2×SD16—2444no—
408C2×D8—21no—
409C2×D8—1221no—
410C2×M16—1221no—
411C2×M16—1221no—
412C22×C8—21no—
413C22×C8—611no—
414C22×C8—611no—
415C22×C8—1221no—
416C4⋊Q8—611no—
417C4⋊Q8—1221no—
418C4⋊Q8—1221no—
419C4⋊Q8—1221no—
420C4⋊Q8C22×D6111yescharacteristic
421#127—611no—
422#127—1221no—
423#127—1221no—
424#127—1221no—
425#127C22×D6111yescharacteristic
426C4×Q8—611no—
427C4×Q8—611no—
428C4×D4—611no—
429C4×D4—1221no—
430C4×D4—1221no—
431C4×D4C22×D6111yescharacteristic
432#117—611no—
433#117—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C42—611no—
437C2×C42—1221no—
438#107—21no—
439#107—1221no—
440#106—1221no—
441#106—1221no—
442C4⋊C8—611no—
443C4⋊C8—611no—
444C4⋊C8—611no—
445C4⋊C8—1221no—
446C4⋊C8—1221no—
447C4⋊C8—1221no—
448C4⋊C8—1221no—
449C4⋊C8C22×D6111yescharacteristic
450C4⋊C8C22×D6111yescharacteristic
451C4⋊C8C22×D6111yescharacteristic
452#103—1221no—
453#103—1221no—
454#102—21no—
455#102—21no—
456#102—1221no—
457#102—1221no—
458#102—1221no—
459#102—1221no—
460C22⋊C8—21no—
461C22⋊C8—1221no—
462C22⋊C8—1221no—
463C4×C8—611no—
464C4×C8—611no—
465C4×C8—611no—
466C4×C8—1221no—
467C4×C8—1221no—
468C4×C8—1221no—
469C4×C8C22×D6111yescharacteristic
470C24—2441no—
471#41—614no—
472#41—1222no—
473C2×Q8—21no—
474C2×Q8—611no—
475C2×Q8—1221no—
476C2×Q8—2441no—
477C2×Q8—2442no—
478C2×Q8—2442no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—1221no—
483C2×D4—1221no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—4882no—
487C2×D4—2442no—
488C2×D4—2442no—
489C2×D4—2441no—
490C2×D4—2441no—
491C2×D4—2441no—
492C2×D4—2442no—
493C2×D4—2442no—
494C2×D4D4×D6111yescharacteristic
495C22×C4—21no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—611no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—2441no—
502SD16—42no—
503SD16—42no—
504SD16—2442no—
505SD16—2442no—
506SD16—2442no—
507SD16—2442no—
508SD16—2442no—
509SD16—2442no—
510SD16—2442no—
511SD16—2442no—
512D8—42no—
513D8—2442no—
514M16—2442no—
515M16—2442no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—42no—
520C2×C8—611no—
521C2×C8—611no—
522C2×C8—1222no—
523C2×C8—1222no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—2442no—
528C2×C8D4×D6111yescharacteristic
529C2×C8D4×D6111yescharacteristic
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—611no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—2441no—
538C4⋊C4—2441no—
539C4⋊C4D4×D6111yescharacteristic
540C22⋊C4—21no—
541C22⋊C4—1221no—
542C22⋊C4—1221no—
543C42—611no—
544C42—611no—
545C42—611no—
546C42—1221no—
547C42—1221no—
548C42—1221no—
549C42C23×D6111yescharacteristic
550C23—21no—
551C23—41no—
552C23—611no—
553C23—4882no—
554C23—4882no—
555C23—4882no—
556C23—2441no—
557Q8—41no—
558Q8—41no—
559Q8—612no—
560Q8—1221no—
561Q8—4882no—
562Q8—2441no—
563Q8—2441no—
564D4—21no—
565D4—41no—
566D4—41no—
567D4—41no—
568D4—41no—
569D4—81no—
570D4—1224no—
571D4—4881no—
572D4—4882no—
573D4—4882no—
574D4—4882no—
575D4—2441no—
576D4—2441no—
577D4—2441no—
578D4—2441no—
579D4SD16×D6112yescomplemented
580C2×C4—21no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—611no—
585C2×C4—611no—
586C2×C4—612no—
587C2×C4—612no—
588C2×C4—611no—
589C2×C4—1222no—
590C2×C4—1224no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—4882no—
596C2×C4—2442no—
597C2×C4—2441no—
598C2×C4C2×D4×D6111yescharacteristic
599C2×C4C2×D4×D6111yescharacteristic
600C2×C4#41×D6111yescharacteristic
601C8—21no—
602C8—41no—
603C8—41no—
604C8—41no—
605C8—612no—
606C8—1221no—
607C8—2441no—
608C8—2441no—
609C8—2441no—
610C8#1451980112yescomplemented
611C22—22no—
612C22—41no—
613C22—41no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—612no—
618C22—1221no—
619C22—1221no—
620C22—9764no—
621C22—4881no—
622C22—4881no—
623C22—4881no—
624C22#121×D6111yescharacteristic
625C4—21no—
626C4—41no—
627C4—81no—
628C4—612no—
629C4—612no—
630C4—1222no—
631C4—1221no—
632C4—4881no—
633C4—2441no—
634C4C2×SD16×D6111yescharacteristic
635C4#135×D6111yescharacteristic
636C4AGL(1,ℤ/8)×D6111yescharacteristic
637C4C2×#1451980111yescharacteristic
638C2—42no—
639C2—161no—
640C2—1222no—
641C2—9761no—
642C2#2145529411yescharacteristic
643C2#465×D6111yescharacteristic
644C2#471×D6111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1665C2
socleC22×C61C2×#121
prime radical#21455294C2
hypercenter#1665D61
hypocenterC61#19266
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1665

Complements

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

NH ≅ G/Ntype
1. #492×D61C2semidirect
2. #21455627C2semidirect
4. C8×#11758224C2semidirect
5. #21445170C2semidirect
6. #21445168C2semidirect
7. #21445166C2semidirect
8. #21445164C2semidirect
9. #21445162C2semidirect
10. #21445160C2semidirect
11. #21445158C2semidirect
12. #21444573C2semidirect
13. #21444502C2semidirect
14. #21444485C2semidirect
15. C61×#1665C2semidirect
29. #16840048C22semidirect
30. #16840044C22semidirect
31. #16840042C22semidirect
37. C8×Dic122C22semidirect
38. #16839886C22semidirect
39. D4×#92418C22semidirect
51. #16839513C22semidirect
52. #16839512C22semidirect
53. #16839512C22semidirect
54. #16839509C22semidirect
55. #16839508C22semidirect
56. #16839507C22semidirect
61. C61×#474C22semidirect
63. C8×D4×C61C22semidirect
64. C61×#334C22semidirect
65. C61×#332C22semidirect
163. #11758224SD16semidirect
242. D4×C61C2×SD16semidirect
277. Dic122C2×SD16semidirect
282. C8×C61AGL(1,ℤ/8)semidirect
287. #92418AGL(1,ℤ/8)semidirect
327. #1665D61semidirect
355. #474C2×D61semidirect
367. C8×D4C2×D61semidirect
380. #334C2×D61semidirect
384. #332C2×D61semidirect
389. C61#19266semidirect
579. D4SD16×D61semidirect
610. C8#14519801semidirect

Maximal subgroups

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

subgroupindex
#492×D612
#214556272
#214552942
C8×#117582242
#214451702
#214451682
#214451662
#214451642
#214451622
#214451602
#214451582
#214445732
#214445022
#214444852
C61×#16652
#1926661