Finite Group Databasebeta

← #29662785

Subgroups — #29662785

Order 15616.

760 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456273C211yesmaximal, characteristic, complemented
2#21455818C211yesmaximal, characteristic
3#474×D61C211yesmaximal, characteristic, complemented
4D4×#11758189C211yesmaximal, characteristic, complemented
5#21445309C211yesmaximal, characteristic, complemented
6#21445307C211yesmaximal, characteristic, complemented
7#21445283C211yesmaximal, characteristic, complemented
8#21445281C211yesmaximal, characteristic, complemented
9#21445184C211yesmaximal, characteristic, complemented
10#21445182C211yesmaximal, characteristic, complemented
11#21445158C211yesmaximal, characteristic, complemented
12#21444568C211yesmaximal, characteristic, complemented
13#21444493C211yesmaximal, characteristic, complemented
14#21444489C211yesmaximal, characteristic, complemented
15C61×#1665C211yesmaximal, characteristic, complemented
16C2×#14519806—21no—
17C2×#14519806—21no—
18C22×#11758189—21no—
19#16840310C2211yescharacteristic
20C4⋊Q8×D61C2211yescharacteristic
21C4×D4×D61C2211yescharacteristic
22#16840153—21no—
23#16840153—21no—
24#16840142C2211yescharacteristic, complemented
25#16840141C2211yescharacteristic, complemented
26#16840134C2211yescharacteristic, complemented
27#16840074—21no—
28#16840043C2211yescharacteristic
29#16840043C2211yescharacteristic
30#16840042C2211yescharacteristic
31#16840042C2211yescharacteristic
32#16840041—21no—
33#16840040—21no—
34#16840039C2211yescharacteristic
35#16839954C2211yescharacteristic
36#16839894—21no—
37D8×Dic61—21no—
38#16839842—21no—
39#16839841C2211yescharacteristic
40#16839839C2211yescharacteristic
41#16839833—21no—
42#16839830—21no—
43D4×#92418C2211yescharacteristic
44#16839814C2211yescharacteristic
45#107×D61—21no—
46#16839784C2211yescharacteristic
47C4⋊C8×D61C2211yescharacteristic
48#102×D61—21no—
49#16839724—21no—
50C4×#11758189C2211yescharacteristic
51#16839515C2211yescharacteristic, complemented
52#16839514C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839509C2211yescharacteristic, complemented
55#16839509C2211yescharacteristic, complemented
56#16839508C2211yescharacteristic, complemented
57#16839507C2211yescharacteristic, complemented
58#16839507C2211yescharacteristic, complemented
59#16839506C2211yescharacteristic, complemented
60C61×#492C2211yescharacteristic, complemented
61C61×#474C2211yescharacteristic
62C61×#459C2211yescharacteristic
63C8×D4×C61C2211yescharacteristic
64C61×#334C2211yescharacteristic, complemented
65C61×#332C2211yescharacteristic, complemented
66C61×#331C2211yescharacteristic, complemented
67C2×#11758224—21no—
68C2×Q8×D61—21no—
69C2×D4×D61D411yescharacteristic
70C22×C4×D61—21no—
71#14519806—44no—
72#14519806—44no—
73C2×Dic244—21no—
74C2×Dic244—21no—
75C2×#11758202—21no—
76C2×#11758202—21no—
77C2×#11758200—21no—
78C2×#11758200—21no—
79C2×#11758199—21no—
80C2×#11758189—21no—
81C2×#11758189—21no—
82C2×#11758189—21no—
83C2×#11758189—44no—
84C2×#11758189D411yescharacteristic
85C2×#11758189D411yescharacteristic
86C22×#92418—21no—
87C2×C8×D61—21no—
88#14519777C2311yescharacteristic
89#14519777C2311yescharacteristic
90#14519776—21no—
91#14519776—21no—
92#14519775—21no—
93#14519764—21no—
94#14519748—21no—
95#14519748—21no—
96C4×D244C2311yescharacteristic
97D4×Dic61—21no—
98D4×Dic61—21no—
99D4×Dic61D411yescharacteristic
100#14519739D411yescharacteristic
101C4×C22⋊D61—21no—
102#14519737—21no—
103C4⋊C4×D61—21no—
104C4⋊C4×D61—21no—
105C4⋊C4×D61D411yescharacteristic
106C22⋊C4×D61—21no—
107C42×D61C2311yescharacteristic
108#14519711—21no—
109#14519711—21no—
110#14519710—21no—
111#14519710—21no—
112#14519710—21no—
113C4⋊#92418C2311yescharacteristic
114C4⋊#92418C2311yescharacteristic
115C4⋊#92418C2311yescharacteristic
116#14519706—21no—
117#14519706—21no—
118#14519699—21no—
119#14519699—21no—
120#14519698—21no—
121#14519698—21no—
122#14519698—21no—
123#14519697—21no—
124C22⋊#92418—21no—
125#14519689—21no—
126#14519689—21no—
127#14519689—21no—
128#14519687D411yescharacteristic
129#14519687D411yescharacteristic
130C4×#92418C2311yescharacteristic
131C8×Dic61—21no—
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—21no—
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—
156C22×Dic61—21no—
157C22×Dic61—41no—
158Q8×D61—42no—
159Q8×D61—42no—
160#11758224—42no—
161#11758224—42no—
162D4×D61—22no—
163D4×D61—41no—
164C2×D244C2×D411yescharacteristic
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—21no—
168C2×C4×D61—42no—
169C2×C4×D61—42no—
170C2×C4×D61—41no—
171C2×C4×D61C2×D411yescharacteristic
172C2×C4×D61C2×D411yescharacteristic
173C2×C4×D61#4111yescharacteristic
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—41no—
177SD16×C61—42no—
178SD16×C61—42no—
179D8×C61—42no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—42no—
184C2×C8×C61C2×D411yescharacteristic
185C2×C8×C61C2×D411yescharacteristic
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61C2×D411yescharacteristic
189C22⋊C4×C61—21no—
190C42×C61C2411yescharacteristic
191C22⋊Dic61—21no—
192C22⋊Dic61—41no—
193C22⋊Dic61—41no—
194#11758202—42no—
195#11758202—42no—
196#11758200—42no—
197#11758200—42no—
198#11758199—42no—
199#11758198—21no—
200#11758198—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61C2×D411yescharacteristic
206#11758196—21no—
207#11758196—41no—
208#11758196—41no—
209C4×Dic61C2×D411yescharacteristic
210C4×Dic61C2×D411yescharacteristic
211C4×Dic61#4111yescharacteristic
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—42no—
216C2×#92418C2×D411yescharacteristic
217C2×#92418C2×D411yescharacteristic
218Dic244—42no—
219Dic244—42no—
220#11758189—22no—
221#11758189—42no—
222#11758189—42no—
223#11758189—42no—
224#11758189D814yescomplemented
225C8×D61—42no—
226C23×C61—21no—
227C23×C61—41no—
228C22×D61—21no—
229C22×D61—42no—
230C22×D61—42no—
231C22×D61—81no—
232C22×D61#12111yescharacteristic
233Q8×C61—41no—
234Q8×C61—41no—
235D4×C61—21no—
236D4×C61—41no—
237D4×C61—41no—
238D4×C61—41no—
239D4×C61—41no—
240D4×C61—81no—
241D4×C61#13712yescomplemented
242C2×C4×C61—21no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C22⋊D61—42no—
250C22⋊D61—82no—
251C2×Dic61—21no—
252C2×Dic61—42no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—82no—
256C2×Dic61#12111yescharacteristic
257C2×Dic61#12111yescharacteristic
258C2×Dic61#12111yescharacteristic
259D244—21no—
260D244—21no—
261C4×D61—21no—
262C4×D61—21no—
263C4×D61—41no—
264C4×D61—42no—
265C4×D61—41no—
266C4×D61—41no—
267C4×D61—42no—
268C4×D61—82no—
269C4×D61C2×D812yes—
270C4×D61AGL(1,ℤ/8)12yes—
271Dic122—41no—
272Dic122—41no—
273Dic122—41no—
274Dic122—41no—
275Dic122—82no—
276C8×C61—21no—
277C8×C61—41no—
278C8×C61—41no—
279C8×C61—41no—
280C8×C61C2×D812yes—
281#92418—21no—
282#92418—41no—
283#92418—41no—
284#92418—41no—
285#92418C2×D812yes—
286#19163—611nomaximal, Sylow
287C22×C61—22no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61C2×#12111yescharacteristic
294C2×D61—21no—
295C2×D61—21no—
296C2×D61—41no—
297C2×D61—41no—
298C2×D61—41no—
299C2×D61—84no—
300C2×D61#45912yes—
301C4×C61—21no—
302C4×C61—41no—
303C4×C61—81no—
304C4×C61C22×D811yescharacteristic
305C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
306C4×C61C2×#13711yescharacteristic
307C4×C61#57511yescharacteristic
308Dic61—21no—
309Dic61—21no—
310Dic61—41no—
311Dic61—81no—
312Dic61—81no—
313Dic61#45912yes—
314#3152—611no—
315#3100—611no—
316C2×#474—611no—
317D4×M16—611no—
318#1689—611no—
319#1689—611no—
320#1685—611no—
321#1685—611no—
322#1669—611no—
323#1669—611no—
324#1665—611no—
325#1665D6111yescharacteristic, complemented
326#1565—611no—
327#1547—611no—
328#1546—611no—
329C2×C61—42no—
330C2×C61—161no—
331C2×C61C2×#45911yescharacteristic
332C2×C61#305611yescharacteristic
333C2×C61#306911yescharacteristic
334D61—22no—
335D61—81no—
336C2×#137—1221no—
337C2×#137—1221no—
338C22×M16—1221no—
339#532—611no—
340C2×C4⋊Q8—611no—
341C2×C4×D4—611no—
342#497—1221no—
343#497—1221no—
344#494—611no—
345#494—611no—
346#492—611no—
347#492C2×D6111yescharacteristic, complemented
348#480—1221no—
349#474—611no—
350#474—611no—
351#474—611no—
352#474—611no—
353#474—611no—
354#474—1221no—
355#474—1221no—
356#474C2×D6111yescharacteristic
357#459—611no—
358#459C2×D6111yescharacteristic
359#447—1221no—
360C4×D8—1221no—
361C4×D8—1221no—
362#436—611no—
363#436—611no—
364#435—1221no—
365#435—1221no—
366C8×D4—611no—
367C8×D4C2×D6111yescharacteristic
368#431—611no—
369C2×#107—1221no—
370#423—611no—
371C2×C4⋊C8—611no—
372C2×#102—1221no—
373#407—1221no—
374C4×M16—611no—
375#335—611no—
376#335—611no—
377#334—611no—
378#334C2×D6111yescharacteristic, complemented
379#332—611no—
380#332—611no—
381#332—611no—
382#332—611no—
383#332—611no—
384#332C2×D6111yescharacteristic, complemented
385#331—611no—
386#331C2×D6111yescharacteristic, complemented
387C61#1916311yescharacteristic, complemented, Sylow
388C2×#41—1221no—
389C22×Q8—1221no—
390C22×D4—611no—
391C23×C4—1221no—
392#137—2444no—
393#137—2444no—
394C2×Q16—1221no—
395C2×Q16—1221no—
396C2×Q16—1221no—
397C2×Q16—1221no—
398C2×SD16—21no—
399C2×SD16—21no—
400C2×SD16—1221no—
401C2×SD16—1221no—
402C2×D8—21no—
403C2×D8—1221no—
404C2×M16—611no—
405C2×M16—611no—
406C2×M16—1221no—
407C2×M16—1221no—
408C2×M16—1221no—
409C2×M16—2444no—
410C22×C8—21no—
411C22×C8—1221no—
412C22×C8—1221no—
413C4⋊Q8—611no—
414C4⋊Q8—611no—
415C4⋊Q8—1221no—
416C4⋊Q8—1221no—
417C4⋊Q8—1221no—
418C4⋊Q8C22×D6111yescharacteristic
419#127C22×D6111yescharacteristic
420#124—1221no—
421#121—1221no—
422#121—1221no—
423C4×D4—611no—
424C4×D4—611no—
425C4×D4—611no—
426C4×D4—1221no—
427C4×D4—1221no—
428C4×D4—1221no—
429C4×D4—1221no—
430C4×D4C22×D6111yescharacteristic
431C2×C4⋊C4—611no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C22⋊C4—1221no—
435C2×C42—611no—
436#107—21no—
437#107—1221no—
438#107—1221no—
439#107—1221no—
440#107—1221no—
441#107—1221no—
442C4⋊C8—611no—
443C4⋊C8—611no—
444C4⋊C8—611no—
445C4⋊C8—1221no—
446C4⋊C8—1221no—
447C4⋊C8C22×D6111yescharacteristic
448C4⋊C8C22×D6111yescharacteristic
449C4⋊C8C22×D6111yescharacteristic
450#102—21no—
451#102—21no—
452#102—1221no—
453#102—1221no—
454#102—1221no—
455#102—1221no—
456#102—1221no—
457#102—1221no—
458C22⋊C8—21no—
459C22⋊C8—1221no—
460C22⋊C8—1221no—
461C22⋊C8—1221no—
462C22⋊C8—1221no—
463#97—611no—
464#97—611no—
465C4×C8—611no—
466C4×C8—1221no—
467C4×C8C22×D6111yescharacteristic
468C24—1221no—
469#41—2442no—
470#41—2442no—
471C2×Q8—21no—
472C2×Q8—1221no—
473C2×Q8—1221no—
474C2×Q8—2441no—
475C2×Q8—2442no—
476C2×Q8—2442no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—611no—
481C2×D4—1221no—
482C2×D4—1222no—
483C2×D4—2441no—
484C2×D4—2441no—
485C2×D4D4×D6111yescharacteristic
486C22×C4—21no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—2442no—
495C22×C4—2442no—
496C22×C4—2441no—
497C22×C4—2441no—
498Q16—2442no—
499Q16—2442no—
500Q16—2442no—
501Q16—2442no—
502SD16—42no—
503SD16—42no—
504SD16—2442no—
505SD16—2442no—
506D8—42no—
507D8—2442no—
508M16—614no—
509M16—1222no—
510M16—2442no—
511M16—2442no—
512M16—2442no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—42no—
517C2×C8—611no—
518C2×C8—611no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—2442no—
523C2×C8—2442no—
524C2×C8D4×D6111yescharacteristic
525C2×C8D4×D6111yescharacteristic
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—611no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—2441no—
535C4⋊C4—2441no—
536C4⋊C4D4×D6111yescharacteristic
537C22⋊C4—21no—
538C22⋊C4—1221no—
539C22⋊C4—1221no—
540C22⋊C4—1221no—
541C22⋊C4—2441no—
542C22⋊C4—2441no—
543C42—611no—
544C42—611no—
545C42—611no—
546C42C23×D6111yescharacteristic
547C23—21no—
548C23—41no—
549C23—611no—
550C23—1221no—
551C23—4881no—
552C23—2442no—
553C23—2442no—
554Q8—41no—
555Q8—41no—
556Q8—4882no—
557Q8—2441no—
558Q8—2441no—
559Q8—2441no—
560Q8—2441no—
561D4—21no—
562D4—41no—
563D4—41no—
564D4—41no—
565D4—41no—
566D4—81no—
567D4—1221no—
568D4—1221no—
569D4—4882no—
570D4—2442no—
571D4#1451980612yescomplemented
572C2×C4—21no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—611no—
577C2×C4—612no—
578C2×C4—612no—
579C2×C4—611no—
580C2×C4—611no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—4882no—
585C2×C4—4882no—
586C2×C4—2441no—
587C2×C4—2442no—
588C2×C4—2442no—
589C2×C4—2442no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4C2×D4×D6111yescharacteristic
595C2×C4C2×D4×D6111yescharacteristic
596C2×C4#41×D6111yescharacteristic
597C8—21no—
598C8—41no—
599C8—41no—
600C8—41no—
601C8—612no—
602C8—1221no—
603C8—2441no—
604C8—2441no—
605C8—2441no—
606C8D8×D6112yes—
607C22—22no—
608C22—41no—
609C22—41no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—612no—
614C22—1221no—
615C22—1221no—
616C22—4884no—
617C22—2441no—
618C22—2441no—
619C22—2441no—
620C22#121×D6111yescharacteristic
621C4—21no—
622C4—41no—
623C4—81no—
624C4—612no—
625C4—1221no—
626C4—1221no—
627C4—4881no—
628C4—4881no—
629C4—2441no—
630C4C2×D8×D6111yescharacteristic
631C4AGL(1,ℤ/8)×D6111yescharacteristic
632C4C2×#1451980611yescharacteristic
633C4#1684055911yescharacteristic
634C2—42no—
635C2—161no—
636C2—1222no—
637C2—4881no—
638C2#459×D6111yescharacteristic
639C2#2145544211yescharacteristic
640C2#2145555111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1665C2
socleC22×C61C2×#121
prime radical#21455818C2
hypercenter#1665D61
hypocenterC61#19163
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. #21456273C2semidirect
3. #474×D61C2semidirect
4. D4×#11758189C2semidirect
5. #21445309C2semidirect
6. #21445307C2semidirect
7. #21445283C2semidirect
8. #21445281C2semidirect
9. #21445184C2semidirect
10. #21445182C2semidirect
11. #21445158C2semidirect
12. #21444568C2semidirect
13. #21444493C2semidirect
14. #21444489C2semidirect
15. C61×#1665C2semidirect
24. #16840142C22semidirect
25. #16840141C22semidirect
26. #16840134C22semidirect
51. #16839515C22semidirect
52. #16839514C22semidirect
53. #16839512C22semidirect
54. #16839509C22semidirect
55. #16839509C22semidirect
56. #16839508C22semidirect
57. #16839507C22semidirect
58. #16839507C22semidirect
59. #16839506C22semidirect
60. C61×#492C22semidirect
64. C61×#334C22semidirect
65. C61×#332C22semidirect
66. C61×#331C22semidirect
224. #11758189D8semidirect
241. D4×C61#137semidirect
325. #1665D61semidirect
347. #492C2×D61semidirect
378. #334C2×D61semidirect
384. #332C2×D61semidirect
386. #331C2×D61semidirect
387. C61#19163semidirect
571. D4#14519806semidirect

Maximal subgroups

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

subgroupindex
#214562732
#214558182
#474×D612
D4×#117581892
#214453092
#214453072
#214452832
#214452812
#214451842
#214451822
#214451582
#214445682
#214444932
#214444892
C61×#16652
#1916361