Finite Group Databasebeta

← #29662868

Subgroups — #29662868

Order 15616.

766 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456270C211yesmaximal, characteristic
2#21455849C211yesmaximal, characteristic, complemented
3#460×D61C211yesmaximal, characteristic, complemented
4#21454771C211yesmaximal, characteristic, complemented
5#21445301C211yesmaximal, characteristic, complemented
6#21445291C211yesmaximal, characteristic, complemented
7#21445289C211yesmaximal, characteristic, complemented
8#21445178C211yesmaximal, characteristic, complemented
9#21445176C211yesmaximal, characteristic, complemented
10#21445168C211yesmaximal, characteristic, complemented
11#21445166C211yesmaximal, characteristic, complemented
12#21444568C211yesmaximal, characteristic, complemented
13#21444493C211yesmaximal, characteristic, complemented
14#21444485C211yesmaximal, characteristic, complemented
15C61×#1688C211yesmaximal, characteristic, complemented
16C2×#14519801—21no—
17C2×#14519801—21no—
18C2×SD16×D61—21no—
19#16840310C2211yescharacteristic
20#127×D61C2211yescharacteristic
21C4×Q8×D61C2211yescharacteristic
22#16840150—21no—
23#16840150—21no—
24#16840138C2211yescharacteristic
25#16840135C2211yescharacteristic
26#16840134C2211yescharacteristic
27#16840079—21no—
28#16840048C2211yescharacteristic, complemented
29#16840047C2211yescharacteristic, complemented
30#16839963C2211yescharacteristic
31#16839962C2211yescharacteristic
32#16839961—21no—
33#16839960—21no—
34#16839959C2211yescharacteristic
35#16839959C2211yescharacteristic, complemented
36#16839899—21no—
37#16839893C2211yescharacteristic, complemented
38#16839891—21no—
39#16839890—21no—
40C8×Dic122C2211yescharacteristic, complemented
41#16839886C2211yescharacteristic, complemented
42#16839853—21no—
43SD16×Dic61—21no—
44#16839814C2211yescharacteristic
45#16839784—21no—
46#16839784C2211yescharacteristic
47C4⋊C8×D61C2211yescharacteristic
48#102×D61—21no—
49#16839714—21no—
50C4×#11758189C2211yescharacteristic
51#16839515C2211yescharacteristic, complemented
52#16839514C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839509C2211yescharacteristic, complemented
55#16839507C2211yescharacteristic, complemented
56#16839506C2211yescharacteristic
57#16839506C2211yescharacteristic, complemented
58#16839505C2211yescharacteristic
59#16839505C2211yescharacteristic
60C61×#493C2211yescharacteristic
61C61×#460C2211yescharacteristic
62C61×#460C2211yescharacteristic, complemented
63C61×#446C2211yescharacteristic, complemented
64C61×#334C2211yescharacteristic, complemented
65C61×#331C2211yescharacteristic
66C61×#331C2211yescharacteristic, complemented
67C2×#11758224—21no—
68C2×Q8×D61D411yescharacteristic
69C2×D4×D61—21no—
70C2×D4×D61—21no—
71C2×D4×D61—21no—
72#14519801—44no—
73#14519801—44no—
74C2×#11758190—21no—
75C2×#11758190—21no—
76C2×#11758190—21no—
77C2×#11758200—21no—
78C2×#11758200—21no—
79C2×#11758200—21no—
80C2×#11758200—21no—
81C2×#11758201—21no—
82SD16×D61—44no—
83C2×#11758199—21no—
84C2×#11758199—21no—
85C2×#11758189—21no—
86C2×#11758189D411yescharacteristic
87C2×#11758189D411yescharacteristic
88C2×C8×D61—21no—
89#14519777C2311yescharacteristic
90#14519774C2311yescharacteristic
91#14519773—21no—
92#14519773—21no—
93#14519773—21no—
94#14519764—21no—
95#14519748—21no—
96#14519748—21no—
97Q8×Dic61D411yescharacteristic
98C4×Dic122—21no—
99C4×Dic122C2311yescharacteristic
100#14519742—21no—
101#14519742D411yescharacteristic
102D4×Dic61—21no—
103D4×Dic61—21no—
104C4⋊C4×D61—21no—
105C4⋊C4×D61D411yescharacteristic
106C42×D61—21no—
107C42×D61C2311yescharacteristic
108#14519709—21no—
109#14519708—21no—
110C4⋊#92418—21no—
111C4⋊#92418C2311yescharacteristic
112C4⋊#92418C2311yescharacteristic
113C4⋊#92418C2311yescharacteristic
114#14519706—21no—
115#14519706—21no—
116#14519706—21no—
117#14519706—21no—
118#14519702—21no—
119#14519701—21no—
120#14519699—21no—
121#14519698—21no—
122#14519698—21no—
123#14519697—21no—
124#14519689—21no—
125#14519688—21no—
126#14519687D411yescharacteristic
127#14519687D411yescharacteristic
128C4×#92418C2311yescharacteristic
129C8×Dic61—21no—
130C8×Dic61—21no—
131C8×Dic61—21no—
132C2×SD16×C61—21no—
133C2×SD16×C61—21no—
134C2×D8×C61—21no—
135C2×D8×C61—21no—
136#127×C61C2311yescharacteristic
137#127×C61C2311yescharacteristic
138C4×Q8×C61C2311yescharacteristic
139C4⋊C8×C61—21no—
140C4⋊C8×C61C2311yescharacteristic
141C4⋊C8×C61C2311yescharacteristic
142C4⋊C8×C61C2311yescharacteristic
143#102×C61—21no—
144#102×C61—21no—
145#97×C61—21no—
146C4×C8×C61C2311yescharacteristic
147C23×D61—41no—
148C2×Q8×C61C2×D411yescharacteristic
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×C22⋊D61—41no—
156C2×C22⋊D61—41no—
157C2×C22⋊D61—41no—
158C2×C22⋊D61—41no—
159C22×Dic61—41no—
160Q8×D61—22no—
161Q8×D61—41no—
162#11758224—42no—
163#11758224—42no—
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—21no—
173C2×C4×D61—21no—
174C2×C4×D61C2×D411yescharacteristic
175C2×C4×D61C2×D411yescharacteristic
176C2×C4×D61#4111yescharacteristic
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122C2×D411yescharacteristic
180SD16×C61—42no—
181SD16×C61—42no—
182D8×C61—42no—
183D8×C61—42no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61C2×D411yescharacteristic
188C2×C8×C61C2×D411yescharacteristic
189C4⋊C4×C61—21no—
190C4⋊C4×C61C2×D411yescharacteristic
191C42×C61—21no—
192C42×C61C2411yescharacteristic
193C22⋊Dic61—41no—
194C22⋊Dic61—41no—
195#11758201—42no—
196#11758200—42no—
197#11758200—42no—
198#11758200—42no—
199#11758200—42no—
200#11758199—42no—
201#11758199—42no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61C2×D411yescharacteristic
206#11758196—21no—
207#11758196—21no—
208#11758196—21no—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61C2×D411yescharacteristic
213C4×Dic61C2×D411yescharacteristic
214C4×Dic61#4111yescharacteristic
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418C2×D411yescharacteristic
219C2×#92418C2×D411yescharacteristic
220#11758190—42no—
221#11758190—42no—
222#11758190—42no—
223#11758189—22no—
224#11758189—42no—
225#11758189SD1614yescomplemented
226C8×D61—42no—
227C23×C61—41no—
228C23×C61—41no—
229C22×D61—41no—
230C22×D61—82no—
231C22×D61—82no—
232C22×D61—82no—
233C22×D61#12111yescharacteristic
234Q8×C61—21no—
235Q8×C61—21no—
236D4×C61—41no—
237D4×C61—41no—
238D4×C61—41no—
239D4×C61—41no—
240D4×C61—41no—
241D4×C61—41no—
242D4×C61—41no—
243D4×C61—41no—
244D4×C61—81no—
245D4×C61—81no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C22⋊D61—82no—
252C22⋊D61—82no—
253C22⋊D61—82no—
254C22⋊D61—82no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—41no—
258C2×Dic61—82no—
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—41no—
270C4×D61—42no—
271C4×D61—81no—
272C4×D61C2×SD1612yes—
273C4×D61AGL(1,ℤ/8)12yes—
274Dic122—21no—
275Dic122—42no—
276Dic122—41no—
277Dic122—41no—
278Dic122AGL(1,ℤ/8)12yescomplemented
279C8×C61—21no—
280C8×C61—41no—
281C8×C61—41no—
282C8×C61—41no—
283C8×C61C2×SD1612yescomplemented
284#92418—21no—
285#92418—41no—
286#92418—41no—
287#92418—41no—
288#92418C2×SD1612yescomplemented
289#19170—611nomaximal, Sylow
290C22×C61—81no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61—81no—
294C22×C61—81no—
295C22×C61—81no—
296C22×C61C2×#12111yescharacteristic
297C2×D61—21no—
298C2×D61—21no—
299C2×D61—81no—
300C2×D61—81no—
301C2×D61—81no—
302C2×D61—164no—
303C2×D61#46012yes—
304C4×C61—21no—
305C4×C61—41no—
306C4×C61—41no—
307C4×C61C22×SD1611yescharacteristic
308C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
309C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
310C4×C61#57511yescharacteristic
311Dic61—22no—
312Dic61—21no—
313Dic61—21no—
314Dic61—41no—
315Dic61—81no—
316Dic61#46012yes—
317#3151—611no—
318#3103—611no—
319C2×#460—611no—
320Q8×M16—611no—
321#1688—611no—
322#1688D6111yescharacteristic, complemented
323#1686—611no—
324#1686—611no—
325#1668—611no—
326#1668—611no—
327#1666—611no—
328#1666—611no—
329#1565—611no—
330#1547—611no—
331#1545—611no—
332C2×C61—161no—
333C2×C61—161no—
334C2×C61C2×#46011yescharacteristic
335C2×C61#305511yescharacteristic
336C2×C61#306811yescharacteristic
337D61—22no—
338D61—161no—
339C2×AGL(1,ℤ/8)—1221no—
340C2×AGL(1,ℤ/8)—1221no—
341C22×SD16—1221no—
342#532—611no—
343C2×#127—611no—
344C2×C4×Q8—611no—
345#496—1221no—
346#496—1221no—
347#493—611no—
348#493C2×D6111yescharacteristic
349#492—611no—
350#492—611no—
351#481—1221no—
352#475—611no—
353#475—611no—
354#460—611no—
355#460—611no—
356#460—611no—
357#460—611no—
358#460—1221no—
359#460—1221no—
360#460C2×D6111yescharacteristic
361#460C2×D6111yescharacteristic, complemented
362#448—1221no—
363#446—611no—
364#446—1221no—
365#446—1221no—
366#446C2×D6111yescharacteristic, complemented
367C8×Q8—611no—
368C8×Q8—611no—
369C4×SD16—1221no—
370C4×SD16—1221no—
371#431—611no—
372#423—611no—
373#423—1221no—
374C2×C4⋊C8—611no—
375C2×#102—1221no—
376C4×M16—611no—
377C4×M16—1221no—
378#335—611no—
379#335—611no—
380#334—611no—
381#334C2×D6111yescharacteristic, complemented
382#332—611no—
383#332—611no—
384#331—611no—
385#331—611no—
386#331—611no—
387#331—611no—
388#331C2×D6111yescharacteristic
389#331C2×D6111yescharacteristic, complemented
390C61#1917011yescharacteristic, complemented, Sylow
391C2×#41—1221no—
392C22×Q8—611no—
393C22×D4—1221no—
394C22×D4—1221no—
395C22×D4—1221no—
396AGL(1,ℤ/8)—2444no—
397AGL(1,ℤ/8)—2444no—
398C2×SD16—21no—
399C2×SD16—21no—
400C2×SD16—1221no—
401C2×SD16—1221no—
402C2×SD16—1221no—
403C2×SD16—1221no—
404C2×SD16—1221no—
405C2×SD16—1221no—
406C2×SD16—1221no—
407C2×SD16—1221no—
408C2×SD16—2444no—
409C2×D8—21no—
410C2×D8—21no—
411C2×D8—1221no—
412C2×D8—1221no—
413C2×M16—611no—
414C2×M16—611no—
415C2×M16—1221no—
416C22×C8—1221no—
417C4⋊Q8—611no—
418#127—611no—
419#127—1221no—
420#127—1221no—
421#127—1221no—
422#127C22×D6111yescharacteristic
423#127C22×D6111yescharacteristic
424#124—1221no—
425#121—1221no—
426#121—1221no—
427C4×Q8—611no—
428C4×Q8—611no—
429C4×Q8—611no—
430C4×Q8—1221no—
431C4×Q8—1221no—
432C4×Q8C22×D6111yescharacteristic
433C4×D4—1221no—
434C4×D4—1221no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—1221no—
437C2×C42—611no—
438C2×C42—1221no—
439#106—1221no—
440#106—1221no—
441C4⋊C8—21no—
442C4⋊C8—611no—
443C4⋊C8—611no—
444C4⋊C8—611no—
445C4⋊C8—1221no—
446C4⋊C8—1221no—
447C4⋊C8—1221no—
448C4⋊C8—1221no—
449C4⋊C8—1221no—
450C4⋊C8C22×D6111yescharacteristic
451C4⋊C8C22×D6111yescharacteristic
452C4⋊C8C22×D6111yescharacteristic
453#103—1221no—
454#103—1221no—
455#102—21no—
456#102—21no—
457#102—1221no—
458#102—1221no—
459#102—1221no—
460#102—1221no—
461C22⋊C8—1221no—
462#97—21no—
463#97—611no—
464#97—611no—
465#97—1221no—
466C4×C8—611no—
467C4×C8—1221no—
468C4×C8—1221no—
469C4×C8—1221no—
470C4×C8C22×D6111yescharacteristic
471C24—2441no—
472#41—2442no—
473#41—2442no—
474C2×Q8—611no—
475C2×Q8—1221no—
476C2×Q8—1222no—
477C2×Q8—1221no—
478C2×Q8—2441no—
479C2×Q8D4×D6111yescharacteristic
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—1221no—
487C2×D4—1221no—
488C2×D4—1221no—
489C2×D4—4882no—
490C2×D4—2442no—
491C2×D4—2442no—
492C2×D4—2441no—
493C2×D4—2441no—
494C2×D4—2441no—
495C2×D4—2442no—
496C2×D4—2442no—
497C2×D4—2441no—
498C22×C4—611no—
499C22×C4—611no—
500C22×C4—611no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—2441no—
504SD16—42no—
505SD16—42no—
506SD16—2442no—
507SD16—2442no—
508SD16—2442no—
509SD16—2442no—
510SD16—2442no—
511SD16—2442no—
512SD16—2442no—
513SD16—2442no—
514D8—42no—
515D8—42no—
516D8—2442no—
517D8—2442no—
518M16—614no—
519M16—1222no—
520M16—2442no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—611no—
525C2×C8—611no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—2442no—
530C2×C8D4×D6111yescharacteristic
531C2×C8D4×D6111yescharacteristic
532C4⋊C4—21no—
533C4⋊C4—611no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4D4×D6111yescharacteristic
541C22⋊C4—2441no—
542C22⋊C4—2441no—
543C42—21no—
544C42—611no—
545C42—611no—
546C42—611no—
547C42—1221no—
548C42—1221no—
549C42—1221no—
550C42C23×D6111yescharacteristic
551C23—41no—
552C23—41no—
553C23—611no—
554C23—4882no—
555C23—4882no—
556C23—4882no—
557C23—2441no—
558Q8—21no—
559Q8—21no—
560Q8—612no—
561Q8—1221no—
562Q8—2442no—
563Q8—2441no—
564Q8—2441no—
565D4—41no—
566D4—41no—
567D4—41no—
568D4—41no—
569D4—41no—
570D4—41no—
571D4—41no—
572D4—41no—
573D4—81no—
574D4—81no—
575D4—4881no—
576D4—4882no—
577D4—4882no—
578D4—4882no—
579D4—4882no—
580D4—2441no—
581D4—2441no—
582D4—2441no—
583D4—2441no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—611no—
587C2×C4—612no—
588C2×C4—612no—
589C2×C4—611no—
590C2×C4—611no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—4881no—
596C2×C4—4882no—
597C2×C4—2441no—
598C2×C4—2442no—
599C2×C4—2441no—
600C2×C4C2×D4×D6111yescharacteristic
601C2×C4C2×D4×D6111yescharacteristic
602C2×C4#41×D6111yescharacteristic
603C8—21no—
604C8—41no—
605C8—41no—
606C8—41no—
607C8—612no—
608C8—1221no—
609C8—2441no—
610C8—2441no—
611C8—2441no—
612C8SD16×D6112yescomplemented
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—612no—
620C22—1221no—
621C22—1221no—
622C22—9764no—
623C22—4881no—
624C22—4881no—
625C22—4881no—
626C22#121×D6111yescharacteristic
627C4—21no—
628C4—41no—
629C4—41no—
630C4—612no—
631C4—1221no—
632C4—1222no—
633C4—1221no—
634C4—4881no—
635C4—2441no—
636C4C2×SD16×D6111yescharacteristic
637C4AGL(1,ℤ/8)×D6111yescharacteristic
638C4AGL(1,ℤ/8)×D6111yescharacteristic
639C4#1684055811yescharacteristic
640C2—161no—
641C2—161no—
642C2—1222no—
643C2—9761no—
644C2#460×D6111yescharacteristic
645C2#2145543711yescharacteristic
646C2#2145554511yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1688C2
socleC22×C61C2×#121
prime radical#21456270C2
hypercenter#1688D61
hypocenterC61#19170
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1688

Complements

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

NH ≅ G/Ntype
2. #21455849C2semidirect
3. #460×D61C2semidirect
4. #21454771C2semidirect
5. #21445301C2semidirect
6. #21445291C2semidirect
7. #21445289C2semidirect
8. #21445178C2semidirect
9. #21445176C2semidirect
10. #21445168C2semidirect
11. #21445166C2semidirect
12. #21444568C2semidirect
13. #21444493C2semidirect
14. #21444485C2semidirect
15. C61×#1688C2semidirect
28. #16840048C22semidirect
29. #16840047C22semidirect
35. #16839959C22semidirect
37. #16839893C22semidirect
40. C8×Dic122C22semidirect
41. #16839886C22semidirect
51. #16839515C22semidirect
52. #16839514C22semidirect
53. #16839512C22semidirect
54. #16839509C22semidirect
55. #16839507C22semidirect
57. #16839506C22semidirect
62. C61×#460C22semidirect
63. C61×#446C22semidirect
64. C61×#334C22semidirect
66. C61×#331C22semidirect
225. #11758189SD16semidirect
278. Dic122AGL(1,ℤ/8)semidirect
283. C8×C61C2×SD16semidirect
288. #92418C2×SD16semidirect
322. #1688D61semidirect
361. #460C2×D61semidirect
366. #446C2×D61semidirect
381. #334C2×D61semidirect
389. #331C2×D61semidirect
390. C61#19170semidirect
612. C8SD16×D61semidirect

Maximal subgroups

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

subgroupindex
#214562702
#214558492
#460×D612
#214547712
#214453012
#214452912
#214452892
#214451782
#214451762
#214451682
#214451662
#214445682
#214444932
#214444852
C61×#16882
#1917061