Finite Group Databasebeta

← #29671387

Subgroups — #29671387

Order 15616.

720 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458800C211yesmaximal, characteristic, complemented
2#21458738C211yesmaximal, characteristic, complemented
3D4×#11758200C211yesmaximal, characteristic, complemented
4#21445408C211yesmaximal, characteristic, complemented
5#21445332C211yesmaximal, characteristic, complemented
6#21445300C211yesmaximal, characteristic, complemented
7#21445165C211yesmaximal, characteristic, complemented
8#21445158C211yesmaximal, characteristic, complemented
9#21445019C211yesmaximal, characteristic, complemented
10#21444998C211yesmaximal, characteristic, complemented
11#21444921C211yesmaximal, characteristic, complemented
12#21444914C211yesmaximal, characteristic, complemented
13#21444707C211yesmaximal, characteristic, complemented
14SD16×#92418C211yesmaximal, characteristic, complemented
15C61×#3274C211yesmaximal, characteristic
16C22×#11758200—21no—
17#16840491C2211yescharacteristic, complemented
18D4×Dic122C2211yescharacteristic
19#16840169—21no—
20#16840167C2211yescharacteristic, complemented
21#16840134—21no—
22#16840134C2211yescharacteristic
23#16840134C2211yescharacteristic, complemented
24#16840108C2211yescharacteristic
25#16840108C2211yescharacteristic, complemented
26#16840060—21no—
27#16840055—21no—
28#16840047C2211yescharacteristic, complemented
29#16840042C2211yescharacteristic
30#16839989—21no—
31#16839967—21no—
32#16839967C2211yescharacteristic
33#16839967C2211yescharacteristic
34#16839959—21no—
35#16839959C2211yescharacteristic
36#16839959C2211yescharacteristic, complemented
37#16839913—21no—
38#16839900—21no—
39#16839892C2211yescharacteristic, complemented
40Q8×#92418C2211yescharacteristic, complemented
41#16839863—21no—
42C4×#11758200—21no—
43C4×#11758200C2211yescharacteristic
44D4×#92418C2211yescharacteristic
45#16839513C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839507C2211yescharacteristic, complemented
48#16839507C2211yescharacteristic, complemented
49#16839506C2211yescharacteristic
50#16839506C2211yescharacteristic, complemented
51Q8⋊#92418C2211yescharacteristic, complemented
52#16839492C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55C8×#92418C2211yescharacteristic, complemented
56C2×AGL(1,ℤ/8)×C61—21no—
57C61×#550C2211yescharacteristic
58D42×C61C2211yescharacteristic
59C61×#493C2211yescharacteristic
60C61×#469—21no—
61C61×#460C2211yescharacteristic
62C61×#459C2211yescharacteristic
63C61×#449—21no—
64C61×#447—21no—
65C4×SD16×C61C2211yescharacteristic
66C61×#436C2211yescharacteristic
67C22×Dic122—21no—
68C2×#11758200—21no—
69C2×#11758200—21no—
70C2×#11758200—21no—
71C2×#11758200—21no—
72C2×#11758200—21no—
73C2×#11758200—21no—
74C2×#11758200—21no—
75C2×#11758200—21no—
76C2×#11758200—44no—
77C2×#11758200D411yescharacteristic
78C22×#92418—21no—
79#14519777—21no—
80#14519777C2311yescharacteristic
81#14519777C2311yescharacteristic
82#14519776—21no—
83#14519768—21no—
84#14519768—41no—
85#14519756—21no—
86#14519754—41no—
87Q8×Dic61—21no—
88C4×Dic122—21no—
89C4×Dic122C2311yescharacteristic
90D4×Dic61—21no—
91#14519711—21no—
92#14519711—21no—
93#14519710—21no—
94#14519708—21no—
95#14519708D411yescharacteristic
96C4⋊#92418—21no—
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100#14519706—21no—
101#14519701—21no—
102#14519700—21no—
103#14519700—21no—
104#14519700D411yescharacteristic
105#14519698—21no—
106#14519698—21no—
107#14519698—21no—
108#14519698—41no—
109#14519698—41no—
110#14519698D411yescharacteristic
111C22⋊#92418—21no—
112#14519687—21no—
113C4×#92418—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C2×#41×C61—21no—
117C22×D4×C61—21no—
118C22×D4×C61—21no—
119AGL(1,ℤ/8)×C61—44no—
120C2×SD16×C61—21no—
121C2×SD16×C61D411yescharacteristic
122C2×D8×C61—21no—
123C2×D8×C61—21no—
124C2×D8×C61—21no—
125C2×M16×C61—21no—
126#127×C61—21no—
127#127×C61C2311yescharacteristic
128#127×C61C2311yescharacteristic
129#121×C61—21no—
130#121×C61—21no—
131#121×C61—21no—
132#121×C61—41no—
133#120×C61—41no—
134C4×Q8×C61C2311yescharacteristic
135C4×D4×C61—21no—
136C4×D4×C61C2311yescharacteristic
137C4×D4×C61C2311yescharacteristic
138#106×C61D411yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140#103×C61D411yescharacteristic
141#102×C61—21no—
142#102×C61—21no—
143#102×C61D411yescharacteristic
144C22⋊C8×C61—21no—
145C4×C8×C61C2311yescharacteristic
146C24×C61—41no—
147#41×C61—42no—
148#41×C61—81no—
149C2×Q8×C61C2×D411yescharacteristic
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×D4×C61—21no—
156C2×D4×C61—41no—
157C2×D4×C61—41no—
158C2×D4×C61—41no—
159C2×D4×C61—41no—
160C2×D4×C61—42no—
161C2×D4×C61—42no—
162C2×D4×C61—81no—
163C2×D4×C61—81no—
164C2×D4×C61C2×D411yescharacteristic
165C2×D4×C61C2×D411yescharacteristic
166C22×C4×C61—21no—
167C22×C4×C61—21no—
168C22×C4×C61—41no—
169C22×Dic61—41no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—42no—
173C2×Dic122—41no—
174C2×Dic122—81no—
175C2×Dic122C2×D411yescharacteristic
176SD16×C61—42no—
177SD16×C61SD1614yescomplemented
178D8×C61—42no—
179D8×C61—42no—
180D8×C61—81no—
181M16×C61—42no—
182C2×C8×C61—21no—
183C2×C8×C61C2×D411yescharacteristic
184C2×C8×C61C2×D411yescharacteristic
185C4⋊C4×C61—21no—
186C4⋊C4×C61C2×D411yescharacteristic
187C4⋊C4×C61C2×D411yescharacteristic
188C4⋊C4×C61C2×D411yescharacteristic
189C22⋊C4×C61—21no—
190C22⋊C4×C61—21no—
191C22⋊C4×C61—41no—
192C22⋊C4×C61—41no—
193C42×C61—21no—
194C42×C61C2411yescharacteristic
195C22⋊Dic61—41no—
196#11758200—22no—
197#11758200—42no—
198#11758200—42no—
199#11758200—42no—
200#11758200—42no—
201#11758200—81no—
202#11758200—81no—
203#11758200—81no—
204#11758200—81no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—41no—
210C4⋊Dic61—41no—
211C4⋊Dic61C2×D411yescharacteristic
212#11758196—21no—
213#11758196—41no—
214#11758196—41no—
215C4×Dic61—21no—
216C4×Dic61—41no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—42no—
222C2×#92418C2×D411yescharacteristic
223C2×#92418C2×D411yescharacteristic
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—82no—
229C23×C61—82no—
230C23×C61—161no—
231Q8×C61—21no—
232Q8×C61C2×SD1612yescomplemented
233D4×C61—21no—
234D4×C61—21no—
235D4×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—82no—
245D4×C61—81no—
246D4×C61—81no—
247D4×C61—82no—
248D4×C61—82no—
249D4×C61—161no—
250D4×C61C2×SD1612yes—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—82no—
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C22×D411yescharacteristic
262C2×C4×C61H(5,𝔽2)11yescharacteristic
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—82no—
268Dic122—21no—
269Dic122—21no—
270Dic122—41no—
271Dic122—41no—
272Dic122—41no—
273Dic122—41no—
274Dic122—82no—
275C8×C61—21no—
276C8×C61—41no—
277C8×C61C2×SD1612yescomplemented
278#92418—21no—
279#92418—41no—
280#92418—41no—
281#92418—41no—
282#92418—41no—
283#92418AGL(1,ℤ/8)12yescomplemented
284#19892—611nomaximal, Sylow
285C22×C61—22no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61—41no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61—81no—
294C22×C61—81no—
295C22×C61—81no—
296C22×C61—81no—
297C22×C61—164no—
298C22×C61D4211yescharacteristic
299C4×C61—22no—
300C4×C61—21no—
301C4×C61—41no—
302C4×C61—41no—
303C4×C61—41no—
304C4×C61C22×SD1611yescharacteristic
305C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
306C4×C61#57611yescharacteristic
307C4×C61#57711yescharacteristic
308Dic61—41no—
309Dic61—41no—
310Dic61—81no—
311Dic61—81no—
312#3392—611no—
313#3386—611no—
314D4×SD16—611no—
315#3274D6111yescharacteristic
316#1706—611no—
317#1693—611no—
318#1688—611no—
319#1666—611no—
320#1665—611no—
321#1642—611no—
322#1639—611no—
323#1627—611no—
324#1626—611no—
325#1590—611no—
326C8×SD16—611no—
327C2×C61—42no—
328C2×C61—81no—
329C2×C61—161no—
330C2×C61—161no—
331C2×C61#327411yescharacteristic
332C2×C61D4×SD1611yescharacteristic
333C2×C61#329011yescharacteristic
334C2×AGL(1,ℤ/8)—21no—
335C22×SD16—1221no—
336#557—611no—
337#550C2×D6111yescharacteristic
338D4×Q8—611no—
339D42C2×D6111yescharacteristic
340#501—1221no—
341#500—611no—
342#493C2×D6111yescharacteristic
343#492—611no—
344#492—611no—
345#492—1221no—
346#486—611no—
347#486—611no—
348#478—1221no—
349#477—1221no—
350#475—611no—
351#474—611no—
352#469—21no—
353#465—1221no—
354#461—611no—
355#461—611no—
356#461—1221no—
357#460—611no—
358#460—611no—
359#460—1221no—
360#460C2×D6111yescharacteristic
361#459C2×D6111yescharacteristic
362#450—1221no—
363#449—21no—
364#448—1221no—
365#447—21no—
366#446—611no—
367C8×Q8—611no—
368#440—1221no—
369C4×SD16—611no—
370C4×SD16—1221no—
371C4×SD16C2×D6111yescharacteristic
372#436C2×D6111yescharacteristic
373C8×D4—611no—
374#334—611no—
375#334—611no—
376#332—611no—
377#332—611no—
378#331—611no—
379#331—611no—
380Q8⋊C8—611no—
381Q8⋊C8—611no—
382D4⋊C8—611no—
383D4⋊C8—611no—
384C82—611no—
385C61#1989211yescharacteristic, complemented, Sylow
386C2×#41—21no—
387C22×Q8—1221no—
388C22×D4—21no—
389C22×D4—21no—
390AGL(1,ℤ/8)—44no—
391C2×SD16—21no—
392C2×SD16—611no—
393C2×SD16—1221no—
394C2×SD16—1221no—
395C2×SD16—1221no—
396C2×SD16—1221no—
397C2×SD16—1221no—
398C2×SD16—1221no—
399C2×SD16—1221no—
400C2×SD16—1221no—
401C2×SD16—2444no—
402C2×SD16C22⋊D6111yescharacteristic
403C2×D8—21no—
404C2×D8—21no—
405C2×D8—21no—
406C2×M16—21no—
407C22×C8—1221no—
408C4⋊Q8—611no—
409C4⋊Q8—611no—
410C4⋊Q8—1221no—
411C4⋊Q8—1221no—
412#127—21no—
413#127C22×D6111yescharacteristic
414#127C22×D6111yescharacteristic
415#125—1221no—
416#125—2441no—
417C22⋊Q8—1221no—
418C22⋊Q8—2441no—
419#121—21no—
420#121—21no—
421#121—21no—
422#121—41no—
423#120—41no—
424C4×Q8—611no—
425C4×Q8—1221no—
426C4×Q8—1221no—
427C4×Q8C22×D6111yescharacteristic
428C4×D4—21no—
429C4×D4—1221no—
430C4×D4C22×D6111yescharacteristic
431C4×D4C22×D6111yescharacteristic
432#107—1221no—
433#107—1221no—
434#107—1221no—
435#106—611no—
436#106—1221no—
437#106C22⋊D6111yescharacteristic
438C4⋊C8—611no—
439C4⋊C8—611no—
440C4⋊C8—611no—
441C4⋊C8—1221no—
442C4⋊C8—1221no—
443C4⋊C8C22×D6111yescharacteristic
444#103—611no—
445#103—1221no—
446#103—1221no—
447#103—1221no—
448#103C22⋊D6111yescharacteristic
449#102—21no—
450#102—21no—
451#102—611no—
452#102—1221no—
453#102—1221no—
454#102—1221no—
455#102—2441no—
456#102—2441no—
457#102C22⋊D6111yescharacteristic
458C22⋊C8—21no—
459C22⋊C8—1221no—
460#97—1221no—
461C4×C8—611no—
462C4×C8—611no—
463C4×C8—1221no—
464C4×C8C22×D6111yescharacteristic
465C24—41no—
466#41—42no—
467#41—81no—
468C2×Q8—611no—
469C2×Q8—1221no—
470C2×Q8—1221no—
471C2×Q8—4881no—
472C2×Q8—2441no—
473C2×Q8—2442no—
474C2×Q8C2×C22⋊D6111yescharacteristic
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—41no—
482C2×D4—41no—
483C2×D4—41no—
484C2×D4—41no—
485C2×D4—42no—
486C2×D4—42no—
487C2×D4—81no—
488C2×D4—81no—
489C2×D4D4×D6111yescharacteristic
490C2×D4C2×C22⋊D6111yescharacteristic
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—41no—
494C22×C4—2441no—
495SD16—42no—
496SD16—1222no—
497SD16—4881no—
498SD16—4881no—
499SD16—4881no—
500SD16—4881no—
501SD16—2442no—
502SD16—2442no—
503SD16—2442no—
504SD16—2442no—
505SD16#1175820014yescomplemented
506D8—42no—
507D8—42no—
508D8—81no—
509M16—42no—
510C2×C8—21no—
511C2×C8—611no—
512C2×C8—611no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—2442no—
518C2×C8C2×C22⋊D6111yescharacteristic
519C2×C8C2×C22⋊D6111yescharacteristic
520C4⋊C4—21no—
521C4⋊C4—611no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—2441no—
528C4⋊C4—2441no—
529C4⋊C4—2441no—
530C4⋊C4—2441no—
531C4⋊C4D4×D6111yescharacteristic
532C4⋊C4C2×C22⋊D6111yescharacteristic
533C4⋊C4C2×C22⋊D6111yescharacteristic
534C22⋊C4—21no—
535C22⋊C4—21no—
536C22⋊C4—41no—
537C22⋊C4—41no—
538C22⋊C4—2441no—
539C42—21no—
540C42—1221no—
541C42—2441no—
542C42C23×D6111yescharacteristic
543C23—21no—
544C23—21no—
545C23—41no—
546C23—41no—
547C23—82no—
548C23—82no—
549C23—161no—
550Q8—21no—
551Q8—1221no—
552Q8—1221no—
553Q8—4882no—
554Q8—2441no—
555Q8—2441no—
556Q8—2441no—
557Q8—2441no—
558Q8C2×#1175820012yescomplemented
559D4—21no—
560D4—21no—
561D4—21no—
562D4—41no—
563D4—41no—
564D4—41no—
565D4—41no—
566D4—41no—
567D4—41no—
568D4—41no—
569D4—41no—
570D4—82no—
571D4—81no—
572D4—81no—
573D4—82no—
574D4—82no—
575D4—161no—
576D4C2×#1175820012yes—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—82no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—4882no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4C2×D4×D6111yescharacteristic
592C2×C4C22×C22⋊D6111yescharacteristic
593C2×C4#1451982211yescharacteristic
594C8—21no—
595C8—41no—
596C8—612no—
597C8—1221no—
598C8—2441no—
599C8—2441no—
600C8—2441no—
601C8—2441no—
602C8C2×#1175820012yescomplemented
603C22—22no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—164no—
616C22D4×C22⋊D6111yescharacteristic
617C4—22no—
618C4—21no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—4881no—
623C4—4881no—
624C4—2441no—
625C4—2441no—
626C4C22×#1175820011yescharacteristic
627C4AGL(1,ℤ/8)×D6111yescharacteristic
628C4#1684056611yescharacteristic
629C4#1684057311yescharacteristic
630C2—42no—
631C2—81no—
632C2—161no—
633C2—161no—
634C2#2145743311yescharacteristic
635C2D4×#1175820011yescharacteristic
636C2#2145760111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3274C2
socleC22×C61D42
hypercenter#3274D61
hypocenterC61#19892
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3274

Complements

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

NH ≅ G/Ntype
1. #21458800C2semidirect
2. #21458738C2semidirect
3. D4×#11758200C2semidirect
4. #21445408C2semidirect
5. #21445332C2semidirect
6. #21445300C2semidirect
7. #21445165C2semidirect
8. #21445158C2semidirect
9. #21445019C2semidirect
10. #21444998C2semidirect
11. #21444921C2semidirect
12. #21444914C2semidirect
13. #21444707C2semidirect
14. SD16×#92418C2semidirect
17. #16840491C22semidirect
20. #16840167C22semidirect
23. #16840134C22semidirect
25. #16840108C22semidirect
28. #16840047C22semidirect
36. #16839959C22semidirect
39. #16839892C22semidirect
40. Q8×#92418C22semidirect
45. #16839513C22semidirect
46. #16839512C22semidirect
47. #16839507C22semidirect
48. #16839507C22semidirect
50. #16839506C22semidirect
51. Q8⋊#92418C22semidirect
52. #16839492C22semidirect
55. C8×#92418C22semidirect
177. SD16×C61SD16semidirect
232. Q8×C61C2×SD16semidirect
277. C8×C61C2×SD16semidirect
283. #92418AGL(1,ℤ/8)semidirect
385. C61#19892semidirect
505. SD16#11758200semidirect
558. Q8C2×#11758200semidirect
602. C8C2×#11758200semidirect

Maximal subgroups

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

subgroupindex
#214588002
#214587382
D4×#117582002
#214454082
#214453322
#214453002
#214451652
#214451582
#214450192
#214449982
#214449212
#214449142
#214447072
SD16×#924182
C61×#32742
#1989261