Finite Group Databasebeta

← #29677206

Subgroups — #29677206

Order 15616.

716 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#21458416C211yesmaximal, characteristic, complemented
2#21458414C211yesmaximal, characteristic, complemented
3#21457415C211yesmaximal, characteristic, complemented
4#21445405C211yesmaximal, characteristic, complemented
5#21445318C211yesmaximal, characteristic, complemented
6#21445316C211yesmaximal, characteristic, complemented
7#21445282C211yesmaximal, characteristic, complemented
8#21445281C211yesmaximal, characteristic, complemented
9#21445103C211yesmaximal, characteristic, complemented
10#21445101C211yesmaximal, characteristic, complemented
11#21444935C211yesmaximal, characteristic, complemented
12#21444901C211yesmaximal, characteristic, complemented
13#21444704C211yesmaximal, characteristic, complemented
14#21444702C211yesmaximal, characteristic, complemented
15C61×#3272C211yesmaximal, characteristic, complemented
16C2×#14519803—21no—
17#16840477C2211yescharacteristic, complemented
18#16840417C2211yescharacteristic
19#16840160C2211yescharacteristic, complemented
20#16840158C2211yescharacteristic, complemented
21#16840158C2211yescharacteristic, complemented
22#16840134C2211yescharacteristic
23#16840134C2211yescharacteristic
24#16840085—21no—
25#16840083—21no—
26#16840080—21no—
27#16840079—21no—
28#16840052C2211yescharacteristic
29#16840052C2211yescharacteristic
30#16840042C2211yescharacteristic, complemented
31#16840039C2211yescharacteristic, complemented
32#16840013—21no—
33#16840012—21no—
34#16840006—21no—
35#16839967C2211yescharacteristic
36#16839954C2211yescharacteristic
37#16839913—21no—
38#16839905—21no—
39C4×#11758199C2211yescharacteristic
40#16839841C2211yescharacteristic, complemented
41#16839840C2211yescharacteristic
42#16839840C2211yescharacteristic, complemented
43#16839803—21no—
44#16839802—21no—
45#16839515C2211yescharacteristic, complemented
46#16839514C2211yescharacteristic, complemented
47#16839510C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839507C2211yescharacteristic, complemented
50#16839506C2211yescharacteristic
51D4⋊#92418C2211yescharacteristic
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic, complemented
54#16839489C2211yescharacteristic, complemented
55C8×#92418C2211yescharacteristic, complemented
56C2×AGL(1,ℤ/8)×C61—21no—
57C61×#547C2211yescharacteristic
58D42×C61C2211yescharacteristic
59C61×#492C2211yescharacteristic
60C61×#468—21no—
61C61×#461C2211yescharacteristic
62C61×#459C2211yescharacteristic
63C61×#450—21no—
64C61×#449—21no—
65C4×D8×C61C2211yescharacteristic
66C61×#436C2211yescharacteristic
67C2×#11758222—21no—
68#14519803—44no—
69C2×#11758200—21no—
70C2×#11758200—21no—
71C2×#11758200—21no—
72C2×#11758200—41no—
73C2×#11758199—21no—
74C2×#11758199D411yescharacteristic
75C2×#11758194—21no—
76C2×#11758194—21no—
77C2×#11758189—21no—
78#14519777C2311yescharacteristic
79#14519777C2311yescharacteristic
80#14519763—41no—
81#14519759—41no—
82#14519756—21no—
83#14519751—21no—
84#14519748—21no—
85#14519748—21no—
86C4×D244C2311yescharacteristic
87#14519736—21no—
88#14519735—21no—
89C4⋊C4×D61—21no—
90#14519710—21no—
91#14519710—21no—
92#14519710D411yescharacteristic
93#14519709—21no—
94#14519709—21no—
95#14519708—21no—
96#14519708—21no—
97#14519708—21no—
98#14519708—21no—
99#14519708—41no—
100C4⋊#92418C2311yescharacteristic
101C4⋊#92418C2311yescharacteristic
102C4⋊#92418C2311yescharacteristic
103#14519700—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—21no—
108#14519698D411yescharacteristic
109#14519697—21no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112C22⋊#92418—21no—
113#14519689—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×C61—21no—
122C2×SD16×C61—21no—
123C2×D8×C61—21no—
124C2×D8×C61D411yescharacteristic
125C2×M16×C61—21no—
126C4⋊Q8×C61C2311yescharacteristic
127#127×C61C2311yescharacteristic
128#123×C61—41no—
129C22⋊Q8×C61—21no—
130#121×C61—21no—
131#121×C61—21no—
132#121×C61—21no—
133#120×C61—41no—
134C4×D4×C61C2311yescharacteristic
135C4×D4×C61C2311yescharacteristic
136C4×D4×C61C2311yescharacteristic
137C2×C4⋊C4×C61—21no—
138#107×C61D411yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140#103×C61—21no—
141#102×C61—21no—
142#102×C61D411yescharacteristic
143#102×C61D411yescharacteristic
144C22⋊C8×C61—21no—
145C4×C8×C61C2311yescharacteristic
146C24×C61—41no—
147#41×C61—42no—
148#41×C61—81no—
149C2×Q8×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—41no—
154C2×D4×C61—41no—
155C2×D4×C61—42no—
156C2×D4×C61—42no—
157C2×D4×C61—41no—
158C2×D4×C61—81no—
159C2×D4×C61—81no—
160C2×D4×C61C2×D411yescharacteristic
161C2×D4×C61C2×D411yescharacteristic
162C2×D4×C61C2×D411yescharacteristic
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61—21no—
166C22×C4×C61—41no—
167C2×C22⋊D61—41no—
168C2×C22⋊D61—41no—
169#11758222—42no—
170#11758222—81no—
171C2×D244C2×D411yescharacteristic
172C2×C4×D61—21no—
173C2×C4×D61—41no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176SD16×C61—42no—
177SD16×C61—42no—
178SD16×C61—81no—
179D8×C61—22no—
180D8×C61—42no—
181M16×C61—42no—
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
191C22⋊C4×C61—21no—
192C22⋊C4×C61—21no—
193C22⋊C4×C61—21no—
194C22⋊C4×C61—41no—
195C22⋊C4×C61—41no—
196C42×C61C2411yescharacteristic
197C22⋊Dic61—41no—
198C22⋊Dic61—41no—
199#11758200—42no—
200#11758200—42no—
201#11758200—81no—
202#11758200—82no—
203#11758199—22no—
204#11758199—42no—
205#11758198—21no—
206#11758198—41no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61C2×D411yescharacteristic
212#11758196—41no—
213C4×Dic61—41no—
214#11758194—42no—
215#11758194—42no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418C2×D411yescharacteristic
221C2×#92418C2×D411yescharacteristic
222#11758189—42no—
223C23×C61—21no—
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—41no—
227C23×C61—82no—
228C23×C61—82no—
229C23×C61—161no—
230C22×D61—21no—
231Q8×C61—41no—
232Q8×C61—41no—
233D4×C61—21no—
234D4×C61—21no—
235D4×C61—21no—
236D4×C61—21no—
237D4×C61—21no—
238D4×C61—21no—
239D4×C61—41no—
240D4×C61—41no—
241D4×C61—41no—
242D4×C61—41no—
243D4×C61—81no—
244D4×C61—82no—
245D4×C61—82no—
246D4×C61—82no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—82no—
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61H(5,𝔽2)11yescharacteristic
261C22⋊D61—82no—
262C22⋊D61—161no—
263C2×Dic61—21no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266D244—21no—
267D244—21no—
268C4×D61—41no—
269C4×D61—41no—
270C4×D61—82no—
271Dic122—41no—
272Dic122—41no—
273Dic122—81no—
274C8×C61—21no—
275C8×C61—41no—
276C8×C61AGL(1,ℤ/8)12yescomplemented
277#92418—22no—
278#92418—21no—
279#92418—41no—
280#92418—41no—
281#92418—41no—
282#92418AGL(1,ℤ/8)12yescomplemented
283#20314—611nomaximal, Sylow
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61—81no—
294C22×C61—81no—
295C22×C61—81no—
296C22×C61—164no—
297C22×C61D4211yescharacteristic
298C2×D61—41no—
299C2×D61—41no—
300C2×D61—41no—
301C4×C61—21no—
302C4×C61—41no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61—81no—
306C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
307C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
308C4×C61#57711yescharacteristic
309C4×C61#57711yescharacteristic
310Dic61—41no—
311Dic61—81no—
312Dic61—81no—
313#3354—611no—
314#3354—611no—
315#3272—611no—
316#3272D6111yescharacteristic, complemented
317#1705—611no—
318#1691—611no—
319#1691—611no—
320#1685—611no—
321#1685—611no—
322#1654—611no—
323#1654—611no—
324#1629—611no—
325#1624—611no—
326#1589—611no—
327#1589—611no—
328C2×C61—81no—
329C2×C61—81no—
330C2×C61—81no—
331C2×C61—161no—
332C2×C61#327211yescharacteristic
333C2×C61#327211yescharacteristic
334C2×C61#329111yescharacteristic
335D61—81no—
336C2×AGL(1,ℤ/8)—21no—
337C2×AGL(1,ℤ/8)—1221no—
338#555—611no—
339#547—611no—
340#547C2×D6111yescharacteristic
341D42C2×D6111yescharacteristic
342#498—611no—
343#498—611no—
344#498—611no—
345#492—611no—
346#492—611no—
347#492C2×D6111yescharacteristic
348#482—1221no—
349#482—1221no—
350#481—1221no—
351#481—1221no—
352#476—611no—
353#476—611no—
354#474—611no—
355#474—611no—
356#469—1221no—
357#469—1221no—
358#468—21no—
359#468—1221no—
360#461—611no—
361#461C2×D6111yescharacteristic
362#459—611no—
363#459C2×D6111yescharacteristic
364#450—21no—
365#450—1221no—
366#449—21no—
367#449—1221no—
368C4×D8—611no—
369C4×D8C2×D6111yescharacteristic
370#436—611no—
371#436—611no—
372#436—611no—
373#436C2×D6111yescharacteristic
374#428—1221no—
375#428—1221no—
376#335—611no—
377#335—611no—
378#333—611no—
379#332—611no—
380#332—611no—
381#331—611no—
382D4⋊C8—611no—
383D4⋊C8—611no—
384D4⋊C8—611no—
385D4⋊C8—611no—
386C82—611no—
387C61#2031411yescharacteristic, complemented, Sylow
388C2×#41—21no—
389C2×#41—1221no—
390C22×D4—21no—
391C22×D4—21no—
392AGL(1,ℤ/8)—44no—
393AGL(1,ℤ/8)—2444no—
394C2×SD16—21no—
395C2×SD16—21no—
396C2×SD16—21no—
397C2×SD16—1221no—
398C2×SD16—1221no—
399C2×SD16—1221no—
400C2×SD16—2441no—
401C2×D8—21no—
402C2×D8—611no—
403C2×D8—1221no—
404C2×D8C22⋊D6111yescharacteristic
405C2×M16—21no—
406C2×M16—1221no—
407C2×M16—1221no—
408C2×M16—1221no—
409C4⋊Q8—611no—
410C4⋊Q8—611no—
411C4⋊Q8C22×D6111yescharacteristic
412#127C22×D6111yescharacteristic
413#124—2441no—
414#123—41no—
415#123—2441no—
416C22⋊Q8—21no—
417C22⋊Q8—1221no—
418#121—21no—
419#121—21no—
420#121—21no—
421#121—1221no—
422#121—1221no—
423#121—1221no—
424#120—41no—
425C4×D4—611no—
426C4×D4C22×D6111yescharacteristic
427C4×D4C22×D6111yescharacteristic
428C4×D4C22×D6111yescharacteristic
429#117—1221no—
430#117—1221no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—1221no—
433#107—611no—
434#107—1221no—
435#107—1221no—
436#107C22⋊D6111yescharacteristic
437#106—1221no—
438#106—1221no—
439#106—1221no—
440#106—1221no—
441#106—1221no—
442#106—1221no—
443#106—2441no—
444C4⋊C8—611no—
445C4⋊C8—611no—
446C4⋊C8—611no—
447C4⋊C8C22×D6111yescharacteristic
448#103—21no—
449#103—1221no—
450#102—21no—
451#102—611no—
452#102—611no—
453#102—1221no—
454#102—1221no—
455#102—1221no—
456#102—1221no—
457#102—1221no—
458#102C22⋊D6111yescharacteristic
459#102C22⋊D6111yescharacteristic
460C22⋊C8—21no—
461C22⋊C8—1221no—
462C22⋊C8—1221no—
463C22⋊C8—1221no—
464C4×C8—611no—
465C4×C8—611no—
466C4×C8C22×D6111yescharacteristic
467C24—41no—
468#41—42no—
469#41—81no—
470#41—4881no—
471#41—2442no—
472C2×Q8—21no—
473C2×Q8—1221no—
474C2×Q8—1221no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—41no—
479C2×D4—41no—
480C2×D4—42no—
481C2×D4—42no—
482C2×D4—41no—
483C2×D4—81no—
484C2×D4—81no—
485C2×D4—611no—
486C2×D4—2441no—
487C2×D4—2441no—
488C2×D4D4×D6111yescharacteristic
489C2×D4C2×C22⋊D6111yescharacteristic
490C2×D4C2×C22⋊D6111yescharacteristic
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—41no—
495C22×C4—1221no—
496C22×C4—2441no—
497SD16—42no—
498SD16—42no—
499SD16—81no—
500SD16—4881no—
501SD16—4882no—
502SD16—2442no—
503SD16—2442no—
504D8—22no—
505D8—42no—
506D8—1222no—
507D8—2442no—
508M16—42no—
509M16—2442no—
510M16—2442no—
511M16—2442no—
512C2×C8—21no—
513C2×C8—611no—
514C2×C8—611no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8C2×C22⋊D6111yescharacteristic
520C2×C8C2×C22⋊D6111yescharacteristic
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—41no—
524C4⋊C4—611no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—2441no—
530C4⋊C4D4×D6111yescharacteristic
531C4⋊C4C2×C22⋊D6111yescharacteristic
532C4⋊C4C2×C22⋊D6111yescharacteristic
533C22⋊C4—21no—
534C22⋊C4—21no—
535C22⋊C4—21no—
536C22⋊C4—41no—
537C22⋊C4—41no—
538C22⋊C4—1221no—
539C22⋊C4—2441no—
540C22⋊C4—2441no—
541C22⋊C4—2441no—
542C42—2441no—
543C42C23×D6111yescharacteristic
544C23—21no—
545C23—21no—
546C23—21no—
547C23—41no—
548C23—82no—
549C23—82no—
550C23—161no—
551C23—1221no—
552Q8—41no—
553Q8—41no—
554Q8—4881no—
555Q8—2441no—
556Q8—2441no—
557D4—21no—
558D4—21no—
559D4—21no—
560D4—21no—
561D4—21no—
562D4—21no—
563D4—41no—
564D4—41no—
565D4—41no—
566D4—41no—
567D4—81no—
568D4—82no—
569D4—82no—
570D4—82no—
571D4—1221no—
572D4—1221no—
573D4—9761no—
574D4—4882no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—82no—
586C2×C4—1221no—
587C2×C4—4882no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4C2×D4×D6111yescharacteristic
593C2×C4C22×C22⋊D6111yescharacteristic
594C2×C4#1451982211yescharacteristic
595C8—21no—
596C8—41no—
597C8—612no—
598C8—1222no—
599C8—1221no—
600C8—2441no—
601C8—2441no—
602C8—2441no—
603C8#1451980312yescomplemented
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—164no—
617C22—2441no—
618C22—2441no—
619C22—2441no—
620C22D4×C22⋊D6111yescharacteristic
621C4—21no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—81no—
626C4—4881no—
627C4—4881no—
628C4—2441no—
629C4AGL(1,ℤ/8)×D6111yescharacteristic
630C4C2×#1451980311yescharacteristic
631C4#1684057211yescharacteristic
632C4#1684057311yescharacteristic
633C2—81no—
634C2—81no—
635C2—81no—
636C2—161no—
637C2—4881no—
638C2#2145741311yescharacteristic
639C2#2145741511yescharacteristic
640C2#2145761011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3272C2
socleC22×C61D42
hypercenter#3272D61
hypocenterC61#20314
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3272

Complements

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

NH ≅ G/Ntype
1. #21458416C2semidirect
2. #21458414C2semidirect
3. #21457415C2semidirect
4. #21445405C2semidirect
5. #21445318C2semidirect
6. #21445316C2semidirect
7. #21445282C2semidirect
8. #21445281C2semidirect
9. #21445103C2semidirect
10. #21445101C2semidirect
11. #21444935C2semidirect
12. #21444901C2semidirect
13. #21444704C2semidirect
14. #21444702C2semidirect
15. C61×#3272C2semidirect
17. #16840477C22semidirect
19. #16840160C22semidirect
20. #16840158C22semidirect
21. #16840158C22semidirect
30. #16840042C22semidirect
31. #16840039C22semidirect
40. #16839841C22semidirect
42. #16839840C22semidirect
45. #16839515C22semidirect
46. #16839514C22semidirect
47. #16839510C22semidirect
48. #16839508C22semidirect
49. #16839507C22semidirect
53. D4⋊#92418C22semidirect
54. #16839489C22semidirect
55. C8×#92418C22semidirect
276. C8×C61AGL(1,ℤ/8)semidirect
282. #92418AGL(1,ℤ/8)semidirect
316. #3272D61semidirect
387. C61#20314semidirect
603. C8#14519803semidirect

Maximal subgroups

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

subgroupindex
#214584162
#214584142
#214574152
#214454052
#214453182
#214453162
#214452822
#214452812
#214451032
#214451012
#214449352
#214449012
#214447042
#214447022
C61×#32722
#2031461