Finite Group Databasebeta

← #29677094

Subgroups — #29677094

Order 15616.

604 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458416C211yesmaximal, characteristic
2#21458129C211yesmaximal, characteristic, complemented
3#21458113C211yesmaximal, characteristic, complemented
4#21445414C211yesmaximal, characteristic, complemented
5#21445396C211yesmaximal, characteristic, complemented
6#21445392C211yesmaximal, characteristic, complemented
7#21445325C211yesmaximal, characteristic, complemented
8#21445315C211yesmaximal, characteristic, complemented
9#21445068C211yesmaximal, characteristic, complemented
10#21445053C211yesmaximal, characteristic, complemented
11#21444963C211yesmaximal, characteristic, complemented
12#21444953C211yesmaximal, characteristic, complemented
13#21444709C211yesmaximal, characteristic, complemented
14#21444704C211yesmaximal, characteristic, complemented
15C61×#3384C211yesmaximal, characteristic, complemented
16#16840477C2211yescharacteristic
17#16840417C2211yescharacteristic
18#16840165C2211yescharacteristic
19#16840160C2211yescharacteristic
20#16840141C2211yescharacteristic
21#16840114C2211yescharacteristic
22#16840112C2211yescharacteristic
23#16840100—21no—
24#16840092—21no—
25#16840085—21no—
26#16840079—21no—
27#16840057C2211yescharacteristic
28#16840052C2211yescharacteristic
29#16840049C2211yescharacteristic
30#16840042C2211yescharacteristic
31#16840018—21no—
32#16840012—21no—
33#16840010—21no—
34#16839972C2211yescharacteristic
35#16839964C2211yescharacteristic
36#16839922—21no—
37#16839901—21no—
38C4×#11758201C2211yescharacteristic
39#16839841C2211yescharacteristic
40#16839840C2211yescharacteristic
41#16839840C2211yescharacteristic
42#16839803—21no—
43#16839771—21no—
44#16839761—21no—
45#16839515C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839507C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55C8×#92418C2211yescharacteristic
56C61×#554C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic
58C61×#501—21no—
59C61×#498C2211yescharacteristic
60C61×#478—21no—
61C61×#476—21no—
62C61×#476C2211yescharacteristic
63C61×#474C2211yescharacteristic
64C61×#446C2211yescharacteristic
65C61×#442—21no—
66C4×D8×C61C2211yescharacteristic
67C2×#11758222—21no—
68C2×#11758202—21no—
69C2×#11758202—41no—
70C2×#11758201—21no—
71C2×#11758201D411yescharacteristic
72C2×#11758194—21no—
73C2×#11758194—21no—
74C2×#11758189—21no—
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519763—41no—
78#14519759—41no—
79#14519756—21no—
80#14519751—21no—
81#14519748—21no—
82#14519748—21no—
83C4×D244C2311yescharacteristic
84#14519736—21no—
85#14519735—21no—
86C4⋊C4×D61—21no—
87#14519710—21no—
88#14519710—21no—
89#14519710—41no—
90#14519709—21no—
91#14519709—21no—
92#14519708—21no—
93#14519708—21no—
94#14519708D411yescharacteristic
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519701—21no—
99#14519701—21no—
100#14519701—21no—
101#14519701—21no—
102#14519701D411yescharacteristic
103#14519700—21no—
104#14519700—21no—
105#14519700—21no—
106#14519698—21no—
107#14519698—21no—
108#14519697—21no—
109#14519697D411yescharacteristic
110C22⋊#92418—21no—
111C22⋊#92418—21no—
112#14519689—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C22×Q8×C61—21no—
116C2×D8×C61D411yescharacteristic
117C4⋊Q8×C61—21no—
118C4⋊Q8×C61C2311yescharacteristic
119C4⋊Q8×C61C2311yescharacteristic
120#125×C61—21no—
121C22⋊Q8×C61—21no—
122C22⋊Q8×C61—21no—
123C22⋊Q8×C61—41no—
124C22⋊Q8×C61—41no—
125C4×Q8×C61C2311yescharacteristic
126C4×D4×C61—21no—
127C4×D4×C61—21no—
128C4×D4×C61C2311yescharacteristic
129C4×D4×C61C2311yescharacteristic
130C2×C4⋊C4×C61—21no—
131#107×C61—21no—
132#107×C61D411yescharacteristic
133#106×C61—21no—
134#106×C61—21no—
135#106×C61—21no—
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2311yescharacteristic
138#102×C61—21no—
139#102×C61—21no—
140#102×C61D411yescharacteristic
141#102×C61D411yescharacteristic
142#97×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—41no—
147C2×Q8×C61—42no—
148C2×Q8×C61—81no—
149C2×Q8×C61C2×D411yescharacteristic
150C2×D4×C61C2×D411yescharacteristic
151C2×D4×C61C2×D411yescharacteristic
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—41no—
155C22×C4×C61—41no—
156C2×C22⋊D61—41no—
157C2×C22⋊D61—41no—
158#11758222—42no—
159#11758222—81no—
160C2×D244C2×D411yescharacteristic
161C2×C4×D61—21no—
162C2×C4×D61—41no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165D8×C61—22no—
166C2×C8×C61—21no—
167C2×C8×C61C2×D411yescharacteristic
168C2×C8×C61C2×D411yescharacteristic
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—41no—
175C4⋊C4×C61—41no—
176C4⋊C4×C61—41no—
177C4⋊C4×C61C2×D411yescharacteristic
178C4⋊C4×C61C2×D411yescharacteristic
179C4⋊C4×C61C2×D411yescharacteristic
180C22⋊C4×C61—21no—
181C22⋊C4×C61—21no—
182C22⋊C4×C61—41no—
183C22⋊C4×C61—41no—
184C42×C61—21no—
185C42×C61C2411yescharacteristic
186C22⋊Dic61—41no—
187C22⋊Dic61—41no—
188#11758202—82no—
189#11758202—81no—
190#11758201—41no—
191#11758201—81no—
192#11758198—21no—
193#11758198—41no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61C2×D411yescharacteristic
199#11758196—41no—
200C4×Dic61—41no—
201#11758194—81no—
202#11758194—81no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209#11758189—42no—
210C23×C61—21no—
211C23×C61—21no—
212C22×D61—21no—
213Q8×C61—21no—
214Q8×C61—21no—
215Q8×C61—41no—
216Q8×C61—41no—
217Q8×C61—81no—
218Q8×C61—82no—
219D4×C61—21no—
220D4×C61—21no—
221D4×C61—21no—
222D4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—82no—
234C2×C4×C61—161no—
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61H(5,𝔽2)11yescharacteristic
238C22⋊D61—82no—
239C22⋊D61—161no—
240C2×Dic61—21no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243D244—21no—
244D244—21no—
245C4×D61—41no—
246C4×D61—41no—
247C4×D61—82no—
248Dic122—41no—
249Dic122—41no—
250Dic122—81no—
251C8×C61—21no—
252C8×C61—41no—
253C8×C61#13712yes—
254#92418—22no—
255#92418—21no—
256#92418—21no—
257#92418—41no—
258#92418—41no—
259#92418—41no—
260#20305—611nomaximal, Sylow
261C22×C61—41no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61—41no—
265C22×C61—41no—
266C22×C61—41no—
267C22×C61D4211yescharacteristic
268C2×D61—41no—
269C2×D61—41no—
270C2×D61—41no—
271C4×C61—21no—
272C4×C61—41no—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61—81no—
277C4×C61—81no—
278C4×C61C2×#13711yescharacteristic
279C4×C61#57511yescharacteristic
280C4×C61#57611yescharacteristic
281C4×C61#57811yescharacteristic
282Dic61—41no—
283Dic61—81no—
284Dic61—81no—
285#3384D6111yescharacteristic, complemented
286#3354—611no—
287#3331—611no—
288#3329—611no—
289#1707—611no—
290#1703—611no—
291#1703—611no—
292#1692—611no—
293#1691—611no—
294#1649—611no—
295#1647—611no—
296#1634—611no—
297#1632—611no—
298#1590—611no—
299#1589—611no—
300C2×C61—81no—
301C2×C61—81no—
302C2×C61#327311yescharacteristic
303C2×C61#327611yescharacteristic
304C2×C61#329111yescharacteristic
305D61—81no—
306#555—611no—
307#554C2×D6111yescharacteristic
308D4×Q8C2×D6111yescharacteristic
309#547—611no—
310#501—21no—
311#500—611no—
312#498—611no—
313#498C2×D6111yescharacteristic
314#494—611no—
315#487—611no—
316#487—611no—
317#485—1221no—
318#483—1221no—
319#482—1221no—
320#481—1221no—
321#478—21no—
322#477—611no—
323#476—21no—
324#476—611no—
325#476—611no—
326#476C2×D6111yescharacteristic
327#474—611no—
328#474C2×D6111yescharacteristic
329#470—1221no—
330#469—1221no—
331#468—1221no—
332#462—611no—
333#461—611no—
334#452—1221no—
335#448—1221no—
336#446C2×D6111yescharacteristic
337#442—21no—
338C4×SD16—611no—
339C4×D8C2×D6111yescharacteristic
340#436—611no—
341#436—611no—
342#436—611no—
343#428—1221no—
344#419—1221no—
345#417—1221no—
346#335—611no—
347#334—611no—
348#333—611no—
349#333—611no—
350#332—611no—
351#332—611no—
352Q8⋊C8—611no—
353D4⋊C8—611no—
354D4⋊C8—611no—
355D4⋊C8—611no—
356C82—611no—
357C61#2030511yescharacteristic, complemented, Sylow
358C2×#41—1221no—
359C22×Q8—21no—
360C2×Q16—1221no—
361C2×Q16—2441no—
362C2×SD16—611no—
363C2×SD16—1221no—
364C2×D8C22⋊D6111yescharacteristic
365C2×M16—1221no—
366C2×M16—1221no—
367C2×M16—1221no—
368C4⋊Q8—21no—
369C4⋊Q8—611no—
370C4⋊Q8—611no—
371C4⋊Q8C22×D6111yescharacteristic
372C4⋊Q8C22×D6111yescharacteristic
373#125—21no—
374#124—2441no—
375#123—2441no—
376C22⋊Q8—21no—
377C22⋊Q8—21no—
378C22⋊Q8—41no—
379C22⋊Q8—41no—
380C22⋊Q8—1221no—
381#121—1221no—
382#121—1221no—
383#121—1221no—
384C4×Q8C22×D6111yescharacteristic
385C4×D4—21no—
386C4×D4—21no—
387C4×D4—611no—
388C4×D4C22×D6111yescharacteristic
389C4×D4C22×D6111yescharacteristic
390#117—1221no—
391#117—1221no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—1221no—
394#107—21no—
395#107—1221no—
396#107—1221no—
397#107—2441no—
398#107C22⋊D6111yescharacteristic
399#106—21no—
400#106—21no—
401#106—21no—
402#106—611no—
403#106—1221no—
404#106—1221no—
405#106—1221no—
406#106—1221no—
407C4⋊C8—21no—
408C4⋊C8—611no—
409C4⋊C8—611no—
410C4⋊C8—611no—
411C4⋊C8C22×D6111yescharacteristic
412#103—611no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#103—1221no—
417#103—1221no—
418#103—1221no—
419#103—1221no—
420#102—21no—
421#102—21no—
422#102—611no—
423#102—1221no—
424#102—1221no—
425#102—1221no—
426#102C22⋊D6111yescharacteristic
427#102C22⋊D6111yescharacteristic
428C22⋊C8—1221no—
429C22⋊C8—1221no—
430C22⋊C8—1221no—
431#97—21no—
432C4×C8—611no—
433C4×C8—611no—
434C4×C8C22×D6111yescharacteristic
435#41—4881no—
436#41—2442no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—41no—
440C2×Q8—42no—
441C2×Q8—81no—
442C2×Q8—1221no—
443C2×Q8—1221no—
444C2×Q8D4×D6111yescharacteristic
445C2×D4—611no—
446C2×D4—2441no—
447C2×D4—2441no—
448C2×D4C2×C22⋊D6111yescharacteristic
449C2×D4C2×C22⋊D6111yescharacteristic
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—1221no—
455C22×C4—2441no—
456Q16—4882no—
457Q16—4881no—
458SD16—4881no—
459SD16—2441no—
460D8—22no—
461M16—4881no—
462M16—4881no—
463M16—2442no—
464C2×C8—21no—
465C2×C8—611no—
466C2×C8—611no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8C2×C22⋊D6111yescharacteristic
472C2×C8C2×C22⋊D6111yescharacteristic
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—611no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—2441no—
487C4⋊C4D4×D6111yescharacteristic
488C4⋊C4C2×C22⋊D6111yescharacteristic
489C4⋊C4C2×C22⋊D6111yescharacteristic
490C22⋊C4—21no—
491C22⋊C4—21no—
492C22⋊C4—41no—
493C22⋊C4—41no—
494C22⋊C4—1221no—
495C22⋊C4—2441no—
496C22⋊C4—2441no—
497C22⋊C4—2441no—
498C42—21no—
499C42—2441no—
500C42C23×D6111yescharacteristic
501C23—21no—
502C23—21no—
503C23—1221no—
504Q8—21no—
505Q8—21no—
506Q8—41no—
507Q8—41no—
508Q8—81no—
509Q8—82no—
510Q8—4881no—
511Q8—2441no—
512Q8—2441no—
513D4—21no—
514D4—21no—
515D4—21no—
516D4—21no—
517D4—1221no—
518D4—1221no—
519D4—9761no—
520D4—4882no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—82no—
532C2×C4—161no—
533C2×C4—1221no—
534C2×C4—4882no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4C2×D4×D6111yescharacteristic
540C2×C4C22×C22⋊D6111yescharacteristic
541C2×C4#1451982211yescharacteristic
542C8—21no—
543C8—41no—
544C8—1222no—
545C8—1221no—
546C8—1221no—
547C8—2441no—
548C8—2441no—
549C8—2441no—
550C8#1451980812yes—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—2441no—
558C22—2441no—
559C22—2441no—
560C22D4×C22⋊D6111yescharacteristic
561C4—21no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—81no—
567C4—81no—
568C4—4881no—
569C4—4881no—
570C4—2441no—
571C4C2×#1451980811yescharacteristic
572C4#1684055811yescharacteristic
573C4#1684056611yescharacteristic
574C4#1684057711yescharacteristic
575C2—81no—
576C2—81no—
577C2—4881no—
578C2#2145742411yescharacteristic
579C2#2145746511yescharacteristic
580C2#2145761011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3384C2
socleC22×C61D42
prime radical#21458416C2
hypercenter#3384D61
hypocenterC61#20305
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3384

Complements

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

NH ≅ G/Ntype
2. #21458129C2semidirect
3. #21458113C2semidirect
4. #21445414C2semidirect
5. #21445396C2semidirect
6. #21445392C2semidirect
7. #21445325C2semidirect
8. #21445315C2semidirect
9. #21445068C2semidirect
10. #21445053C2semidirect
11. #21444963C2semidirect
12. #21444953C2semidirect
13. #21444709C2semidirect
14. #21444704C2semidirect
15. C61×#3384C2semidirect
285. #3384D61semidirect
357. C61#20305semidirect

Maximal subgroups

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

subgroupindex
#214584162
#214581292
#214581132
#214454142
#214453962
#214453922
#214453252
#214453152
#214450682
#214450532
#214449632
#214449532
#214447092
#214447042
C61×#33842
#2030561