Finite Group Databasebeta

← #29669761

Subgroups — #29669761

Order 15616.

598 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458466C211yesmaximal, characteristic
2#21458102C211yesmaximal, characteristic, complemented
3#21457695C211yesmaximal, characteristic, complemented
4#21445464C211yesmaximal, characteristic, complemented
5#21445392C211yesmaximal, characteristic, complemented
6#21445277C211yesmaximal, characteristic, complemented
7#21445240C211yesmaximal, characteristic, complemented
8#21445210C211yesmaximal, characteristic, complemented
9#21445088C211yesmaximal, characteristic, complemented
10#21445082C211yesmaximal, characteristic, complemented
11#21445032C211yesmaximal, characteristic, complemented
12#21444984C211yesmaximal, characteristic, complemented
13#21444649C211yesmaximal, characteristic
14#21444636C211yesmaximal, characteristic, complemented
15Q8×D8×C61C211yesmaximal, characteristic
16#16840482C2211yescharacteristic
17#16840469C2211yescharacteristic
18#16840158C2211yescharacteristic
19#16840115—21no—
20#16840114C2211yescharacteristic
21#16840113C2211yescharacteristic
22#16840112C2211yescharacteristic
23#16840112C2211yescharacteristic
24#16840100—21no—
25#16840095—21no—
26#16840068—21no—
27#16840057C2211yescharacteristic
28#16840052C2211yescharacteristic
29#16840047C2211yescharacteristic
30#16840018—21no—
31#16840001—21no—
32#16839975—21no—
33#16839969C2211yescharacteristic
34#16839969C2211yescharacteristic
35#16839967C2211yescharacteristic
36#16839918—21no—
37#16839918—21no—
38C8×Dic122C2211yescharacteristic
39C4×#11758202C2211yescharacteristic
40#16839853—21no—
41#16839840C2211yescharacteristic
42D4×#92418C2211yescharacteristic
43#16839771—21no—
44C2×#14519701—21no—
45#16839515C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839510C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839508C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52#16839492C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic
56D4×Q8×C61C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic
58C61×#500—21no—
59C61×#500C2211yescharacteristic
60C61×#474—21no—
61C61×#474—21no—
62C61×#474C2211yescharacteristic
63C61×#474C2211yescharacteristic
64C8×Q8×C61C2211yescharacteristic
65C4×D8×C61—21no—
66C4×D8×C61C2211yescharacteristic
67C2×#11758202—21no—
68C2×#11758202—21no—
69C2×#11758202D411yescharacteristic
70C2×#11758200—21no—
71C2×#11758194—21no—
72C22×#92418—21no—
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519770—41no—
76#14519768—21no—
77#14519764—21no—
78#14519756—21no—
79#14519756—21no—
80#14519754—41no—
81C4×Dic122C2311yescharacteristic
82#14519742—21no—
83D4×Dic61—21no—
84#14519736—21no—
85C2×C4⋊Dic61—21no—
86#14519710—21no—
87#14519710D411yescharacteristic
88#14519708—21no—
89#14519708—21no—
90#14519708—41no—
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94#14519706—21no—
95#14519702—21no—
96#14519702—21no—
97#14519701—21no—
98#14519701—21no—
99#14519701—21no—
100#14519701—21no—
101#14519701—21no—
102#14519701—21no—
103#14519701—21no—
104#14519701—21no—
105#14519701D411yescharacteristic
106#14519700—21no—
107#14519700—41no—
108#14519700D411yescharacteristic
109#14519698—21no—
110C22⋊#92418—21no—
111C22⋊#92418—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
115C22×Q8×C61—21no—
116C22×Q8×C61—21no—
117C2×D8×C61D411yescharacteristic
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61C2311yescharacteristic
121C4⋊Q8×C61C2311yescharacteristic
122C22⋊Q8×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—41no—
125C22⋊Q8×C61—41no—
126C4×Q8×C61C2311yescharacteristic
127C4×D4×C61—21no—
128C4×D4×C61—21no—
129C4×D4×C61C2311yescharacteristic
130C4×D4×C61C2311yescharacteristic
131#107×C61—21no—
132#107×C61—21no—
133#107×C61—21no—
134#107×C61—21no—
135#107×C61D411yescharacteristic
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2311yescharacteristic
138#102×C61—21no—
139#102×C61—21no—
140#102×C61D411yescharacteristic
141#102×C61D411yescharacteristic
142C4×C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—41no—
147C2×Q8×C61—41no—
148C2×Q8×C61—42no—
149C2×Q8×C61—42no—
150C2×Q8×C61—81no—
151C2×Q8×C61—81no—
152C2×Q8×C61C2×D411yescharacteristic
153C2×D4×C61C2×D411yescharacteristic
154C2×D4×C61C2×D411yescharacteristic
155C22×C4×C61—21no—
156C22×C4×C61—21no—
157C22×C4×C61—41no—
158C22×C4×C61—41no—
159C22×Dic61—41no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122C2×D411yescharacteristic
163D8×C61—22no—
164C2×C8×C61—21no—
165C2×C8×C61C2×D411yescharacteristic
166C2×C8×C61C2×D411yescharacteristic
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—41no—
173C4⋊C4×C61—41no—
174C4⋊C4×C61C2×D411yescharacteristic
175C4⋊C4×C61C2×D411yescharacteristic
176C4⋊C4×C61C2×D411yescharacteristic
177C22⋊C4×C61—21no—
178C22⋊C4×C61—21no—
179C22⋊C4×C61—41no—
180C22⋊C4×C61—41no—
181C42×C61—21no—
182C42×C61C2411yescharacteristic
183C22⋊Dic61—41no—
184C22⋊Dic61—41no—
185C22⋊Dic61—41no—
186#11758202—22no—
187#11758202—81no—
188#11758202—81no—
189#11758200—42no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—41no—
195C4⋊Dic61C2×D411yescharacteristic
196#11758196—21no—
197#11758196—41no—
198#11758196—41no—
199C4×Dic61—21no—
200C4×Dic61—41no—
201#11758194—81no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—81no—
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209C23×C61—21no—
210C23×C61—21no—
211Q8×C61—21no—
212Q8×C61—41no—
213Q8×C61—41no—
214Q8×C61—41no—
215Q8×C61—41no—
216Q8×C61—82no—
217Q8×C61—82no—
218Q8×C61#13512yes—
219D4×C61—21no—
220D4×C61—21no—
221D4×C61—21no—
222D4×C61#13712yes—
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—82no—
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61H(5,𝔽2)11yescharacteristic
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—161no—
243Dic122—21no—
244Dic122—21no—
245Dic122—41no—
246Dic122—41no—
247Dic122—81no—
248C8×C61—21no—
249C8×C61—21no—
250C8×C61—41no—
251#92418—21no—
252#92418—21no—
253#92418—41no—
254#92418—41no—
255#92418—41no—
256#92418—41no—
257#19782—611nomaximal, Sylow
258C22×C61—22no—
259C22×C61—41no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61D4211yescharacteristic
265C4×C61—22no—
266C4×C61—21no—
267C4×C61—41no—
268C4×C61—41no—
269C4×C61—41no—
270C4×C61—81no—
271C4×C61—81no—
272C4×C61C2×#13511yescharacteristic
273C4×C61C2×#13711yescharacteristic
274C4×C61#57611yescharacteristic
275C4×C61#57811yescharacteristic
276Dic61—41no—
277Dic61—41no—
278Dic61—81no—
279Dic61—81no—
280Q8×D8D6111yescharacteristic
281#3360—611no—
282#3328—611no—
283#3299—611no—
284#1718—611no—
285#1703—611no—
286#1684—611no—
287#1679—611no—
288#1674—611no—
289#1652—611no—
290#1651—611no—
291#1644—611no—
292#1637—611no—
293#1581—611no—
294#1579—611no—
295C2×C61—42no—
296C2×C61—81no—
297C2×C61#327011yescharacteristic
298C2×C61#328011yescharacteristic
299C2×C61#328911yescharacteristic
300#556—611no—
301#554—611no—
302D4×Q8C2×D6111yescharacteristic
303D4×Q8C2×D6111yescharacteristic
304#500—21no—
305#500C2×D6111yescharacteristic
306#498—611no—
307#488—1221no—
308#487—611no—
309#487—611no—
310#487—611no—
311#487—611no—
312#485—1221no—
313#484—1221no—
314#479—1221no—
315#477—611no—
316#476—611no—
317#475—611no—
318#474—21no—
319#474—21no—
320#474C2×D6111yescharacteristic
321#474C2×D6111yescharacteristic
322#470—1221no—
323#467—1221no—
324#463—1221no—
325#462—611no—
326#462—611no—
327#461—611no—
328#451—1221no—
329#451—1221no—
330C8×Q8—611no—
331C8×Q8C2×D6111yescharacteristic
332C4×Q16—611no—
333C4×SD16—1221no—
334C4×D8—21no—
335C4×D8C2×D6111yescharacteristic
336#436—611no—
337C8×D4—611no—
338#419—1221no—
339C2×#103—1221no—
340#335—611no—
341#334—611no—
342#333—611no—
343#332—611no—
344#332—611no—
345#332—611no—
346Q8⋊C8—611no—
347Q8⋊C8—611no—
348D4⋊C8—611no—
349D4⋊C8—611no—
350#322—611no—
351C61#1978211yescharacteristic, complemented, Sylow
352C22×Q8—21no—
353C22×Q8—21no—
354C2×Q16—611no—
355C2×Q16—1221no—
356C2×Q16—1221no—
357C2×SD16—1221no—
358C2×D8C22⋊D6111yescharacteristic
359C2×M16—1221no—
360C22×C8—1221no—
361C4⋊Q8—21no—
362C4⋊Q8—21no—
363C4⋊Q8—611no—
364C4⋊Q8—611no—
365C4⋊Q8C22×D6111yescharacteristic
366C4⋊Q8C22×D6111yescharacteristic
367#126—2441no—
368#125—1221no—
369#124—1221no—
370C22⋊Q8—21no—
371C22⋊Q8—21no—
372C22⋊Q8—41no—
373C22⋊Q8—41no—
374C22⋊Q8—1221no—
375C22⋊Q8—1221no—
376C22⋊Q8—2441no—
377C4×Q8—611no—
378C4×Q8—1221no—
379C4×Q8C22×D6111yescharacteristic
380C4×D4—21no—
381C4×D4—21no—
382C4×D4—1221no—
383C4×D4C22×D6111yescharacteristic
384C4×D4C22×D6111yescharacteristic
385#117—1221no—
386C2×C4⋊C4—1221no—
387#107—21no—
388#107—21no—
389#107—21no—
390#107—21no—
391#107—611no—
392#107—1221no—
393#107C22⋊D6111yescharacteristic
394#106—1221no—
395#106—1221no—
396#106—2441no—
397C4⋊C8—21no—
398C4⋊C8—611no—
399C4⋊C8—611no—
400C4⋊C8—611no—
401C4⋊C8—1221no—
402C4⋊C8C22×D6111yescharacteristic
403#103—611no—
404#103—611no—
405#103—1221no—
406#103—1221no—
407#103—1221no—
408#103—1221no—
409#103—1221no—
410#103—1221no—
411#103—1221no—
412#103—1221no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#103—2441no—
417#102—21no—
418#102—21no—
419#102—1221no—
420#102C22⋊D6111yescharacteristic
421#102C22⋊D6111yescharacteristic
422C22⋊C8—1221no—
423C22⋊C8—1221no—
424C4×C8—21no—
425C4×C8—611no—
426C4×C8—611no—
427C4×C8—1221no—
428C4×C8C22×D6111yescharacteristic
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—41no—
432C2×Q8—41no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—81no—
436C2×Q8—81no—
437C2×Q8—611no—
438C2×Q8—1221no—
439C2×Q8—1221no—
440C2×Q8D4×D6111yescharacteristic
441C2×D4C2×C22⋊D6111yescharacteristic
442C2×D4C2×C22⋊D6111yescharacteristic
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—2441no—
448Q16—1222no—
449Q16—4881no—
450Q16—4881no—
451SD16—2442no—
452D8—22no—
453M16—4881no—
454C2×C8—21no—
455C2×C8—611no—
456C2×C8—611no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—4881no—
462C2×C8C2×C22⋊D6111yescharacteristic
463C2×C8C2×C22⋊D6111yescharacteristic
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—611no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—2441no—
478C4⋊C4—2441no—
479C4⋊C4—2441no—
480C4⋊C4D4×D6111yescharacteristic
481C4⋊C4C2×C22⋊D6111yescharacteristic
482C4⋊C4C2×C22⋊D6111yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—2441no—
488C22⋊C4—2441no—
489C22⋊C4—2441no—
490C42—21no—
491C42—1221no—
492C42—2441no—
493C42C23×D6111yescharacteristic
494C23—21no—
495C23—21no—
496Q8—21no—
497Q8—41no—
498Q8—41no—
499Q8—41no—
500Q8—41no—
501Q8—82no—
502Q8—82no—
503Q8—1221no—
504Q8—1221no—
505Q8—4881no—
506Q8—2441no—
507Q8—2441no—
508Q8#1451979612yes—
509D4—21no—
510D4—21no—
511D4—21no—
512D4#1451980812yes—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—1221no—
526C2×C4—1221no—
527C2×C4—9761no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4C2×D4×D6111yescharacteristic
531C2×C4C22×C22⋊D6111yescharacteristic
532C2×C4#1451982211yescharacteristic
533C8—21no—
534C8—21no—
535C8—41no—
536C8—1221no—
537C8—1221no—
538C8—2441no—
539C8—2441no—
540C8—2441no—
541C8—2441no—
542C22—22no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22D4×C22⋊D6111yescharacteristic
549C4—22no—
550C4—21no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—81no—
555C4—81no—
556C4—4881no—
557C4—4881no—
558C4—2441no—
559C4—2441no—
560C4C2×#1451979611yescharacteristic
561C4C2×#1451980811yescharacteristic
562C4#1684056311yescharacteristic
563C4#1684057711yescharacteristic
564C2—42no—
565C2—81no—
566C2#2145739011yescharacteristic
567C2#2145750411yescharacteristic
568C2#2145759011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingQ8×D8×C61C2
socleC22×C61D42
prime radicalC2×D8×C61D4
hypercenterQ8×D8D61
hypocenterC61#19782
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ Q8×D8

Complements

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

NH ≅ G/Ntype
2. #21458102C2semidirect
3. #21457695C2semidirect
4. #21445464C2semidirect
5. #21445392C2semidirect
6. #21445277C2semidirect
7. #21445240C2semidirect
8. #21445210C2semidirect
9. #21445088C2semidirect
10. #21445082C2semidirect
11. #21445032C2semidirect
12. #21444984C2semidirect
14. #21444636C2semidirect
351. C61#19782semidirect

Maximal subgroups

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

subgroupindex
#214584662
#214581022
#214576952
#214454642
#214453922
#214452772
#214452402
#214452102
#214450882
#214450822
#214450322
#214449842
#214446492
#214446362
Q8×D8×C612
#1978261