Finite Group Databasebeta

← #29665311

Subgroups — #29665311

Order 15616.

602 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456763C211yesmaximal, characteristic
2#21455619C211yesmaximal, characteristic
3#475×D61C211yesmaximal, characteristic
4#21454898C211yesmaximal, characteristic
5#21445278C211yesmaximal, characteristic, complemented
6#21445277C211yesmaximal, characteristic, complemented
7#21445276C211yesmaximal, characteristic, complemented
8#21445275C211yesmaximal, characteristic, complemented
9#21445188C211yesmaximal, characteristic, complemented
10#21445187C211yesmaximal, characteristic, complemented
11#21445186C211yesmaximal, characteristic, complemented
12#21444592C211yesmaximal, characteristic
13#21444502C211yesmaximal, characteristic
14#21444475C211yesmaximal, characteristic
15C61×#1670C211yesmaximal, characteristic, complemented
16#16840317C2211yescharacteristic
17C4⋊Q8×D61C2211yescharacteristic
18C4×Q8×D61C2211yescharacteristic
19#16840141C2211yescharacteristic
20#16840113C2211yescharacteristic
21#16840112C2211yescharacteristic
22#16840072—21no—
23#16840071—21no—
24#16840058C2211yescharacteristic
25#16840057C2211yescharacteristic
26#16840048C2211yescharacteristic
27#16840047C2211yescharacteristic
28#16840046—21no—
29#16840045—21no—
30#16840044C2211yescharacteristic
31#16840044C2211yescharacteristic
32#16839970—21no—
33#16839970—21no—
34#16839916—21no—
35#16839916—21no—
36Q8×#92418C2211yescharacteristic
37C8×Dic122—21no—
38C8×Dic122C2211yescharacteristic
39#16839886—21no—
40#16839886C2211yescharacteristic
41#16839814—21no—
42#16839814C2211yescharacteristic
43#16839813—21no—
44#16839813C2211yescharacteristic
45#16839801—21no—
46#106×D61—21no—
47#16839784C2211yescharacteristic
48C4⋊C8×D61C2211yescharacteristic
49#16839769—21no—
50#103×D61—21no—
51#16839515C2211yescharacteristic
52#16839514C2211yescharacteristic
53#16839514C2211yescharacteristic
54#16839510C2211yescharacteristic
55#16839510C2211yescharacteristic
56#16839509C2211yescharacteristic
57#16839508C2211yescharacteristic
58#16839508C2211yescharacteristic
59#16839507C2211yescharacteristic
60C61×#494C2211yescharacteristic
61C61×#475C2211yescharacteristic
62C61×#475C2211yescharacteristic
63C8×Q8×C61C2211yescharacteristic
64C61×#335C2211yescharacteristic
65C61×#332C2211yescharacteristic
66C61×#332C2211yescharacteristic
67C2×Q8×D61—21no—
68C2×Q8×D61D411yescharacteristic
69C2×#11758202—21no—
70C2×#11758202—21no—
71C2×#11758189—21no—
72C2×C8×D61—21no—
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519776—21no—
76#14519776—21no—
77#14519775—21no—
78#14519768—21no—
79#14519757—21no—
80#14519753—21no—
81Q8×Dic61—21no—
82Q8×Dic61D411yescharacteristic
83C4×Dic122—21no—
84C4×Dic122C2311yescharacteristic
85#14519742—21no—
86#14519742—21no—
87#14519742D411yescharacteristic
88#14519735—21no—
89C4⋊C4×D61—21no—
90C4⋊C4×D61—21no—
91C4⋊C4×D61—21no—
92C4⋊C4×D61D411yescharacteristic
93C42×D61—21no—
94C42×D61C2311yescharacteristic
95#14519711—21no—
96#14519710—21no—
97#14519709—21no—
98#14519708—21no—
99#14519708—21no—
100#14519708—21no—
101C4⋊#92418—21no—
102C4⋊#92418C2311yescharacteristic
103C4⋊#92418C2311yescharacteristic
104C4⋊#92418C2311yescharacteristic
105#14519706—21no—
106#14519706—21no—
107#14519706—21no—
108#14519706—21no—
109#14519706D411yescharacteristic
110#14519706D411yescharacteristic
111#14519702—21no—
112#14519702—21no—
113#14519702—21no—
114#14519702—21no—
115#14519701—21no—
116#14519701—21no—
117#14519701—21no—
118#14519701—21no—
119#14519700—21no—
120#14519700—21no—
121#14519689—21no—
122#14519689D411yescharacteristic
123#14519689D411yescharacteristic
124C4×#92418—21no—
125C4×#92418C2311yescharacteristic
126C8×Dic61—21no—
127C8×Dic61—21no—
128C8×Dic61—21no—
129C2×Q16×C61—21no—
130C2×Q16×C61—21no—
131C4⋊Q8×C61C2311yescharacteristic
132C4⋊Q8×C61C2311yescharacteristic
133C4×Q8×C61C2311yescharacteristic
134#106×C61—21no—
135#106×C61—21no—
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140#103×C61—21no—
141#103×C61—21no—
142C4×C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147Q8×D61—22no—
148Q8×D61—81no—
149Q8×D61—81no—
150Q8×D61Q1614yescomplemented
151C2×C4×D61—21no—
152C2×C4×D61—21no—
153C2×C4×D61—41no—
154C2×C4×D61C2×D411yescharacteristic
155C2×C4×D61C2×D411yescharacteristic
156C2×C4×D61#4111yescharacteristic
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—41no—
161C2×Dic122C2×D411yescharacteristic
162Q16×C61—81no—
163Q16×C61—81no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
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×C61C2×D411yescharacteristic
175C42×C61—21no—
176C42×C61C2411yescharacteristic
177#11758202—81no—
178#11758202—81no—
179#11758198—41no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—41no—
186C4⋊Dic61C2×D411yescharacteristic
187#11758196—21no—
188#11758196—21no—
189#11758196—21no—
190#11758196—41no—
191#11758196—41no—
192#11758196—41no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C4×Dic61—41no—
197C4×Dic61C2×D411yescharacteristic
198C4×Dic61C2×D411yescharacteristic
199C4×Dic61#4111yescharacteristic
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418C2×D411yescharacteristic
204C2×#92418C2×D411yescharacteristic
205#11758189—42no—
206C8×D61—42no—
207C22×D61#12111yescharacteristic
208Q8×C61—21no—
209Q8×C61—41no—
210Q8×C61—41no—
211Q8×C61—41no—
212Q8×C61—41no—
213Q8×C61C2×Q1612yes—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C2×Dic61—21no—
222C2×Dic61—21no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61#12111yescharacteristic
226C2×Dic61#12111yescharacteristic
227C2×Dic61#12111yescharacteristic
228C4×D61—24no—
229C4×D61—21no—
230C4×D61—21no—
231C4×D61—41no—
232C4×D61—81no—
233C4×D61—82no—
234C4×D61C2×Q1612yes—
235C4×D61#13512yes—
236Dic122—21no—
237Dic122—24no—
238Dic122—41no—
239Dic122—41no—
240Dic122—82no—
241Dic122—81no—
242Dic122C2×Q1612yes—
243C8×C61—21no—
244C8×C61—21no—
245C8×C61—41no—
246C8×C61—41no—
247C8×C61—41no—
248#92418—21no—
249#92418—21no—
250#92418—41no—
251#92418—41no—
252#92418—41no—
253#19399—611nomaximal, Sylow
254C22×C61C2×#12111yescharacteristic
255C2×D61—21no—
256C2×D61—21no—
257C2×D61#46712yes—
258C4×C61—22no—
259C4×C61—21no—
260C4×C61—41no—
261C4×C61—81no—
262C4×C61—81no—
263C4×C61C22×Q1611yescharacteristic
264C4×C61C2×#13511yescharacteristic
265C4×C61#57711yescharacteristic
266C4×C61#57711yescharacteristic
267Dic61—21no—
268Dic61—22no—
269Dic61—21no—
270Dic61—41no—
271Dic61—81no—
272Dic61—81no—
273Dic61#46712yes—
274#3209—611no—
275#3078—611no—
276C2×#475—611no—
277#2982—611no—
278#1684—611no—
279#1684—611no—
280#1684—611no—
281#1684—611no—
282#1670—611no—
283#1670—611no—
284#1670—611no—
285#1670D6111yescharacteristic, complemented
286#1571—611no—
287#1549—611no—
288#1542—611no—
289C2×C61#304311yescharacteristic
290C2×C61C2×#46711yescharacteristic
291C2×C61#306911yescharacteristic
292D61—22no—
293#533—611no—
294C2×C4⋊Q8—611no—
295C2×C4×Q8—611no—
296#494—611no—
297#494C2×D6111yescharacteristic
298#487—611no—
299#487—611no—
300#479—1221no—
301#479—1221no—
302#477—611no—
303#477—611no—
304#475—611no—
305#475—611no—
306#475—611no—
307#475—611no—
308#475—1221no—
309#475—1221no—
310#475C2×D6111yescharacteristic
311#475C2×D6111yescharacteristic
312#462—1221no—
313#462—1221no—
314#451—1221no—
315#451—1221no—
316C8×Q8—611no—
317C8×Q8—611no—
318C8×Q8—611no—
319C8×Q8—1221no—
320C8×Q8—1221no—
321C8×Q8C2×D6111yescharacteristic
322#431—611no—
323#431—611no—
324#431—1221no—
325#431—1221no—
326#428—1221no—
327C2×#106—1221no—
328#423—611no—
329C2×C4⋊C8—611no—
330#419—1221no—
331C2×#103—1221no—
332#335—611no—
333#335—611no—
334#335—611no—
335#335C2×D6111yescharacteristic
336#333—611no—
337#333—611no—
338#332—611no—
339#332—611no—
340#332—611no—
341#332—611no—
342#332C2×D6111yescharacteristic
343#332C2×D6111yescharacteristic
344C61#1939911yescharacteristic, complemented, Sylow
345C22×Q8—611no—
346C22×Q8—1221no—
347C2×Q16—21no—
348C2×Q16—21no—
349C2×Q16—1221no—
350C2×Q16—1221no—
351C2×M16—1221no—
352C22×C8—1221no—
353C4⋊Q8—611no—
354C4⋊Q8—611no—
355C4⋊Q8—1221no—
356C4⋊Q8—1221no—
357C4⋊Q8—1221no—
358C4⋊Q8C22×D6111yescharacteristic
359C4⋊Q8C22×D6111yescharacteristic
360#125—1221no—
361C22⋊Q8—1221no—
362C22⋊Q8—1221no—
363C4×Q8—611no—
364C4×Q8—611no—
365C4×Q8—611no—
366C4×Q8—1221no—
367C4×Q8—1221no—
368C4×Q8—1221no—
369C4×Q8—1221no—
370C4×Q8C22×D6111yescharacteristic
371#117—1221no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C42—611no—
377C2×C42—1221no—
378#107—1221no—
379#107—1221no—
380#106—21no—
381#106—21no—
382#106—1221no—
383#106—1221no—
384#106—1221no—
385#106—1221no—
386C4⋊C8—21no—
387C4⋊C8—611no—
388C4⋊C8—611no—
389C4⋊C8—611no—
390C4⋊C8—611no—
391C4⋊C8—611no—
392C4⋊C8—1221no—
393C4⋊C8—1221no—
394C4⋊C8—1221no—
395C4⋊C8—1221no—
396C4⋊C8—1221no—
397C4⋊C8C22×D6111yescharacteristic
398C4⋊C8C22×D6111yescharacteristic
399C4⋊C8C22×D6111yescharacteristic
400#103—21no—
401#103—21no—
402#103—1221no—
403#103—1221no—
404#103—1221no—
405#103—1221no—
406#103—1221no—
407#103—1221no—
408#103—1221no—
409#103—1221no—
410#103—1221no—
411#103—1221no—
412C22⋊C8—611no—
413C22⋊C8—611no—
414C22⋊C8—1221no—
415C4×C8—21no—
416C4×C8—611no—
417C4×C8—1221no—
418C4×C8—1221no—
419C4×C8—1221no—
420C4×C8—1221no—
421C4×C8C22×D6111yescharacteristic
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—614no—
425C2×Q8—611no—
426C2×Q8—1221no—
427C2×Q8—1222no—
428C2×Q8—1221no—
429C2×Q8—1221no—
430C2×Q8—4881no—
431C2×Q8—4881no—
432C2×Q8—2441no—
433C2×Q8D4×D6111yescharacteristic
434C22×C4—611no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—1221no—
438C22×C4—1221no—
439C22×C4—2441no—
440Q16—81no—
441Q16—81no—
442Q16—4881no—
443Q16—4881no—
444M16—2442no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—611no—
449C2×C8—611no—
450C2×C8—1221no—
451C2×C8—1221no—
452C2×C8—1221no—
453C2×C8—2442no—
454C2×C8D4×D6111yescharacteristic
455C2×C8D4×D6111yescharacteristic
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—611no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—2441no—
471C4⋊C4—2441no—
472C4⋊C4—2441no—
473C4⋊C4—2441no—
474C4⋊C4D4×D6111yescharacteristic
475C22⋊C4—2441no—
476C42—21no—
477C42—611no—
478C42—611no—
479C42—611no—
480C42—1221no—
481C42—1221no—
482C42—1221no—
483C42—2441no—
484C42C23×D6111yescharacteristic
485C23—611no—
486Q8—21no—
487Q8—41no—
488Q8—41no—
489Q8—41no—
490Q8—41no—
491Q8—612no—
492Q8—1224no—
493Q8—1221no—
494Q8—4881no—
495Q8—4882no—
496Q8—2441no—
497Q8—2441no—
498Q8Q16×D6112yes—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—611no—
504C2×C4—612no—
505C2×C4—612no—
506C2×C4—611no—
507C2×C4—611no—
508C2×C4—1224no—
509C2×C4—1221no—
510C2×C4—1221no—
511C2×C4—1221no—
512C2×C4—1221no—
513C2×C4—4881no—
514C2×C4—4882no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4C2×D4×D6111yescharacteristic
519C2×C4C2×D4×D6111yescharacteristic
520C2×C4#41×D6111yescharacteristic
521C8—21no—
522C8—21no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—1221no—
527C8—1221no—
528C8—2441no—
529C8—2441no—
530C8—2441no—
531C22—612no—
532C22—1221no—
533C22—1221no—
534C22#121×D6111yescharacteristic
535C4—22no—
536C4—21no—
537C4—41no—
538C4—81no—
539C4—81no—
540C4—612no—
541C4—1222no—
542C4—1221no—
543C4—1221no—
544C4—4881no—
545C4—4881no—
546C4—2441no—
547C4C2×Q16×D6111yescharacteristic
548C4#135×D6111yescharacteristic
549C4#1684056811yescharacteristic
550C4#1684057211yescharacteristic
551C2—1222no—
552C2#2145533511yescharacteristic
553C2#467×D6111yescharacteristic
554C2#2145555411yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1670C2
socleC22×C61C2×#121
prime radicalC22×D61#121
hypercenter#1670D61
hypocenterC61#19399
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1670

Complements

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

NH ≅ G/Ntype
5. #21445278C2semidirect
6. #21445277C2semidirect
7. #21445276C2semidirect
8. #21445275C2semidirect
9. #21445188C2semidirect
10. #21445187C2semidirect
11. #21445186C2semidirect
15. C61×#1670C2semidirect
150. Q8×D61Q16semidirect
285. #1670D61semidirect
344. C61#19399semidirect

Maximal subgroups

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

subgroupindex
#214567632
#214556192
#475×D612
#214548982
#214452782
#214452772
#214452762
#214452752
#214451882
#214451872
#214451862
#214445922
#214445022
#214444752
C61×#16702
#1939961