Finite Group Databasebeta

← #29665369

Subgroups — #29665369

Order 15616.

590 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456971C211yesmaximal, characteristic
2#21455625C211yesmaximal, characteristic
3#477×D61C211yesmaximal, characteristic
4#21454898C211yesmaximal, characteristic
5#21445280C211yesmaximal, characteristic, complemented
6#21445279C211yesmaximal, characteristic, complemented
7#21445278C211yesmaximal, characteristic, complemented
8#21445277C211yesmaximal, characteristic, complemented
9#21445222C211yesmaximal, characteristic, complemented
10#21445221C211yesmaximal, characteristic, complemented
11#21445220C211yesmaximal, characteristic, complemented
12#21444592C211yesmaximal, characteristic
13#21444498C211yesmaximal, characteristic
14#21444473C211yesmaximal, characteristic
15C61×#1676C211yesmaximal, characteristic, complemented
16#16840317C2211yescharacteristic
17C4⋊Q8×D61C2211yescharacteristic
18C4×Q8×D61C2211yescharacteristic
19#16840165C2211yescharacteristic
20#16840114C2211yescharacteristic
21#16840113C2211yescharacteristic
22#16840099—21no—
23#16840099—21no—
24#16840068—21no—
25#16840067—21no—
26#16840058C2211yescharacteristic
27#16840057C2211yescharacteristic
28#16840056—21no—
29#16840056—21no—
30#16840056—21no—
31#16840055—21no—
32#16840054C2211yescharacteristic
33#16840054C2211yescharacteristic
34#16840048C2211yescharacteristic
35#16840047C2211yescharacteristic
36Q8×#92418C2211yescharacteristic
37C8×Dic122—21no—
38C8×Dic122C2211yescharacteristic
39#16839886—21no—
40#16839886C2211yescharacteristic
41#16839814—21no—
42#16839814C2211yescharacteristic
43#16839813—21no—
44#16839813C2211yescharacteristic
45#16839803—21no—
46#107×D61—21no—
47#16839784C2211yescharacteristic
48C4⋊C8×D61C2211yescharacteristic
49#16839769—21no—
50#103×D61—21no—
51#16839515C2211yescharacteristic
52#16839514C2211yescharacteristic
53#16839514C2211yescharacteristic
54#16839511C2211yescharacteristic
55#16839511C2211yescharacteristic
56#16839510C2211yescharacteristic
57#16839510C2211yescharacteristic
58#16839509C2211yescharacteristic
59#16839507C2211yescharacteristic
60C61×#500C2211yescharacteristic
61C61×#477C2211yescharacteristic
62C61×#477C2211yescharacteristic
63C8×Q8×C61C2211yescharacteristic
64C61×#335C2211yescharacteristic
65C61×#333C2211yescharacteristic
66C61×#333C2211yescharacteristic
67C2×Q8×D61—21no—
68C2×Q8×D61D411yescharacteristic
69C2×#11758189—21no—
70C2×C8×D61—21no—
71#14519777C2311yescharacteristic
72#14519777C2311yescharacteristic
73#14519776—21no—
74#14519776—21no—
75#14519775—21no—
76#14519768—21no—
77#14519757—21no—
78#14519753—21no—
79Q8×Dic61—21no—
80Q8×Dic61D411yescharacteristic
81C4×Dic122—21no—
82C4×Dic122C2311yescharacteristic
83#14519742—21no—
84#14519742—21no—
85#14519742D411yescharacteristic
86#14519735—21no—
87C4⋊C4×D61—21no—
88C4⋊C4×D61—21no—
89C4⋊C4×D61—21no—
90C4⋊C4×D61D411yescharacteristic
91C42×D61—21no—
92C42×D61C2311yescharacteristic
93#14519711—21no—
94#14519710—21no—
95#14519710—21no—
96#14519710—21no—
97#14519710—21no—
98#14519710—21no—
99#14519709—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#14519700—21no—
118#14519700—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—
129C4⋊Q8×C61C2311yescharacteristic
130C4⋊Q8×C61C2311yescharacteristic
131C4×Q8×C61C2311yescharacteristic
132#107×C61—21no—
133#107×C61—21no—
134#107×C61—21no—
135#107×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—82no—
149Q8×D61Q1614yes—
150C2×C4×D61—21no—
151C2×C4×D61—21no—
152C2×C4×D61—41no—
153C2×C4×D61C2×D411yescharacteristic
154C2×C4×D61C2×D411yescharacteristic
155C2×C4×D61#4111yescharacteristic
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—41no—
160C2×Dic122C2×D411yescharacteristic
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61C2×D411yescharacteristic
165C2×C8×C61C2×D411yescharacteristic
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61C2×D411yescharacteristic
172C42×C61—21no—
173C42×C61C2411yescharacteristic
174#11758198—41no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—41no—
181C4⋊Dic61C2×D411yescharacteristic
182#11758196—21no—
183#11758196—21no—
184#11758196—21no—
185#11758196—41no—
186#11758196—41no—
187#11758196—41no—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—41no—
192C4×Dic61C2×D411yescharacteristic
193C4×Dic61C2×D411yescharacteristic
194C4×Dic61#4111yescharacteristic
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418C2×D411yescharacteristic
199C2×#92418C2×D411yescharacteristic
200#11758189—42no—
201C8×D61—42no—
202C22×D61#12111yescharacteristic
203Q8×C61—21no—
204Q8×C61—81no—
205Q8×C61—81no—
206Q8×C61C2×Q1612yes—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C2×#4111yescharacteristic
214C2×Dic61—21no—
215C2×Dic61—21no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218C2×Dic61#12111yescharacteristic
219C2×Dic61#12111yescharacteristic
220C2×Dic61#12111yescharacteristic
221C4×D61—24no—
222C4×D61—21no—
223C4×D61—21no—
224C4×D61—41no—
225C4×D61—81no—
226C4×D61—82no—
227C4×D61C2×Q1612yes—
228C4×D61#13512yes—
229Dic122—21no—
230Dic122—24no—
231Dic122—41no—
232Dic122—41no—
233Dic122—82no—
234Dic122—81no—
235Dic122C2×Q1612yes—
236C8×C61—21no—
237C8×C61—21no—
238C8×C61—41no—
239C8×C61—41no—
240C8×C61—41no—
241#92418—21no—
242#92418—21no—
243#92418—41no—
244#92418—41no—
245#92418—41no—
246#19405—611nomaximal, Sylow
247C22×C61C2×#12111yescharacteristic
248C2×D61—21no—
249C2×D61—21no—
250C2×D61#46712yes—
251C4×C61—22no—
252C4×C61—21no—
253C4×C61—41no—
254C4×C61—81no—
255C4×C61—81no—
256C4×C61C22×Q1611yescharacteristic
257C4×C61C2×#13511yescharacteristic
258C4×C61#57811yescharacteristic
259C4×C61#57811yescharacteristic
260Dic61—21no—
261Dic61—22no—
262Dic61—21no—
263Dic61—41no—
264Dic61—81no—
265Dic61—81no—
266Dic61#46712yes—
267#3230—611no—
268#3078—611no—
269C2×#477—611no—
270#2982—611no—
271#1684—611no—
272#1684—611no—
273#1684—611no—
274#1684—611no—
275#1676—611no—
276#1676—611no—
277#1676—611no—
278#1676D6111yescharacteristic, complemented
279#1571—611no—
280#1548—611no—
281#1542—611no—
282C2×C61#304411yescharacteristic
283C2×C61C2×#46711yescharacteristic
284C2×C61#306811yescharacteristic
285D61—22no—
286#533—611no—
287C2×C4⋊Q8—611no—
288C2×C4×Q8—611no—
289#500—611no—
290#500C2×D6111yescharacteristic
291#487—611no—
292#487—611no—
293#484—1221no—
294#484—1221no—
295#479—1221no—
296#479—1221no—
297#477—611no—
298#477—611no—
299#477—611no—
300#477—611no—
301#477—1221no—
302#477—1221no—
303#477—1221no—
304#477—1221no—
305#477C2×D6111yescharacteristic
306#477C2×D6111yescharacteristic
307#475—611no—
308#475—611no—
309C8×Q8—611no—
310C8×Q8—611no—
311C8×Q8—611no—
312C8×Q8—1221no—
313C8×Q8—1221no—
314C8×Q8C2×D6111yescharacteristic
315#431—611no—
316#431—611no—
317#431—1221no—
318#431—1221no—
319#428—1221no—
320C2×#107—1221no—
321#423—611no—
322C2×C4⋊C8—611no—
323#419—1221no—
324C2×#103—1221no—
325#335—611no—
326#335—611no—
327#335—611no—
328#335C2×D6111yescharacteristic
329#333—611no—
330#333—611no—
331#333—611no—
332#333—611no—
333#333C2×D6111yescharacteristic
334#333C2×D6111yescharacteristic
335#332—611no—
336#332—611no—
337C61#1940511yescharacteristic, complemented, Sylow
338C22×Q8—611no—
339C22×Q8—1221no—
340C2×M16—1221no—
341C22×C8—1221no—
342C4⋊Q8—611no—
343C4⋊Q8—611no—
344C4⋊Q8—1221no—
345C4⋊Q8—1221no—
346C4⋊Q8—1221no—
347C4⋊Q8C22×D6111yescharacteristic
348C4⋊Q8C22×D6111yescharacteristic
349#125—1221no—
350C22⋊Q8—1221no—
351C22⋊Q8—1221no—
352C4×Q8—611no—
353C4×Q8—611no—
354C4×Q8—611no—
355C4×Q8—1221no—
356C4×Q8—1221no—
357C4×Q8—1221no—
358C4×Q8—1221no—
359C4×Q8C22×D6111yescharacteristic
360#117—1221no—
361C2×C4⋊C4—611no—
362C2×C4⋊C4—1221no—
363C2×C4⋊C4—1221no—
364C2×C4⋊C4—1221no—
365C2×C42—611no—
366C2×C42—1221no—
367#107—21no—
368#107—21no—
369#107—21no—
370#107—21no—
371#107—1221no—
372#107—1221no—
373#107—1221no—
374#107—1221no—
375#107—1221no—
376#107—1221no—
377#106—1221no—
378#106—1221no—
379C4⋊C8—21no—
380C4⋊C8—611no—
381C4⋊C8—611no—
382C4⋊C8—611no—
383C4⋊C8—611no—
384C4⋊C8—611no—
385C4⋊C8—1221no—
386C4⋊C8—1221no—
387C4⋊C8—1221no—
388C4⋊C8—1221no—
389C4⋊C8—1221no—
390C4⋊C8C22×D6111yescharacteristic
391C4⋊C8C22×D6111yescharacteristic
392C4⋊C8C22×D6111yescharacteristic
393#103—21no—
394#103—21no—
395#103—1221no—
396#103—1221no—
397#103—1221no—
398#103—1221no—
399#103—1221no—
400#103—1221no—
401#103—1221no—
402#103—1221no—
403#103—1221no—
404#103—1221no—
405C22⋊C8—611no—
406C22⋊C8—611no—
407C22⋊C8—1221no—
408C4×C8—21no—
409C4×C8—611no—
410C4×C8—1221no—
411C4×C8—1221no—
412C4×C8—1221no—
413C4×C8—1221no—
414C4×C8C22×D6111yescharacteristic
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—614no—
418C2×Q8—611no—
419C2×Q8—1221no—
420C2×Q8—1222no—
421C2×Q8—1221no—
422C2×Q8—1221no—
423C2×Q8—4882no—
424C2×Q8—2441no—
425C2×Q8D4×D6111yescharacteristic
426C22×C4—611no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—1221no—
430C22×C4—1221no—
431C22×C4—2441no—
432M16—2442no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—611no—
437C2×C8—611no—
438C2×C8—1221no—
439C2×C8—1221no—
440C2×C8—1221no—
441C2×C8—2442no—
442C2×C8D4×D6111yescharacteristic
443C2×C8D4×D6111yescharacteristic
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—611no—
450C4⋊C4—1221no—
451C4⋊C4—1221no—
452C4⋊C4—1221no—
453C4⋊C4—1221no—
454C4⋊C4—1221no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—1221no—
458C4⋊C4—2441no—
459C4⋊C4—2441no—
460C4⋊C4—2441no—
461C4⋊C4—2441no—
462C4⋊C4D4×D6111yescharacteristic
463C22⋊C4—2441no—
464C42—21no—
465C42—611no—
466C42—611no—
467C42—611no—
468C42—1221no—
469C42—1221no—
470C42—1221no—
471C42—2441no—
472C42C23×D6111yescharacteristic
473C23—611no—
474Q8—21no—
475Q8—81no—
476Q8—81no—
477Q8—612no—
478Q8—1224no—
479Q8—1221no—
480Q8—4881no—
481Q8—4882no—
482Q8—2441no—
483Q8—2441no—
484Q8Q16×D6112yes—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—611no—
490C2×C4—612no—
491C2×C4—612no—
492C2×C4—611no—
493C2×C4—611no—
494C2×C4—1224no—
495C2×C4—1221no—
496C2×C4—1221no—
497C2×C4—1221no—
498C2×C4—1221no—
499C2×C4—4881no—
500C2×C4—4882no—
501C2×C4—2441no—
502C2×C4—2441no—
503C2×C4—2441no—
504C2×C4C2×D4×D6111yescharacteristic
505C2×C4C2×D4×D6111yescharacteristic
506C2×C4#41×D6111yescharacteristic
507C8—21no—
508C8—21no—
509C8—41no—
510C8—41no—
511C8—41no—
512C8—1221no—
513C8—1221no—
514C8—2441no—
515C8—2441no—
516C8—2441no—
517C22—612no—
518C22—1221no—
519C22—1221no—
520C22#121×D6111yescharacteristic
521C4—22no—
522C4—21no—
523C4—41no—
524C4—81no—
525C4—81no—
526C4—612no—
527C4—1222no—
528C4—1221no—
529C4—1221no—
530C4—4881no—
531C4—4881no—
532C4—2441no—
533C4C2×Q16×D6111yescharacteristic
534C4#135×D6111yescharacteristic
535C4#1684057511yescharacteristic
536C4#1684057811yescharacteristic
537C2—1222no—
538C2#2145534411yescharacteristic
539C2#467×D6111yescharacteristic
540C2#2145554511yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1676C2
socleC22×C61C2×#121
prime radicalC22×D61#121
hypercenter#1676D61
hypocenterC61#19405
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1676

Complements

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

NH ≅ G/Ntype
5. #21445280C2semidirect
6. #21445279C2semidirect
7. #21445278C2semidirect
8. #21445277C2semidirect
9. #21445222C2semidirect
10. #21445221C2semidirect
11. #21445220C2semidirect
15. C61×#1676C2semidirect
278. #1676D61semidirect
337. C61#19405semidirect

Maximal subgroups

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

subgroupindex
#214569712
#214556252
#477×D612
#214548982
#214452802
#214452792
#214452782
#214452772
#214452222
#214452212
#214452202
#214445922
#214444982
#214444732
C61×#16762
#1940561