Finite Group Databasebeta

← #29651203

Subgroups — #29651203

Order 15616.

600 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455897C211yesmaximal, characteristic
2#21454974C211yesmaximal, characteristic
3#21454905C211yesmaximal, characteristic, complemented
4#21444714C211yesmaximal, characteristic, complemented
5#21444710C211yesmaximal, characteristic, complemented
6#21444700C211yesmaximal, characteristic, complemented
7#21444693C211yesmaximal, characteristic, complemented
8#21444692C211yesmaximal, characteristic, complemented
9#21444625C211yesmaximal, characteristic, complemented
10C8×#11758200C211yesmaximal, characteristic, complemented
11#21444603C211yesmaximal, characteristic, complemented
12#21444276C211yesmaximal, characteristic
13Q8⋊C8×D61C211yesmaximal, characteristic, complemented
14#21444104C211yesmaximal, characteristic, complemented
15C61×#1585C211yesmaximal, characteristic, complemented
16#16840294C2211yescharacteristic
17C4×Q8×D61C2211yescharacteristic
18#16840094C411yescharacteristic
19#16840074C411yescharacteristic
20#16840072C411yescharacteristic
21#16840071C411yescharacteristic
22#16839983C411yescharacteristic
23#16839982C411yescharacteristic
24#16839919C411yescharacteristic
25#16839899C411yescharacteristic
26#16839893C2211yescharacteristic
27#16839892C2211yescharacteristic
28#16839891—21no—
29#16839890—21no—
30C8×Dic122C2211yescharacteristic
31#16839886C2211yescharacteristic
32#16839886C2211yescharacteristic
33C4×Dic244C2211yescharacteristic
34C4×#11758202C2211yescharacteristic
35C4×#11758200C2211yescharacteristic
36#16839840C2211yescharacteristic
37#16839836—21no—
38C8×C22⋊D61—21no—
39#16839820—21no—
40#16839818C2211yescharacteristic
41#16839817—21no—
42#16839814—21no—
43#16839813C2211yescharacteristic
44#16839813C2211yescharacteristic
45#16839788—21no—
46C4⋊C8×D61C2211yescharacteristic
47C4×C8×D61C2211yescharacteristic
48#16839514C2211yescharacteristic
49#16839513C2211yescharacteristic
50#16839512C2211yescharacteristic
51#16839494C2211yescharacteristic
52#16839494C2211yescharacteristic
53Q8⋊#92418C2211yescharacteristic
54#16839492C2211yescharacteristic
55#16839492C2211yescharacteristic
56D4⋊#92418C2211yescharacteristic
57#16839483C2211yescharacteristic
58C8×#92418C2211yescharacteristic
59C8×#92418C2211yescharacteristic
60C61×#446C2211yescharacteristic
61C4×SD16×C61C2211yescharacteristic
62C8×D4×C61C2211yescharacteristic
63C61×#335C2211yescharacteristic
64C61×Q8⋊C8C2211yescharacteristic
65C61×D4⋊C8C2211yescharacteristic
66C61×#322C2211yescharacteristic
67C2×Q8×D61C2×C411yescharacteristic
68C2×Dic244C2×C411yescharacteristic
69C2×#11758202C2×C411yescharacteristic
70C2×#11758200C2×C411yescharacteristic
71C2×#11758194—21no—
72C2×C8×D61—21no—
73C2×C8×D61—21no—
74#14519770—21no—
75#14519768C2×C411yescharacteristic
76#14519764C2×C411yescharacteristic
77#14519759—21no—
78#14519757C2×C411yescharacteristic
79#14519748C2×C411yescharacteristic
80Q8×Dic61C2×C411yescharacteristic
81C4×Dic122—21no—
82C4×Dic122C2311yescharacteristic
83C4×Dic122C2311yescharacteristic
84#14519742—21no—
85#14519742C2×C411yescharacteristic
86C4×C22⋊D61—21no—
87#14519736—21no—
88C4⋊C4×D61—21no—
89C4⋊C4×D61C2×C411yescharacteristic
90C42×D61—21no—
91C42×D61C2311yescharacteristic
92#14519711C2×C411yescharacteristic
93#14519710C2×C411yescharacteristic
94#14519708C2×C411yescharacteristic
95C4⋊#92418—21no—
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519706—21no—
99#14519706—21no—
100#14519706—21no—
101#14519706—21no—
102#14519706—21no—
103#14519706C2×C411yescharacteristic
104#14519706C2×C411yescharacteristic
105#14519706D411yescharacteristic
106#14519706D411yescharacteristic
107#14519702C2×C411yescharacteristic
108#14519702C2×C411yescharacteristic
109#14519701C2×C411yescharacteristic
110#14519700C2×C411yescharacteristic
111#14519700C2×C411yescharacteristic
112#14519698C2×C411yescharacteristic
113C22⋊#92418—21no—
114#14519689—21no—
115#14519689C2×C411yescharacteristic
116#14519689C2×C411yescharacteristic
117#14519689D411yescharacteristic
118#14519689D411yescharacteristic
119#14519688—21no—
120#14519687—21no—
121#14519687—21no—
122C4×#92418C2311yescharacteristic
123C4×#92418C2311yescharacteristic
124C4×#92418C2311yescharacteristic
125C8×Dic61—21no—
126C8×Dic61—21no—
127C8×Dic61—21no—
128C8×Dic61—21no—
129C2×SD16×C61C2×C411yescharacteristic
130C22×C8×C61—21no—
131C4×Q8×C61C2311yescharacteristic
132C4×D4×C61C2311yescharacteristic
133#106×C61C2×C411yescharacteristic
134C4⋊C8×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137#103×C61C2×C411yescharacteristic
138#102×C61C2×C411yescharacteristic
139C22⋊C8×C61—21no—
140#97×C61—21no—
141C4×C8×C61C2311yescharacteristic
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61C22×C411yescharacteristic
145C2×D4×C61C22×C411yescharacteristic
146C22×C4×C61—21no—
147C2×C22⋊D61—21no—
148Q8×D61—22no—
149Q8×D61—41no—
150C2×C4×D61—21no—
151C2×C4×D61—21no—
152C2×C4×D61C22×C411yescharacteristic
153C2×C4×D61C2×D411yescharacteristic
154C2×C4×D61#4111yescharacteristic
155C2×Dic122—21no—
156C2×Dic122C22×C411yescharacteristic
157C2×Dic122C22×C411yescharacteristic
158SD16×C61—41no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—81no—
163C2×C8×C61C22×C411yescharacteristic
164C2×C8×C61C22×C411yescharacteristic
165C2×C8×C61C2×D411yescharacteristic
166C2×C8×C61C2×D411yescharacteristic
167C4⋊C4×C61—21no—
168C4⋊C4×C61C22×C411yescharacteristic
169C4⋊C4×C61C22×C411yescharacteristic
170C22⋊C4×C61—21no—
171C42×C61—21no—
172C42×C61C2411yescharacteristic
173C22⋊Dic61—21no—
174C22⋊Dic61—21no—
175C22⋊Dic61—21no—
176#11758202—41no—
177#11758200—22no—
178#11758198—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61C22×C411yescharacteristic
183C4⋊Dic61C22×C411yescharacteristic
184#11758196—21no—
185#11758196—21no—
186#11758196—21no—
187#11758196—21no—
188#11758196—21no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—21no—
193C4×Dic61C22×C411yescharacteristic
194C4×Dic61C2×D411yescharacteristic
195C4×Dic61#4111yescharacteristic
196#11758194—81no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418C22×C411yescharacteristic
201C2×#92418C22×C411yescharacteristic
202C2×#92418C2×D411yescharacteristic
203C2×#92418C2×D411yescharacteristic
204Dic244—22no—
205C8×D61—24no—
206C8×D61—81no—
207C23×C61—21no—
208C22×D61C4×D411yescharacteristic
209Q8×C61—21no—
210Q8×C61—21no—
211D4×C61—21no—
212D4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61C23×C411yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C22⋊D61—42no—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61C4×D411yescharacteristic
227C2×Dic61C4×D411yescharacteristic
228C2×Dic61C4×D411yescharacteristic
229C4×D61—22no—
230C4×D61—21no—
231C4×D61—21no—
232C4×D61—21no—
233C4×D61—21no—
234C4×D61—42no—
235C4×D61—42no—
236Dic122—21no—
237Dic122—21no—
238Dic122—21no—
239Dic122—42no—
240Dic122#13112yes—
241C8×C61—22no—
242C8×C61—21no—
243C8×C61—21no—
244C8×C61—21no—
245C8×C61—21no—
246C8×C61—41no—
247C8×C61—41no—
248#92418—22no—
249#92418—21no—
250#92418—21no—
251#92418—21no—
252#92418—21no—
253#92418—41no—
254#92418—41no—
255#18017—611nomaximal, Sylow
256C22×C61—41no—
257C22×C61—41no—
258C22×C61—41no—
259C22×C61C2×C4×D411yescharacteristic
260C2×D61—21no—
261C2×D61—21no—
262C2×D61#44312yes—
263C4×C61—21no—
264C4×C61—41no—
265C4×C61—41no—
266C4×C61—41no—
267C4×C61C2×#13111yescharacteristic
268C4×C61#56811yescharacteristic
269C4×C61#57711yescharacteristic
270C4×C61#57811yescharacteristic
271Dic61—22no—
272Dic61—21no—
273Dic61—21no—
274Dic61—21no—
275Dic61—41no—
276Dic61—41no—
277Dic61—41no—
278#3108—611no—
279#2994—611no—
280#2984—611no—
281#1591—611no—
282#1590—611no—
283#1588—611no—
284#1587—611no—
285#1587—611no—
286#1585D6111yescharacteristic, complemented
287C8×Q16—611no—
288C8×SD16—611no—
289#1573—611no—
290#1494—611no—
291C2×Q8⋊C8—611no—
292#1450—611no—
293C2×C61—81no—
294C2×C61#293311yescharacteristic
295C2×C61#294711yescharacteristic
296C2×C61C2×#44311yescharacteristic
297D61—22no—
298#529—611no—
299C2×C4×Q8—611no—
300#483—611no—
301#480—611no—
302#479—611no—
303#479—611no—
304#464—611no—
305#464—611no—
306#451—611no—
307#448—611no—
308#446—611no—
309#446—611no—
310#446—1221no—
311#446—1221no—
312#446C2×D6111yescharacteristic
313C8×Q8—611no—
314C8×Q8—611no—
315C8×Q8—611no—
316C4×Q16—611no—
317C4×Q16—611no—
318C4×SD16—611no—
319C4×SD16C2×D6111yescharacteristic
320#436—611no—
321#435—1221no—
322C8×D4—1221no—
323C8×D4C2×D6111yescharacteristic
324#433—1221no—
325#432—611no—
326#432—1221no—
327#431—611no—
328#431—611no—
329#431—1221no—
330#424—1221no—
331C2×C4⋊C8—611no—
332C2×C4×C8—611no—
333#335—611no—
334#335C2×D6111yescharacteristic
335#334—611no—
336#334—611no—
337Q8⋊C8—611no—
338Q8⋊C8—611no—
339Q8⋊C8—611no—
340Q8⋊C8—611no—
341Q8⋊C8—611no—
342Q8⋊C8C2×D6111yescharacteristic
343D4⋊C8—611no—
344D4⋊C8C2×D6111yescharacteristic
345#322—611no—
346#322C2×D6111yescharacteristic
347C82—611no—
348C82—611no—
349C61#1801711yescharacteristic, complemented, Sylow
350C22×Q8—611no—
351C2×Q16—611no—
352C2×Q16—611no—
353C2×SD16—611no—
354C2×SD16C4×D6111yescharacteristic
355C2×M16—1221no—
356C22×C8—21no—
357C22×C8—1221no—
358C22×C8—1221no—
359#126—1221no—
360#125—611no—
361#124—611no—
362#123—1221no—
363C22⋊Q8—611no—
364#121—611no—
365C4×Q8—611no—
366C4×Q8—611no—
367C4×Q8—611no—
368C4×Q8—611no—
369C4×Q8—1221no—
370C4×Q8—1221no—
371C4×Q8C22×D6111yescharacteristic
372C4×D4—1221no—
373C4×D4C22×D6111yescharacteristic
374#117—1221no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—1221no—
377C2×C42—611no—
378C2×C42—1221no—
379#107—611no—
380#107—611no—
381#106—611no—
382#106C4×D6111yescharacteristic
383C4⋊C8—21no—
384C4⋊C8—611no—
385C4⋊C8—611no—
386C4⋊C8—611no—
387C4⋊C8—611no—
388C4⋊C8—611no—
389C4⋊C8—611no—
390C4⋊C8—1221no—
391C4⋊C8—1221no—
392C4⋊C8—1221no—
393C4⋊C8—1221no—
394C4⋊C8—1221no—
395C4⋊C8—1221no—
396C4⋊C8C22×D6111yescharacteristic
397C4⋊C8C22×D6111yescharacteristic
398#103—611no—
399#103—611no—
400#103—611no—
401#103—611no—
402#103—611no—
403#103C4×D6111yescharacteristic
404#102—611no—
405#102C4×D6111yescharacteristic
406C22⋊C8—21no—
407C22⋊C8—611no—
408C22⋊C8—611no—
409C22⋊C8—611no—
410C22⋊C8—611no—
411C22⋊C8—1221no—
412C22⋊C8—1221no—
413#97—21no—
414#97—1221no—
415#97—1221no—
416#97—1221no—
417C4×C8—611no—
418C4×C8—611no—
419C4×C8—611no—
420C4×C8—1221no—
421C4×C8—1221no—
422C4×C8—1221no—
423C4×C8—1221no—
424C4×C8C22×D6111yescharacteristic
425C4×C8C22×D6111yescharacteristic
426C4×C8C22×D6111yescharacteristic
427C2×Q8—611no—
428C2×Q8—611no—
429C2×Q8—1221no—
430C2×Q8—1222no—
431C2×Q8—2441no—
432C2×Q8C2×C4×D6111yescharacteristic
433C2×D4—1221no—
434C2×D4C2×C4×D6111yescharacteristic
435C22×C4—21no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—611no—
439C22×C4—1221no—
440C22×C4—1221no—
441Q16—1222no—
442Q16—2441no—
443SD16—41no—
444SD16—1222no—
445M16—4881no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—81no—
450C2×C8—611no—
451C2×C8—611no—
452C2×C8—611no—
453C2×C8—611no—
454C2×C8—1224no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—4881no—
459C2×C8C2×C4×D6111yescharacteristic
460C2×C8C2×C4×D6111yescharacteristic
461C2×C8D4×D6111yescharacteristic
462C2×C8D4×D6111yescharacteristic
463C4⋊C4—21no—
464C4⋊C4—611no—
465C4⋊C4—611no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4C2×C4×D6111yescharacteristic
475C4⋊C4C2×C4×D6111yescharacteristic
476C22⋊C4—21no—
477C22⋊C4—1221no—
478C22⋊C4—1221no—
479C22⋊C4—1221no—
480C22⋊C4—1221no—
481C42—21no—
482C42—611no—
483C42—611no—
484C42—611no—
485C42—1221no—
486C42—1221no—
487C42—1221no—
488C42—1221no—
489C42C23×D6111yescharacteristic
490C23—21no—
491C23—611no—
492Q8—21no—
493Q8—21no—
494Q8—612no—
495Q8—1221no—
496Q8—1221no—
497Q8—1221no—
498Q8—2442no—
499D4—21no—
500D4—21no—
501D4—2442no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—611no—
508C2×C4—611no—
509C2×C4—611no—
510C2×C4—1222no—
511C2×C4—1221no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—1221no—
515C2×C4—1221no—
516C2×C4—1221no—
517C2×C4—1221no—
518C2×C4—1221no—
519C2×C4—2442no—
520C2×C4—2442no—
521C2×C4C22×C4×D6111yescharacteristic
522C2×C4C2×D4×D6111yescharacteristic
523C2×C4#41×D6111yescharacteristic
524C8—22no—
525C8—21no—
526C8—21no—
527C8—21no—
528C8—21no—
529C8—41no—
530C8—41no—
531C8—1222no—
532C8—1221no—
533C8—1221no—
534C8—1221no—
535C8—1221no—
536C8—2441no—
537C8—2441no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—612no—
542C22—1221no—
543C22—1221no—
544C22C4×D4×D6111yescharacteristic
545C4—21no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—1221no—
550C4—1221no—
551C4—1221no—
552C4—1222no—
553C4—2441no—
554C4—2441no—
555C4—2441no—
556C4#131×D6111yescharacteristic
557C4#1684052811yescharacteristic
558C4#1684057211yescharacteristic
559C4#1684057811yescharacteristic
560C2—81no—
561C2—1222no—
562C2#2145454211yescharacteristic
563C2#2145465911yescharacteristic
564C2#443×D6111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1585C2
socleC22×C61C2×C4×D4
prime radical#14519748C2×C4
cosocleC42×C61C24
hypercenter#1585D61
hypocenterC61#18017
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1585

Complements

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

NH ≅ G/Ntype
3. #21454905C2semidirect
4. #21444714C2semidirect
5. #21444710C2semidirect
6. #21444700C2semidirect
7. #21444693C2semidirect
8. #21444692C2semidirect
9. #21444625C2semidirect
10. C8×#11758200C2semidirect
11. #21444603C2semidirect
13. Q8⋊C8×D61C2semidirect
14. #21444104C2semidirect
15. C61×#1585C2semidirect
286. #1585D61semidirect
349. C61#18017semidirect

Maximal subgroups

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

subgroupindex
#214558972
#214549742
#214549052
#214447142
#214447102
#214447002
#214446932
#214446922
#214446252
C8×#117582002
#214446032
#214442762
Q8⋊C8×D612
#214441042
C61×#15852
#1801761