Finite Group Databasebeta

← #30613700

Subgroups — #30613700

Order 15616.

588 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#21462651C211yesmaximal, characteristic
2#21462615C211yesmaximal, characteristic, complemented
3#21461460C211yesmaximal, characteristic, complemented
4#21461457C211yesmaximal, characteristic, complemented
5#21461449C211yesmaximal, characteristic, complemented
6#21461442C211yesmaximal, characteristic, complemented
7#21461433C211yesmaximal, characteristic, complemented
8#21461422C211yesmaximal, characteristic, complemented
9#21461413C211yesmaximal, characteristic, complemented
10#21461402C211yesmaximal, characteristic, complemented
11#21461371C211yesmaximal, characteristic, complemented
12#21461143C211yesmaximal, characteristic, complemented
13#21461135C211yesmaximal, characteristic
14#21461043C211yesmaximal, characteristic, complemented
15#21456428C211yesmaximal, characteristic
16#16840813—21no—
17#16840813—21no—
18#16840813C2211yescharacteristic
19#16840812C2211yescharacteristic
20#16840812C2211yescharacteristic
21#16840811C2211yescharacteristic
22#16840811C2211yescharacteristic
23Q8×#92420C2211yescharacteristic
24#16840797C2211yescharacteristic
25#16840795—21no—
26#16840791—21no—
27D4×#92420C2211yescharacteristic
28#16840786—21no—
29#16840781C2211yescharacteristic
30#16840769—21no—
31D61⋊(C4⋊C8)C2211yescharacteristic
32#16840727—21no—
33#16840726—21no—
34C4×#11758213C2211yescharacteristic
35C4×#11758213C2211yescharacteristic
36#16840626C2211yescharacteristic
37#16840625C2211yescharacteristic
38#16840624C2211yescharacteristic
39#16840623C2211yescharacteristic
40#16840612C2211yescharacteristic
41#16840612C2211yescharacteristic
42#16840612C2211yescharacteristic
43#16840612C2211yescharacteristic
44Q8⋊#92420C2211yescharacteristic
45Q8⋊#92420C2211yescharacteristic
46D4⋊#92420C2211yescharacteristic
47D4⋊#92420C2211yescharacteristic
48#92420⋊C8C2211yescharacteristic
49#92420⋊C8C2211yescharacteristic
50#16840603C2211yescharacteristic
51C8×#92420C2211yescharacteristic
52#16840317C2211yescharacteristic
53#16840291C2211yescharacteristic
54#16840168C411yescharacteristic
55#16840157C411yescharacteristic
56#16840131C411yescharacteristic
57#16840019C411yescharacteristic
58#16840019C411yescharacteristic
59#16840007C411yescharacteristic
60#16840007C411yescharacteristic
61#16839854C2211yescharacteristic
62#16839854C2211yescharacteristic
63#16839853C2211yescharacteristic
64#16839853C2211yescharacteristic
65#16839714C2211yescharacteristic
66C61×#489C411yescharacteristic
67C2×#11758213—21no—
68C2×#11758213D411yescharacteristic
69C2×#11758213D411yescharacteristic
70C2×#11758217—21no—
71C2×D61⋊C8—21no—
72C22×#92420—21no—
73#14519844—21no—
74#14519844—21no—
75#14519844—21no—
76#14519844—21no—
77C4⋊#92420—21no—
78C4⋊#92420—21no—
79C4⋊#92420—21no—
80C4⋊#92420C2311yescharacteristic
81C4⋊#92420C2311yescharacteristic
82C4⋊#92420C2311yescharacteristic
83C4⋊#92420C2311yescharacteristic
84#14519834—21no—
85C22⋊#92420—21no—
86C22⋊#92420—21no—
87#14519832—21no—
88#14519832—21no—
89#14519832—21no—
90#14519832—21no—
91#14519832D411yescharacteristic
92#14519832D411yescharacteristic
93C4×#92420—21no—
94C4×#92420—21no—
95C4×#92420—21no—
96C4×#92420—21no—
97C4×#92420C2311yescharacteristic
98C4×#92420C2311yescharacteristic
99C4×#92420C2311yescharacteristic
100C4×#92420C2311yescharacteristic
101C2×#11758202C2×C411yescharacteristic
102C2×#11758202C2×C411yescharacteristic
103C2×#11758200C2×C411yescharacteristic
104C2×#11758200C2×C411yescharacteristic
105C2×#11758189C2×C411yescharacteristic
106C2×#11758189C2×C411yescharacteristic
107#14519777C2×C411yescharacteristic
108#14519770—21no—
109#14519769C2×C411yescharacteristic
110#14519768—21no—
111#14519759—21no—
112#14519757C2×C411yescharacteristic
113#14519757C2×C411yescharacteristic
114#14519753—21no—
115#14519753C2×C411yescharacteristic
116#14519748C2×C411yescharacteristic
117Q8×Dic61—21no—
118Q8×Dic61C2311yescharacteristic
119#14519742C2311yescharacteristic
120#14519742C2311yescharacteristic
121D4×Dic61C2311yescharacteristic
122#14519737—21no—
123#14519735—21no—
124#14519733—21no—
125C42×D61C2311yescharacteristic
126#14519710C2×C411yescharacteristic
127#14519710C2×C411yescharacteristic
128#14519708C2×C411yescharacteristic
129#14519708C2×C411yescharacteristic
130#14519702C2×C411yescharacteristic
131#14519702C2×C411yescharacteristic
132#14519702C2×C411yescharacteristic
133#14519702C2×C411yescharacteristic
134#14519688C2×C411yescharacteristic
135C8×Dic61C2311yescharacteristic
136C8×Dic61C2311yescharacteristic
137C4⋊Q8×C61C2×C411yescharacteristic
138#124×C61C2×C411yescharacteristic
139#103×C61C2×C411yescharacteristic
140#103×C61C2×C411yescharacteristic
141#102×C61C2×C411yescharacteristic
142#102×C61C2×C411yescharacteristic
143#97×C61C2×C411yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61C22×C411yescharacteristic
146C2×D4×C61C22×C411yescharacteristic
147C2×C22⋊D61—21no—
148C22×Dic61—21no—
149C2×C4×D61C22×C411yescharacteristic
150C2×C4×D61C2×D411yescharacteristic
151C2×C4×D61#4111yescharacteristic
152C2×Dic122C22×C411yescharacteristic
153C2×Dic122C22×C411yescharacteristic
154#11758217—42no—
155C2×#92420—21no—
156C2×#92420—21no—
157C2×#92420—21no—
158C2×#92420—21no—
159C2×#92420—21no—
160C2×#92420—21no—
161C2×#92420—42no—
162C2×#92420C2×D411yescharacteristic
163C2×#92420C2×D411yescharacteristic
164C2×#92420C2×D411yescharacteristic
165C2×#92420C2×D411yescharacteristic
166#11758213—22no—
167#11758213—41no—
168#11758213—42no—
169D61⋊C8—42no—
170C2×C8×C61C22×C411yescharacteristic
171C2×C8×C61C22×C411yescharacteristic
172C4⋊C4×C61—21no—
173C4⋊C4×C61C22×C411yescharacteristic
174C4⋊C4×C61C22×C411yescharacteristic
175C22⋊C4×C61—21no—
176C22⋊C4×C61—21no—
177C42×C61C22×C411yescharacteristic
178C22⋊Dic61—21no—
179C22⋊Dic61—21no—
180#11758202—22no—
181#11758202—41no—
182#11758200—22no—
183#11758200—41no—
184#11758198—21no—
185#11758198—21no—
186#11758198—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61C22×C411yescharacteristic
193C4⋊Dic61C22×C411yescharacteristic
194#11758196—21no—
195#11758196—21no—
196#11758196—21no—
197#11758196—21no—
198#11758196—21no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61C2×D411yescharacteristic
203C4×Dic61#4111yescharacteristic
204C4×Dic61C2411yescharacteristic
205C2×#92418C22×C411yescharacteristic
206C2×#92418C22×C411yescharacteristic
207#11758189—22no—
208#11758189—41no—
209C23×C61—21no—
210C22×D61C4×D411yescharacteristic
211Q8×C61—21no—
212Q8×C61—21no—
213Q8×C61—81no—
214D4×C61—21no—
215D4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61C4×D411yescharacteristic
220C2×C4×C61C4×D411yescharacteristic
221C2×C4×C61C23×C411yescharacteristic
222C22⋊D61—81no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61C4×D411yescharacteristic
230C2×Dic61C22×D411yescharacteristic
231C2×Dic61C2×#4111yescharacteristic
232C4×D61—21no—
233C4×D61—21no—
234C4×D61—21no—
235C4×D61—21no—
236C4×D61—41no—
237Dic122—21no—
238Dic122—21no—
239Dic122—21no—
240Dic122—21no—
241#92420—22no—
242#92420—21no—
243#92420—21no—
244#92420—21no—
245#92420—41no—
246#92420—41no—
247#92420—41no—
248#92420—41no—
249#92420—41no—
250#92420#13712yes—
251C8×C61—21no—
252C8×C61—21no—
253#92418—21no—
254#92418#13112yes—
255#18077—611nomaximal, Sylow
256C22×C61—41no—
257C22×C61—41no—
258C22×C61—41no—
259C22×C61C2×C4×D411yescharacteristic
260C2×D61—21no—
261C2×D61—21no—
262C2×D61—21no—
263C4×C61—21no—
264C4×C61—21no—
265C4×C61—41no—
266C4×C61—41no—
267C4×C61—41no—
268C4×C61C2×#13111yescharacteristic
269C4×C61#56811yescharacteristic
270Dic61—21no—
271Dic61—21no—
272Dic61—41no—
273Dic61—41no—
274Dic61—41no—
275Dic61—41no—
276Dic61C2×#13711yescharacteristic
277Dic61#57511yescharacteristic
278#3172—611no—
279#2969—611no—
280#2962—611no—
281#1592—611no—
282#1591—611no—
283#1588—611no—
284#1587—611no—
285#1584—611no—
286#1582—611no—
287#1579—611no—
288C8×SD16—611no—
289#1565—611no—
290#1485—611no—
291#1483—611no—
292#1455—611no—
293C2×C61—81no—
294C2×C61#293511yescharacteristic
295C2×C61C4×#13711yescharacteristic
296C2×C61#295611yescharacteristic
297D61—41no—
298#533—611no—
299#529—611no—
300#501—611no—
301#497—611no—
302#491—611no—
303#489#692311yescharacteristic
304#470—611no—
305#470—611no—
306#468—611no—
307#468—611no—
308#446—611no—
309#446—611no—
310#446—611no—
311#446—1221no—
312#446—1221no—
313C8×Q8—611no—
314C8×Q8—611no—
315C8×Q8—611no—
316C4×Q16—611no—
317C4×Q16—611no—
318C4×SD16—611no—
319C4×SD16—611no—
320#436—611no—
321#436—1221no—
322#435—1221no—
323C8×D4—611no—
324#433—1221no—
325#431—611no—
326#424—1221no—
327C2×C4⋊C8—611no—
328#405—1221no—
329#405—1221no—
330C4×M16—611no—
331C4×M16—611no—
332C4×M16—611no—
333#335—611no—
334#335—611no—
335#334—611no—
336#334—611no—
337Q8⋊C8—611no—
338Q8⋊C8—611no—
339Q8⋊C8—611no—
340Q8⋊C8—611no—
341Q8⋊C8—611no—
342Q8⋊C8—611no—
343D4⋊C8—611no—
344D4⋊C8—611no—
345#322—611no—
346#322—611no—
347#322—611no—
348C82—611no—
349C61#1807711yescharacteristic, complemented, Sylow
350C2×Q16—611no—
351C2×Q16—611no—
352C2×SD16—611no—
353C2×SD16—611no—
354C2×M16—611no—
355C2×M16—611no—
356C2×M16—611no—
357C2×M16—611no—
358C2×M16—1221no—
359C2×M16—1221no—
360C22×C8—1221no—
361C22×C8—1221no—
362C4⋊Q8—611no—
363C4⋊Q8C2×#692311yescharacteristic
364#126—1221no—
365#125—611no—
366#125—1221no—
367#124C2×#692311yescharacteristic
368#123—1221no—
369C22⋊Q8—611no—
370C22⋊Q8—611no—
371C22⋊Q8—611no—
372C22⋊Q8—1221no—
373#121—611no—
374C4×Q8—611no—
375C4×Q8—611no—
376C4×Q8—611no—
377C4×Q8—1221no—
378C4×D4—611no—
379C4×D4—1221no—
380#117—1221no—
381#117—1221no—
382C2×C42—611no—
383#107—611no—
384#107—611no—
385#106—611no—
386#106—611no—
387C4⋊C8—611no—
388C4⋊C8—611no—
389C4⋊C8—611no—
390C4⋊C8—611no—
391C4⋊C8—1221no—
392C4⋊C8—1221no—
393C4⋊C8—1221no—
394C4⋊C8—1221no—
395C4⋊C8—1221no—
396C4⋊C8—1221no—
397C4⋊C8—1221no—
398#103—611no—
399#103—611no—
400#103—611no—
401#103—611no—
402#103C2×#692311yescharacteristic
403#103C2×#692311yescharacteristic
404#102C2×#692311yescharacteristic
405#102C2×#692311yescharacteristic
406C22⋊C8—1221no—
407C22⋊C8—1221no—
408C22⋊C8—1221no—
409#97—611no—
410#97—611no—
411#97—611no—
412#97—1221no—
413#97—1221no—
414#97—1221no—
415#97—1221no—
416#97C2×#692311yescharacteristic
417C4×C8—611no—
418C4×C8—611no—
419C4×C8—611no—
420C4×C8—611no—
421C4×C8—611no—
422C4×C8—611no—
423C4×C8—1221no—
424C4×C8—1221no—
425C4×C8—1221no—
426C4×C8—1221no—
427C2×Q8—21no—
428C2×Q8—611no—
429C2×Q8—611no—
430C2×Q8C22×#692311yescharacteristic
431C2×D4—1221no—
432C2×D4C22×#692311yescharacteristic
433C22×C4—611no—
434C22×C4—611no—
435C22×C4—611no—
436C22×C4—1221no—
437Q16—1222no—
438Q16—2441no—
439SD16—1222no—
440SD16—2441no—
441M16—1222no—
442M16—1222no—
443M16—2441no—
444M16—2441no—
445M16—2442no—
446M16—2442no—
447C2×C8—611no—
448C2×C8—611no—
449C2×C8—611no—
450C2×C8—611no—
451C2×C8—611no—
452C2×C8—611no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—2442no—
460C2×C8—2442no—
461C2×C8C22×#692311yescharacteristic
462C2×C8C22×#692311yescharacteristic
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⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4C22×#692311yescharacteristic
477C4⋊C4C22×#692311yescharacteristic
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—1221no—
481C22⋊C4—1221no—
482C22⋊C4—1221no—
483C22⋊C4—1221no—
484C22⋊C4—1221no—
485C42—611no—
486C42—611no—
487C42—611no—
488C42—1221no—
489C42—1221no—
490C42—1221no—
491C42C22×#692311yescharacteristic
492C23—21no—
493C23—611no—
494Q8—21no—
495Q8—21no—
496Q8—81no—
497Q8—1221no—
498Q8—1221no—
499Q8—1221no—
500Q8—1221no—
501D4—21no—
502D4—21no—
503D4—4881no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—611no—
508C2×C4—611no—
509C2×C4—611no—
510C2×C4—1221no—
511C2×C4—1221no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—1221no—
515C2×C4—1221no—
516C2×C4—1221no—
517C2×C4—1221no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4—2441no—
521C2×C4D4×#692311yescharacteristic
522C2×C4D4×#692311yescharacteristic
523C2×C4C23×#692311yescharacteristic
524C8—21no—
525C8—21no—
526C8—612no—
527C8—612no—
528C8—1221no—
529C8—1221no—
530C8—1221no—
531C8—1222no—
532C8—1221no—
533C8—2441no—
534C8—2441no—
535C8—2441no—
536C8—2441no—
537C8—2441no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—1221no—
542C22—1221no—
543C22—1221no—
544C22C2×D4×#692311yescharacteristic
545C4—21no—
546C4—21no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—611no—
551C4—611no—
552C4—1221no—
553C4—1221no—
554C4—2441no—
555C4—2441no—
556C4—2441no—
557C4—2441no—
558C4D61⋊#13111yescharacteristic
559C4#1684084111yescharacteristic
560C2—81no—
561C2—2441no—
562C2#2146256411yescharacteristic
563C2#137×#692311yescharacteristic
564C2#2146259311yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#489C4
socleC22×C61C2×C4×D4
prime radical#14519748C2×C4
cosocleC4×Dic61C24
hypercenter#489#6923
hypocenterC61#18077
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
2. #21462615C2semidirect
3. #21461460C2semidirect
4. #21461457C2semidirect
5. #21461449C2semidirect
6. #21461442C2semidirect
7. #21461433C2semidirect
8. #21461422C2semidirect
9. #21461413C2semidirect
10. #21461402C2semidirect
11. #21461371C2semidirect
12. #21461143C2semidirect
14. #21461043C2semidirect
349. C61#18077semidirect

Maximal subgroups

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

subgroupindex
#214626512
#214626152
#214614602
#214614572
#214614492
#214614422
#214614332
#214614222
#214614132
#214614022
#214613712
#214611432
#214611352
#214610432
#214564282
#1807761