Finite Group Databasebeta

← #30611981

Subgroups — #30611981

Order 15616.

868 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462751C211yesmaximal, characteristic
2#21462745C211yesmaximal, characteristic
3#21461358C211yesmaximal, characteristic, complemented
4#21461355C212yesmaximal, complemented
5#21461345C212yesmaximal, complemented
6#21461337C212yesmaximal, complemented
7#21461332C211yesmaximal, characteristic, complemented
8#21461332C211yesmaximal, characteristic, complemented
9#21461324C211yesmaximal, characteristic, complemented
10#21461321C212yesmaximal, complemented
11#21459704C211yesmaximal, characteristic
12#16840813—21no—
13Q8×#92420—21no—
14#16840797—22no—
15#16840797—21no—
16#16840793—22no—
17D4×#92420—21no—
18#16840782C2212yes—
19#16840779C2212yes—
20#16840779C2211yescharacteristic
21#16840779C2211yescharacteristic
22#16840765C2211yescharacteristic
23#16840765C2211yescharacteristic
24#16840765C2212yes—
25C2×#14519844C2212yes—
26#16840622C2212yes—
27#16840622C2212yes—
28#16840622C2212yes—
29#16840622C2212yes—
30#16840621C2211yescharacteristic
31#16840621C2212yes—
32#16840621C2211yescharacteristic
33#16840621C2211yescharacteristic
34#16840621C2211yescharacteristic
35#16840621C2212yes—
36#16840487—21no—
37#16840487—21no—
38#16840481—21no—
39#16840481—21no—
40#16840449—22no—
41#16840449—22no—
42#16840372C412yes—
43#16840372C412yes—
44#16840333C411yescharacteristic
45#16840315C2211yescharacteristic
46#16840314C2212yes—
47#16840314C2212yes—
48C2×#14519775C2211yescharacteristic
49#16840240C411yescharacteristic
50#16840239C411yescharacteristic
51#41×Dic61C2211yescharacteristic
52C61×#536C411yescharacteristic
53C2×#11758217—21no—
54C2×#11758217—21no—
55C2×#11758217—22no—
56C22×#92420—22no—
57#14519844—22no—
58#14519844C2311yescharacteristic
59#14519844C2312yes—
60#14519844C2312yes—
61#14519844C2311yescharacteristic
62#14519844C2311yescharacteristic
63#14519844C2311yescharacteristic
64C4⋊#92420—21no—
65C4⋊#92420—22no—
66C4⋊#92420—21no—
67C22⋊#92420—21no—
68C22⋊#92420—21no—
69C22⋊#92420—22no—
70C22⋊#92420—22no—
71#14519832—22no—
72C4×#92420—21no—
73C4×#92420—22no—
74C4×#92420—21no—
75C22×Dic122C2×C411yescharacteristic
76#14519776C2311yescharacteristic
77#14519776C2311yescharacteristic
78#14519775—21no—
79#14519775—21no—
80#14519775—21no—
81#14519775C2×C412yes—
82#14519775C2312yes—
83#14519770—22no—
84#14519770—22no—
85#14519770—22no—
86#14519770—22no—
87#14519768—22no—
88#14519768—22no—
89#14519768C2×C411yescharacteristic
90#14519768C2×C411yescharacteristic
91#14519767—22no—
92#14519767—22no—
93#14519767—22no—
94#14519767—22no—
95#14519764—21no—
96#14519758—22no—
97#14519756C2×C412yes—
98#14519756C2×C412yes—
99#14519754—21no—
100#14519754—21no—
101#14519754—22no—
102#14519754—22no—
103Q8×Dic61—21no—
104Q8×Dic61—21no—
105Q8×Dic61C2×C412yes—
106Q8×Dic61C2×C412yes—
107Q8×Dic61D411yescharacteristic
108Q8×Dic61D411yescharacteristic
109C4×Dic122—21no—
110C4×Dic122—21no—
111C4×Dic122—21no—
112C4×Dic122—21no—
113#14519742—22no—
114#14519742—22no—
115#14519742C2×C412yes—
116#14519742C2×C412yes—
117D4×Dic61D412yes—
118D4×Dic61D412yes—
119D4×Dic61D411yescharacteristic
120D4×Dic61D411yescharacteristic
121#14519736—22no—
122#14519736C2×C411yescharacteristic
123#14519736C2×C411yescharacteristic
124#14519736C2×C412yes—
125#14519733—22no—
126#14519733—22no—
127#14519733—22no—
128C2×#11758196—21no—
129C2×#11758196—21no—
130C2×C4×Dic61C2312yes—
131C2×C4×Dic61C2311yescharacteristic
132C2×#41×C61C2×C411yescharacteristic
133C22×Q8×C61C2×C411yescharacteristic
134C4⋊Q8×C61—21no—
135#124×C61—21no—
136#123×C61—22no—
137C22⋊Q8×C61C2×C412yes—
138C22⋊Q8×C61C2×C412yes—
139#117×C61C2×C411yescharacteristic
140#41×C61—42no—
141C2×Q8×C61—21no—
142C2×Q8×C61—42no—
143C2×Q8×C61C22⋊C411yescharacteristic
144C2×Q8×C61C22×C412yes—
145C2×D4×C61C22⋊C411yescharacteristic
146C2×D4×C61C22⋊C412yes—
147C22×C4×C61—21no—
148C22×C4×C61C22×C411yescharacteristic
149C22×C4×C61C22×C412yes—
150C22×Dic61—21no—
151C22×Dic61C2×D412yes—
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D412yes—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—42no—
159C2×Dic122C22×C412yes—
160#11758217—81no—
161#11758217—81no—
162#11758217—82no—
163C2×#92420—21no—
164C2×#92420—21no—
165C2×#92420—22no—
166C2×#92420—21no—
167C2×#92420—21no—
168C2×#92420—22no—
169C2×#92420—82no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61C22×C411yescharacteristic
173C4⋊C4×C61C22×C411yescharacteristic
174C22⋊C4×C61—21no—
175C22⋊C4×C61—22no—
176C22⋊C4×C61—22no—
177C42×C61—21no—
178C22⋊Dic61—22no—
179C22⋊Dic61—22no—
180C22⋊Dic61—21no—
181C22⋊Dic61C22⋊C412yes—
182C22⋊Dic61C22⋊C412yes—
183C22⋊Dic61C22⋊C412yes—
184C4⋊Dic61—22no—
185C4⋊Dic61—22no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61C22⋊C411yescharacteristic
189C4⋊Dic61C22⋊C411yescharacteristic
190C4⋊Dic61C22⋊C412yes—
191C4⋊Dic61C22⋊C412yes—
192C4⋊Dic61C22×C411yescharacteristic
193C4⋊Dic61C22×C411yescharacteristic
194#11758196—22no—
195#11758196—22no—
196#11758196—22no—
197#11758196—22no—
198#11758196—22no—
199#11758196—22no—
200#11758196—21no—
201#11758196—21no—
202#11758196—22no—
203#11758196—22no—
204#11758196—21no—
205#11758196—21no—
206C4×Dic61—21no—
207C4×Dic61—22no—
208C4×Dic61—22no—
209C4×Dic61—22no—
210C4×Dic61—22no—
211C4×Dic61C22×C412yes—
212C4×Dic61C22×C411yescharacteristic
213C4×Dic61C2×D412yes—
214C4×Dic61C2×D411yescharacteristic
215C4×Dic61C2×D411yescharacteristic
216C4×Dic61C2×D412yes—
217C4×Dic61C2411yescharacteristic
218C23×C61C2×C22⋊C412yes—
219C23×C61C2×C22⋊C411yescharacteristic
220Q8×C61—22no—
221Q8×C61—42no—
222Q8×C61—42no—
223D4×C61—22no—
224D4×C61—42no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61C2×C22⋊C412yes—
232C2×C4×C61C2×C22⋊C411yescharacteristic
233C2×C4×C61C23×C411yescharacteristic
234C2×Dic61—22no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—22no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—22no—
245C2×Dic61—42no—
246C2×Dic61C2×C22⋊C412yes—
247C2×Dic61C2×C22⋊C412yes—
248C2×Dic61C2×C22⋊C411yescharacteristic
249C2×Dic61C2×C22⋊C411yescharacteristic
250C2×Dic61C22×D411yescharacteristic
251C2×Dic61C22×D411yescharacteristic
252Dic122—42no—
253Dic122—41no—
254Dic122—41no—
255Dic122—41no—
256Dic122—41no—
257Dic122—42no—
258#92420—41no—
259#92420—42no—
260#92420—42no—
261#92420—41no—
262#92420—41no—
263#92420—41no—
264#17778—611nomaximal, Sylow
265C22×C61—21no—
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—22no—
269C22×C61—21no—
270C22×C61—21no—
271C22×C61C22×C22⋊C411yescharacteristic
272C4×C61—21no—
273C4×C61—22no—
274C4×C61—42no—
275C4×C61—41no—
276C4×C61—41no—
277C4×C61#41312yes—
278Dic61—22no—
279Dic61—22no—
280Dic61—21no—
281Dic61—21no—
282Dic61—42no—
283Dic61—41no—
284Dic61—41no—
285Dic61#57711yescharacteristic
286Dic61#57711yescharacteristic
287Dic61#57811yescharacteristic
288Dic61#57811yescharacteristic
289#3509—611no—
290#2991—611no—
291#2990—611no—
292#1561—611no—
293#1559—612no—
294#1554—612no—
295#1550—612no—
296#1548—611no—
297#1548—611no—
298#1544—611no—
299#1542—612no—
300C2×C61—41no—
301C2×C61—42no—
302C2×C61C2×#41311yescharacteristic
303C2×C61#289811yescharacteristic
304C2×C61#289811yescharacteristic
305#557—1221no—
306#557—1221no—
307#556—1221no—
308#556—1221no—
309#552—1222no—
310#552—1222no—
311#541—612no—
312#541—612no—
313#536—611no—
314#536#692311yescharacteristic
315#533—611no—
316#533—612no—
317#533—612no—
318C2×C4⋊Q8—611no—
319#519—611no—
320#519—611no—
321C4×#41—611no—
322#446—1221no—
323C8×Q8—1221no—
324#436—1222no—
325#436—1221no—
326#435—1222no—
327C8×D4—1221no—
328#432—612no—
329#431—612no—
330#431—611no—
331#431—611no—
332#423—611no—
333#423—611no—
334#423—612no—
335C2×C4⋊C8—612no—
336#333—612no—
337#333—612no—
338#333—612no—
339#333—612no—
340#332—611no—
341#332—612no—
342#332—611no—
343#332—611no—
344#332—611no—
345#332—612no—
346C61#1777811yescharacteristic, complemented, Sylow
347C2×#41C2×#692311yescharacteristic
348C22×Q8—611no—
349C22×Q8C2×#692311yescharacteristic
350C2×M16—1221no—
351C2×M16—1221no—
352C2×M16—1222no—
353C22×C8—1222no—
354C4⋊Q8—21no—
355C4⋊Q8—611no—
356C4⋊Q8—611no—
357C4⋊Q8—612no—
358C4⋊Q8—612no—
359C4⋊Q8—1221no—
360C4⋊Q8—1221no—
361C4⋊Q8—1221no—
362#126—1222no—
363#126—1222no—
364#126—1222no—
365#126—1222no—
366#125—611no—
367#125—611no—
368#125—1222no—
369#125—1222no—
370#125—1222no—
371#125—1222no—
372#125—1222no—
373#125—1222no—
374#124—21no—
375#124—1221no—
376#123—22no—
377#123—1222no—
378C22⋊Q8—612no—
379C22⋊Q8—612no—
380C22⋊Q8—1221no—
381C22⋊Q8—1221no—
382C22⋊Q8—1222no—
383C22⋊Q8—1222no—
384C22⋊Q8C2×#692312yes—
385C22⋊Q8C2×#692312yes—
386C4×Q8—611no—
387C4×Q8—611no—
388C4×Q8—612no—
389C4×Q8—612no—
390C4×Q8—612no—
391C4×Q8—612no—
392C4×Q8—1221no—
393C4×Q8—1221no—
394C4×Q8—1221no—
395C4×Q8—1221no—
396C4×Q8—1222no—
397C4×Q8—1222no—
398C4×Q8—1221no—
399C4×Q8—1221no—
400C4×D4—612no—
401C4×D4—612no—
402C4×D4—611no—
403C4×D4—611no—
404#117—611no—
405#117—612no—
406#117—611no—
407#117—1222no—
408#117—1222no—
409#117—1222no—
410#117—1222no—
411#117C2×#692311yescharacteristic
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C42—611no—
415C2×C42—612no—
416C4⋊C8—611no—
417C4⋊C8—612no—
418C4⋊C8—612no—
419C4⋊C8—611no—
420C4⋊C8—611no—
421C4⋊C8—611no—
422C4⋊C8—1221no—
423C4⋊C8—1222no—
424C4⋊C8—1221no—
425C4⋊C8—1222no—
426C22⋊C8—1221no—
427C22⋊C8—1221no—
428C22⋊C8—1222no—
429C22⋊C8—1222no—
430#97—1222no—
431C4×C8—1221no—
432C4×C8—1222no—
433C4×C8—1221no—
434#41—42no—
435C2×Q8—21no—
436C2×Q8—42no—
437C2×Q8—612no—
438C2×Q8—1221no—
439C2×Q8—1221no—
440C2×Q8—1221no—
441C2×Q8—2442no—
442C2×Q8C22⋊#692311yescharacteristic
443C2×Q8C22×#692312yes—
444C2×D4C22⋊#692311yescharacteristic
445C2×D4C22⋊#692312yes—
446C22×C4—21no—
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—612no—
450C22×C4—612no—
451C22×C4—1221no—
452C22×C4C22×#692311yescharacteristic
453C22×C4C22×#692312yes—
454M16—4881no—
455M16—4881no—
456M16—4882no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1222no—
461C2×C8—1221no—
462C2×C8—1222no—
463C2×C8—4882no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—611no—
467C4⋊C4—611no—
468C4⋊C4—612no—
469C4⋊C4—612no—
470C4⋊C4—611no—
471C4⋊C4—611no—
472C4⋊C4—1222no—
473C4⋊C4—1222no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1222no—
479C4⋊C4—1222no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1222no—
487C4⋊C4—1222no—
488C4⋊C4C22×#692311yescharacteristic
489C4⋊C4C22×#692311yescharacteristic
490C22⋊C4—21no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—612no—
494C22⋊C4—612no—
495C22⋊C4—612no—
496C22⋊C4—1221no—
497C22⋊C4—1222no—
498C22⋊C4—1222no—
499C42—21no—
500C42—612no—
501C42—611no—
502C42—611no—
503C42—612no—
504C42—611no—
505C42—612no—
506C42—611no—
507C42—1222no—
508C42—1222no—
509C42—1222no—
510C42—1222no—
511C42—1221no—
512C23C2×C22⋊#692312yes—
513C23C2×C22⋊#692311yescharacteristic
514Q8—22no—
515Q8—42no—
516Q8—42no—
517Q8—2442no—
518Q8—2441no—
519Q8—2441no—
520Q8—2441no—
521Q8—2441no—
522Q8—2442no—
523D4—22no—
524D4—42no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—611no—
532C2×C4—611no—
533C2×C4—612no—
534C2×C4—612no—
535C2×C4—611no—
536C2×C4—611no—
537C2×C4—1222no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1222no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1222no—
548C2×C4—2442no—
549C2×C4C2×C22⋊#692312yes—
550C2×C4C2×C22⋊#692311yescharacteristic
551C2×C4C23×#692311yescharacteristic
552C8—2441no—
553C8—2441no—
554C8—2441no—
555C8—2442no—
556C8—2442no—
557C8—2441no—
558C22—21no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—21no—
563C22—21no—
564C22C22×C22⋊#692311yescharacteristic
565C4—21no—
566C4—22no—
567C4—42no—
568C4—41no—
569C4—41no—
570C4—611no—
571C4—611no—
572C4—611no—
573C4—611no—
574C4—1222no—
575C4—1222no—
576C4—1221no—
577C4—1221no—
578C4—2442no—
579C4—2441no—
580C4—2441no—
581C4#1684074912yes—
582C2—41no—
583C2—42no—
584C2C2×#1684074911yescharacteristic
585C2#2146244111yescharacteristic
586C2#2146244111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#536C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#41×C61C2×C4
cosocleC4×Dic61C24
hypercenter#536#6923
hypocenterC61#17778
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
3. #21461358C2semidirect
4. #21461355C2semidirect
5. #21461345C2semidirect
6. #21461337C2semidirect
7. #21461332C2semidirect
8. #21461332C2semidirect
9. #21461324C2semidirect
10. #21461321C2semidirect
346. C61#17778semidirect

Maximal subgroups

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

subgroupindex
#214627512
#214627452
#214613582
#214613552
#214613452
#214613372
#214613322
#214613242
#214613212
#214597042
#1777861