Finite Group Databasebeta

← #30611926

Subgroups — #30611926

Order 15616.

896 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462742C211yesmaximal, characteristic
2#21462516C211yesmaximal, characteristic
3#21461355C212yesmaximal, complemented
4#21461348C212yesmaximal, complemented
5#21461345C212yesmaximal, complemented
6#21461336C211yesmaximal, characteristic, complemented
7#21461335C211yesmaximal, characteristic, complemented
8#21461328C212yesmaximal, complemented
9#21461320C211yesmaximal, characteristic, complemented
10D61⋊#332C211yesmaximal, characteristic, complemented
11#21459633C211yesmaximal, characteristic
12#16840812—21no—
13#16840797—21no—
14#16840794—22no—
15#16840786—22no—
16#16840782C2212yes—
17#16840782C2212yes—
18#16840780C2211yescharacteristic
19#16840779C2212yes—
20#16840767C2211yescharacteristic
21#16840765C2212yes—
22D61⋊(C4⋊C8)C2211yescharacteristic
23D61⋊(C4⋊C8)C2211yescharacteristic
24#16840736—21no—
25#16840727—21no—
26#16840622C2211yescharacteristic
27#16840622C2211yescharacteristic
28#16840622C2212yes—
29#16840621C2212yes—
30#16840621C2212yes—
31#16840621C2211yescharacteristic
32#16840621C2211yescharacteristic
33#16840621C2212yes—
34#16840621C2212yes—
35#16840621C2212yes—
36Q8×Dic122C411yescharacteristic
37#16840492C411yescharacteristic
38#16840489—21no—
39#16840470C412yes—
40#16840469C411yescharacteristic
41#16840467C412yes—
42D4×Dic122—21no—
43#16840425—22no—
44#16840392—21no—
45#16840391—22no—
46#16840316C2211yescharacteristic
47#16840314C2212yes—
48C2×#14519776C2212yes—
49C4⋊Q8×D61C2211yescharacteristic
50#16840241—21no—
51C4×#11758224C2211yescharacteristic
52C61×#554C411yescharacteristic
53C2×#11758213—21no—
54C2×#11758217—22no—
55C2×D61⋊C8—21no—
56C2×D61⋊C8—21no—
57#14519844—21no—
58#14519844C2312yes—
59#14519844C2311yescharacteristic
60#14519844C2311yescharacteristic
61#14519844C2311yescharacteristic
62#14519844C2312yes—
63#14519844C2311yescharacteristic
64C4⋊#92420—21no—
65C4⋊#92420—21no—
66C4⋊#92420—22no—
67#14519834—21no—
68C22⋊#92420—22no—
69C22⋊#92420—22no—
70C22⋊#92420—22no—
71#14519832—22no—
72#14519832—22no—
73C4×#92420—22no—
74C4×#92420—21no—
75C2×#11758224D411yescharacteristic
76C22×Dic122—22no—
77C2×Q8×D61—21no—
78#14519777—21no—
79#14519777C2×C411yescharacteristic
80#14519776—21no—
81#14519776—22no—
82#14519776—21no—
83#14519776—22no—
84#14519776—22no—
85#14519776C2×C412yes—
86#14519776C2×C412yes—
87#14519776C2312yes—
88#14519776C2311yescharacteristic
89#14519775—22no—
90#14519775—21no—
91#14519775C2311yescharacteristic
92#14519768—21no—
93#14519767—22no—
94#14519767—22no—
95#14519767—22no—
96#14519767—21no—
97#14519757—22no—
98#14519757C2×C411yescharacteristic
99#14519756—22no—
100#14519756C2×C412yes—
101#14519755—21no—
102#14519755—22no—
103#14519754—22no—
104#14519754—22no—
105#14519754—22no—
106#14519753C2×C411yescharacteristic
107Q8×Dic61C2×C412yes—
108C4×Dic122—21no—
109C4×Dic122—21no—
110C4×Dic122C2×C411yescharacteristic
111C4×Dic122C2×C411yescharacteristic
112#14519742—21no—
113#14519742—22no—
114#14519742C2×C412yes—
115#14519742D411yescharacteristic
116D4×Dic61—21no—
117D4×Dic61D411yescharacteristic
118C4×C22⋊D61—22no—
119C4×C22⋊D61C2×C412yes—
120#14519737—22no—
121#14519737D412yes—
122#14519735D411yescharacteristic
123#14519734—21no—
124#14519733—22no—
125#14519733D412yes—
126C2×C4⋊Dic61C2×C412yes—
127C2×#11758196—22no—
128C4⋊C4×D61—21no—
129C4⋊C4×D61C2×C412yes—
130C2×C4×Dic61C2312yes—
131C42×D61C2311yescharacteristic
132C4⋊Q8×C61C2×C411yescharacteristic
133#125×C61—21no—
134C22⋊Q8×C61—22no—
135C22⋊Q8×C61C2×C412yes—
136C4×Q8×C61C2×C411yescharacteristic
137C4×D4×C61—21no—
138C4×D4×C61C2×C411yescharacteristic
139C2×C4⋊C4×C61C2×C412yes—
140C2×Q8×C61—21no—
141C2×Q8×C61C22×C411yescharacteristic
142C2×D4×C61C22⋊C411yescharacteristic
143C22×C4×C61—22no—
144C22×C4×C61C22×C412yes—
145C2×C22⋊D61C22⋊C412yes—
146C22×Dic61—22no—
147C22×Dic61C2×D412yes—
148C22×Dic61C2×D412yes—
149Q8×D61—81no—
150Q8×D61—81no—
151#11758224—22no—
152#11758224—22no—
153C2×C4×D61—22no—
154C2×C4×D61C22×C411yescharacteristic
155C2×C4×D61C2×D411yescharacteristic
156C2×C4×D61C2×D411yescharacteristic
157C2×Dic122—22no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—22no—
161C2×Dic122—22no—
162C2×Dic122—82no—
163C2×Dic122—82no—
164C2×Dic122C22⋊C411yescharacteristic
165C2×Dic122C22×C411yescharacteristic
166#11758217—82no—
167C2×#92420—21no—
168C2×#92420—21no—
169C2×#92420—22no—
170C2×#92420—21no—
171C2×#92420—21no—
172C2×#92420—22no—
173#11758213—42no—
174D61⋊C8—42no—
175D61⋊C8—81no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61C22⋊C411yescharacteristic
181C4⋊C4×C61C22×C411yescharacteristic
182C4⋊C4×C61C22×C412yes—
183C22⋊C4×C61—22no—
184C22⋊C4×C61C22⋊C412yes—
185C42×C61—21no—
186C42×C61C22×C411yescharacteristic
187C22⋊Dic61—22no—
188C22⋊Dic61C22⋊C412yes—
189#11758198—21no—
190#11758198—21no—
191#11758198C22⋊C412yes—
192C4⋊Dic61—21no—
193C4⋊Dic61—22no—
194C4⋊Dic61—21no—
195C4⋊Dic61—22no—
196C4⋊Dic61C22⋊C411yescharacteristic
197C4⋊Dic61C22×C412yes—
198C4⋊Dic61C22×C411yescharacteristic
199#11758196—22no—
200#11758196—21no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—21no—
206#11758196—22no—
207#11758196—21no—
208#11758196—21no—
209#11758196—21no—
210#11758196C22⋊C412yes—
211#11758196C22⋊C412yes—
212C4×Dic61—21no—
213C4×Dic61—22no—
214C4×Dic61—22no—
215C4×Dic61C22×C412yes—
216C4×Dic61C2×D412yes—
217C4×Dic61C2×D411yescharacteristic
218C4×Dic61C2×D412yes—
219C4×Dic61C2×D411yescharacteristic
220C4×Dic61C2411yescharacteristic
221C23×C61C2×C22⋊C412yes—
222C22×D61C2×C22⋊C411yescharacteristic
223Q8×C61—41no—
224Q8×C61—81no—
225D4×C61—22no—
226C2×C4×C61—21no—
227C2×C4×C61—22no—
228C2×C4×C61—21no—
229C2×C4×C61—42no—
230C2×C4×C61—82no—
231C2×C4×C61C2×C22⋊C412yes—
232C2×C4×C61C2×C22⋊C411yescharacteristic
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C2×C4×C61C23×C411yescharacteristic
235C22⋊D61—24no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—22no—
243C2×Dic61—82no—
244C2×Dic61C2×C22⋊C412yes—
245C2×Dic61C2×C22⋊C412yes—
246C2×Dic61C2×C22⋊C411yescharacteristic
247C2×Dic61C22×D411yescharacteristic
248C2×Dic61C22×D411yescharacteristic
249C4×D61—21no—
250C4×D61—21no—
251C4×D61—41no—
252C4×D61—82no—
253C4×D61#13712yes—
254C4×D61#13712yes—
255Dic122—22no—
256Dic122—41no—
257Dic122—42no—
258Dic122—41no—
259Dic122—42no—
260Dic122—41no—
261Dic122—82no—
262Dic122—82no—
263Dic122—81no—
264#92420—41no—
265#92420—41no—
266#92420—42no—
267#92420—42no—
268#92420—41no—
269#92420—41no—
270#17761—611nomaximal, Sylow
271C22×C61—22no—
272C22×C61—22no—
273C22×C61—22no—
274C22×C61C22×C22⋊C411yescharacteristic
275C2×D61—21no—
276C2×D61—21no—
277C2×D61#41912yes—
278C4×C61—21no—
279C4×C61—22no—
280C4×C61—21no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—42no—
284C4×C61#41912yes—
285Dic61—21no—
286Dic61—22no—
287Dic61—22no—
288Dic61—41no—
289Dic61—42no—
290Dic61—41no—
291Dic61C2×#13711yescharacteristic
292Dic61C2×#13711yescharacteristic
293Dic61#57611yescharacteristic
294Dic61#57711yescharacteristic
295#3503—611no—
296#2989—611no—
297#2925—611no—
298#1559—612no—
299#1556—612no—
300#1554—612no—
301#1550—611no—
302#1549—611no—
303#1546—612no—
304#1542—611no—
305C2×#332—611no—
306C2×C61—42no—
307C2×C61#288911yescharacteristic
308C2×C61C2×#41911yescharacteristic
309C2×C61#289711yescharacteristic
310D61—22no—
311Q82—611no—
312Q82—611no—
313#557—1221no—
314#554—612no—
315#554—611no—
316#554—612no—
317#554#692311yescharacteristic
318D4×Q8—1221no—
319D4×Q8—1222no—
320#544—1222no—
321#544—1221no—
322#533—612no—
323#533—611no—
324C2×C4⋊Q8—611no—
325C2×C4⋊Q8—612no—
326#520—1221no—
327C4×#41—611no—
328#446—1221no—
329#436—1221no—
330#435—1222no—
331#433—1222no—
332#432—612no—
333#432—612no—
334#431—611no—
335#431—612no—
336#423—612no—
337#423—611no—
338C2×C4⋊C8—611no—
339C2×C4⋊C8—611no—
340#408—1221no—
341#405—1221no—
342#333—611no—
343#333—611no—
344#333—612no—
345#332—612no—
346#332—612no—
347#332—611no—
348#332—611no—
349#332—612no—
350#332—612no—
351#332—612no—
352C61#1776111yescharacteristic, complemented, Sylow
353C2×#41—611no—
354C22×Q8—1222no—
355C22×Q8—1221no—
356C2×M16—1222no—
357C2×M16—1221no—
358C22×C8—1221no—
359C22×C8—1221no—
360C4⋊Q8—612no—
361C4⋊Q8—611no—
362C4⋊Q8—612no—
363C4⋊Q8—612no—
364C4⋊Q8—611no—
365C4⋊Q8—611no—
366C4⋊Q8—1221no—
367C4⋊Q8—1222no—
368C4⋊Q8—1221no—
369C4⋊Q8—1222no—
370C4⋊Q8—1221no—
371C4⋊Q8—1222no—
372C4⋊Q8—1221no—
373C4⋊Q8—1222no—
374C4⋊Q8C2×#692311yescharacteristic
375#125—21no—
376#125—1222no—
377#125—1221no—
378#125—1222no—
379#125—1222no—
380#125—1221no—
381C22⋊Q8—22no—
382C22⋊Q8—611no—
383C22⋊Q8—612no—
384C22⋊Q8—611no—
385C22⋊Q8—1222no—
386C22⋊Q8—1222no—
387C22⋊Q8—1222no—
388C22⋊Q8—1222no—
389C22⋊Q8—1222no—
390C22⋊Q8—1221no—
391C22⋊Q8—1222no—
392C22⋊Q8C2×#692312yes—
393C4×Q8—611no—
394C4×Q8—611no—
395C4×Q8—612no—
396C4×Q8—612no—
397C4×Q8—611no—
398C4×Q8—1221no—
399C4×Q8—1221no—
400C4×Q8—1222no—
401C4×Q8—1221no—
402C4×Q8C2×#692311yescharacteristic
403C4×D4—21no—
404C4×D4—611no—
405C4×D4—612no—
406C4×D4—612no—
407C4×D4—1222no—
408C4×D4—1222no—
409C4×D4—1221no—
410C4×D4C2×#692311yescharacteristic
411#117—611no—
412#117—612no—
413#117—1221no—
414#117—1222no—
415C2×C4⋊C4—612no—
416C2×C4⋊C4—612no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1222no—
419C2×C4⋊C4C2×#692312yes—
420C2×C42—612no—
421C2×C42—611no—
422C4⋊C8—612no—
423C4⋊C8—611no—
424C4⋊C8—611no—
425C4⋊C8—611no—
426C4⋊C8—612no—
427C4⋊C8—611no—
428C4⋊C8—1221no—
429C4⋊C8—1221no—
430C4⋊C8—1222no—
431C4⋊C8—1221no—
432C22⋊C8—1222no—
433C22⋊C8—1222no—
434C22⋊C8—1222no—
435C22⋊C8—1221no—
436#97—1222no—
437#97—1222no—
438C4×C8—1222no—
439C4×C8—1221no—
440#41—1222no—
441#41—1222no—
442C2×Q8—21no—
443C2×Q8—611no—
444C2×Q8—611no—
445C2×Q8—1222no—
446C2×Q8—1222no—
447C2×Q8—1221no—
448C2×Q8—1222no—
449C2×Q8—1221no—
450C2×Q8—4882no—
451C2×Q8—4882no—
452C2×Q8—4881no—
453C2×Q8—4881no—
454C2×Q8C22×#692311yescharacteristic
455C2×D4—612no—
456C2×D4C22⋊#692311yescharacteristic
457C22×C4—22no—
458C22×C4—611no—
459C22×C4—612no—
460C22×C4—612no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4C22×#692312yes—
466M16—4882no—
467M16—2442no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—1222no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1222no—
474C2×C8—4881no—
475C2×C8—2442no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—612no—
481C4⋊C4—612no—
482C4⋊C4—611no—
483C4⋊C4—611no—
484C4⋊C4—612no—
485C4⋊C4—1222no—
486C4⋊C4—1221no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C4⋊C4—1222no—
490C4⋊C4—1221no—
491C4⋊C4—1222no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1222no—
495C4⋊C4—1221no—
496C4⋊C4—1222no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1222no—
500C4⋊C4C22⋊#692311yescharacteristic
501C4⋊C4C22×#692312yes—
502C4⋊C4C22×#692311yescharacteristic
503C22⋊C4—22no—
504C22⋊C4—612no—
505C22⋊C4—612no—
506C22⋊C4—1221no—
507C22⋊C4—1221no—
508C22⋊C4—1222no—
509C22⋊C4C22⋊#692312yes—
510C42—21no—
511C42—612no—
512C42—611no—
513C42—612no—
514C42—611no—
515C42—611no—
516C42—612no—
517C42—1222no—
518C42—1222no—
519C42—1221no—
520C42C22×#692311yescharacteristic
521C23—611no—
522C23C2×C22⋊#692312yes—
523Q8—41no—
524Q8—81no—
525Q8—1222no—
526Q8—4881no—
527Q8—4882no—
528Q8—4882no—
529Q8—2441no—
530Q8—2441no—
531Q8—2441no—
532Q8—2442no—
533Q8—2442no—
534D4—22no—
535D4—1224no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—42no—
540C2×C4—82no—
541C2×C4—611no—
542C2×C4—611no—
543C2×C4—612no—
544C2×C4—612no—
545C2×C4—612no—
546C2×C4—611no—
547C2×C4—612no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—1222no—
553C2×C4—1222no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1222no—
557C2×C4—4882no—
558C2×C4—4882no—
559C2×C4—2441no—
560C2×C4C2×C22⋊#692312yes—
561C2×C4C2×C22⋊#692311yescharacteristic
562C2×C4C2×C22⋊#692311yescharacteristic
563C2×C4C23×#692311yescharacteristic
564C8—2441no—
565C8—2442no—
566C8—2441no—
567C8—2441no—
568C8—2441no—
569C8—2442no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—612no—
574C22—1221no—
575C22—1221no—
576C22C22×C22⋊#692311yescharacteristic
577C4—21no—
578C4—22no—
579C4—21no—
580C4—41no—
581C4—41no—
582C4—42no—
583C4—611no—
584C4—611no—
585C4—611no—
586C4—611no—
587C4—1221no—
588C4—1222no—
589C4—1222no—
590C4—2441no—
591C4—2442no—
592C4—2441no—
593C4#1684075512yes—
594C2—42no—
595C2—1222no—
596C2#2146243111yescharacteristic
597C2C2×#1684075511yescharacteristic
598C2#2146244011yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#554C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#11758224D4
cosocleC4×Dic61C24
hypercenter#554#6923
hypocenterC61#17761
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
3. #21461355C2semidirect
4. #21461348C2semidirect
5. #21461345C2semidirect
6. #21461336C2semidirect
7. #21461335C2semidirect
8. #21461328C2semidirect
9. #21461320C2semidirect
10. D61⋊#332C2semidirect
352. C61#17761semidirect

Maximal subgroups

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

subgroupindex
#214627422
#214625162
#214613552
#214613482
#214613452
#214613362
#214613352
#214613282
#214613202
D61⋊#3322
#214596332
#1776161