Finite Group Databasebeta

← #30611967

Subgroups — #30611967

Order 15616.

904 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462745C211yesmaximal, characteristic
2#21462745C211yesmaximal, characteristic
3#21461358C211yesmaximal, characteristic, complemented
4#21461348C212yesmaximal, complemented
5#21461340C212yesmaximal, complemented
6#21461340C212yesmaximal, complemented
7#21461334C211yesmaximal, characteristic, complemented
8#21461328C211yesmaximal, characteristic, complemented
9#21461324C211yesmaximal, characteristic, complemented
10C2×#16840621C212yesmaximal, complemented
11#21459632C211yesmaximal, characteristic
12#16840813—21no—
13#16840813—21no—
14#16840793—22no—
15#16840793—22no—
16D4×#92420—21no—
17D4×#92420—21no—
18#16840782C2212yes—
19#16840782C2212yes—
20#16840779C2211yescharacteristic
21#16840779C2211yescharacteristic
22#16840765C2211yescharacteristic
23#16840765C2211yescharacteristic
24C2×#14519844C2212yes—
25C2×#14519844C2212yes—
26#16840621C2211yescharacteristic
27#16840621C2212yes—
28#16840621C2212yes—
29#16840621C2212yes—
30#16840621C2211yescharacteristic
31#16840621C2211yescharacteristic
32#16840621C2211yescharacteristic
33#16840621C2212yes—
34#16840621C2212yes—
35#16840621C2212yes—
36Q8×Dic122—21no—
37#16840488—21no—
38#16840471—21no—
39#16840471—22no—
40D4×Dic122—21no—
41D4×Dic122—22no—
42#16840393C412yes—
43#16840393C411yescharacteristic
44#16840390C411yescharacteristic
45#16840390C412yes—
46#16840315C2211yescharacteristic
47#16840315C2212yes—
48C2×#14519776C2212yes—
49C2×#14519776C2211yescharacteristic
50#16840245C411yescharacteristic
51#41×Dic61C2211yescharacteristic
52C61×#520C411yescharacteristic
53C2×#11758217—21no—
54C2×#11758217—21no—
55C22×#92420—22no—
56C22×#92420—22no—
57#14519844—22no—
58#14519844—22no—
59#14519844C2311yescharacteristic
60#14519844C2311yescharacteristic
61#14519844C2312yes—
62#14519844C2312yes—
63#14519844C2311yescharacteristic
64#14519844C2311yescharacteristic
65C4⋊#92420—21no—
66C4⋊#92420—21no—
67C22⋊#92420—21no—
68C22⋊#92420—21no—
69C22⋊#92420—22no—
70C22⋊#92420—22no—
71#14519832—22no—
72#14519832—22no—
73C4×#92420—21no—
74C4×#92420—21no—
75C22×Dic122—22no—
76C22×Dic122—21no—
77#14519777—22no—
78#14519777—21no—
79#14519776—22no—
80#14519776—21no—
81#14519776—21no—
82#14519776—22no—
83#14519776—21no—
84#14519776—22no—
85#14519776C2×C412yes—
86#14519776C2×C412yes—
87#14519776C2×C412yes—
88#14519776C2311yescharacteristic
89#14519776C2312yes—
90#14519776C2311yescharacteristic
91#14519769—22no—
92#14519769—21no—
93#14519768—22no—
94#14519768—21no—
95#14519768C2×C411yescharacteristic
96#14519768C2×C412yes—
97#14519768C2×C411yescharacteristic
98#14519768C2×C412yes—
99#14519756—22no—
100#14519756—22no—
101#14519756—22no—
102#14519754—22no—
103#14519754—21no—
104#14519754—22no—
105#14519754—21no—
106#14519754—22no—
107#14519754—22no—
108#14519754—22no—
109Q8×Dic61—21no—
110Q8×Dic61D411yescharacteristic
111Q8×Dic61D411yescharacteristic
112C4×Dic122—21no—
113C4×Dic122—22no—
114C4×Dic122—21no—
115C4×Dic122—22no—
116D4×Dic61—22no—
117D4×Dic61—21no—
118D4×Dic61D411yescharacteristic
119D4×Dic61D412yes—
120D4×Dic61D411yescharacteristic
121D4×Dic61D412yes—
122#14519736C2×C411yescharacteristic
123#14519736C2×C412yes—
124#14519736C2×C411yescharacteristic
125#14519736C2×C412yes—
126C2×C4⋊Dic61C2×C412yes—
127C2×C4⋊Dic61C2×C411yescharacteristic
128C2×#11758196—22no—
129C2×#11758196—21no—
130C2×C4×Dic61C2312yes—
131C2×C4×Dic61C2311yescharacteristic
132C2×#41×C61C2×C411yescharacteristic
133C4×Q8×C61—21no—
134C4×D4×C61—22no—
135C4×D4×C61—21no—
136#117×C61C2×C411yescharacteristic
137#117×C61C2×C412yes—
138C2×C4⋊C4×C61C2×C412yes—
139C2×C4⋊C4×C61C2×C411yescharacteristic
140#41×C61—24no—
141C2×Q8×C61C22⋊C411yescharacteristic
142C2×D4×C61C22⋊C412yes—
143C2×D4×C61C22⋊C411yescharacteristic
144C22×C4×C61—21no—
145C22×C4×C61—22no—
146C22×C4×C61C22×C412yes—
147C22×C4×C61C22×C411yescharacteristic
148C22×Dic61—21no—
149C22×Dic61—22no—
150C22×Dic61C2×D412yes—
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D412yes—
153C22×Dic61C2×D411yescharacteristic
154C2×Dic122—21no—
155C2×Dic122—22no—
156C2×Dic122—21no—
157C2×Dic122—22no—
158C2×Dic122—22no—
159C2×Dic122—22no—
160C2×Dic122—82no—
161C2×Dic122—81no—
162C2×Dic122—82no—
163C2×Dic122—81no—
164#11758217—81no—
165#11758217—81no—
166C2×#92420—21no—
167C2×#92420—21no—
168C2×#92420—21no—
169C2×#92420—21no—
170C2×#92420—22no—
171C2×#92420—22no—
172C2×#92420—82no—
173C2×#92420—82no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61C22×C411yescharacteristic
177C4⋊C4×C61C22×C412yes—
178C4⋊C4×C61C22×C411yescharacteristic
179C22⋊C4×C61—21no—
180C22⋊C4×C61—22no—
181C42×C61—22no—
182C42×C61—21no—
183C22⋊Dic61—21no—
184C22⋊Dic61—22no—
185C22⋊Dic61C22⋊C412yes—
186C22⋊Dic61C22⋊C412yes—
187C22⋊Dic61C22⋊C412yes—
188C4⋊Dic61—22no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—22no—
192C4⋊Dic61—21no—
193C4⋊Dic61—22no—
194C4⋊Dic61C22⋊C412yes—
195C4⋊Dic61C22⋊C411yescharacteristic
196C4⋊Dic61C22⋊C412yes—
197C4⋊Dic61C22⋊C411yescharacteristic
198C4⋊Dic61C22×C411yescharacteristic
199C4⋊Dic61C22×C412yes—
200C4⋊Dic61C22×C411yescharacteristic
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—21no—
205#11758196—22no—
206#11758196—21no—
207#11758196—22no—
208#11758196—21no—
209#11758196—22no—
210#11758196—21no—
211C4×Dic61—22no—
212C4×Dic61—21no—
213C4×Dic61C22×C412yes—
214C4×Dic61C22×C411yescharacteristic
215C4×Dic61C2×D411yescharacteristic
216C4×Dic61C2×D411yescharacteristic
217C4×Dic61C2×D412yes—
218C4×Dic61C2×D412yes—
219C4×Dic61C2411yescharacteristic
220C23×C61C2×C22⋊C411yescharacteristic
221C23×C61C2×C22⋊C412yes—
222Q8×C61—22no—
223D4×C61—24no—
224D4×C61#10114yes—
225C2×C4×C61—24no—
226C2×C4×C61—22no—
227C2×C4×C61—21no—
228C2×C4×C61—22no—
229C2×C4×C61—21no—
230C2×C4×C61—82no—
231C2×C4×C61—81no—
232C2×C4×C61C2×C22⋊C412yes—
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C2×C4×C61C23×C411yescharacteristic
235C2×Dic61—22no—
236C2×Dic61—21no—
237C2×Dic61—22no—
238C2×Dic61—21no—
239C2×Dic61—22no—
240C2×Dic61—21no—
241C2×Dic61—22no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—22no—
245C2×Dic61—21no—
246C2×Dic61—82no—
247C2×Dic61—81no—
248C2×Dic61C2×C22⋊C411yescharacteristic
249C2×Dic61C2×C22⋊C412yes—
250C2×Dic61C2×C22⋊C411yescharacteristic
251C2×Dic61C2×C22⋊C412yes—
252C2×Dic61C22×D411yescharacteristic
253C2×Dic61C22×D411yescharacteristic
254Dic122—41no—
255Dic122—42no—
256Dic122—41no—
257Dic122—41no—
258Dic122—42no—
259Dic122—41no—
260Dic122—82no—
261Dic122—82no—
262Dic122—82no—
263#92420—41no—
264#92420—42no—
265#92420—41no—
266#92420—41no—
267#92420—42no—
268#92420—41no—
269#17774—611nomaximal, Sylow
270C22×C61—22no—
271C22×C61—22no—
272C22×C61—21no—
273C22×C61—21no—
274C22×C61—21no—
275C22×C61C2×#10114yes—
276C22×C61C22×C22⋊C411yescharacteristic
277C4×C61—21no—
278C4×C61—22no—
279C4×C61—41no—
280C4×C61—42no—
281C4×C61—41no—
282C4×C61C2×#10112yes—
283Dic61—22no—
284Dic61—21no—
285Dic61—22no—
286Dic61—21no—
287Dic61—41no—
288Dic61—42no—
289Dic61—41no—
290Dic61#57611yescharacteristic
291Dic61#57711yescharacteristic
292Dic61#57711yescharacteristic
293Dic61#57811yescharacteristic
294#3503—611no—
295#2990—611no—
296#2990—611no—
297#1561—611no—
298#1556—612no—
299#1552—612no—
300#1552—612no—
301#1549—611no—
302#1546—611no—
303#1544—611no—
304C2×#332—612no—
305C2×C61—24no—
306C2×C61—41no—
307C2×C61C22×#10111yescharacteristic
308C2×C61#289711yescharacteristic
309C2×C61#289811yescharacteristic
310Q82—1221no—
311#557—1221no—
312#554—1221no—
313#554—1222no—
314D4×Q8—1221no—
315D4×Q8—1222no—
316#544—612no—
317#544—611no—
318#544—612no—
319#544—611no—
320#533—611no—
321#533—612no—
322C2×C4⋊Q8—612no—
323C2×C4⋊Q8—611no—
324#520—611no—
325#520#692311yescharacteristic
326C4×#41—611no—
327#446—1221no—
328#446—1221no—
329#435—1222no—
330#435—1222no—
331C8×D4—1221no—
332C8×D4—1221no—
333#432—612no—
334#432—612no—
335#431—611no—
336#431—611no—
337#423—611no—
338#423—611no—
339C2×C4⋊C8—612no—
340C2×C4⋊C8—612no—
341#332—611no—
342#332—612no—
343#332—612no—
344#332—612no—
345#332—611no—
346#332—611no—
347#332—611no—
348#332—612no—
349#332—612no—
350#332—612no—
351C61#1777411yescharacteristic, complemented, Sylow
352C2×#41C2×#692311yescharacteristic
353C22×Q8—1222no—
354C22×Q8—1221no—
355C2×M16—1221no—
356C2×M16—1221no—
357C22×C8—1222no—
358C22×C8—1222no—
359C4⋊Q8—611no—
360C4⋊Q8—612no—
361C4⋊Q8—612no—
362C4⋊Q8—612no—
363C4⋊Q8—612no—
364C4⋊Q8—611no—
365C4⋊Q8—1222no—
366C4⋊Q8—1221no—
367C4⋊Q8—1221no—
368C4⋊Q8—1222no—
369C4⋊Q8—1221no—
370C4⋊Q8—1222no—
371C4⋊Q8—1222no—
372C4⋊Q8—1221no—
373#125—611no—
374#125—612no—
375#125—611no—
376#125—612no—
377#125—1222no—
378#125—1221no—
379#125—1222no—
380#125—1221no—
381C22⋊Q8—1222no—
382C22⋊Q8—1222no—
383C22⋊Q8—1222no—
384C22⋊Q8—1222no—
385C22⋊Q8—1221no—
386C22⋊Q8—1222no—
387C22⋊Q8—1221no—
388C22⋊Q8—1222no—
389C22⋊Q8—1222no—
390C22⋊Q8—1222no—
391C4×Q8—21no—
392C4×Q8—611no—
393C4×Q8—611no—
394C4×Q8—1221no—
395C4×Q8—1222no—
396C4×Q8—1221no—
397C4×Q8—1222no—
398C4×Q8—1221no—
399C4×D4—22no—
400C4×D4—21no—
401C4×D4—611no—
402C4×D4—612no—
403C4×D4—611no—
404C4×D4—612no—
405C4×D4—1222no—
406C4×D4—1221no—
407#117—611no—
408#117—612no—
409#117—611no—
410#117—612no—
411#117C2×#692311yescharacteristic
412#117C2×#692312yes—
413C2×C4⋊C4—612no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—1222no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4C2×#692312yes—
418C2×C4⋊C4C2×#692311yescharacteristic
419C2×C42—612no—
420C2×C42—611no—
421C4⋊C8—611no—
422C4⋊C8—611no—
423C4⋊C8—612no—
424C4⋊C8—612no—
425C4⋊C8—611no—
426C4⋊C8—611no—
427C4⋊C8—1221no—
428C4⋊C8—1221no—
429C4⋊C8—1222no—
430C4⋊C8—1222no—
431C22⋊C8—1221no—
432C22⋊C8—1221no—
433C22⋊C8—1222no—
434C22⋊C8—1222no—
435#97—1222no—
436#97—1222no—
437C4×C8—1221no—
438C4×C8—1221no—
439#41—24no—
440C2×Q8—1222no—
441C2×Q8—1222no—
442C2×Q8—1222no—
443C2×Q8—1221no—
444C2×Q8—1221no—
445C2×Q8—1222no—
446C2×Q8—4882no—
447C2×Q8—4881no—
448C2×Q8—4882no—
449C2×Q8—4881no—
450C2×Q8C22⋊#692311yescharacteristic
451C2×D4C22⋊#692312yes—
452C2×D4C22⋊#692311yescharacteristic
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—612no—
456C22×C4—611no—
457C22×C4—612no—
458C22×C4—611no—
459C22×C4—1221no—
460C22×C4—1222no—
461C22×C4C22×#692312yes—
462C22×C4C22×#692311yescharacteristic
463M16—4881no—
464M16—4881no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8—1222no—
470C2×C8—1222no—
471C2×C8—4882no—
472C2×C8—4882no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—612no—
476C4⋊C4—611no—
477C4⋊C4—612no—
478C4⋊C4—611no—
479C4⋊C4—611no—
480C4⋊C4—612no—
481C4⋊C4—611no—
482C4⋊C4—1221no—
483C4⋊C4—1222no—
484C4⋊C4—1222no—
485C4⋊C4—1222no—
486C4⋊C4—1222no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1222no—
490C4⋊C4—1221no—
491C4⋊C4—1222no—
492C4⋊C4—1221no—
493C4⋊C4—1222no—
494C4⋊C4—1221no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—1221no—
498C4⋊C4C22×#692311yescharacteristic
499C4⋊C4C22×#692312yes—
500C4⋊C4C22×#692311yescharacteristic
501C22⋊C4—21no—
502C22⋊C4—22no—
503C22⋊C4—612no—
504C22⋊C4—612no—
505C22⋊C4—612no—
506C22⋊C4—1221no—
507C22⋊C4—1222no—
508C42—22no—
509C42—21no—
510C42—611no—
511C42—611no—
512C42—612no—
513C42—612no—
514C42—611no—
515C42—612no—
516C42—611no—
517C42—1222no—
518C42—1221no—
519C23C2×C22⋊#692311yescharacteristic
520C23C2×C22⋊#692312yes—
521Q8—22no—
522Q8—4882no—
523Q8—4882no—
524Q8—4882no—
525Q8—2441no—
526Q8—2441no—
527Q8—2441no—
528Q8—2442no—
529Q8—2441no—
530Q8—2442no—
531D4—24no—
532D4#1451983814yes—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—82no—
539C2×C4—81no—
540C2×C4—611no—
541C2×C4—611no—
542C2×C4—611no—
543C2×C4—612no—
544C2×C4—611no—
545C2×C4—612no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1222no—
551C2×C4—1221no—
552C2×C4—1222no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1222no—
556C2×C4—1221no—
557C2×C4—4882no—
558C2×C4—4881no—
559C2×C4C2×C22⋊#692312yes—
560C2×C4C2×C22⋊#692311yescharacteristic
561C2×C4C23×#692311yescharacteristic
562C8—2441no—
563C8—2441no—
564C8—2441no—
565C8—2441no—
566C8—2442no—
567C8—2442no—
568C22—22no—
569C22—22no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22C2×#1451983814yes—
574C22C22×C22⋊#692311yescharacteristic
575C4—21no—
576C4—22no—
577C4—41no—
578C4—42no—
579C4—41no—
580C4—611no—
581C4—611no—
582C4—611no—
583C4—611no—
584C4—1222no—
585C4—1221no—
586C4—1222no—
587C4—1221no—
588C4—2441no—
589C4—2442no—
590C4—2441no—
591C4C2×#1451983812yes—
592C2—24no—
593C2—41no—
594C2C22×#1451983811yescharacteristic
595C2#2146244011yescharacteristic
596C2#2146244111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#520C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#41×C61C2×C4
cosocleC4×Dic61C24
hypercenter#520#6923
hypocenterC61#17774
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×#41 ◁ #520

Complements

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

NH ≅ G/Ntype
3. #21461358C2semidirect
4. #21461348C2semidirect
5. #21461340C2semidirect
6. #21461340C2semidirect
7. #21461334C2semidirect
8. #21461328C2semidirect
9. #21461324C2semidirect
10. C2×#16840621C2semidirect
351. C61#17774semidirect

Maximal subgroups

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

subgroupindex
#214627452
#214613582
#214613482
#214613402
#214613342
#214613282
#214613242
C2×#168406212
#214596322
#1777461