Finite Group Databasebeta

← #29709141

Subgroups — #29709141

Order 15616.

774 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454406C211yesmaximal, characteristic
2C2×#16839801C211yesmaximal, characteristic
3#428×D61C211yesmaximal, characteristic
4#21454380C211yesmaximal, characteristic
5#21447333C211yesmaximal, characteristic, complemented
6#21447332C211yesmaximal, characteristic, complemented
7#21447330C211yesmaximal, characteristic, complemented
8#21447328C211yesmaximal, characteristic, complemented
9#21447327C211yesmaximal, characteristic, complemented
10#21447326C211yesmaximal, characteristic, complemented
11#21447324C211yesmaximal, characteristic, complemented
12#21445556C211yesmaximal, characteristic
13#343×D61C211yesmaximal, characteristic
14#21445544C211yesmaximal, characteristic
15C61×#2090C211yesmaximal, characteristic, complemented
16C22×#11758189C2211yescharacteristic
17C2×M16×D61C2211yescharacteristic
18C2×M16×D61C2211yescharacteristic
19C2×M16×D61C2211yescharacteristic
20C2×#14519735C2211yescharacteristic
21#117×D61C2211yescharacteristic
22C2×C4⋊C4×D61C2211yescharacteristic
23#16839805C2211yescharacteristic
24#16839805C2211yescharacteristic
25#16839804C2211yescharacteristic
26#16839804C2211yescharacteristic
27#16839803—21no—
28#16839803—21no—
29#16839802C2211yescharacteristic
30#16839802C2211yescharacteristic
31#16839801—22no—
32#16839801—22no—
33#16839801—21no—
34#16839801—21no—
35#16839799—21no—
36#16839797—21no—
37C2×#14519711C2211yescharacteristic
38C2×#14519710C2211yescharacteristic
39#107×D61—21no—
40C2×#14519708C2211yescharacteristic
41#106×D61—21no—
42#16839790C2211yescharacteristic
43#16839789C2211yescharacteristic
44#16839788C2211yescharacteristic
45#16839787—22no—
46#16839784—22no—
47#16839531C2211yescharacteristic
48#16839531C2211yescharacteristic
49#16839530C2211yescharacteristic
50#16839530C2211yescharacteristic
51#16839530C2211yescharacteristic
52#16839530C2211yescharacteristic
53#16839527C2211yescharacteristic
54#16839526C2211yescharacteristic
55#16839526C2211yescharacteristic
56C61×#428C2211yescharacteristic
57C61×#428C2211yescharacteristic
58C2×#106×C61C2211yescharacteristic
59C61×#424C2211yescharacteristic
60C61×#343C2211yescharacteristic
61C61×#343C2211yescharacteristic
62C61×#341C2211yescharacteristic
63C22×C4×D61—21no—
64C22×C4×D61C2311yescharacteristic
65C2×#11758194C2311yescharacteristic
66C2×#11758194C2311yescharacteristic
67C2×#11758194C2311yescharacteristic
68C2×#11758189—21no—
69C2×#11758189—21no—
70C2×#11758189—22no—
71C2×#11758189—22no—
72C2×#11758189—21no—
73C2×#11758189D412yes—
74C2×#11758189D412yes—
75M16×D61—22no—
76M16×D61—22no—
77M16×D61—22no—
78M16×D61—22no—
79M16×D61Q814yes—
80M16×D61Q814yes—
81C22×#92418C2311yescharacteristic
82C2×C8×D61—21no—
83C2×C8×D61—21no—
84C2×C8×D61—21no—
85#14519736C2311yescharacteristic
86#14519735—21no—
87#14519735—21no—
88#14519735—21no—
89#14519735—21no—
90#14519735D411yescharacteristic
91#14519735D411yescharacteristic
92#14519734—21no—
93#14519733—21no—
94C2×C4⋊Dic61C2311yescharacteristic
95C2×C4⋊Dic61C2311yescharacteristic
96C2×#11758196—21no—
97C4⋊C4×D61—21no—
98C4⋊C4×D61—21no—
99C4⋊C4×D61—21no—
100C4⋊C4×D61—21no—
101C4⋊C4×D61D411yescharacteristic
102C4⋊C4×D61D411yescharacteristic
103C2×#11758198—21no—
104C22⋊C4×D61—21no—
105C2×C4×Dic61—21no—
106C42×D61—21no—
107#14519711—22no—
108#14519711—21no—
109#14519711—21no—
110#14519710—21no—
111#14519710—21no—
112#14519710—21no—
113#14519710—21no—
114#14519710—22no—
115#14519709—21no—
116#14519709—21no—
117#14519708—21no—
118#14519708—21no—
119#14519708—21no—
120#14519708—21no—
121#14519708—22no—
122C4⋊#92418—22no—
123#14519706—22no—
124#14519706—22no—
125C2×M16×C61C2311yescharacteristic
126C2×M16×C61C2311yescharacteristic
127C2×M16×C61C2311yescharacteristic
128C22×C8×C61C2311yescharacteristic
129#117×C61C2311yescharacteristic
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C4⋊C4×C61C2311yescharacteristic
132#107×C61—21no—
133#107×C61—21no—
134#106×C61—22no—
135#106×C61—21no—
136#106×C61—21no—
137C4⋊C8×C61—22no—
138C23×D61#4111yescharacteristic
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61C2411yescharacteristic
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61#4111yescharacteristic
145C2×C4×D61—21no—
146C2×C4×D61—21no—
147C2×C4×D61—41no—
148C2×C4×D61—41no—
149C2×C4×D61—82no—
150C2×C4×D61C2×D411yescharacteristic
151C2×C4×D61C2×D411yescharacteristic
152C2×C4×D61C2×D411yescharacteristic
153C2×C4×D61C2×D411yescharacteristic
154C2×C4×D61C2×Q812yes—
155C2×C4×D61C2×Q812yes—
156C2×C4×D61#4112yes—
157C2×C4×D61#4112yes—
158M16×C61—21no—
159M16×C61—21no—
160M16×C61—21no—
161M16×C61—21no—
162M16×C61C2×Q812yes—
163M16×C61C2×Q812yes—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61C2×D412yes—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61C2×D411yescharacteristic
175C4⋊C4×C61C2×D411yescharacteristic
176C22⋊C4×C61—21no—
177C42×C61—21no—
178C22⋊Dic61—21no—
179#11758198—21no—
180#11758198—21no—
181#11758198—41no—
182#11758198—41no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61C2×D411yescharacteristic
188C4⋊Dic61C2×D411yescharacteristic
189#11758196—21no—
190#11758196—21no—
191#11758196—41no—
192#11758196—41no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C4×Dic61—41no—
197C4×Dic61—41no—
198#11758194—21no—
199#11758194—21no—
200#11758194—21no—
201#11758194—21no—
202#11758194C2×Q812yes—
203#11758194C2×Q812yes—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418C2×D412yes—
210#11758189—24no—
211#11758189—24no—
212#11758189—24no—
213#11758189—24no—
214#11758189—42no—
215C8×D61—24no—
216C8×D61—24no—
217C8×D61—42no—
218C23×C61C2×#4111yescharacteristic
219C22×D61—41no—
220C22×D61—41no—
221C22×D61#12111yescharacteristic
222C22×D61#12111yescharacteristic
223C22×D61#12512yes—
224C22×D61#12512yes—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—81no—
230C2×C4×C61—81no—
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×Q811yescharacteristic
234C2×C4×C61C22×Q811yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—81no—
242C2×Dic61—81no—
243C2×Dic61#12111yescharacteristic
244C2×Dic61#12111yescharacteristic
245C2×Dic61#12512yes—
246C2×Dic61#12512yes—
247C4×D61—81no—
248C4×D61—81no—
249C4×D61—82no—
250C4×D61—82no—
251C4×D61C22⋊Q812yes—
252C4×D61C22⋊Q812yes—
253C4×D61C22⋊Q812yes—
254C4×D61C22⋊Q812yes—
255C4×D61#12312yes—
256C4×D61#12312yes—
257C4×D61C4⋊Q812yes—
258C4×D61C4⋊Q812yes—
259C8×C61—22no—
260C8×C61—22no—
261C8×C61—22no—
262C8×C61—41no—
263#92418—22no—
264#92418—22no—
265#92418—22no—
266#92418—41no—
267#23675—611nomaximal, Sylow
268C22×C61—21no—
269C22×C61—21no—
270C22×C61C2×#12111yescharacteristic
271C22×C61C2×#12511yescharacteristic
272C22×C61C2×#12511yescharacteristic
273C2×D61—21no—
274C2×D61—21no—
275C2×D61—21no—
276C2×D61—21no—
277C2×D61—42no—
278C2×D61—42no—
279C2×D61#39812yes—
280C2×D61#39812yes—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—81no—
284C4×C61—81no—
285C4×C61C2×C22⋊Q811yescharacteristic
286C4×C61C2×C22⋊Q811yescharacteristic
287C4×C61C2×#12311yescharacteristic
288C4×C61C2×C4⋊Q811yescharacteristic
289Dic61—41no—
290Dic61—41no—
291Dic61—81no—
292Dic61—81no—
293Dic61#39812yes—
294Dic61#39812yes—
295#2911—611no—
296C2×#428—611no—
297C2×#428—611no—
298#2905—611no—
299#2090—611no—
300#2090—611no—
301#2090—611no—
302#2090D6111yescharacteristic, complemented
303#2089—611no—
304#2089—611no—
305#2089—611no—
306#2089—611no—
307#1742—611no—
308C2×#343—611no—
309#1738—611no—
310C2×C61—21no—
311C2×C61—21no—
312C2×C61C2×#39811yescharacteristic
313C2×C61#306811yescharacteristic
314C2×C61#306911yescharacteristic
315D61—22no—
316D61—22no—
317C22×M16—611no—
318C22×M16—611no—
319C22×M16—611no—
320C22×M16—611no—
321C2×#117—611no—
322C2×#117—611no—
323C22×C4⋊C4—611no—
324#428—611no—
325#428—611no—
326#428—611no—
327#428—611no—
328#428—611no—
329#428—611no—
330#428—1221no—
331#428—1221no—
332#428—1222no—
333#428—1222no—
334#428—1221no—
335#428—1221no—
336#428C2×D6111yescharacteristic
337#428C2×D6111yescharacteristic
338#427—1221no—
339#427—1221no—
340C2×#107—611no—
341C2×#107—611no—
342C2×#107—1221no—
343C2×#106—611no—
344C2×#106—1221no—
345C2×#106C2×D6111yescharacteristic
346#424—611no—
347#424—611no—
348#424—611no—
349#424—1222no—
350#424C2×D6111yescharacteristic
351#423—1222no—
352#343—611no—
353#343—611no—
354#343—611no—
355#343—611no—
356#343—611no—
357#343—611no—
358#343C2×D6111yescharacteristic
359#343C2×D6111yescharacteristic
360#341—611no—
361#341—611no—
362#341—611no—
363#341C2×D6111yescharacteristic
364C61#2367511yescharacteristic, complemented, Sylow
365C23×C4—611no—
366C23×C4—1221no—
367C2×M16—612no—
368C2×M16—612no—
369C2×M16—611no—
370C2×M16—611no—
371C2×M16—614no—
372C2×M16—614no—
373C2×M16—611no—
374C2×M16—1221no—
375C2×M16—1221no—
376C2×M16—1222no—
377C2×M16—1222no—
378C2×M16—1221no—
379C2×M16—1222no—
380C2×M16—1222no—
381C2×M16—1222no—
382C2×M16—1222no—
383C2×M16C22×D6111yescharacteristic
384C2×M16C22×D6111yescharacteristic
385C2×M16C22×D6111yescharacteristic
386C22×C8—611no—
387C22×C8—1221no—
388C22×C8—1221no—
389C22×C8—1221no—
390C22×C8C22×D6111yescharacteristic
391#117—611no—
392#117—611no—
393#117—611no—
394#117—1221no—
395#117—1221no—
396#117—1221no—
397#117—1221no—
398#117—1221no—
399#117—1221no—
400#117C22×D6111yescharacteristic
401C2×C4⋊C4—611no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4C22×D6111yescharacteristic
411C2×C4⋊C4C22×D6111yescharacteristic
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C42—1221no—
415C2×C42—1221no—
416#107—21no—
417#107—21no—
418#107—1221no—
419#107—1221no—
420#107—1222no—
421#107—1221no—
422#107—1221no—
423#107—1222no—
424#107—1221no—
425#107—1221no—
426#106—22no—
427#106—21no—
428#106—21no—
429#106—1221no—
430#106—1221no—
431#106—1222no—
432#106—1221no—
433#106—1221no—
434#106—1221no—
435#106—1221no—
436C4⋊C8—22no—
437C4⋊C8—1222no—
438C4⋊C8—1222no—
439C4⋊C8—1222no—
440C24—611no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—612no—
449C22×C4—612no—
450C22×C4—612no—
451C22×C4—612no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—4882no—
457C22×C4—2441no—
458C22×C4—2441no—
459C22×C4C23×D6111yescharacteristic
460M16—21no—
461M16—21no—
462M16—21no—
463M16—21no—
464M16—612no—
465M16—612no—
466M16—1224no—
467M16—1224no—
468M16—1224no—
469M16—1224no—
470M16—1221no—
471M16—1221no—
472M16—1221no—
473M16—1221no—
474M16—2442no—
475M16Q8×D6112yes—
476M16Q8×D6112yes—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—612no—
483C2×C8—1224no—
484C2×C8—1224no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—2442no—
491C2×C8D4×D6112yes—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—611no—
497C4⋊C4—611no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—2441no—
505C4⋊C4—2441no—
506C4⋊C4D4×D6111yescharacteristic
507C4⋊C4D4×D6111yescharacteristic
508C22⋊C4—21no—
509C22⋊C4—1221no—
510C22⋊C4—1221no—
511C22⋊C4—1221no—
512C22⋊C4—2441no—
513C22⋊C4—2441no—
514C42—21no—
515C42—1221no—
516C42—1221no—
517C42—1221no—
518C42—2441no—
519C42—2441no—
520C23—611no—
521C23—611no—
522C23—612no—
523C23—612no—
524C23—2441no—
525C23—2441no—
526C23#41×D6111yescharacteristic
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—611no—
534C2×C4—611no—
535C2×C4—612no—
536C2×C4—612no—
537C2×C4—612no—
538C2×C4—612no—
539C2×C4—612no—
540C2×C4—612no—
541C2×C4—612no—
542C2×C4—612no—
543C2×C4—612no—
544C2×C4—612no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—4881no—
548C2×C4—4881no—
549C2×C4—4882no—
550C2×C4—4882no—
551C2×C4—4881no—
552C2×C4—4881no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4C2×D4×D6111yescharacteristic
556C2×C4C2×D4×D6111yescharacteristic
557C2×C4C2×Q8×D6111yescharacteristic
558C2×C4C2×Q8×D6111yescharacteristic
559C2×C4#41×D6111yescharacteristic
560C2×C4#41×D6111yescharacteristic
561C8—22no—
562C8—22no—
563C8—22no—
564C8—41no—
565C8—1222no—
566C8—1222no—
567C8—1222no—
568C8—2441no—
569C22—21no—
570C22—21no—
571C22—612no—
572C22—612no—
573C22—1221no—
574C22—1221no—
575C22—1221no—
576C22—1221no—
577C22—2442no—
578C22—2442no—
579C22#121×D6111yescharacteristic
580C22#125×D6111yescharacteristic
581C22#125×D6111yescharacteristic
582C4—41no—
583C4—41no—
584C4—81no—
585C4—81no—
586C4—612no—
587C4—612no—
588C4—4881no—
589C4—4881no—
590C4—2441no—
591C4—2441no—
592C4C22⋊Q8×D6111yescharacteristic
593C4C22⋊Q8×D6111yescharacteristic
594C4#123×D6111yescharacteristic
595C4C4⋊Q8×D6111yescharacteristic
596C2—21no—
597C2—21no—
598C2—1222no—
599C2—1222no—
600C2#398×D6111yescharacteristic
601C2#2145554211yescharacteristic
602C2#2145555111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2090C2
socleC22×C61C2×#121
prime radicalC23×D61#41
hypercenter#2090D61
hypocenterC61#23675
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2090

Complements

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

NH ≅ G/Ntype
5. #21447333C2semidirect
6. #21447332C2semidirect
7. #21447330C2semidirect
8. #21447328C2semidirect
9. #21447327C2semidirect
10. #21447326C2semidirect
11. #21447324C2semidirect
15. C61×#2090C2semidirect
302. #2090D61semidirect
364. C61#23675semidirect

Maximal subgroups

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

subgroupindex
#214544062
C2×#168398012
#428×D612
#214543802
#214473332
#214473322
#214473302
#214473282
#214473272
#214473262
#214473242
#214455562
#343×D612
#214455442
C61×#20902
#2367561