Finite Group Databasebeta

← #29667621

Subgroups — #29667621

Order 15616.

580 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456967C211yesmaximal, characteristic, complemented
2#21456843C211yesmaximal, characteristic
3#21454979C211yesmaximal, characteristic
4#21445371C211yesmaximal, characteristic, complemented
5#21445364C211yesmaximal, characteristic, complemented
6#21445335C211yesmaximal, characteristic, complemented
7#21445330C211yesmaximal, characteristic, complemented
8#21445247C211yesmaximal, characteristic, complemented
9#21445240C211yesmaximal, characteristic, complemented
10#21445211C211yesmaximal, characteristic, complemented
11#21445206C211yesmaximal, characteristic, complemented
12#21444602C211yesmaximal, characteristic, complemented
13#21444519C211yesmaximal, characteristic, complemented
14C2×#16839509C211yesmaximal, characteristic, complemented
15C61×#3248C211yesmaximal, characteristic
16C2×#14519777C2211yescharacteristic
17#16840165C2211yescharacteristic
18#16840165C2211yescharacteristic
19#16840157—21no—
20#16840132—21no—
21#16840112C2211yescharacteristic
22#16840112C2211yescharacteristic
23#16840095—21no—
24#16840095—21no—
25#16840057—21no—
26#16840057—21no—
27#16839972C2211yescharacteristic
28#16839972C2211yescharacteristic
29#16839967C2211yescharacteristic
30#16839967C2211yescharacteristic
31#16839917—21no—
32#16839900—21no—
33#16839893C2211yescharacteristic
34Q8×#92418C2211yescharacteristic
35#16839840C2211yescharacteristic
36#16839834—21no—
37D4×#92418—21no—
38D4×#92418C2211yescharacteristic
39#16839823—21no—
40#16839819C2211yescharacteristic
41#16839815C2211yescharacteristic
42#16839789—21no—
43C2×C4⋊#92418C2211yescharacteristic
44#16839514C2211yescharacteristic
45#16839514C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839512C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839509C2211yescharacteristic
50#16839509C2211yescharacteristic
51#16839509C2211yescharacteristic
52#16839508C2211yescharacteristic
53#16839508C2211yescharacteristic
54#16839508C2211yescharacteristic
55#16839508C2211yescharacteristic
56C61×#533C2211yescharacteristic
57C61×#529C2211yescharacteristic
58C61×#479—21no—
59C61×#475C2211yescharacteristic
60C61×#474C2211yescharacteristic
61C61×#467—21no—
62C61×#465—21no—
63C61×#464—21no—
64C61×#462C2211yescharacteristic
65C61×#461C2211yescharacteristic
66C2×C4⋊C8×C61C2211yescharacteristic
67C22×Dic122—21no—
68C2×#11758202—21no—
69C2×#11758202—21no—
70C2×#11758200—21no—
71C2×#11758200—21no—
72C2×#11758194—21no—
73C22×#92418—21no—
74C22×#92418—21no—
75#14519777—21no—
76#14519777—21no—
77#14519777—21no—
78#14519777C2311yescharacteristic
79#14519777C2311yescharacteristic
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61—21no—
82#14519710—21no—
83#14519710—21no—
84#14519710—21no—
85#14519710—21no—
86C4⋊#92418—21no—
87C4⋊#92418—21no—
88C4⋊#92418—21no—
89C4⋊#92418D411yescharacteristic
90C4⋊#92418D411yescharacteristic
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95#14519701—21no—
96#14519701—21no—
97#14519701—21no—
98#14519701—21no—
99#14519700—21no—
100#14519700—21no—
101#14519700—21no—
102#14519700—21no—
103C22⋊#92418—21no—
104C22⋊#92418—21no—
105C22⋊#92418—21no—
106C22⋊#92418D411yescharacteristic
107C22⋊#92418D411yescharacteristic
108#14519688—21no—
109#14519688—21no—
110C4×#92418—21no—
111C4×#92418C2311yescharacteristic
112C4×#92418C2311yescharacteristic
113C2×Q16×C61—21no—
114C2×SD16×C61—21no—
115C22×C8×C61—21no—
116C4⋊Q8×C61C2311yescharacteristic
117C4⋊Q8×C61C2311yescharacteristic
118#126×C61—21no—
119#125×C61—21no—
120#125×C61D411yescharacteristic
121#124×C61D411yescharacteristic
122#123×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61D411yescharacteristic
126#121×C61D411yescharacteristic
127C4×Q8×C61—21no—
128C4×Q8×C61—21no—
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61C2311yescharacteristic
132#117×C61—21no—
133#117×C61—21no—
134C2×C42×C61C2311yescharacteristic
135#107×C61—21no—
136#106×C61—21no—
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140#103×C61—21no—
141#103×C61—21no—
142#103×C61—21no—
143#102×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×D4×C61—21no—
148C2×D4×C61C2×D411yescharacteristic
149C22×C4×C61—21no—
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61#4111yescharacteristic
153C22×Dic61—41no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—41no—
157C2×Dic122—81no—
158C2×Dic122—81no—
159Q16×C61—81no—
160SD16×C61—81no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—81no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—41no—
173C4⋊C4×C61—41no—
174C4⋊C4×C61C2×D411yescharacteristic
175C4⋊C4×C61C2×D411yescharacteristic
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C22⋊C4×C61—21no—
179C22⋊C4×C61—21no—
180C22⋊C4×C61—21no—
181C22⋊C4×C61—41no—
182C42×C61—21no—
183C42×C61—41no—
184C42×C61C2×D411yescharacteristic
185C42×C61C2×D411yescharacteristic
186C42×C61#4111yescharacteristic
187C42×C61C2411yescharacteristic
188#11758202—81no—
189#11758202—81no—
190#11758200—81no—
191#11758200—81no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—41no—
197C4⋊Dic61—41no—
198#11758194—81no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—81no—
204C2×#92418—81no—
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209C23×C61—21no—
210C23×C61#12111yescharacteristic
211Q8×C61—21no—
212Q8×C61—21no—
213Q8×C61—41no—
214Q8×C61—41no—
215Q8×C61—41no—
216Q8×C61—41no—
217D4×C61—21no—
218D4×C61—21no—
219D4×C61—42no—
220C2×C4×C61—22no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61#12111yescharacteristic
233C2×C4×C61#12111yescharacteristic
234C2×C4×C61#12111yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—82no—
241Dic122—41no—
242Dic122—41no—
243Dic122—41no—
244Dic122—41no—
245Dic122—82no—
246C8×C61—41no—
247C8×C61—41no—
248#92418—21no—
249#92418—21no—
250#92418—21no—
251#92418—21no—
252#92418—41no—
253#92418—41no—
254#92418—41no—
255#92418—41no—
256#19590—611nomaximal, Sylow
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—41no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61#47312yes—
263C22×C61C2×#12111yescharacteristic
264C4×C61—21no—
265C4×C61—21no—
266C4×C61—21no—
267C4×C61—21no—
268C4×C61—41no—
269C4×C61—41no—
270C4×C61—41no—
271C4×C61—81no—
272C4×C61—81no—
273C4×C61#57611yescharacteristic
274C4×C61#57711yescharacteristic
275C4×C61#57711yescharacteristic
276C4×C61#57811yescharacteristic
277Dic61—81no—
278Dic61—81no—
279#3248D6111yescharacteristic
280#3230—611no—
281#3218—611no—
282#2994—611no—
283#1700—611no—
284#1699—611no—
285#1694—611no—
286#1693—611no—
287#1680—611no—
288#1679—611no—
289#1674—611no—
290#1673—611no—
291#1573—611no—
292#1552—611no—
293C2×#332—611no—
294C2×C61—22no—
295C2×C61—81no—
296C2×C61#304611yescharacteristic
297C2×C61#306211yescharacteristic
298C2×C61C2×#47311yescharacteristic
299#533C2×D6111yescharacteristic
300C2×C4⋊Q8—611no—
301#529C2×D6111yescharacteristic
302#500—611no—
303#500—611no—
304#497—1221no—
305#491—1221no—
306#487—611no—
307#487—611no—
308#484—1221no—
309#484—1221no—
310#479—21no—
311#477—1221no—
312#477—1221no—
313#475C2×D6111yescharacteristic
314#474C2×D6111yescharacteristic
315#467—21no—
316#465—21no—
317#464—21no—
318#462—611no—
319#462—611no—
320#462C2×D6111yescharacteristic
321#461—611no—
322#461—611no—
323#461C2×D6111yescharacteristic
324#451—1221no—
325#448—1221no—
326#446—611no—
327C8×Q8—611no—
328#436—611no—
329#435—1221no—
330C8×D4—611no—
331C8×D4—1221no—
332#433—1221no—
333#432—611no—
334#431—611no—
335#424—1221no—
336C2×C4⋊C8—611no—
337C2×C4⋊C8C2×D6111yescharacteristic
338#335—611no—
339#335—611no—
340#334—611no—
341#334—611no—
342#332—611no—
343#332—611no—
344#332—611no—
345#332—611no—
346#332—611no—
347#332—611no—
348#332—611no—
349#332—611no—
350C61#1959011yescharacteristic, complemented, Sylow
351C22×Q8—1221no—
352C2×Q16—21no—
353C2×Q16—1221no—
354C2×Q16—1221no—
355C2×SD16—21no—
356C2×SD16—1221no—
357C2×SD16—1221no—
358C2×M16—1221no—
359C22×C8—21no—
360C22×C8—1221no—
361C22×C8—1221no—
362C4⋊Q8—611no—
363C4⋊Q8—611no—
364C4⋊Q8—1221no—
365C4⋊Q8—1221no—
366C4⋊Q8—1221no—
367C4⋊Q8C22×D6111yescharacteristic
368C4⋊Q8C22×D6111yescharacteristic
369#126—21no—
370#125—21no—
371#125C22⋊D6111yescharacteristic
372#124C22⋊D6111yescharacteristic
373#123—21no—
374C22⋊Q8—21no—
375C22⋊Q8—21no—
376C22⋊Q8C22⋊D6111yescharacteristic
377#121C22⋊D6111yescharacteristic
378C4×Q8—21no—
379C4×Q8—21no—
380C4×Q8C22×D6111yescharacteristic
381C4×D4—21no—
382C4×D4C22×D6111yescharacteristic
383#117—21no—
384#117—21no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C42C22×D6111yescharacteristic
388#107—21no—
389#107—1221no—
390#107—1221no—
391#107—1221no—
392#107—1221no—
393#106—21no—
394C4⋊C8—21no—
395C4⋊C8—611no—
396C4⋊C8—611no—
397C4⋊C8—611no—
398C4⋊C8—611no—
399C4⋊C8—611no—
400C4⋊C8—611no—
401C4⋊C8—1221no—
402C4⋊C8—1221no—
403C4⋊C8—1221no—
404C4⋊C8C22×D6111yescharacteristic
405C4⋊C8C22×D6111yescharacteristic
406#103—21no—
407#103—21no—
408#103—21no—
409#103—1221no—
410#103—1221no—
411#103—1221no—
412#103—1221no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#103—1221no—
417#102—21no—
418C22⋊C8—611no—
419C22⋊C8—611no—
420C22⋊C8—1221no—
421C22⋊C8—1221no—
422C22⋊C8—1221no—
423#97—1221no—
424#97—1221no—
425C4×C8—611no—
426C4×C8—611no—
427C4×C8—1221no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—1221no—
431C2×Q8—1221no—
432C2×Q8—4881no—
433C2×Q8—4881no—
434C2×Q8—2441no—
435C2×Q8C2×C22⋊D6111yescharacteristic
436C2×D4—21no—
437C2×D4C2×C22⋊D6111yescharacteristic
438C22×C4—21no—
439C22×C4—2441no—
440C22×C4D4×D6111yescharacteristic
441C22×C4#1175822411yescharacteristic
442C22×C4C2×C22⋊D6111yescharacteristic
443Q16—81no—
444Q16—4881no—
445Q16—4881no—
446SD16—81no—
447SD16—4881no—
448SD16—4881no—
449M16—4881no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—81no—
453C2×C8—611no—
454C2×C8—611no—
455C2×C8—611no—
456C2×C8—611no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—4881no—
462C2×C8—4881no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—2441no—
478C4⋊C4—2441no—
479C4⋊C4C2×C22⋊D6111yescharacteristic
480C4⋊C4C2×C22⋊D6111yescharacteristic
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—41no—
487C42—21no—
488C42—41no—
489C42D4×D6111yescharacteristic
490C42#1175822411yescharacteristic
491C42C2×C22⋊D6111yescharacteristic
492C42C23×D6111yescharacteristic
493C23—21no—
494C23#1451974911yescharacteristic
495Q8—21no—
496Q8—21no—
497Q8—41no—
498Q8—41no—
499Q8—41no—
500Q8—41no—
501Q8—4882no—
502Q8—2441no—
503Q8—2441no—
504Q8—2441no—
505Q8—2441no—
506D4—21no—
507D4—21no—
508D4—42no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—4882no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4#1451974911yescharacteristic
525C2×C4#1451974911yescharacteristic
526C2×C4#1451974911yescharacteristic
527C2×C4C2×D4×D6111yescharacteristic
528C2×C4C22×C22⋊D6111yescharacteristic
529C2×C4C2×#1175822411yescharacteristic
530C8—41no—
531C8—41no—
532C8—1221no—
533C8—1221no—
534C8—1221no—
535C8—1221no—
536C8—2441no—
537C8—2441no—
538C8—2441no—
539C8—2441no—
540C22—21no—
541C22—21no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22#1684003512yes—
546C22C2×#1451974911yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—81no—
555C4—81no—
556C4—4881no—
557C4—4881no—
558C4#1684056511yescharacteristic
559C4#1684056911yescharacteristic
560C4#1684057011yescharacteristic
561C4#1684057611yescharacteristic
562C2—22no—
563C2—81no—
564C2#2145536711yescharacteristic
565C2#2145550111yescharacteristic
566C2C2×#1684003511yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3248C2
socleC22×C61C2×#121
prime radical#121×C61D4
hypercenter#3248D61
hypocenterC61#19590
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3248

Complements

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

NH ≅ G/Ntype
1. #21456967C2semidirect
4. #21445371C2semidirect
5. #21445364C2semidirect
6. #21445335C2semidirect
7. #21445330C2semidirect
8. #21445247C2semidirect
9. #21445240C2semidirect
10. #21445211C2semidirect
11. #21445206C2semidirect
12. #21444602C2semidirect
13. #21444519C2semidirect
14. C2×#16839509C2semidirect
350. C61#19590semidirect

Maximal subgroups

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

subgroupindex
#214569672
#214568432
#214549792
#214453712
#214453642
#214453352
#214453302
#214452472
#214452402
#214452112
#214452062
#214446022
#214445192
C2×#168395092
C61×#32482
#1959061