Finite Group Databasebeta

← #29663419

Subgroups — #29663419

Order 15616.

596 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456994C211yesmaximal, characteristic, complemented
2#21456374C211yesmaximal, characteristic, complemented
3#21454795C211yesmaximal, characteristic, complemented
4#21445323C211yesmaximal, characteristic, complemented
5#21445316C211yesmaximal, characteristic, complemented
6#21445289C211yesmaximal, characteristic, complemented
7#21445282C211yesmaximal, characteristic, complemented
8#21445251C211yesmaximal, characteristic, complemented
9#21445244C211yesmaximal, characteristic, complemented
10#21445233C211yesmaximal, characteristic, complemented
11#21445226C211yesmaximal, characteristic, complemented
12#21444565C211yesmaximal, characteristic, complemented
13#21444527C211yesmaximal, characteristic, complemented
14#21444524C211yesmaximal, characteristic, complemented
15C61×#3212C211yesmaximal, characteristic
16#16840319C2211yescharacteristic
17#16840161—21no—
18#16840158C2211yescharacteristic
19#16840134C2211yescharacteristic
20#16840116—21no—
21#16840114C2211yescharacteristic
22#16840108C2211yescharacteristic
23#16840100—21no—
24#16840083—21no—
25#16840057C2211yescharacteristic
26#16840052C2211yescharacteristic
27#16840047C2211yescharacteristic
28#16840042C2211yescharacteristic
29#16839989—21no—
30#16839989—21no—
31#16839893C2211yescharacteristic
32Q8×#92418C2211yescharacteristic
33C4×#11758200—21no—
34C4×#11758200—21no—
35#16839840C2211yescharacteristic
36#16839834—21no—
37D4×#92418—21no—
38D4×#92418C2211yescharacteristic
39#16839823—21no—
40#16839819C2211yescharacteristic
41#16839815C2211yescharacteristic
42#16839718—21no—
43C2×C4×#92418C2211yescharacteristic
44#16839514C2211yescharacteristic
45#16839514C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839512C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839510C2211yescharacteristic
50#16839508C2211yescharacteristic
51#16839508C2211yescharacteristic
52#16839507C2211yescharacteristic
53#16839507C2211yescharacteristic
54#16839506C2211yescharacteristic
55#16839506C2211yescharacteristic
56C61×#532C2211yescharacteristic
57C61×#529C2211yescharacteristic
58C61×#495—21no—
59C61×#488—21no—
60C61×#462C2211yescharacteristic
61C61×#461C2211yescharacteristic
62C61×#460C2211yescharacteristic
63C61×#459C2211yescharacteristic
64C61×#452—21no—
65C61×#449—21no—
66C61×#431C2211yescharacteristic
67C2×#11758200—21no—
68C2×#11758200—21no—
69C2×#11758194—21no—
70C22×#92418—21no—
71C22×#92418D411yescharacteristic
72C22×#92418D411yescharacteristic
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519769—21no—
76#14519756—21no—
77#14519756—21no—
78C4×Dic122—21no—
79C4×Dic122—21no—
80#14519736—21no—
81#14519710—21no—
82#14519710—21no—
83#14519708—21no—
84#14519708—21no—
85#14519708—21no—
86#14519708—21no—
87C4⋊#92418—21no—
88C4⋊#92418—21no—
89C4⋊#92418C2311yescharacteristic
90C4⋊#92418C2311yescharacteristic
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93#14519701—21no—
94#14519701—21no—
95#14519700—21no—
96#14519700—21no—
97#14519698—21no—
98#14519698—21no—
99#14519698—21no—
100#14519698—21no—
101C22⋊#92418—21no—
102C22⋊#92418—21no—
103C22⋊#92418—21no—
104C22⋊#92418—21no—
105#14519688—21no—
106#14519688—21no—
107C4×#92418—21no—
108C4×#92418—21no—
109C4×#92418D411yescharacteristic
110C4×#92418D411yescharacteristic
111C4×#92418C2311yescharacteristic
112C4×#92418C2311yescharacteristic
113C2×#41×C61—21no—
114C2×Q16×C61—21no—
115C2×SD16×C61—21no—
116C2×SD16×C61—21no—
117C2×D8×C61—21no—
118C4⋊Q8×C61C2311yescharacteristic
119#127×C61C2311yescharacteristic
120#126×C61—21no—
121#125×C61D411yescharacteristic
122#124×C61—21no—
123#124×C61D411yescharacteristic
124#123×C61—21no—
125C22⋊Q8×C61D411yescharacteristic
126#121×C61—21no—
127#121×C61—21no—
128#121×C61D411yescharacteristic
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61—21no—
132C4×D4×C61—21no—
133C4×D4×C61C2311yescharacteristic
134#117×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142C22⋊C8×C61—21no—
143C4×C8×C61—21no—
144#41×C61—81no—
145#41×C61—81no—
146C2×Q8×C61—21no—
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—41no—
153C2×D4×C61C2×D411yescharacteristic
154C22×C4×C61—21no—
155C22×C4×C61—41no—
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61#4111yescharacteristic
159C2×Dic122—21no—
160C2×Dic122—21no—
161Q16×C61—81no—
162SD16×C61—81no—
163SD16×C61—81no—
164D8×C61—81no—
165C2×C8×C61—21no—
166C2×C8×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—21no—
173C4⋊C4×C61C2×D411yescharacteristic
174C4⋊C4×C61C2×D411yescharacteristic
175C22⋊C4×C61—21no—
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C22⋊C4×C61—21no—
179C22⋊C4×C61—21no—
180C22⋊C4×C61—41no—
181C22⋊C4×C61—41no—
182C42×C61—21no—
183C42×C61C2×D411yescharacteristic
184C42×C61C2×D411yescharacteristic
185C42×C61#4111yescharacteristic
186C42×C61C2411yescharacteristic
187C22⋊Dic61—41no—
188#11758200—42no—
189#11758200—42no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194#11758196—41no—
195#11758196—41no—
196C4×Dic61—41no—
197#11758194—42no—
198C2×#92418—22no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—41no—
204C2×#92418—42no—
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209C23×C61—21no—
210C23×C61—41no—
211C23×C61#12111yescharacteristic
212Q8×C61—21no—
213Q8×C61—21no—
214Q8×C61—41no—
215Q8×C61—41no—
216D4×C61—21no—
217D4×C61—21no—
218D4×C61—42no—
219D4×C61—41no—
220D4×C61—41no—
221D4×C61—41no—
222D4×C61—41no—
223D4×C61—81no—
224D4×C61—82no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—82no—
237C2×C4×C61#12111yescharacteristic
238C2×C4×C61#12111yescharacteristic
239C2×C4×C61#12111yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×Dic61—41no—
244C2×Dic61—41no—
245Dic122—41no—
246Dic122—41no—
247Dic122—81no—
248C8×C61—41no—
249C8×C61—41no—
250#92418—21no—
251#92418—21no—
252#92418—21no—
253#92418—41no—
254#92418—41no—
255#92418—41no—
256#92418—41no—
257#92418#13512yes—
258#19224—611nomaximal, Sylow
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61—41no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61C2×#12111yescharacteristic
269C4×C61—21no—
270C4×C61—21no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61—81no—
277C4×C61C2×#13511yescharacteristic
278C4×C61#57511yescharacteristic
279C4×C61#57711yescharacteristic
280C4×C61#57711yescharacteristic
281Dic61—81no—
282Dic61—81no—
283#3233—611no—
284#3212D6111yescharacteristic
285#3167—611no—
286#2966—611no—
287#1692—611no—
288#1691—611no—
289#1686—611no—
290#1685—611no—
291#1680—611no—
292#1679—611no—
293#1678—611no—
294#1677—611no—
295#1564—611no—
296#1554—611no—
297#1553—611no—
298C2×C61—41no—
299C2×C61—81no—
300C2×C61—161no—
301C2×C61#303511yescharacteristic
302C2×C61#305711yescharacteristic
303C2×C61#306711yescharacteristic
304#533—611no—
305#532C2×D6111yescharacteristic
306#529C2×D6111yescharacteristic
307#499—1221no—
308#498—611no—
309#495—21no—
310#492—611no—
311#488—21no—
312#488—1221no—
313#487—611no—
314#486—611no—
315#485—1221no—
316#482—1221no—
317#477—611no—
318#476—611no—
319#475—611no—
320#474—611no—
321#465—1221no—
322#465—1221no—
323#462C2×D6111yescharacteristic
324#461C2×D6111yescharacteristic
325#460C2×D6111yescharacteristic
326#459C2×D6111yescharacteristic
327#452—21no—
328#449—21no—
329#446—611no—
330C8×Q8—611no—
331C4×SD16—1221no—
332C4×SD16—1221no—
333#436—611no—
334#435—1221no—
335C8×D4—611no—
336C8×D4—1221no—
337#433—1221no—
338#432—611no—
339#431—611no—
340#431C2×D6111yescharacteristic
341#405—1221no—
342C2×C4×C8—611no—
343#335—611no—
344#335—611no—
345#334—611no—
346#334—611no—
347#333—611no—
348#333—611no—
349#332—611no—
350#332—611no—
351#332—611no—
352#332—611no—
353#331—611no—
354#331—611no—
355C61#1922411yescharacteristic, complemented, Sylow
356C2×#41—21no—
357C2×Q16—21no—
358C2×SD16—21no—
359C2×SD16—21no—
360C2×SD16—1221no—
361C2×SD16—1221no—
362C2×D8—21no—
363C2×M16—1221no—
364C22×C8—611no—
365C22×C8—611no—
366C22×C8—1221no—
367C4⋊Q8—611no—
368C4⋊Q8—611no—
369C4⋊Q8C22×D6111yescharacteristic
370#127C22×D6111yescharacteristic
371#126—21no—
372#125—1221no—
373#125C22⋊D6111yescharacteristic
374#124—21no—
375#124C22⋊D6111yescharacteristic
376#123—21no—
377C22⋊Q8—1221no—
378C22⋊Q8—1221no—
379C22⋊Q8C22⋊D6111yescharacteristic
380#121—21no—
381#121—21no—
382#121C22⋊D6111yescharacteristic
383C4×Q8—1221no—
384C4×Q8—1221no—
385C4×Q8C22×D6111yescharacteristic
386C4×D4—21no—
387C4×D4—21no—
388C4×D4—21no—
389C4×D4C22×D6111yescharacteristic
390#117—21no—
391#117—1221no—
392C2×C42C22×D6111yescharacteristic
393#107—1221no—
394#107—1221no—
395#106—1221no—
396#106—1221no—
397#106—1221no—
398#106—1221no—
399C4⋊C8—611no—
400C4⋊C8—611no—
401C4⋊C8—611no—
402C4⋊C8—611no—
403C4⋊C8—1221no—
404C4⋊C8—1221no—
405C4⋊C8C22×D6111yescharacteristic
406C4⋊C8C22×D6111yescharacteristic
407#103—21no—
408#103—21no—
409#103—1221no—
410#103—1221no—
411#103—1221no—
412#103—1221no—
413#102—21no—
414#102—21no—
415#102—1221no—
416#102—1221no—
417#102—1221no—
418#102—1221no—
419C22⋊C8—21no—
420C22⋊C8—1221no—
421C22⋊C8—1221no—
422C22⋊C8—1221no—
423C22⋊C8—1221no—
424#97—1221no—
425#97—1221no—
426C4×C8—21no—
427C4×C8—611no—
428C4×C8—611no—
429C4×C8—611no—
430C4×C8—611no—
431C4×C8—1221no—
432C4×C8—1221no—
433#41—81no—
434#41—81no—
435C2×Q8—21no—
436C2×Q8—1221no—
437C2×Q8—1221no—
438C2×Q8C2×C22⋊D6111yescharacteristic
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—41no—
444C2×D4C2×C22⋊D6111yescharacteristic
445C22×C4—21no—
446C22×C4—41no—
447C22×C4D4×D6111yescharacteristic
448C22×C4#1175822411yescharacteristic
449C22×C4C2×C22⋊D6111yescharacteristic
450Q16—81no—
451SD16—81no—
452SD16—81no—
453SD16—2442no—
454SD16—2442no—
455D8—81no—
456M16—2442no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—611no—
460C2×C8—611no—
461C2×C8—611no—
462C2×C8—611no—
463C2×C8—1222no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—2441no—
469C2×C8—2442no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—2441no—
481C4⋊C4—2441no—
482C4⋊C4C2×C22⋊D6111yescharacteristic
483C4⋊C4C2×C22⋊D6111yescharacteristic
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—41no—
490C22⋊C4—41no—
491C22⋊C4—2441no—
492C42—21no—
493C42—2441no—
494C42D4×D6111yescharacteristic
495C42#1175822411yescharacteristic
496C42C2×C22⋊D6111yescharacteristic
497C42C23×D6111yescharacteristic
498C23—21no—
499C23—41no—
500C23#1451974911yescharacteristic
501Q8—21no—
502Q8—21no—
503Q8—41no—
504Q8—41no—
505Q8—4881no—
506Q8—2441no—
507Q8—2441no—
508D4—21no—
509D4—21no—
510D4—42no—
511D4—41no—
512D4—41no—
513D4—41no—
514D4—41no—
515D4—81no—
516D4—82no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—82no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4#1451974911yescharacteristic
532C2×C4#1451974911yescharacteristic
533C2×C4#1451974911yescharacteristic
534C2×C4C2×D4×D6111yescharacteristic
535C2×C4C22×C22⋊D6111yescharacteristic
536C2×C4C2×#1175822411yescharacteristic
537C8—41no—
538C8—41no—
539C8—612no—
540C8—1221no—
541C8—1221no—
542C8—1221no—
543C8—2441no—
544C8—2441no—
545C8—2441no—
546C8—2441no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22C2×#1451974911yescharacteristic
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—81no—
565C4—4881no—
566C4—4881no—
567C4#135×D6111yescharacteristic
568C4#1684055911yescharacteristic
569C4#1684057311yescharacteristic
570C4#1684057311yescharacteristic
571C2—41no—
572C2—81no—
573C2—161no—
574C2#2145527011yescharacteristic
575C2#2145545511yescharacteristic
576C2#2145553611yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3212C2
socleC22×C61C2×#121
hypercenter#3212D61
hypocenterC61#19224
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3212

Complements

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

NH ≅ G/Ntype
1. #21456994C2semidirect
2. #21456374C2semidirect
3. #21454795C2semidirect
4. #21445323C2semidirect
5. #21445316C2semidirect
6. #21445289C2semidirect
7. #21445282C2semidirect
8. #21445251C2semidirect
9. #21445244C2semidirect
10. #21445233C2semidirect
11. #21445226C2semidirect
12. #21444565C2semidirect
13. #21444527C2semidirect
14. #21444524C2semidirect
355. C61#19224semidirect

Maximal subgroups

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

subgroupindex
#214569942
#214563742
#214547952
#214453232
#214453162
#214452892
#214452822
#214452512
#214452442
#214452332
#214452262
#214445652
#214445272
#214445242
C61×#32122
#1922461