Finite Group Databasebeta

← #29663058

Subgroups — #29663058

Order 15616.

592 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456560C211yesmaximal, characteristic, complemented
2#21455890C211yesmaximal, characteristic
3#21454839C211yesmaximal, characteristic
4#21445396C211yesmaximal, characteristic, complemented
5#21445364C211yesmaximal, characteristic, complemented
6#21445320C211yesmaximal, characteristic, complemented
7#21445306C211yesmaximal, characteristic, complemented
8#21445279C211yesmaximal, characteristic, complemented
9#21445247C211yesmaximal, characteristic, complemented
10#21445203C211yesmaximal, characteristic, complemented
11#21445187C211yesmaximal, characteristic, complemented
12#21444569C211yesmaximal, characteristic, complemented
13#21444529C211yesmaximal, characteristic, complemented
14#21444517C211yesmaximal, characteristic, complemented
15C61×#3076C211yesmaximal, characteristic
16#16840319C2211yescharacteristic
17#16840158C2211yescharacteristic
18#16840141C2211yescharacteristic
19#16840114C2211yescharacteristic
20#16840112C2211yescharacteristic
21#16840103—21no—
22#16840047C2211yescharacteristic
23#16840042C2211yescharacteristic
24#16840018—21no—
25#16840018—21no—
26#16839972C2211yescharacteristic
27#16839967C2211yescharacteristic
28#16839924—21no—
29Q8×#92418C2211yescharacteristic
30Q8×#92418C2211yescharacteristic
31#16839867—21no—
32#16839867—21no—
33C4×#11758202—21no—
34C4×#11758200—21no—
35#16839840—21no—
36#16839840—21no—
37#16839840C2211yescharacteristic
38#16839840C2211yescharacteristic
39#16839819—21no—
40#16839819C2211yescharacteristic
41#16839815C2211yescharacteristic
42C4×#11758194—21no—
43C4×#11758194C2211yescharacteristic
44#16839514C2211yescharacteristic
45#16839514C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839512C2211yescharacteristic
48#16839511C2211yescharacteristic
49#16839511C2211yescharacteristic
50#16839509C2211yescharacteristic
51#16839509C2211yescharacteristic
52#16839508C2211yescharacteristic
53#16839508C2211yescharacteristic
54#16839507C2211yescharacteristic
55#16839507C2211yescharacteristic
56C61×#533C2211yescharacteristic
57C4×#41×C61C2211yescharacteristic
58C61×#479—21no—
59C61×#478—21no—
60C61×#477C2211yescharacteristic
61C61×#476C2211yescharacteristic
62C61×#475C2211yescharacteristic
63C61×#474C2211yescharacteristic
64C61×#427—21no—
65C2×C4⋊C8×C61C2211yescharacteristic
66C61×#416—21no—
67C2×#11758202—21no—
68C2×#11758202—21no—
69C2×#11758202—21no—
70C2×#11758200—21no—
71C2×#11758194—21no—
72C2×#11758194—21no—
73C2×#11758194D411yescharacteristic
74C2×#11758194D411yescharacteristic
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519769—21no—
78#14519756—21no—
79#14519756—21no—
80C4×Dic122—21no—
81C4×Dic122—21no—
82#14519736—21no—
83#14519710—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#14519701—21no—
96#14519700—21no—
97#14519700—21no—
98#14519700—21no—
99#14519700—21no—
100#14519698—21no—
101C22⋊#92418—21no—
102C22⋊#92418—21no—
103C22⋊#92418—21no—
104C22⋊#92418—21no—
105#14519688—21no—
106#14519688D411yescharacteristic
107#14519688D411yescharacteristic
108C4×#92418—21no—
109C4×#92418—21no—
110C4×#92418—21no—
111C4×#92418C2311yescharacteristic
112C4×#92418C2311yescharacteristic
113C2×#41×C61D411yescharacteristic
114C22×C8×C61—21no—
115C4⋊Q8×C61C2311yescharacteristic
116C4⋊Q8×C61C2311yescharacteristic
117#125×C61—21no—
118C22⋊Q8×C61—21no—
119C22⋊Q8×C61—21no—
120C4×Q8×C61—21no—
121C4×Q8×C61—21no—
122C4×Q8×C61D411yescharacteristic
123C4×Q8×C61C2311yescharacteristic
124C4×D4×C61—21no—
125C4×D4×C61—21no—
126C4×D4×C61D411yescharacteristic
127C4×D4×C61C2311yescharacteristic
128#117×C61—21no—
129#117×C61—21no—
130#117×C61D411yescharacteristic
131C2×C42×C61—21no—
132C2×C42×C61C2311yescharacteristic
133#107×C61—21no—
134#107×C61—21no—
135#106×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#102×C61—21no—
143#102×C61—21no—
144#41×C61—22no—
145#41×C61—41no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61C2×D411yescharacteristic
149C2×D4×C61—21no—
150C2×D4×C61C2×D411yescharacteristic
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61#4111yescharacteristic
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—42no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—41no—
169C4⋊C4×C61—41no—
170C4⋊C4×C61C2×D411yescharacteristic
171C4⋊C4×C61C2×D411yescharacteristic
172C22⋊C4×C61—21no—
173C22⋊C4×C61—21no—
174C22⋊C4×C61—21no—
175C22⋊C4×C61—41no—
176C42×C61—21no—
177C42×C61—21no—
178C42×C61—21no—
179C42×C61—41no—
180C42×C61C2×D411yescharacteristic
181C42×C61C2×D411yescharacteristic
182C42×C61#4111yescharacteristic
183C42×C61C2411yescharacteristic
184C22⋊Dic61—41no—
185#11758202—42no—
186#11758202—42no—
187#11758202—42no—
188#11758200—42no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193#11758196—41no—
194#11758196—41no—
195C4×Dic61—41no—
196#11758194—22no—
197#11758194—41no—
198#11758194—42no—
199#11758194—42no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418C2×D411yescharacteristic
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207C2×#92418C2×D411yescharacteristic
208C23×C61—21no—
209C23×C61#12111yescharacteristic
210Q8×C61—21no—
211Q8×C61—41no—
212Q8×C61—41no—
213Q8×C61—81no—
214Q8×C61#13712yes—
215D4×C61—21no—
216D4×C61—21no—
217D4×C61—42no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—41no—
226C2×C4×C61—42no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—81no—
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—
240Dic122—41no—
241Dic122—41no—
242Dic122—41no—
243Dic122—41no—
244C8×C61—41no—
245C8×C61—41no—
246#92418—21no—
247#92418—21no—
248#92418—21no—
249#92418—41no—
250#92418—41no—
251#92418—41no—
252#92418—41no—
253#92418#13512yes—
254#19185—611nomaximal, Sylow
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—41no—
259C22×C61—41no—
260C22×C61—41no—
261C22×C61C2×#12111yescharacteristic
262C4×C61—21no—
263C4×C61—21no—
264C4×C61—22no—
265C4×C61—21no—
266C4×C61—21no—
267C4×C61—41no—
268C4×C61—41no—
269C4×C61—81no—
270C4×C61—81no—
271C4×C61C2×#13511yescharacteristic
272C4×C61C2×#13711yescharacteristic
273C4×C61#57511yescharacteristic
274C4×C61#57511yescharacteristic
275Dic61—81no—
276Dic61—81no—
277#3185—611no—
278#3107—611no—
279#3076D6111yescharacteristic
280#2971—611no—
281#1703—611no—
282#1699—611no—
283#1691—611no—
284#1689—611no—
285#1684—611no—
286#1680—611no—
287#1672—611no—
288#1670—611no—
289#1565—611no—
290#1554—611no—
291#1552—611no—
292C2×C61—41no—
293C2×C61—81no—
294C2×C61#303511yescharacteristic
295C2×C61#305611yescharacteristic
296C2×C61#306911yescharacteristic
297#533—611no—
298#533C2×D6111yescharacteristic
299C4×#41C2×D6111yescharacteristic
300#498—611no—
301#494—611no—
302#487—611no—
303#487—611no—
304#485—1221no—
305#479—21no—
306#478—21no—
307#477C2×D6111yescharacteristic
308#476C2×D6111yescharacteristic
309#475—611no—
310#475C2×D6111yescharacteristic
311#474—611no—
312#474C2×D6111yescharacteristic
313#470—1221no—
314#470—1221no—
315#462—611no—
316#461—611no—
317#452—1221no—
318C8×Q8—611no—
319C8×Q8—611no—
320#441—1221no—
321#441—1221no—
322C4×Q16—1221no—
323C4×SD16—1221no—
324#436—611no—
325#436—611no—
326#436—1221no—
327#436—1221no—
328#432—611no—
329#432—1221no—
330#431—611no—
331#427—21no—
332C2×C4⋊C8C2×D6111yescharacteristic
333#416—21no—
334C4×M16—611no—
335C4×M16—1221no—
336#335—611no—
337#335—611no—
338#334—611no—
339#334—611no—
340#333—611no—
341#333—611no—
342#332—611no—
343#332—611no—
344#332—611no—
345#332—611no—
346#332—611no—
347#332—611no—
348C61#1918511yescharacteristic, complemented, Sylow
349C2×#41C22⋊D6111yescharacteristic
350C2×Q16—1221no—
351C2×Q16—1221no—
352C2×Q16—1221no—
353C2×SD16—1221no—
354C2×M16—611no—
355C2×M16—611no—
356C2×M16—1221no—
357C2×M16—1221no—
358C22×C8—21no—
359C4⋊Q8—611no—
360C4⋊Q8—611no—
361C4⋊Q8C22×D6111yescharacteristic
362C4⋊Q8C22×D6111yescharacteristic
363#125—21no—
364#125—1221no—
365C22⋊Q8—21no—
366C22⋊Q8—21no—
367C22⋊Q8—1221no—
368C22⋊Q8—1221no—
369C4×Q8—21no—
370C4×Q8—21no—
371C4×Q8—1221no—
372C4×Q8—1221no—
373C4×Q8C22⋊D6111yescharacteristic
374C4×Q8C22×D6111yescharacteristic
375C4×D4—21no—
376C4×D4—21no—
377C4×D4C22⋊D6111yescharacteristic
378C4×D4C22×D6111yescharacteristic
379#117—21no—
380#117—21no—
381#117—1221no—
382#117C22⋊D6111yescharacteristic
383C2×C42—21no—
384C2×C42C22×D6111yescharacteristic
385#107—21no—
386#107—21no—
387#107—1221no—
388#106—21no—
389#106—21no—
390#106—1221no—
391#106—1221no—
392#106—1221no—
393C4⋊C8—21no—
394C4⋊C8—611no—
395C4⋊C8—611no—
396C4⋊C8—611no—
397C4⋊C8—611no—
398C4⋊C8—1221no—
399C4⋊C8—1221no—
400C4⋊C8C22×D6111yescharacteristic
401C4⋊C8C22×D6111yescharacteristic
402#103—21no—
403#103—21no—
404#103—1221no—
405#103—1221no—
406#103—1221no—
407#103—1221no—
408#103—1221no—
409#103—1221no—
410#103—1221no—
411#102—21no—
412#102—21no—
413#102—1221no—
414C22⋊C8—1221no—
415C22⋊C8—1221no—
416C22⋊C8—1221no—
417C22⋊C8—1221no—
418#97—611no—
419#97—611no—
420#97—1221no—
421C4×C8—611no—
422C4×C8—611no—
423C4×C8—1221no—
424C4×C8—1221no—
425C4×C8—1221no—
426#41—22no—
427#41—41no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—1221no—
431C2×Q8—1221no—
432C2×Q8C2×C22⋊D6111yescharacteristic
433C2×D4—21no—
434C2×D4C2×C22⋊D6111yescharacteristic
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4D4×D6111yescharacteristic
439C22×C4#1175822411yescharacteristic
440C22×C4C2×C22⋊D6111yescharacteristic
441Q16—2442no—
442Q16—2442no—
443Q16—2442no—
444SD16—2442no—
445M16—1222no—
446M16—2442no—
447M16—2441no—
448M16—2442no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—42no—
452C2×C8—611no—
453C2×C8—611no—
454C2×C8—611no—
455C2×C8—611no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—2441no—
473C4⋊C4—2441no—
474C4⋊C4C2×C22⋊D6111yescharacteristic
475C4⋊C4C2×C22⋊D6111yescharacteristic
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—41no—
480C22⋊C4—2441no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—41no—
485C42—2441no—
486C42D4×D6111yescharacteristic
487C42#1175822411yescharacteristic
488C42C2×C22⋊D6111yescharacteristic
489C42C23×D6111yescharacteristic
490C23—21no—
491C23#1451974911yescharacteristic
492Q8—21no—
493Q8—41no—
494Q8—41no—
495Q8—81no—
496Q8—2441no—
497Q8—2441no—
498Q8—2441no—
499Q8—2441no—
500Q8#1451980812yes—
501D4—21no—
502D4—21no—
503D4—42no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—81no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4#1451974911yescharacteristic
521C2×C4#1451974911yescharacteristic
522C2×C4#1451974911yescharacteristic
523C2×C4C2×D4×D6111yescharacteristic
524C2×C4C22×C22⋊D6111yescharacteristic
525C2×C4C2×#1175822411yescharacteristic
526C8—41no—
527C8—41no—
528C8—612no—
529C8—1221no—
530C8—1221no—
531C8—1221no—
532C8—2441no—
533C8—2441no—
534C8—2441no—
535C8—2441no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22C2×#1451974911yescharacteristic
543C4—21no—
544C4—22no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—41no—
549C4—41no—
550C4—81no—
551C4—81no—
552C4—4881no—
553C4—4881no—
554C4#135×D6111yescharacteristic
555C4C2×#1451980811yescharacteristic
556C4#1684055911yescharacteristic
557C4#1684056011yescharacteristic
558C2—41no—
559C2—81no—
560C2#2145527011yescharacteristic
561C2#2145544511yescharacteristic
562C2#2145555811yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3076C2
socleC22×C61C2×#121
prime radicalC2×#41×C61D4
hypercenter#3076D61
hypocenterC61#19185
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3076

Complements

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

NH ≅ G/Ntype
1. #21456560C2semidirect
4. #21445396C2semidirect
5. #21445364C2semidirect
6. #21445320C2semidirect
7. #21445306C2semidirect
8. #21445279C2semidirect
9. #21445247C2semidirect
10. #21445203C2semidirect
11. #21445187C2semidirect
12. #21444569C2semidirect
13. #21444529C2semidirect
14. #21444517C2semidirect
348. C61#19185semidirect

Maximal subgroups

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

subgroupindex
#214565602
#214558902
#214548392
#214453962
#214453642
#214453202
#214453062
#214452792
#214452472
#214452032
#214451872
#214445692
#214445292
#214445172
C61×#30762
#1918561