Finite Group Databasebeta

← #29662729

Subgroups — #29662729

Order 15616.

596 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456401C211yesmaximal, characteristic, complemented
2#21455607C211yesmaximal, characteristic
3#21454786C211yesmaximal, characteristic
4#21445403C211yesmaximal, characteristic, complemented
5#21445399C211yesmaximal, characteristic, complemented
6#21445396C211yesmaximal, characteristic, complemented
7#21445392C211yesmaximal, characteristic, complemented
8#21445345C211yesmaximal, characteristic, complemented
9#21445340C211yesmaximal, characteristic, complemented
10#21445311C211yesmaximal, characteristic, complemented
11#21445306C211yesmaximal, characteristic, complemented
12C2×#16839512C211yesmaximal, characteristic, complemented
13#21444477C211yesmaximal, characteristic, complemented
14#21444476C211yesmaximal, characteristic, complemented
15C61×#3076C211yesmaximal, characteristic
16#16840319C2211yescharacteristic
17#16840165C2211yescharacteristic
18#16840141C2211yescharacteristic
19#16840114C2211yescharacteristic
20#16840112C2211yescharacteristic
21#16840069—21no—
22#16840062—21no—
23#16840057C2211yescharacteristic
24#16840052C2211yescharacteristic
25#16840047C2211yescharacteristic
26#16840042C2211yescharacteristic
27#16840001—21no—
28#16840001—21no—
29#16839893C2211yescharacteristic
30#16839893C2211yescharacteristic
31C4×#11758202—21no—
32C4×#11758202—21no—
33#16839840C2211yescharacteristic
34#16839840C2211yescharacteristic
35#16839834—21no—
36#16839834—21no—
37#16839815—21no—
38#16839815C2211yescharacteristic
39#16839798—21no—
40C2×C4⋊#92418C2211yescharacteristic
41#16839758—21no—
42C4×#11758194—21no—
43C2×C4×#92418C2211yescharacteristic
44#16839512C2211yescharacteristic
45#16839512C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839512C2211yescharacteristic
48#16839511C2211yescharacteristic
49#16839511C2211yescharacteristic
50#16839510C2211yescharacteristic
51#16839510C2211yescharacteristic
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×#11758194—21no—
70C2×#11758194—21no—
71C22×#92418—21no—
72C22×#92418D411yescharacteristic
73C22×#92418D411yescharacteristic
74#14519777C2311yescharacteristic
75#14519777C2311yescharacteristic
76#14519769—21no—
77#14519756—21no—
78#14519756—21no—
79C4×Dic122—21no—
80C4×Dic122—21no—
81#14519736—21no—
82#14519710—21no—
83#14519710—21no—
84#14519710—21no—
85#14519710—21no—
86#14519708—21no—
87#14519708—21no—
88C4⋊#92418—21no—
89C4⋊#92418—21no—
90C4⋊#92418—21no—
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#14519698—21no—
102#14519698—21no—
103C22⋊#92418—21no—
104C22⋊#92418—21no—
105C22⋊#92418—21no—
106#14519688—21no—
107#14519688—21no—
108C4×#92418—21no—
109C4×#92418D411yescharacteristic
110C4×#92418D411yescharacteristic
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—22no—
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—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191#11758196—41no—
192#11758196—41no—
193C4×Dic61—41no—
194#11758194—42no—
195#11758194—42no—
196C2×#92418—22no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—41no—
202C2×#92418—42no—
203C2×#92418C2×D411yescharacteristic
204C2×#92418C2×D411yescharacteristic
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207C23×C61—21no—
208C23×C61#12111yescharacteristic
209Q8×C61—21no—
210Q8×C61—21no—
211Q8×C61—41no—
212Q8×C61—41no—
213Q8×C61—81no—
214D4×C61—21no—
215D4×C61—21no—
216D4×C61—42no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—41no—
223C2×C4×C61—42no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—42no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61#12111yescharacteristic
230C2×C4×C61#12111yescharacteristic
231C2×C4×C61#12111yescharacteristic
232C2×C4×C61#13712yes—
233C2×C4×C61#13712yes—
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×Dic61—41no—
238C2×Dic61—41no—
239Dic122—41no—
240Dic122—41no—
241Dic122—81no—
242C8×C61—41no—
243C8×C61—41no—
244#92418—21no—
245#92418—21no—
246#92418—21no—
247#92418—41no—
248#92418—41no—
249#92418—41no—
250#92418—41no—
251#92418#13512yes—
252#19158—611nomaximal, Sylow
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—41no—
256C22×C61—41no—
257C22×C61—41no—
258C22×C61#47012yes—
259C22×C61C2×#12111yescharacteristic
260C4×C61—21no—
261C4×C61—21no—
262C4×C61—21no—
263C4×C61—41no—
264C4×C61—41no—
265C4×C61—41no—
266C4×C61—81no—
267C4×C61—81no—
268C4×C61#47012yes—
269C4×C61C2×#13511yescharacteristic
270C4×C61C2×#13711yescharacteristic
271C4×C61C2×#13711yescharacteristic
272C4×C61#57511yescharacteristic
273Dic61—81no—
274Dic61—81no—
275#3170—611no—
276#3076—611no—
277#3076D6111yescharacteristic
278#2965—611no—
279#1704—611no—
280#1704—611no—
281#1703—611no—
282#1703—611no—
283#1696—611no—
284#1695—611no—
285#1690—611no—
286#1689—611no—
287C2×#334—611no—
288#1542—611no—
289#1542—611no—
290C2×C61—22no—
291C2×C61—81no—
292C2×C61#303511yescharacteristic
293C2×C61C2×#47011yescharacteristic
294C2×C61#306711yescharacteristic
295#533—611no—
296#533C2×D6111yescharacteristic
297C4×#41C2×D6111yescharacteristic
298#500—611no—
299#494—611no—
300#487—611no—
301#487—611no—
302#479—21no—
303#479—1221no—
304#478—21no—
305#478—1221no—
306#477—611no—
307#477C2×D6111yescharacteristic
308#476—611no—
309#476C2×D6111yescharacteristic
310#475—611no—
311#475C2×D6111yescharacteristic
312#474—611no—
313#474C2×D6111yescharacteristic
314#467—1221no—
315#467—1221no—
316#446—611no—
317#446—611no—
318C4×Q16—1221no—
319C4×Q16—1221no—
320#436—611no—
321#436—611no—
322#435—1221no—
323#435—1221no—
324#431—611no—
325#431—1221no—
326#427—21no—
327#427—1221no—
328C2×C4⋊C8—611no—
329C2×C4⋊C8C2×D6111yescharacteristic
330#416—21no—
331#416—1221no—
332C4×M16—1221no—
333C2×C4×C8—611no—
334#334—611no—
335#334—611no—
336#334—611no—
337#334—611no—
338#333—611no—
339#333—611no—
340#333—611no—
341#333—611no—
342#332—611no—
343#332—611no—
344#332—611no—
345#332—611no—
346C61#1915811yescharacteristic, complemented, Sylow
347C2×#41C22⋊D6111yescharacteristic
348C2×Q16—1221no—
349C2×Q16—1221no—
350C2×M16—1221no—
351C2×M16—1221no—
352C22×C8—21no—
353C22×C8—611no—
354C22×C8—611no—
355C22×C8—1221no—
356C4⋊Q8—611no—
357C4⋊Q8—611no—
358C4⋊Q8C22×D6111yescharacteristic
359C4⋊Q8C22×D6111yescharacteristic
360#125—21no—
361#125—1221no—
362C22⋊Q8—21no—
363C22⋊Q8—21no—
364C22⋊Q8—1221no—
365C22⋊Q8—1221no—
366C4×Q8—21no—
367C4×Q8—21no—
368C4×Q8—1221no—
369C4×Q8—1221no—
370C4×Q8C22⋊D6111yescharacteristic
371C4×Q8C22×D6111yescharacteristic
372C4×D4—21no—
373C4×D4—21no—
374C4×D4C22⋊D6111yescharacteristic
375C4×D4C22×D6111yescharacteristic
376#117—21no—
377#117—21no—
378#117—1221no—
379#117C22⋊D6111yescharacteristic
380C2×C42—21no—
381C2×C42C22×D6111yescharacteristic
382#107—21no—
383#107—21no—
384#107—1221no—
385#107—1221no—
386#107—1221no—
387#107—1221no—
388#106—21no—
389#106—21no—
390#106—1221no—
391#106—1221no—
392C4⋊C8—21no—
393C4⋊C8—611no—
394C4⋊C8—611no—
395C4⋊C8—611no—
396C4⋊C8—611no—
397C4⋊C8—1221no—
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#102—21no—
411#102—21no—
412#102—1221no—
413#102—1221no—
414C22⋊C8—1221no—
415C22⋊C8—1221no—
416C22⋊C8—1221no—
417#97—1221no—
418#97—1221no—
419C4×C8—611no—
420C4×C8—611no—
421C4×C8—611no—
422C4×C8—611no—
423C4×C8—1221no—
424#41—22no—
425#41—22no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—1221no—
429C2×Q8—1221no—
430C2×Q8C2×C22⋊D6111yescharacteristic
431C2×D4—21no—
432C2×D4C2×C22⋊D6111yescharacteristic
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4D4×D6111yescharacteristic
437C22×C4#1175822411yescharacteristic
438C22×C4C2×C22⋊D6111yescharacteristic
439Q16—2442no—
440Q16—2442no—
441M16—2442no—
442M16—2442no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—42no—
446C2×C8—611no—
447C2×C8—611no—
448C2×C8—611no—
449C2×C8—611no—
450C2×C8—1222no—
451C2×C8—1221no—
452C2×C8—1221no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8—2441no—
456C2×C8—2442no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—2441no—
470C4⋊C4—2441no—
471C4⋊C4C2×C22⋊D6111yescharacteristic
472C4⋊C4C2×C22⋊D6111yescharacteristic
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—41no—
477C22⋊C4—2441no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—41no—
482C42—2441no—
483C42D4×D6111yescharacteristic
484C42#1175822411yescharacteristic
485C42C2×C22⋊D6111yescharacteristic
486C42C23×D6111yescharacteristic
487C23—21no—
488C23#1451974911yescharacteristic
489Q8—21no—
490Q8—21no—
491Q8—41no—
492Q8—41no—
493Q8—81no—
494Q8—4881no—
495Q8—2441no—
496Q8—2441no—
497D4—21no—
498D4—21no—
499D4—42no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4#1451974911yescharacteristic
515C2×C4#1451974911yescharacteristic
516C2×C4#1451974911yescharacteristic
517C2×C4#1451980712yes—
518C2×C4#1451980812yes—
519C2×C4C2×D4×D6111yescharacteristic
520C2×C4C22×C22⋊D6111yescharacteristic
521C2×C4C2×#1175822411yescharacteristic
522C8—41no—
523C8—41no—
524C8—612no—
525C8—1221no—
526C8—1221no—
527C8—1221no—
528C8—2441no—
529C8—2441no—
530C8—2441no—
531C8—2441no—
532C22—21no—
533C22—21no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22#1684001812yes—
538C22C2×#1451974911yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—81no—
546C4—81no—
547C4—4881no—
548C4—4881no—
549C4#1684001812yes—
550C4#135×D6111yescharacteristic
551C4C2×#1451980711yescharacteristic
552C4C2×#1451980811yescharacteristic
553C4#1684055911yescharacteristic
554C2—22no—
555C2—81no—
556C2#2145527011yescharacteristic
557C2C2×#1684001811yescharacteristic
558C2#2145553611yescharacteristic
subgroupG/H
centerC2×C4#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3076C2
socleC22×C61C2×#121
prime radicalC2×#41×C61D4
hypercenter#3076D61
hypocenterC61#19158
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3076

Complements

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

NH ≅ G/Ntype
1. #21456401C2semidirect
4. #21445403C2semidirect
5. #21445399C2semidirect
6. #21445396C2semidirect
7. #21445392C2semidirect
8. #21445345C2semidirect
9. #21445340C2semidirect
10. #21445311C2semidirect
11. #21445306C2semidirect
12. C2×#16839512C2semidirect
13. #21444477C2semidirect
14. #21444476C2semidirect
346. C61#19158semidirect

Maximal subgroups

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

subgroupindex
#214564012
#214556072
#214547862
#214454032
#214453992
#214453962
#214453922
#214453452
#214453402
#214453112
#214453062
C2×#168395122
#214444772
#214444762
C61×#30762
#1915861