Finite Group Databasebeta

← #29665094

Subgroups — #29665094

Order 15616.

572 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457186C211yesmaximal, characteristic
2#21457040C211yesmaximal, characteristic
3#21454823C211yesmaximal, characteristic
4#21445344C211yesmaximal, characteristic, complemented
5#21445339C211yesmaximal, characteristic, complemented
6#21445322C211yesmaximal, characteristic, complemented
7#21445315C211yesmaximal, characteristic, complemented
8#21445220C211yesmaximal, characteristic, complemented
9#21445215C211yesmaximal, characteristic, complemented
10#21445198C211yesmaximal, characteristic, complemented
11#21445191C211yesmaximal, characteristic, complemented
12#21444586C211yesmaximal, characteristic, complemented
13#21444529C211yesmaximal, characteristic, complemented
14#21444500C211yesmaximal, characteristic, complemented
15C61×#3165C211yesmaximal, characteristic, complemented
16#16840296C2211yescharacteristic
17#16840168—21no—
18#16840124—21no—
19#16840092—21no—
20#16840075—21no—
21#16840069—21no—
22#16840054C2211yescharacteristic
23#16840054C2211yescharacteristic
24#16840054C2211yescharacteristic
25#16840049C2211yescharacteristic
26#16840049C2211yescharacteristic
27#16840049C2211yescharacteristic
28#16840018—21no—
29#16840010—21no—
30#16839984—21no—
31#16839969C2211yescharacteristic
32#16839964C2211yescharacteristic
33#16839892C2211yescharacteristic
34C8×Dic122C2211yescharacteristic
35#16839841C2211yescharacteristic
36#16839838—21no—
37#16839831—21no—
38C8×D244C2211yescharacteristic
39#16839824—21no—
40#16839819C2211yescharacteristic
41#16839815C2211yescharacteristic
42#16839786C2211yescharacteristic
43#16839719—21no—
44#16839515C2211yescharacteristic
45#16839515C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839513C2211yescharacteristic
48#16839511C2211yescharacteristic
49#16839511C2211yescharacteristic
50#16839511C2211yescharacteristic
51#16839511C2211yescharacteristic
52#16839511C2211yescharacteristic
53#16839511C2211yescharacteristic
54#16839508C2211yescharacteristic
55#16839508C2211yescharacteristic
56C61×#533C2211yescharacteristic
57C2×C4⋊Q8×C61C2211yescharacteristic
58C61×#501—21no—
59C61×#501—21no—
60C61×#500C2211yescharacteristic
61C61×#500C2211yescharacteristic
62C61×#498C2211yescharacteristic
63C61×#498C2211yescharacteristic
64C61×#428—21no—
65C61×#428—21no—
66C4×M16×C61C2211yescharacteristic
67C2×#11758202—21no—
68C2×#11758201—21no—
69C2×#11758194—21no—
70C2×#11758189—21no—
71C2×C8×D61—21no—
72#14519771—21no—
73#14519769D411yescharacteristic
74#14519765D411yescharacteristic
75#14519760—21no—
76#14519756D411yescharacteristic
77#14519751D411yescharacteristic
78C4×Dic122C2311yescharacteristic
79C4×D244C2311yescharacteristic
80C4×C22⋊D61—21no—
81#14519734—21no—
82#14519710—21no—
83#14519710—21no—
84#14519710—21no—
85#14519708—21no—
86#14519708—21no—
87#14519708—21no—
88C4⋊#92418—21no—
89C4⋊#92418C2311yescharacteristic
90C4⋊#92418C2311yescharacteristic
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95#14519706—21no—
96#14519706—21no—
97#14519701—21no—
98#14519701—21no—
99#14519700—21no—
100#14519700—21no—
101#14519700—21no—
102#14519697—21no—
103#14519697—21no—
104#14519697—21no—
105C22⋊#92418—21no—
106C22⋊#92418—21no—
107#14519689—21no—
108#14519689—21no—
109#14519688—21no—
110#14519687—21no—
111C4×#92418—21no—
112C8×Dic61—21no—
113C22×Q8×C61—21no—
114C2×M16×C61D411yescharacteristic
115C2×M16×C61D411yescharacteristic
116C4⋊Q8×C61—21no—
117C4⋊Q8×C61—21no—
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61C2311yescharacteristic
120C4⋊Q8×C61C2311yescharacteristic
121C4⋊Q8×C61C2311yescharacteristic
122C4⋊Q8×C61C2311yescharacteristic
123#125×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—21no—
126C4×Q8×C61—21no—
127C4×Q8×C61—21no—
128#117×C61—21no—
129C2×C4⋊C4×C61—21no—
130C2×C4⋊C4×C61—21no—
131C2×C42×C61C2311yescharacteristic
132#107×C61—21no—
133#107×C61—21no—
134#107×C61—21no—
135#107×C61—21no—
136#106×C61—21no—
137#106×C61—21no—
138#106×C61—21no—
139#106×C61—21no—
140#97×C61D411yescharacteristic
141#97×C61D411yescharacteristic
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61—41no—
149C2×Q8×C61—82no—
150C22×C4×C61—41no—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61#4111yescharacteristic
154C2×C22⋊D61—21no—
155C2×D244C2×D411yescharacteristic
156C2×C4×D61—21no—
157C2×Dic122C2×D411yescharacteristic
158M16×C61—22no—
159M16×C61—22no—
160C2×C8×C61C2×D411yescharacteristic
161C2×C8×C61C2×D411yescharacteristic
162C2×C8×C61C2×D411yescharacteristic
163C2×C8×C61C2×D411yescharacteristic
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×C61—41no—
175C4⋊C4×C61—41no—
176C22⋊C4×C61—41no—
177C42×C61—41no—
178C42×C61C2×D411yescharacteristic
179C42×C61C2×D411yescharacteristic
180C42×C61#4111yescharacteristic
181C42×C61C2411yescharacteristic
182C22⋊Dic61—21no—
183#11758202—81no—
184#11758201—81no—
185#11758198—21no—
186#11758198—21no—
187#11758198—21no—
188#11758198—21no—
189C4⋊Dic61C2×D411yescharacteristic
190C4⋊Dic61C2×D411yescharacteristic
191#11758196—21no—
192#11758196—21no—
193#11758196—21no—
194#11758196—21no—
195C4×Dic61—21no—
196#11758194—81no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203#11758189—42no—
204C8×D61—42no—
205C23×C61#12111yescharacteristic
206C22×D61—21no—
207Q8×C61—41no—
208Q8×C61—41no—
209Q8×C61—82no—
210Q8×C61—81no—
211Q8×C61—81no—
212Q8×C61—81no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—82no—
223C2×C4×C61#12111yescharacteristic
224C2×C4×C61#12111yescharacteristic
225C2×C4×C61#12111yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C22⋊D61—81no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233D244—21no—
234D244—21no—
235C4×D61—41no—
236C4×D61—41no—
237Dic122—21no—
238Dic122—21no—
239C8×C61—21no—
240C8×C61—21no—
241C8×C61#13712yes—
242C8×C61#13712yes—
243#92418—41no—
244#92418—41no—
245#92418—41no—
246#92418—41no—
247#92418—41no—
248#92418—41no—
249#19379—611nomaximal, Sylow
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—21no—
253C22×C61C2×#12111yescharacteristic
254C2×D61—41no—
255C2×D61—41no—
256C2×D61—41no—
257C4×C61—21no—
258C4×C61—21no—
259C4×C61—21no—
260C4×C61—21no—
261C4×C61—81no—
262C4×C61—81no—
263C4×C61—81no—
264C4×C61—81no—
265C4×C61C2×#13711yescharacteristic
266C4×C61C2×#13711yescharacteristic
267C4×C61#57711yescharacteristic
268C4×C61#57811yescharacteristic
269Dic61—41no—
270Dic61—41no—
271Dic61—41no—
272#3252—611no—
273#3238—611no—
274#3165D6111yescharacteristic, complemented
275#2969—611no—
276#1696—611no—
277#1695—611no—
278#1692—611no—
279#1691—611no—
280#1676—611no—
281#1675—611no—
282#1672—611no—
283#1671—611no—
284#1569—611no—
285#1554—611no—
286#1548—611no—
287C2×C61—41no—
288C2×C61#303911yescharacteristic
289C2×C61#306311yescharacteristic
290C2×C61#306811yescharacteristic
291D61—81no—
292#533C2×D6111yescharacteristic
293C2×C4⋊Q8C2×D6111yescharacteristic
294#529—611no—
295#501—21no—
296#501—21no—
297#501—1221no—
298#500C2×D6111yescharacteristic
299#500C2×D6111yescharacteristic
300#498C2×D6111yescharacteristic
301#498C2×D6111yescharacteristic
302#489—1221no—
303#483—1221no—
304#480—1221no—
305#479—1221no—
306#477—611no—
307#477—611no—
308#477—611no—
309#476—611no—
310#476—611no—
311#476—611no—
312#470—1221no—
313#468—1221no—
314#464—1221no—
315#462—611no—
316#461—611no—
317#446—611no—
318C8×Q8—611no—
319#436—611no—
320#436—1221no—
321#435—1221no—
322C8×D4—611no—
323#433—1221no—
324#432—611no—
325#431—611no—
326#428—21no—
327#428—21no—
328#423—611no—
329#405—1221no—
330C4×M16C2×D6111yescharacteristic
331#335—611no—
332#335—611no—
333#334—611no—
334#334—611no—
335#333—611no—
336#333—611no—
337#333—611no—
338#333—611no—
339#333—611no—
340#333—611no—
341#332—611no—
342#332—611no—
343C61#1937911yescharacteristic, complemented, Sylow
344C22×Q8—21no—
345C2×Q16—1221no—
346C2×SD16—1221no—
347C2×M16—1221no—
348C2×M16—1221no—
349C2×M16C22⋊D6111yescharacteristic
350C2×M16C22⋊D6111yescharacteristic
351C22×C8—1221no—
352C4⋊Q8—21no—
353C4⋊Q8—21no—
354C4⋊Q8—21no—
355C4⋊Q8C22×D6111yescharacteristic
356C4⋊Q8C22×D6111yescharacteristic
357C4⋊Q8C22×D6111yescharacteristic
358C4⋊Q8C22×D6111yescharacteristic
359#126—1221no—
360#125—21no—
361#125—611no—
362#124—611no—
363#123—1221no—
364C22⋊Q8—21no—
365C22⋊Q8—21no—
366C22⋊Q8—611no—
367#121—611no—
368C4×Q8—21no—
369C4×Q8—21no—
370C4×Q8—611no—
371C4×D4—611no—
372C4×D4—1221no—
373#117—21no—
374#117—1221no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C42C22×D6111yescharacteristic
378#107—21no—
379#107—21no—
380#107—21no—
381#107—21no—
382#107—1221no—
383#107—1221no—
384#107—1221no—
385#106—21no—
386#106—21no—
387#106—21no—
388#106—21no—
389#106—1221no—
390#106—1221no—
391#106—1221no—
392C4⋊C8—611no—
393C4⋊C8—611no—
394C4⋊C8—611no—
395C4⋊C8—611no—
396C4⋊C8—611no—
397C4⋊C8—611no—
398C4⋊C8—1221no—
399C4⋊C8—1221no—
400C4⋊C8—1221no—
401#103—1221no—
402#103—1221no—
403#103—1221no—
404#103—1221no—
405#103—1221no—
406#102—1221no—
407#102—1221no—
408#102—1221no—
409C22⋊C8—1221no—
410C22⋊C8—1221no—
411C22⋊C8—1221no—
412C22⋊C8—1221no—
413#97—1221no—
414#97—1221no—
415#97C22⋊D6111yescharacteristic
416#97C22⋊D6111yescharacteristic
417C4×C8—1221no—
418C4×C8—1221no—
419C4×C8C22×D6111yescharacteristic
420C4×C8C22×D6111yescharacteristic
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—41no—
426C2×Q8—82no—
427C2×Q8—611no—
428C2×D4—611no—
429C2×D4—1221no—
430C22×C4—41no—
431C22×C4—1221no—
432C22×C4D4×D6111yescharacteristic
433C22×C4#1175822411yescharacteristic
434C22×C4C2×C22⋊D6111yescharacteristic
435Q16—4881no—
436SD16—4881no—
437M16—22no—
438M16—22no—
439M16—4881no—
440M16—2442no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8—1221no—
446C2×C8—1221no—
447C2×C8—2442no—
448C2×C8C2×C22⋊D6111yescharacteristic
449C2×C8C2×C22⋊D6111yescharacteristic
450C2×C8C2×C22⋊D6111yescharacteristic
451C2×C8C2×C22⋊D6111yescharacteristic
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—611no—
465C4⋊C4—611no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C22⋊C4—41no—
471C22⋊C4—1221no—
472C22⋊C4—1221no—
473C22⋊C4—1221no—
474C22⋊C4—1221no—
475C22⋊C4—1221no—
476C42—41no—
477C42—1221no—
478C42D4×D6111yescharacteristic
479C42#1175822411yescharacteristic
480C42C2×C22⋊D6111yescharacteristic
481C42C23×D6111yescharacteristic
482C23—1221no—
483C23#1451974811yescharacteristic
484Q8—41no—
485Q8—41no—
486Q8—81no—
487Q8—81no—
488Q8—81no—
489Q8—82no—
490Q8—1221no—
491Q8—1221no—
492D4—1221no—
493D4—1221no—
494D4—4881no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—82no—
505C2×C4—1221no—
506C2×C4—1221no—
507C2×C4—1221no—
508C2×C4—2441no—
509C2×C4—2441no—
510C2×C4#1451974811yescharacteristic
511C2×C4#1451974811yescharacteristic
512C2×C4#1451974811yescharacteristic
513C2×C4C2×D4×D6111yescharacteristic
514C2×C4C22×C22⋊D6111yescharacteristic
515C2×C4C2×#1175822411yescharacteristic
516C8—21no—
517C8—21no—
518C8—2441no—
519C8—2441no—
520C8—2441no—
521C8—2441no—
522C8—2441no—
523C8—2441no—
524C8#1451980812yes—
525C8#1451980812yes—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—2441no—
530C22—2441no—
531C22—2441no—
532C22C2×#1451974811yescharacteristic
533C4—21no—
534C4—21no—
535C4—21no—
536C4—21no—
537C4—81no—
538C4—81no—
539C4—81no—
540C4—81no—
541C4—2441no—
542C4—2441no—
543C4—2441no—
544C4C2×#1451980811yescharacteristic
545C4C2×#1451980811yescharacteristic
546C4#1684057111yescharacteristic
547C4#1684057811yescharacteristic
548C2—41no—
549C2—4881no—
550C2#2145530211yescharacteristic
551C2#2145550411yescharacteristic
552C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3165C2
socleC22×C61C2×#121
prime radical#14519751D4
hypercenter#3165D61
hypocenterC61#19379
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
4. #21445344C2semidirect
5. #21445339C2semidirect
6. #21445322C2semidirect
7. #21445315C2semidirect
8. #21445220C2semidirect
9. #21445215C2semidirect
10. #21445198C2semidirect
11. #21445191C2semidirect
12. #21444586C2semidirect
13. #21444529C2semidirect
14. #21444500C2semidirect
15. C61×#3165C2semidirect
274. #3165D61semidirect
343. C61#19379semidirect

Maximal subgroups

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

subgroupindex
#214571862
#214570402
#214548232
#214453442
#214453392
#214453222
#214453152
#214452202
#214452152
#214451982
#214451912
#214445862
#214445292
#214445002
C61×#31652
#1937961