Finite Group Databasebeta

← #29663371

Subgroups — #29663371

Order 15616.

562 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456994C211yesmaximal, characteristic
2#21456401C211yesmaximal, characteristic, complemented
3#21454795C211yesmaximal, characteristic
4#21445403C211yesmaximal, characteristic, complemented
5#21445399C211yesmaximal, characteristic, complemented
6#21445396C211yesmaximal, characteristic, complemented
7#21445392C211yesmaximal, characteristic, complemented
8#21445221C211yesmaximal, characteristic, complemented
9#21445216C211yesmaximal, characteristic, complemented
10#21445187C211yesmaximal, characteristic, complemented
11#21445182C211yesmaximal, characteristic, complemented
12#21444565C211yesmaximal, characteristic, complemented
13#21444529C211yesmaximal, characteristic, complemented
14#21444527C211yesmaximal, characteristic, complemented
15C61×#3233C211yesmaximal, characteristic
16#16840319C2211yescharacteristic
17#16840165C2211yescharacteristic
18#16840161—21no—
19#16840141C2211yescharacteristic
20#16840116—21no—
21#16840114C2211yescharacteristic
22#16840112C2211yescharacteristic
23#16840100—21no—
24#16840083—21no—
25#16840057C2211yescharacteristic
26#16840052C2211yescharacteristic
27#16840047C2211yescharacteristic
28#16840042C2211yescharacteristic
29#16840001—21no—
30#16840001—21no—
31#16839893C2211yescharacteristic
32Q8×#92418C2211yescharacteristic
33C4×#11758202—21no—
34C4×#11758202—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#16839511C2211yescharacteristic
49#16839511C2211yescharacteristic
50#16839510C2211yescharacteristic
51#16839510C2211yescharacteristic
52#16839508C2211yescharacteristic
53#16839508C2211yescharacteristic
54#16839507C2211yescharacteristic
55#16839507C2211yescharacteristic
56C61×#533C2211yescharacteristic
57C61×#529C2211yescharacteristic
58C61×#499—21no—
59C61×#488—21no—
60C61×#485—21no—
61C61×#482—21no—
62C61×#477C2211yescharacteristic
63C61×#476C2211yescharacteristic
64C61×#475C2211yescharacteristic
65C61×#474C2211yescharacteristic
66C61×#431C2211yescharacteristic
67C2×#11758202—21no—
68C2×#11758202—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#14519710—21no—
84#14519710—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#14519701—21no—
97#14519700—21no—
98#14519700—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
113C4⋊Q8×C61C2311yescharacteristic
114C4⋊Q8×C61C2311yescharacteristic
115#126×C61—21no—
116#125×C61—21no—
117#125×C61D411yescharacteristic
118#124×C61D411yescharacteristic
119#123×C61—21no—
120C22⋊Q8×C61—21no—
121C22⋊Q8×C61—21no—
122C22⋊Q8×C61D411yescharacteristic
123#121×C61D411yescharacteristic
124C4×Q8×C61—21no—
125C4×Q8×C61—21no—
126C4×Q8×C61C2311yescharacteristic
127C4×D4×C61—21no—
128C4×D4×C61C2311yescharacteristic
129#117×C61—21no—
130#117×C61—21no—
131C2×C42×C61C2311yescharacteristic
132#107×C61—21no—
133#107×C61—21no—
134#106×C61—21no—
135#106×C61—21no—
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—
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
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—41no—
166C4⋊C4×C61—41no—
167C4⋊C4×C61C2×D411yescharacteristic
168C4⋊C4×C61C2×D411yescharacteristic
169C22⋊C4×C61—21no—
170C22⋊C4×C61—21no—
171C22⋊C4×C61—21no—
172C22⋊C4×C61—21no—
173C22⋊C4×C61—21no—
174C22⋊C4×C61—41no—
175C42×C61—21no—
176C42×C61—41no—
177C42×C61C2×D411yescharacteristic
178C42×C61C2×D411yescharacteristic
179C42×C61#4111yescharacteristic
180C42×C61C2411yescharacteristic
181C22⋊Dic61—41no—
182#11758202—42no—
183#11758202—42no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188#11758196—41no—
189#11758196—41no—
190C4×Dic61—41no—
191#11758194—42no—
192C2×#92418—22no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—41no—
198C2×#92418—42no—
199C2×#92418C2×D411yescharacteristic
200C2×#92418C2×D411yescharacteristic
201C2×#92418C2×D411yescharacteristic
202C2×#92418C2×D411yescharacteristic
203C23×C61—21no—
204C23×C61#12111yescharacteristic
205Q8×C61—21no—
206Q8×C61—21no—
207Q8×C61—41no—
208Q8×C61—41no—
209Q8×C61—81no—
210D4×C61—21no—
211D4×C61—21no—
212D4×C61—42no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61#12111yescharacteristic
226C2×C4×C61#12111yescharacteristic
227C2×C4×C61#12111yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×Dic61—41no—
232C2×Dic61—41no—
233Dic122—41no—
234Dic122—41no—
235Dic122—81no—
236C8×C61—41no—
237C8×C61—41no—
238#92418—21no—
239#92418—21no—
240#92418—21no—
241#92418—41no—
242#92418—41no—
243#92418—41no—
244#92418—41no—
245#92418#13512yes—
246#19220—611nomaximal, Sylow
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—21no—
250C22×C61—41no—
251C22×C61—41no—
252C22×C61—41no—
253C22×C61C2×#12111yescharacteristic
254C4×C61—21no—
255C4×C61—21no—
256C4×C61—21no—
257C4×C61—21no—
258C4×C61—41no—
259C4×C61—41no—
260C4×C61—41no—
261C4×C61—81no—
262C4×C61—81no—
263C4×C61C2×#13511yescharacteristic
264C4×C61#57511yescharacteristic
265C4×C61#57811yescharacteristic
266C4×C61#57811yescharacteristic
267Dic61—81no—
268Dic61—81no—
269#3233—611no—
270#3233D6111yescharacteristic
271#3170—611no—
272#2966—611no—
273#1704—611no—
274#1704—611no—
275#1703—611no—
276#1703—611no—
277#1676—611no—
278#1675—611no—
279#1670—611no—
280#1669—611no—
281#1564—611no—
282#1554—611no—
283#1554—611no—
284C2×C61—41no—
285C2×C61—81no—
286C2×C61#303511yescharacteristic
287C2×C61#305911yescharacteristic
288C2×C61#306711yescharacteristic
289#533—611no—
290#533C2×D6111yescharacteristic
291#529C2×D6111yescharacteristic
292#500—611no—
293#499—21no—
294#499—1221no—
295#494—611no—
296#488—21no—
297#488—1221no—
298#487—611no—
299#487—611no—
300#485—21no—
301#485—1221no—
302#482—21no—
303#482—1221no—
304#477—611no—
305#477C2×D6111yescharacteristic
306#476—611no—
307#476C2×D6111yescharacteristic
308#475—611no—
309#475C2×D6111yescharacteristic
310#474—611no—
311#474C2×D6111yescharacteristic
312#467—1221no—
313#467—1221no—
314#446—611no—
315C8×Q8—611no—
316C4×Q16—1221no—
317C4×Q16—1221no—
318#436—611no—
319#435—1221no—
320C8×D4—611no—
321C8×D4—1221no—
322#433—1221no—
323#432—611no—
324#431—611no—
325#431C2×D6111yescharacteristic
326#405—1221no—
327C2×C4×C8—611no—
328#335—611no—
329#335—611no—
330#334—611no—
331#334—611no—
332#333—611no—
333#333—611no—
334#333—611no—
335#333—611no—
336#332—611no—
337#332—611no—
338#332—611no—
339#332—611no—
340C61#1922011yescharacteristic, complemented, Sylow
341C2×Q16—1221no—
342C2×Q16—1221no—
343C2×M16—1221no—
344C22×C8—611no—
345C22×C8—611no—
346C22×C8—1221no—
347C4⋊Q8—611no—
348C4⋊Q8—611no—
349C4⋊Q8C22×D6111yescharacteristic
350C4⋊Q8C22×D6111yescharacteristic
351#126—21no—
352#125—21no—
353#125—1221no—
354#125C22⋊D6111yescharacteristic
355#124C22⋊D6111yescharacteristic
356#123—21no—
357C22⋊Q8—21no—
358C22⋊Q8—21no—
359C22⋊Q8—1221no—
360C22⋊Q8—1221no—
361C22⋊Q8C22⋊D6111yescharacteristic
362#121C22⋊D6111yescharacteristic
363C4×Q8—21no—
364C4×Q8—21no—
365C4×Q8—1221no—
366C4×Q8—1221no—
367C4×Q8C22×D6111yescharacteristic
368C4×D4—21no—
369C4×D4C22×D6111yescharacteristic
370#117—21no—
371#117—21no—
372#117—1221no—
373C2×C42C22×D6111yescharacteristic
374#107—21no—
375#107—21no—
376#107—1221no—
377#107—1221no—
378#107—1221no—
379#107—1221no—
380#106—21no—
381#106—21no—
382#106—1221no—
383#106—1221no—
384C4⋊C8—611no—
385C4⋊C8—611no—
386C4⋊C8—611no—
387C4⋊C8—611no—
388C4⋊C8—1221no—
389C4⋊C8—1221no—
390C4⋊C8C22×D6111yescharacteristic
391C4⋊C8C22×D6111yescharacteristic
392#103—21no—
393#103—21no—
394#103—1221no—
395#103—1221no—
396#103—1221no—
397#103—1221no—
398#103—1221no—
399#103—1221no—
400#102—21no—
401#102—21no—
402#102—1221no—
403#102—1221no—
404C22⋊C8—21no—
405C22⋊C8—1221no—
406C22⋊C8—1221no—
407C22⋊C8—1221no—
408C22⋊C8—1221no—
409#97—1221no—
410#97—1221no—
411C4×C8—21no—
412C4×C8—611no—
413C4×C8—611no—
414C4×C8—611no—
415C4×C8—611no—
416C4×C8—1221no—
417C4×C8—1221no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—1221no—
421C2×Q8—1221no—
422C2×Q8C2×C22⋊D6111yescharacteristic
423C2×D4—21no—
424C2×D4C2×C22⋊D6111yescharacteristic
425C22×C4—21no—
426C22×C4D4×D6111yescharacteristic
427C22×C4#1175822411yescharacteristic
428C22×C4C2×C22⋊D6111yescharacteristic
429Q16—2442no—
430Q16—2442no—
431M16—2442no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—611no—
435C2×C8—611no—
436C2×C8—611no—
437C2×C8—611no—
438C2×C8—1222no—
439C2×C8—1221no—
440C2×C8—1221no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8—2442no—
444C2×C8—2441no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
459C4⋊C4—2441no—
460C4⋊C4—2441no—
461C4⋊C4C2×C22⋊D6111yescharacteristic
462C4⋊C4C2×C22⋊D6111yescharacteristic
463C22⋊C4—21no—
464C22⋊C4—21no—
465C22⋊C4—21no—
466C22⋊C4—21no—
467C22⋊C4—21no—
468C22⋊C4—41no—
469C22⋊C4—2441no—
470C42—21no—
471C42—41no—
472C42—2441no—
473C42D4×D6111yescharacteristic
474C42#1175822411yescharacteristic
475C42C2×C22⋊D6111yescharacteristic
476C42C23×D6111yescharacteristic
477C23—21no—
478C23#1451974911yescharacteristic
479Q8—21no—
480Q8—21no—
481Q8—41no—
482Q8—41no—
483Q8—81no—
484Q8—4881no—
485Q8—2441no—
486Q8—2441no—
487D4—21no—
488D4—21no—
489D4—42no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—2441no—
503C2×C4—2441no—
504C2×C4#1451974911yescharacteristic
505C2×C4#1451974911yescharacteristic
506C2×C4#1451974911yescharacteristic
507C2×C4C2×D4×D6111yescharacteristic
508C2×C4C22×C22⋊D6111yescharacteristic
509C2×C4C2×#1175822411yescharacteristic
510C8—41no—
511C8—41no—
512C8—612no—
513C8—1221no—
514C8—1221no—
515C8—1221no—
516C8—2441no—
517C8—2441no—
518C8—2441no—
519C8—2441no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22C2×#1451974911yescharacteristic
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—41no—
532C4—41no—
533C4—41no—
534C4—81no—
535C4—81no—
536C4—4881no—
537C4—4881no—
538C4#135×D6111yescharacteristic
539C4#1684055911yescharacteristic
540C4#1684057711yescharacteristic
541C4#1684057711yescharacteristic
542C2—41no—
543C2—81no—
544C2#2145527011yescharacteristic
545C2#2145547111yescharacteristic
546C2#2145553611yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3233C2
socleC22×C61C2×#121
prime radical#121×C61D4
hypercenter#3233D61
hypocenterC61#19220
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3233

Complements

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

NH ≅ G/Ntype
2. #21456401C2semidirect
4. #21445403C2semidirect
5. #21445399C2semidirect
6. #21445396C2semidirect
7. #21445392C2semidirect
8. #21445221C2semidirect
9. #21445216C2semidirect
10. #21445187C2semidirect
11. #21445182C2semidirect
12. #21444565C2semidirect
13. #21444529C2semidirect
14. #21444527C2semidirect
340. C61#19220semidirect

Maximal subgroups

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

subgroupindex
#214569942
#214564012
#214547952
#214454032
#214453992
#214453962
#214453922
#214452212
#214452162
#214451872
#214451822
#214445652
#214445292
#214445272
C61×#32332
#1922061