Finite Group Databasebeta

← #29676993

Subgroups — #29676993

Order 15616.

572 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458738C211yesmaximal, characteristic
2#21458622C211yesmaximal, characteristic, complemented
3#21458096C211yesmaximal, characteristic
4#21445413C211yesmaximal, characteristic, complemented
5#21445375C211yesmaximal, characteristic, complemented
6#21445368C211yesmaximal, characteristic
7#21445337C211yesmaximal, characteristic, complemented
8#21445310C211yesmaximal, characteristic, complemented
9#21445096C211yesmaximal, characteristic, complemented
10#21445094C211yesmaximal, characteristic, complemented
11#21445089C211yesmaximal, characteristic
12#21445080C211yesmaximal, characteristic
13#21444714C211yesmaximal, characteristic, complemented
14#21444710C211yesmaximal, characteristic
15C61×#3398C211yesmaximal, characteristic
16#16840491C2211yescharacteristic
17D4×Dic122C2211yescharacteristic
18#16840169—21no—
19#16840167C2211yescharacteristic
20#16840158C2211yescharacteristic
21#16840158C2211yescharacteristic
22#16840132—21no—
23#16840114C2211yescharacteristic
24#16840114C2211yescharacteristic
25#16840103—21no—
26#16840066—21no—
27#16840060—21no—
28#16840055—21no—
29#16840047C2211yescharacteristic
30#16840044C2211yescharacteristic
31#16840044C2211yescharacteristic
32#16840042C2211yescharacteristic
33#16840018—21no—
34#16839972—21no—
35#16839972C2211yescharacteristic
36#16839972C2211yescharacteristic
37#16839917—21no—
38#16839893C2211yescharacteristic
39#16839892C2211yescharacteristic
40#16839867—21no—
41#16839863—21no—
42C4×#11758202C2211yescharacteristic
43#16839840C2211yescharacteristic
44#16839770—21no—
45#16839514C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839508C2211yescharacteristic
48#16839508C2211yescharacteristic
49#16839507C2211yescharacteristic
50#16839507C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53#16839492C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55C8×#92418C2211yescharacteristic
56C61×#557C2211yescharacteristic
57C61×#556C2211yescharacteristic
58C61×#497—21no—
59C61×#494C2211yescharacteristic
60C61×#464—21no—
61C61×#462—21no—
62C61×#462C2211yescharacteristic
63C61×#461C2211yescharacteristic
64C61×#446C2211yescharacteristic
65C61×#441—21no—
66C4×SD16×C61C2211yescharacteristic
67C22×Dic122—21no—
68C2×#11758202—21no—
69C2×#11758202—21no—
70C2×#11758202D411yescharacteristic
71C2×#11758194—21no—
72#14519777—21no—
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519776—21no—
76#14519768—21no—
77#14519768—41no—
78#14519756—21no—
79#14519754—41no—
80Q8×Dic61—21no—
81C4×Dic122—21no—
82C4×Dic122C2311yescharacteristic
83D4×Dic61—21no—
84#14519711—21no—
85#14519711—21no—
86#14519710—21no—
87#14519710D411yescharacteristic
88#14519708—21no—
89#14519708—21no—
90#14519708—21no—
91#14519708—41no—
92#14519708—41no—
93C4⋊#92418—21no—
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97#14519706—21no—
98#14519701—21no—
99#14519701D411yescharacteristic
100#14519700—21no—
101#14519700—21no—
102#14519700—21no—
103#14519700—21no—
104#14519700—21no—
105#14519700—21no—
106#14519700—21no—
107#14519700—21no—
108#14519700D411yescharacteristic
109#14519698—21no—
110C22⋊#92418—21no—
111#14519688—21no—
112#14519687—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×Q16×C61—21no—
116C2×Q16×C61—21no—
117C2×Q16×C61—21no—
118C2×SD16×C61—21no—
119C2×SD16×C61D411yescharacteristic
120C4⋊Q8×C61—21no—
121C4⋊Q8×C61C2311yescharacteristic
122C4⋊Q8×C61C2311yescharacteristic
123#126×C61—41no—
124#125×C61—21no—
125#125×C61—41no—
126#124×C61—21no—
127C22⋊Q8×C61—21no—
128C4×Q8×C61—21no—
129C4×Q8×C61—21no—
130C4×Q8×C61—21no—
131C4×Q8×C61C2311yescharacteristic
132C4×Q8×C61C2311yescharacteristic
133C4×D4×C61C2311yescharacteristic
134#117×C61—21no—
135#106×C61D411yescharacteristic
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#102×C61D411yescharacteristic
142#97×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61C2×D411yescharacteristic
149C22×C4×C61—21no—
150C22×Dic61—41no—
151C2×Dic122—21no—
152C2×Dic122—21no—
153C2×Dic122—42no—
154C2×Dic122—41no—
155C2×Dic122—81no—
156C2×Dic122C2×D411yescharacteristic
157Q16×C61—42no—
158Q16×C61—81no—
159Q16×C61—81no—
160SD16×C61—22no—
161SD16×C61—81no—
162C2×C8×C61—21no—
163C2×C8×C61C2×D411yescharacteristic
164C2×C8×C61C2×D411yescharacteristic
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—41no—
172C4⋊C4×C61—41no—
173C4⋊C4×C61—41no—
174C4⋊C4×C61—41no—
175C4⋊C4×C61C2×D411yescharacteristic
176C4⋊C4×C61C2×D411yescharacteristic
177C4⋊C4×C61C2×D411yescharacteristic
178C22⋊C4×C61—21no—
179C22⋊C4×C61—41no—
180C22⋊C4×C61—41no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—41no—
184C42×C61—41no—
185C42×C61C2411yescharacteristic
186C22⋊Dic61—41no—
187#11758202—41no—
188#11758202—42no—
189#11758202—81no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—41no—
195C4⋊Dic61—41no—
196C4⋊Dic61C2×D411yescharacteristic
197#11758196—21no—
198#11758196—41no—
199#11758196—41no—
200C4×Dic61—21no—
201C4×Dic61—41no—
202#11758194—81no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209C23×C61—21no—
210Q8×C61—21no—
211Q8×C61—21no—
212Q8×C61—21no—
213Q8×C61—21no—
214Q8×C61—41no—
215Q8×C61—41no—
216Q8×C61—41no—
217Q8×C61—41no—
218D4×C61—21no—
219D4×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—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61H(5,𝔽2)11yescharacteristic
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—82no—
237Dic122—21no—
238Dic122—21no—
239Dic122—41no—
240Dic122—41no—
241Dic122—41no—
242Dic122—41no—
243Dic122—82no—
244C8×C61—21no—
245C8×C61—41no—
246C8×C61#13712yes—
247#92418—22no—
248#92418—21no—
249#92418—21no—
250#92418—41no—
251#92418—41no—
252#92418—41no—
253#20297—611nomaximal, Sylow
254C22×C61—41no—
255C22×C61—41no—
256C22×C61—41no—
257C22×C61D4211yescharacteristic
258C4×C61—21no—
259C4×C61—41no—
260C4×C61—41no—
261C4×C61—41no—
262C4×C61—41no—
263C4×C61—41no—
264C4×C61—81no—
265C4×C61—81no—
266C4×C61C2×#13711yescharacteristic
267C4×C61#57511yescharacteristic
268C4×C61#57711yescharacteristic
269C4×C61#57711yescharacteristic
270Dic61—41no—
271Dic61—41no—
272Dic61—81no—
273Dic61—81no—
274#3398D6111yescharacteristic
275#3386—611no—
276#3374—611no—
277#3328—611no—
278#1707—611no—
279#1700—611no—
280#1699—611no—
281#1694—611no—
282#1690—611no—
283#1653—611no—
284#1653—611no—
285#1652—611no—
286#1651—611no—
287#1591—611no—
288#1590—611no—
289C2×C61—81no—
290C2×C61#327511yescharacteristic
291C2×C61#327811yescharacteristic
292C2×C61#329311yescharacteristic
293#557—611no—
294#557C2×D6111yescharacteristic
295#556C2×D6111yescharacteristic
296D4×Q8—611no—
297#501—1221no—
298#500—611no—
299#498—611no—
300#498—611no—
301#497—21no—
302#494C2×D6111yescharacteristic
303#491—1221no—
304#487—611no—
305#487—611no—
306#485—1221no—
307#479—1221no—
308#478—1221no—
309#477—1221no—
310#475—611no—
311#475—611no—
312#475—611no—
313#474—611no—
314#470—1221no—
315#464—21no—
316#462—21no—
317#462—611no—
318#462—611no—
319#462—1221no—
320#462C2×D6111yescharacteristic
321#461C2×D6111yescharacteristic
322#451—1221no—
323#446—611no—
324#446—611no—
325#446C2×D6111yescharacteristic
326#441—21no—
327#441—1221no—
328#440—1221no—
329C4×Q16—611no—
330C4×SD16C2×D6111yescharacteristic
331#436—611no—
332#419—1221no—
333#335—611no—
334#334—611no—
335#332—611no—
336#332—611no—
337#332—611no—
338#332—611no—
339Q8⋊C8—611no—
340Q8⋊C8—611no—
341Q8⋊C8—611no—
342D4⋊C8—611no—
343C82—611no—
344C61#2029711yescharacteristic, complemented, Sylow
345C22×Q8—1221no—
346C2×Q16—21no—
347C2×Q16—21no—
348C2×Q16—21no—
349C2×Q16—611no—
350C2×Q16—1221no—
351C2×Q16—1221no—
352C2×SD16—21no—
353C2×SD16C22⋊D6111yescharacteristic
354C2×M16—1221no—
355C4⋊Q8—21no—
356C4⋊Q8—611no—
357C4⋊Q8—611no—
358C4⋊Q8—1221no—
359C4⋊Q8—1221no—
360C4⋊Q8C22×D6111yescharacteristic
361C4⋊Q8C22×D6111yescharacteristic
362#126—41no—
363#125—21no—
364#125—41no—
365#125—1221no—
366#125—2441no—
367#124—21no—
368C22⋊Q8—21no—
369C22⋊Q8—1221no—
370C22⋊Q8—2441no—
371C4×Q8—21no—
372C4×Q8—21no—
373C4×Q8—21no—
374C4×Q8—611no—
375C4×Q8—1221no—
376C4×Q8—1221no—
377C4×Q8C22×D6111yescharacteristic
378C4×Q8C22×D6111yescharacteristic
379C4×D4—1221no—
380C4×D4C22×D6111yescharacteristic
381#117—21no—
382#107—611no—
383#107—1221no—
384#107—1221no—
385#107—1221no—
386#106—1221no—
387#106—1221no—
388#106—1221no—
389#106—2441no—
390#106—2441no—
391#106C22⋊D6111yescharacteristic
392C4⋊C8—21no—
393C4⋊C8—611no—
394C4⋊C8—611no—
395C4⋊C8—611no—
396C4⋊C8—1221no—
397C4⋊C8—1221no—
398C4⋊C8C22×D6111yescharacteristic
399#103—21no—
400#103—21no—
401#103—611no—
402#103—611no—
403#103—1221no—
404#103—1221no—
405#103—1221no—
406#103—1221no—
407#103—1221no—
408#103—1221no—
409#103—1221no—
410#103—1221no—
411#103—1221no—
412#103C22⋊D6111yescharacteristic
413#102—1221no—
414#102C22⋊D6111yescharacteristic
415C22⋊C8—1221no—
416#97—21no—
417#97—1221no—
418#97—1221no—
419C4×C8—611no—
420C4×C8—611no—
421C4×C8C22×D6111yescharacteristic
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—611no—
425C2×Q8—1221no—
426C2×Q8—1221no—
427C2×Q8—4881no—
428C2×Q8—2441no—
429C2×Q8—2442no—
430C2×Q8D4×D6111yescharacteristic
431C2×Q8C2×C22⋊D6111yescharacteristic
432C2×D4C2×C22⋊D6111yescharacteristic
433C22×C4—21no—
434C22×C4—2441no—
435Q16—42no—
436Q16—81no—
437Q16—81no—
438Q16—4881no—
439Q16—2441no—
440Q16—2442no—
441SD16—22no—
442SD16—81no—
443M16—4881no—
444C2×C8—21no—
445C2×C8—611no—
446C2×C8—611no—
447C2×C8—1221no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8C2×C22⋊D6111yescharacteristic
452C2×C8C2×C22⋊D6111yescharacteristic
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—611no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—2441no—
470C4⋊C4—2441no—
471C4⋊C4—2441no—
472C4⋊C4—2441no—
473C4⋊C4D4×D6111yescharacteristic
474C4⋊C4C2×C22⋊D6111yescharacteristic
475C4⋊C4C2×C22⋊D6111yescharacteristic
476C22⋊C4—21no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—2441no—
480C42—21no—
481C42—21no—
482C42—41no—
483C42—41no—
484C42—1221no—
485C42—2441no—
486C42C23×D6111yescharacteristic
487C23—21no—
488Q8—21no—
489Q8—21no—
490Q8—21no—
491Q8—21no—
492Q8—41no—
493Q8—41no—
494Q8—41no—
495Q8—41no—
496Q8—1221no—
497Q8—1221no—
498Q8—4882no—
499Q8—2441no—
500Q8—2441no—
501Q8—2441no—
502Q8—2441no—
503D4—21no—
504D4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—1221no—
515C2×C4—1221no—
516C2×C4—4882no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4C2×D4×D6111yescharacteristic
520C2×C4C22×C22⋊D6111yescharacteristic
521C2×C4#1451982211yescharacteristic
522C8—21no—
523C8—41no—
524C8—1222no—
525C8—1221no—
526C8—1221no—
527C8—2441no—
528C8—2441no—
529C8—2441no—
530C8#1451980712yes—
531C22—41no—
532C22—41no—
533C22—41no—
534C22D4×C22⋊D6111yescharacteristic
535C4—21no—
536C4—41no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—41no—
541C4—81no—
542C4—81no—
543C4—4881no—
544C4—4881no—
545C4—2441no—
546C4—2441no—
547C4C2×#1451980711yescharacteristic
548C4#1684056111yescharacteristic
549C4#1684057111yescharacteristic
550C4#1684057311yescharacteristic
551C2—81no—
552C2#2145744411yescharacteristic
553C2#2145748511yescharacteristic
554C2#2145762111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3398C2
socleC22×C61D42
prime radicalC2×D4×C61C2×D4
hypercenter#3398D61
hypocenterC61#20297
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3398

Complements

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

NH ≅ G/Ntype
2. #21458622C2semidirect
4. #21445413C2semidirect
5. #21445375C2semidirect
7. #21445337C2semidirect
8. #21445310C2semidirect
9. #21445096C2semidirect
10. #21445094C2semidirect
13. #21444714C2semidirect
344. C61#20297semidirect

Maximal subgroups

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

subgroupindex
#214587382
#214586222
#214580962
#214454132
#214453752
#214453682
#214453372
#214453102
#214450962
#214450942
#214450892
#214450802
#214447142
#214447102
C61×#33982
#2029761