Finite Group Databasebeta

← #29665136

Subgroups — #29665136

Order 15616.

572 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#21457186C211yesmaximal, characteristic
2#21457051C211yesmaximal, characteristic
3#21456201C211yesmaximal, characteristic, complemented
4#21454817C211yesmaximal, characteristic
5#21445404C211yesmaximal, characteristic, complemented
6#21445397C211yesmaximal, characteristic, complemented
7#21445376C211yesmaximal, characteristic, complemented
8#21445369C211yesmaximal, characteristic, complemented
9#21445280C211yesmaximal, characteristic, complemented
10#21445273C211yesmaximal, characteristic, complemented
11#21445252C211yesmaximal, characteristic, complemented
12#21445245C211yesmaximal, characteristic, complemented
13#21444587C211yesmaximal, characteristic, complemented
14#21444503C211yesmaximal, characteristic, complemented
15C61×#1554C211yesmaximal, characteristic, complemented
16C2×#14519777C2211yescharacteristic
17#16840296C2211yescharacteristic
18#16840171—21no—
19#16840132—21no—
20#16840132—21no—
21#16840122—21no—
22#16840114C2211yescharacteristic
23#16840114C2211yescharacteristic
24#16840114C2211yescharacteristic
25#16840114C2211yescharacteristic
26#16840098—21no—
27#16840081—21no—
28#16840069—21no—
29#16840054C2211yescharacteristic
30#16840049C2211yescharacteristic
31#16840048C2211yescharacteristic
32#16840048C2211yescharacteristic
33#16840043C2211yescharacteristic
34#16840043C2211yescharacteristic
35#16840018—21no—
36#16840010—21no—
37#16839984—21no—
38#16839969C2211yescharacteristic
39#16839964C2211yescharacteristic
40#16839889C2211yescharacteristic
41#16839886C2211yescharacteristic
42#16839839C2211yescharacteristic
43#16839837—21no—
44#16839836—21no—
45#16839827C2211yescharacteristic
46#16839821—21no—
47#16839786C2211yescharacteristic
48#16839770—21no—
49#16839770—21no—
50#16839716—21no—
51C4×#11758194C2211yescharacteristic
52#16839514C2211yescharacteristic
53#16839514C2211yescharacteristic
54#16839512C2211yescharacteristic
55#16839512C2211yescharacteristic
56#16839509C2211yescharacteristic
57#16839509C2211yescharacteristic
58#16839509C2211yescharacteristic
59#16839509C2211yescharacteristic
60C61×#533C2211yescharacteristic
61C61×#432C2211yescharacteristic
62C61×#431C2211yescharacteristic
63C61×#333C2211yescharacteristic
64C61×#333C2211yescharacteristic
65C61×#332C2211yescharacteristic
66C61×#332C2211yescharacteristic
67C22×Dic122—21no—
68C2×#11758202—21no—
69C2×#11758201—21no—
70C2×#11758194—21no—
71C2×#11758194D411yescharacteristic
72C2×#11758194D411yescharacteristic
73C2×#11758189—21no—
74C2×C8×D61—21no—
75#14519777—21no—
76#14519777—21no—
77#14519777—21no—
78#14519777C2311yescharacteristic
79#14519777C2311yescharacteristic
80#14519771—21no—
81#14519769D411yescharacteristic
82#14519765D411yescharacteristic
83#14519760—21no—
84#14519756D411yescharacteristic
85#14519751D411yescharacteristic
86C4×Dic122C2311yescharacteristic
87C4×D244C2311yescharacteristic
88C4×C22⋊D61—21no—
89#14519734—21no—
90C2×C4⋊Dic61—21no—
91C2×C4⋊Dic61—21no—
92#14519711—21no—
93#14519711—21no—
94#14519710—21no—
95#14519709—21no—
96#14519709—21no—
97#14519708—21no—
98C4⋊#92418—21no—
99C4⋊#92418C2311yescharacteristic
100C4⋊#92418C2311yescharacteristic
101#14519706—21no—
102#14519706—21no—
103#14519702—21no—
104#14519702—21no—
105#14519701—21no—
106#14519701—21no—
107#14519700—21no—
108#14519700—21no—
109#14519700—21no—
110#14519700—21no—
111#14519700—21no—
112#14519700—21no—
113#14519700—21no—
114#14519700—21no—
115#14519700—21no—
116#14519699—21no—
117#14519699—21no—
118#14519697—21no—
119#14519689—21no—
120#14519689—21no—
121#14519688D411yescharacteristic
122#14519688D411yescharacteristic
123#14519687—21no—
124C4×#92418C2311yescharacteristic
125C4×#92418C2311yescharacteristic
126C8×Dic61—21no—
127C4⋊Q8×C61C2311yescharacteristic
128C4⋊Q8×C61C2311yescharacteristic
129#125×C61—21no—
130C22⋊Q8×C61—21no—
131C22⋊Q8×C61—21no—
132C4×Q8×C61—21no—
133C4×Q8×C61—21no—
134#117×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140C22⋊C8×C61—21no—
141C22⋊C8×C61—21no—
142#97×C61—21no—
143C4×C8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61#4111yescharacteristic
149C2×C22⋊D61—21no—
150C22×Dic61—41no—
151C2×D244C2×D411yescharacteristic
152C2×C4×D61—21no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—42no—
156C2×Dic122—42no—
157C2×Dic122—41no—
158C2×Dic122C2×D411yescharacteristic
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—41no—
168C4⋊C4×C61—41no—
169C22⋊C4×C61—41no—
170C42×C61—41no—
171C42×C61C2×D411yescharacteristic
172C42×C61C2×D411yescharacteristic
173C42×C61#4111yescharacteristic
174C42×C61C2411yescharacteristic
175C22⋊Dic61—21no—
176#11758202—81no—
177#11758201—81no—
178#11758198—21no—
179#11758198—21no—
180#11758198—21no—
181#11758198—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—41no—
187C4⋊Dic61—41no—
188C4⋊Dic61C2×D411yescharacteristic
189C4⋊Dic61C2×D411yescharacteristic
190#11758196—21no—
191#11758196—21no—
192#11758196—21no—
193#11758196—21no—
194C4×Dic61—21no—
195#11758194—41no—
196#11758194—41no—
197#11758194—81no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418C2×D411yescharacteristic
201C2×#92418C2×D411yescharacteristic
202C2×#92418C2×D411yescharacteristic
203C2×#92418C2×D411yescharacteristic
204#11758189—42no—
205C8×D61—42no—
206C23×C61#12111yescharacteristic
207C22×D61—21no—
208Q8×C61—41no—
209Q8×C61—41no—
210Q8×C61—81no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61#12111yescharacteristic
219C2×C4×C61#12111yescharacteristic
220C2×C4×C61#12111yescharacteristic
221C2×C4×C61C22×D411yescharacteristic
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C2×#4111yescharacteristic
224C22⋊D61—81no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—82no—
231D244—21no—
232D244—21no—
233C4×D61—41no—
234C4×D61—41no—
235Dic122—21no—
236Dic122—21no—
237Dic122—41no—
238Dic122—41no—
239Dic122—41no—
240Dic122—41no—
241Dic122—82no—
242C8×C61—41no—
243C8×C61—41no—
244C8×C61—41no—
245C8×C61—41no—
246#92418—21no—
247#92418—21no—
248#92418—21no—
249#92418—21no—
250#92418—41no—
251#92418—41no—
252#19381—611nomaximal, Sylow
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—21no—
256C22×C61C2×#12111yescharacteristic
257C2×D61—41no—
258C2×D61—41no—
259C2×D61—41no—
260C4×C61—21no—
261C4×C61—21no—
262C4×C61—21no—
263C4×C61—21no—
264C4×C61—81no—
265C4×C61—81no—
266C4×C61#57511yescharacteristic
267C4×C61#57511yescharacteristic
268C4×C61#57711yescharacteristic
269C4×C61#57811yescharacteristic
270Dic61—41no—
271Dic61—41no—
272Dic61—41no—
273Dic61—81no—
274Dic61—81no—
275#3252—611no—
276#3239—611no—
277#3139—611no—
278#2969—611no—
279#1704—611no—
280#1703—611no—
281#1700—611no—
282#1699—611no—
283#1684—611no—
284#1683—611no—
285#1680—611no—
286#1679—611no—
287#1569—611no—
288#1554D6111yescharacteristic, complemented
289#1549—611no—
290C2×C61—41no—
291C2×C61#304011yescharacteristic
292C2×C61#306311yescharacteristic
293C2×C61#306911yescharacteristic
294D61—81no—
295#533C2×D6111yescharacteristic
296C2×C4⋊Q8—611no—
297#529—611no—
298#501—1221no—
299#491—1221no—
300#491—1221no—
301#489—1221no—
302#487—611no—
303#487—611no—
304#487—611no—
305#487—611no—
306#484—1221no—
307#481—1221no—
308#479—1221no—
309#477—611no—
310#476—611no—
311#475—611no—
312#475—611no—
313#474—611no—
314#474—611no—
315#470—1221no—
316#468—1221no—
317#464—1221no—
318#462—611no—
319#461—611no—
320#446—611no—
321C8×Q8—611no—
322#436—611no—
323#436—1221no—
324#435—1221no—
325C8×D4—611no—
326#433—1221no—
327#432C2×D6111yescharacteristic
328#431C2×D6111yescharacteristic
329#423—611no—
330#419—1221no—
331#419—1221no—
332#405—1221no—
333C4×M16—611no—
334#335—611no—
335#335—611no—
336#334—611no—
337#334—611no—
338#333C2×D6111yescharacteristic
339#333C2×D6111yescharacteristic
340#332—611no—
341#332—611no—
342#332—611no—
343#332—611no—
344#332C2×D6111yescharacteristic
345#332C2×D6111yescharacteristic
346C61#1938111yescharacteristic, complemented, Sylow
347C22×Q8—1221no—
348C2×Q16—1221no—
349C2×SD16—1221no—
350C2×M16—611no—
351C2×M16—611no—
352C2×M16—1221no—
353C2×M16—1221no—
354C22×C8—1221no—
355C4⋊Q8—611no—
356C4⋊Q8—611no—
357C4⋊Q8—1221no—
358C4⋊Q8—1221no—
359C4⋊Q8—1221no—
360C4⋊Q8C22×D6111yescharacteristic
361C4⋊Q8C22×D6111yescharacteristic
362#126—1221no—
363#125—21no—
364#125—611no—
365#124—611no—
366#123—1221no—
367C22⋊Q8—21no—
368C22⋊Q8—21no—
369C22⋊Q8—611no—
370#121—611no—
371C4×Q8—21no—
372C4×Q8—21no—
373C4×Q8—611no—
374C4×D4—611no—
375C4×D4—1221no—
376#117—21no—
377#117—1221no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1221no—
380C2×C42C22×D6111yescharacteristic
381#107—1221no—
382#107—1221no—
383#107—1221no—
384#106—1221no—
385#106—1221no—
386#106—1221no—
387C4⋊C8—611no—
388C4⋊C8—611no—
389C4⋊C8—1221no—
390C4⋊C8—1221no—
391C4⋊C8—1221no—
392C4⋊C8C22×D6111yescharacteristic
393C4⋊C8C22×D6111yescharacteristic
394C4⋊C8C22×D6111yescharacteristic
395C4⋊C8C22×D6111yescharacteristic
396#103—1221no—
397#103—1221no—
398#103—1221no—
399#103—1221no—
400#103—1221no—
401#103—1221no—
402#103—1221no—
403#103—1221no—
404#103—1221no—
405#103—1221no—
406#103—1221no—
407#103—1221no—
408#103—1221no—
409#102—1221no—
410#102—1221no—
411#102—1221no—
412C22⋊C8—21no—
413C22⋊C8—21no—
414C22⋊C8—1221no—
415C22⋊C8—1221no—
416#97—21no—
417#97—611no—
418#97—611no—
419#97—1221no—
420C4×C8—21no—
421C4×C8—611no—
422C4×C8—611no—
423C4×C8—1221no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—611no—
427C2×Q8—1221no—
428C2×Q8—1221no—
429C2×Q8—2442no—
430C2×Q8—2442no—
431C2×Q8—2441no—
432C2×D4—611no—
433C2×D4—1221no—
434C22×C4—1221no—
435C22×C4—2441no—
436C22×C4#1175822211yescharacteristic
437C22×C4D4×D6111yescharacteristic
438C22×C4D4×D6111yescharacteristic
439Q16—4881no—
440SD16—4881no—
441M16—4881no—
442M16—2442no—
443M16—2441no—
444M16—2441no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—611no—
450C2×C8—611no—
451C2×C8—611no—
452C2×C8—611no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8—2442no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—611no—
463C4⋊C4—611no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—2441no—
473C4⋊C4—2441no—
474C22⋊C4—41no—
475C22⋊C4—1221no—
476C22⋊C4—1221no—
477C22⋊C4—1221no—
478C22⋊C4—1221no—
479C22⋊C4—1221no—
480C42—41no—
481C42—1221no—
482C42#1175822211yescharacteristic
483C42D4×D6111yescharacteristic
484C42D4×D6111yescharacteristic
485C42C23×D6111yescharacteristic
486C23—1221no—
487C23#1451975011yescharacteristic
488Q8—41no—
489Q8—41no—
490Q8—81no—
491Q8—1221no—
492Q8—1221no—
493Q8—4882no—
494Q8—2441no—
495Q8—2441no—
496Q8—2441no—
497Q8—2441no—
498D4—1221no—
499D4—1221no—
500D4—4881no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—1221no—
509C2×C4—1221no—
510C2×C4—1221no—
511C2×C4—4882no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4#1451975011yescharacteristic
517C2×C4#1451975011yescharacteristic
518C2×C4#1451975011yescharacteristic
519C2×C4C2×D4×D6111yescharacteristic
520C2×C4C2×D4×D6111yescharacteristic
521C2×C4C2×#1175822211yescharacteristic
522C8—41no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—1221no—
527C8—1221no—
528C8—1221no—
529C8—1221no—
530C8—2441no—
531C8—2441no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—2441no—
536C22—2441no—
537C22—2441no—
538C22C2×#1451975011yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—81no—
544C4—81no—
545C4—4881no—
546C4—4881no—
547C4—2441no—
548C4—2441no—
549C4—2441no—
550C4#1684056111yescharacteristic
551C4#1684056111yescharacteristic
552C4#1684056911yescharacteristic
553C4#1684057611yescharacteristic
554C2—41no—
555C2—4881no—
556C2#2145531611yescharacteristic
557C2#2145551311yescharacteristic
558C2#2145555911yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1554C2
socleC22×C61C2×#121
prime radical#14519751D4
hypercenter#1554D61
hypocenterC61#19381
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1554

Complements

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

NH ≅ G/Ntype
3. #21456201C2semidirect
5. #21445404C2semidirect
6. #21445397C2semidirect
7. #21445376C2semidirect
8. #21445369C2semidirect
9. #21445280C2semidirect
10. #21445273C2semidirect
11. #21445252C2semidirect
12. #21445245C2semidirect
13. #21444587C2semidirect
14. #21444503C2semidirect
15. C61×#1554C2semidirect
288. #1554D61semidirect
346. C61#19381semidirect

Maximal subgroups

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

subgroupindex
#214571862
#214570512
#214562012
#214548172
#214454042
#214453972
#214453762
#214453692
#214452802
#214452732
#214452522
#214452452
#214445872
#214445032
C61×#15542
#1938161