Finite Group Databasebeta

← #29663467

Subgroups — #29663467

Order 15616.

584 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457156C211yesmaximal, characteristic
2#21456401C211yesmaximal, characteristic, complemented
3#21454795C211yesmaximal, characteristic
4#21445335C211yesmaximal, characteristic, complemented
5#21445330C211yesmaximal, characteristic, complemented
6#21445311C211yesmaximal, characteristic, complemented
7#21445306C211yesmaximal, characteristic, complemented
8#21445251C211yesmaximal, characteristic, complemented
9#21445247C211yesmaximal, characteristic, complemented
10#21445244C211yesmaximal, characteristic, complemented
11#21445240C211yesmaximal, characteristic, complemented
12C2×#16839512C211yesmaximal, characteristic, complemented
13#21444519C211yesmaximal, characteristic, complemented
14#21444517C211yesmaximal, characteristic, complemented
15C61×#3248C211yesmaximal, characteristic
16#16840319C2211yescharacteristic
17#16840165C2211yescharacteristic
18#16840141C2211yescharacteristic
19#16840114C2211yescharacteristic
20#16840112C2211yescharacteristic
21#16840066—21no—
22#16840047C2211yescharacteristic
23#16840042C2211yescharacteristic
24#16840001—21no—
25#16840001—21no—
26#16840001—21no—
27#16839989—21no—
28#16839981—21no—
29#16839972C2211yescharacteristic
30#16839967C2211yescharacteristic
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
41C2×C4⋊#92418C2211yescharacteristic
42#16839718—21no—
43C2×C4×#92418C2211yescharacteristic
44#16839512C2211yescharacteristic
45#16839512C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839512C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839509C2211yescharacteristic
50#16839508C2211yescharacteristic
51#16839508C2211yescharacteristic
52#16839508C2211yescharacteristic
53#16839508C2211yescharacteristic
54#16839507C2211yescharacteristic
55#16839507C2211yescharacteristic
56C61×#533C2211yescharacteristic
57C61×#529C2211yescharacteristic
58C61×#479—21no—
59C61×#475C2211yescharacteristic
60C61×#474C2211yescharacteristic
61C61×#467—21no—
62C61×#465—21no—
63C61×#464—21no—
64C61×#462C2211yescharacteristic
65C61×#461C2211yescharacteristic
66C2×C4⋊C8×C61C2211yescharacteristic
67C2×#11758202—21no—
68C2×#11758202—21no—
69C2×#11758202—21no—
70C2×#11758200—21no—
71C2×#11758194—21no—
72C22×#92418—21no—
73C22×#92418—21no—
74C22×#92418D411yescharacteristic
75C22×#92418D411yescharacteristic
76#14519777C2311yescharacteristic
77#14519777C2311yescharacteristic
78#14519769—21no—
79#14519756—21no—
80#14519756—21no—
81C4×Dic122—21no—
82C4×Dic122—21no—
83#14519736—21no—
84#14519710—21no—
85#14519710—21no—
86#14519710—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#14519700—21no—
99#14519700—21no—
100#14519700—21no—
101#14519700—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×Q16×C61—21no—
114C2×SD16×C61—21no—
115C22×C8×C61—21no—
116C4⋊Q8×C61C2311yescharacteristic
117C4⋊Q8×C61C2311yescharacteristic
118#126×C61—21no—
119#125×C61—21no—
120#125×C61D411yescharacteristic
121#124×C61D411yescharacteristic
122#123×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61D411yescharacteristic
126#121×C61D411yescharacteristic
127C4×Q8×C61—21no—
128C4×Q8×C61—21no—
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61C2311yescharacteristic
132#117×C61—21no—
133#117×C61—21no—
134C2×C42×C61C2311yescharacteristic
135#107×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#103×C61—21no—
143#102×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—
155Q16×C61—81no—
156SD16×C61—81no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—81no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
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—21no—
176C22⋊C4×C61—21no—
177C22⋊C4×C61—41no—
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—
187#11758202—81no—
188#11758200—81no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193#11758196—41no—
194#11758196—41no—
195C4×Dic61—41no—
196#11758194—42no—
197C2×#92418—22no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—41no—
203C2×#92418—42no—
204C2×#92418—81no—
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209C23×C61—21no—
210C23×C61#12111yescharacteristic
211Q8×C61—21no—
212Q8×C61—21no—
213Q8×C61—41no—
214Q8×C61—41no—
215Q8×C61—41no—
216Q8×C61—41no—
217D4×C61—21no—
218D4×C61—21no—
219D4×C61—42no—
220C2×C4×C61—22no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61#12111yescharacteristic
233C2×C4×C61#12111yescharacteristic
234C2×C4×C61#12111yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×Dic61—41no—
239C2×Dic61—41no—
240Dic122—41no—
241Dic122—41no—
242Dic122—41no—
243Dic122—41no—
244C8×C61—41no—
245C8×C61—41no—
246#92418—21no—
247#92418—21no—
248#92418—21no—
249#92418—41no—
250#92418—41no—
251#92418—41no—
252#92418—41no—
253#92418#13512yes—
254#19228—611nomaximal, Sylow
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—41no—
258C22×C61—41no—
259C22×C61—41no—
260C22×C61#47312yes—
261C22×C61C2×#12111yescharacteristic
262C4×C61—21no—
263C4×C61—21no—
264C4×C61—21no—
265C4×C61—21no—
266C4×C61—41no—
267C4×C61—41no—
268C4×C61—41no—
269C4×C61—81no—
270C4×C61—81no—
271C4×C61C2×#13511yescharacteristic
272C4×C61#57511yescharacteristic
273C4×C61#57711yescharacteristic
274C4×C61#57711yescharacteristic
275Dic61—81no—
276Dic61—81no—
277#3248—611no—
278#3248D6111yescharacteristic
279#3170—611no—
280#2966—611no—
281#1694—611no—
282#1693—611no—
283#1690—611no—
284#1689—611no—
285#1680—611no—
286#1680—611no—
287#1679—611no—
288#1679—611no—
289C2×#334—611no—
290#1552—611no—
291#1552—611no—
292C2×C61—22no—
293C2×C61—81no—
294C2×C61#303511yescharacteristic
295C2×C61#306011yescharacteristic
296C2×C61C2×#47311yescharacteristic
297#533—611no—
298#533C2×D6111yescharacteristic
299#529C2×D6111yescharacteristic
300#500—611no—
301#494—611no—
302#487—611no—
303#487—611no—
304#479—21no—
305#479—1221no—
306#475—611no—
307#475C2×D6111yescharacteristic
308#474—611no—
309#474C2×D6111yescharacteristic
310#467—21no—
311#467—1221no—
312#467—1221no—
313#467—1221no—
314#465—21no—
315#465—1221no—
316#464—21no—
317#464—1221no—
318#462—611no—
319#462C2×D6111yescharacteristic
320#461—611no—
321#461C2×D6111yescharacteristic
322#446—611no—
323C8×Q8—611no—
324C4×Q16—1221no—
325C4×Q16—1221no—
326#436—611no—
327#435—1221no—
328C8×D4—611no—
329C8×D4—1221no—
330#433—1221no—
331#432—611no—
332C2×C4⋊C8—611no—
333C2×C4⋊C8C2×D6111yescharacteristic
334#405—1221no—
335C2×C4×C8—611no—
336#334—611no—
337#334—611no—
338#334—611no—
339#334—611no—
340#332—611no—
341#332—611no—
342#332—611no—
343#332—611no—
344#332—611no—
345#332—611no—
346#332—611no—
347#332—611no—
348C61#1922811yescharacteristic, complemented, Sylow
349C2×Q16—21no—
350C2×Q16—1221no—
351C2×Q16—1221no—
352C2×Q16—1221no—
353C2×SD16—21no—
354C2×SD16—1221no—
355C2×M16—1221no—
356C22×C8—21no—
357C22×C8—611no—
358C22×C8—611no—
359C22×C8—1221no—
360C22×C8—1221no—
361C4⋊Q8—611no—
362C4⋊Q8—611no—
363C4⋊Q8C22×D6111yescharacteristic
364C4⋊Q8C22×D6111yescharacteristic
365#126—21no—
366#125—21no—
367#125—1221no—
368#125C22⋊D6111yescharacteristic
369#124C22⋊D6111yescharacteristic
370#123—21no—
371C22⋊Q8—21no—
372C22⋊Q8—21no—
373C22⋊Q8—1221no—
374C22⋊Q8—1221no—
375C22⋊Q8C22⋊D6111yescharacteristic
376#121C22⋊D6111yescharacteristic
377C4×Q8—21no—
378C4×Q8—21no—
379C4×Q8—1221no—
380C4×Q8—1221no—
381C4×Q8C22×D6111yescharacteristic
382C4×D4—21no—
383C4×D4C22×D6111yescharacteristic
384#117—21no—
385#117—21no—
386#117—1221no—
387C2×C42C22×D6111yescharacteristic
388#107—21no—
389#107—1221no—
390#107—1221no—
391#107—1221no—
392#106—21no—
393#106—1221no—
394C4⋊C8—21no—
395C4⋊C8—611no—
396C4⋊C8—611no—
397C4⋊C8—611no—
398C4⋊C8—611no—
399C4⋊C8—1221no—
400C4⋊C8—1221no—
401C4⋊C8—1221no—
402C4⋊C8C22×D6111yescharacteristic
403C4⋊C8C22×D6111yescharacteristic
404#103—21no—
405#103—21no—
406#103—21no—
407#103—1221no—
408#103—1221no—
409#103—1221no—
410#103—1221no—
411#103—1221no—
412#103—1221no—
413#103—1221no—
414#102—21no—
415#102—1221no—
416C22⋊C8—1221no—
417C22⋊C8—1221no—
418C22⋊C8—1221no—
419#97—1221no—
420#97—1221no—
421C4×C8—611no—
422C4×C8—611no—
423C4×C8—611no—
424C4×C8—611no—
425C4×C8—1221no—
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×C4D4×D6111yescharacteristic
435C22×C4#1175822411yescharacteristic
436C22×C4C2×C22⋊D6111yescharacteristic
437Q16—81no—
438Q16—4881no—
439Q16—2442no—
440Q16—2442no—
441SD16—81no—
442SD16—4881no—
443M16—2442no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—81no—
447C2×C8—611no—
448C2×C8—611no—
449C2×C8—611no—
450C2×C8—611no—
451C2×C8—1222no—
452C2×C8—1221no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—4881no—
457C2×C8—2441no—
458C2×C8—2442no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—2441no—
474C4⋊C4—2441no—
475C4⋊C4C2×C22⋊D6111yescharacteristic
476C4⋊C4C2×C22⋊D6111yescharacteristic
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—41no—
483C22⋊C4—2441no—
484C42—21no—
485C42—41no—
486C42—2441no—
487C42D4×D6111yescharacteristic
488C42#1175822411yescharacteristic
489C42C2×C22⋊D6111yescharacteristic
490C42C23×D6111yescharacteristic
491C23—21no—
492C23#1451974911yescharacteristic
493Q8—21no—
494Q8—21no—
495Q8—41no—
496Q8—41no—
497Q8—41no—
498Q8—41no—
499Q8—2441no—
500Q8—2441no—
501Q8—2441no—
502Q8—2441no—
503D4—21no—
504D4—21no—
505D4—42no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4#1451974911yescharacteristic
521C2×C4#1451974911yescharacteristic
522C2×C4#1451974911yescharacteristic
523C2×C4C2×D4×D6111yescharacteristic
524C2×C4C22×C22⋊D6111yescharacteristic
525C2×C4C2×#1175822411yescharacteristic
526C8—41no—
527C8—41no—
528C8—612no—
529C8—1221no—
530C8—1221no—
531C8—1221no—
532C8—2441no—
533C8—2441no—
534C8—2441no—
535C8—2441no—
536C22—21no—
537C22—21no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22#1684003512yes—
542C22C2×#1451974911yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—81no—
551C4—81no—
552C4—4881no—
553C4—4881no—
554C4#135×D6111yescharacteristic
555C4#1684055911yescharacteristic
556C4#1684057011yescharacteristic
557C4#1684057311yescharacteristic
558C2—22no—
559C2—81no—
560C2#2145527011yescharacteristic
561C2#2145547811yescharacteristic
562C2C2×#1684003511yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3248C2
socleC22×C61C2×#121
prime radical#121×C61D4
hypercenter#3248D61
hypocenterC61#19228
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3248

Complements

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

NH ≅ G/Ntype
2. #21456401C2semidirect
4. #21445335C2semidirect
5. #21445330C2semidirect
6. #21445311C2semidirect
7. #21445306C2semidirect
8. #21445251C2semidirect
9. #21445247C2semidirect
10. #21445244C2semidirect
11. #21445240C2semidirect
12. C2×#16839512C2semidirect
13. #21444519C2semidirect
14. #21444517C2semidirect
348. C61#19228semidirect

Maximal subgroups

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

subgroupindex
#214571562
#214564012
#214547952
#214453352
#214453302
#214453112
#214453062
#214452512
#214452472
#214452442
#214452402
C2×#168395122
#214445192
#214445172
C61×#32482
#1922861