Finite Group Databasebeta

← #29671433

Subgroups — #29671433

Order 15616.

624 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458725C211yesmaximal, characteristic, complemented
2#21458178C211yesmaximal, characteristic
3#21457465C211yesmaximal, characteristic
4#21445414C211yesmaximal, characteristic, complemented
5#21445340C211yesmaximal, characteristic, complemented
6#21445311C211yesmaximal, characteristic, complemented
7#21445201C211yesmaximal, characteristic, complemented
8#21445191C211yesmaximal, characteristic, complemented
9#21445040C211yesmaximal, characteristic, complemented
10#21445025C211yesmaximal, characteristic, complemented
11#21444984C211yesmaximal, characteristic, complemented
12#21444969C211yesmaximal, characteristic
13#21444710C211yesmaximal, characteristic, complemented
14C8×#11758201C211yesmaximal, characteristic, complemented
15Q8×SD16×C61C211yesmaximal, characteristic, complemented
16C2×#14519808—21no—
17#16840472C2211yescharacteristic
18#16840417C2211yescharacteristic
19#16840168—21no—
20#16840165C2211yescharacteristic
21#16840160C2211yescharacteristic
22#16840141C2211yescharacteristic
23#16840114C2211yescharacteristic
24#16840112C2211yescharacteristic
25#16840091—21no—
26#16840079—21no—
27#16840069—21no—
28#16840057C2211yescharacteristic
29#16840052C2211yescharacteristic
30#16840049C2211yescharacteristic
31#16840047C2211yescharacteristic
32#16840039—21no—
33#16840018—21no—
34#16839988—21no—
35#16839972C2211yescharacteristic
36#16839964C2211yescharacteristic
37#16839922—21no—
38#16839918—21no—
39#16839893C2211yescharacteristic
40#16839860—21no—
41C4×#11758201C2211yescharacteristic
42#16839840C2211yescharacteristic
43C8×D244C2211yescharacteristic
44#106×D61—21no—
45#16839513C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839507C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55C8×#92418C2211yescharacteristic
56Q82×C61C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic
58C61×#498—21no—
59C61×#498C2211yescharacteristic
60C61×#476—21no—
61C61×#476C2211yescharacteristic
62C61×#475—21no—
63C61×#475C2211yescharacteristic
64C8×Q8×C61C2211yescharacteristic
65C4×SD16×C61—21no—
66C4×SD16×C61C2211yescharacteristic
67C2×#11758222—21no—
68#14519808—44no—
69C2×#11758202—21no—
70C2×#11758202—21no—
71C2×#11758202—21no—
72C2×#11758201—21no—
73C2×#11758201D411yescharacteristic
74C2×#11758194—21no—
75C2×C8×D61—21no—
76#14519777C2311yescharacteristic
77#14519777C2311yescharacteristic
78#14519769—21no—
79#14519759—41no—
80#14519756—21no—
81#14519755—41no—
82#14519753—21no—
83#14519748—21no—
84C4×D244—21no—
85C4×D244C2311yescharacteristic
86C4⋊C4×D61—21no—
87C4⋊C4×D61—21no—
88#14519710—21no—
89#14519710—21no—
90#14519710—21no—
91#14519709—21no—
92#14519709—21no—
93#14519708—21no—
94#14519708—21no—
95#14519708—21no—
96#14519708D411yescharacteristic
97C4⋊#92418—21no—
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100C4⋊#92418C2311yescharacteristic
101#14519701—21no—
102#14519701—21no—
103#14519701—21no—
104#14519701—41no—
105#14519701D411yescharacteristic
106#14519700—21no—
107#14519700—41no—
108#14519698—21no—
109#14519697—21no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112#14519689—21no—
113#14519688—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C22×Q8×C61—21no—
117C2×SD16×C61D411yescharacteristic
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61—21no—
121C4⋊Q8×C61—41no—
122C4⋊Q8×C61C2311yescharacteristic
123C4⋊Q8×C61C2311yescharacteristic
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—41no—
126C4×Q8×C61—21no—
127C4×Q8×C61—21no—
128C4×Q8×C61C2311yescharacteristic
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61C2311yescharacteristic
132#106×C61—21no—
133#106×C61—21no—
134#106×C61—21no—
135#106×C61—21no—
136#106×C61D411yescharacteristic
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C4×C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—41no—
148C2×Q8×C61—41no—
149C2×Q8×C61—42no—
150C2×Q8×C61—81no—
151C2×Q8×C61C2×D411yescharacteristic
152C2×Q8×C61C2×D411yescharacteristic
153C2×D4×C61C2×D411yescharacteristic
154C22×C4×C61—21no—
155C22×C4×C61—41no—
156C2×C22⋊D61—41no—
157#11758222—42no—
158#11758222—81no—
159C2×D244C2×D411yescharacteristic
160C2×C4×D61—21no—
161C2×C4×D61—41no—
162C2×C4×D61—41no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165SD16×C61—22no—
166C2×C8×C61—21no—
167C2×C8×C61C2×D411yescharacteristic
168C2×C8×C61C2×D411yescharacteristic
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—41no—
176C4⋊C4×C61—41no—
177C4⋊C4×C61—41no—
178C4⋊C4×C61C2×D411yescharacteristic
179C4⋊C4×C61C2×D411yescharacteristic
180C4⋊C4×C61C2×D411yescharacteristic
181C22⋊C4×C61—21no—
182C22⋊C4×C61—41no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—41no—
186C42×C61C2411yescharacteristic
187C22⋊Dic61—41no—
188#11758202—42no—
189#11758202—42no—
190#11758202—81no—
191#11758201—42no—
192#11758201SD1614yescomplemented
193#11758198—21no—
194#11758198—41no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61C2×D411yescharacteristic
200#11758196—41no—
201#11758196—41no—
202#11758196—41no—
203#11758194—42no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418C2×D411yescharacteristic
209C2×#92418C2×D411yescharacteristic
210C8×D61—42no—
211C23×C61—21no—
212C22×D61—21no—
213Q8×C61—21no—
214Q8×C61—21no—
215Q8×C61—21no—
216Q8×C61—41no—
217Q8×C61—41no—
218Q8×C61—82no—
219Q8×C61—81no—
220Q8×C61—161no—
221Q8×C61C2×SD1612yes—
222D4×C61—21no—
223D4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—82no—
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61H(5,𝔽2)11yescharacteristic
237C22⋊D61—82no—
238C2×Dic61—21no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241D244—21no—
242D244C2×SD1612yes—
243C4×D61—41no—
244C4×D61—41no—
245C4×D61—82no—
246C4×D61—161no—
247Dic122—41no—
248Dic122—41no—
249Dic122—81no—
250C8×C61—21no—
251C8×C61—41no—
252C8×C61#13712yescomplemented
253#92418—21no—
254#92418—41no—
255#92418—41no—
256#92418—41no—
257#92418—41no—
258#92418C2×SD1612yes—
259#19895—611nomaximal, Sylow
260C22×C61—41no—
261C22×C61—41no—
262C22×C61—41no—
263C22×C61D4211yescharacteristic
264C2×D61—22no—
265C2×D61—41no—
266C2×D61—41no—
267C4×C61—22no—
268C4×C61—21no—
269C4×C61—41no—
270C4×C61—41no—
271C4×C61—41no—
272C4×C61—41no—
273C4×C61—81no—
274C4×C61—81no—
275C4×C61C22×SD1611yescharacteristic
276C4×C61C2×#13711yescharacteristic
277C4×C61#57711yescharacteristic
278C4×C61#57811yescharacteristic
279Dic61—41no—
280Dic61—81no—
281Dic61—81no—
282#3385—611no—
283Q8×SD16D6111yescharacteristic, complemented
284#3335—611no—
285#3276—611no—
286#1707—611no—
287#1695—611no—
288#1690—611no—
289#1672—611no—
290#1671—611no—
291#1645—611no—
292#1643—611no—
293#1637—611no—
294#1635—611no—
295#1590—611no—
296C8×SD16—611no—
297C2×C61—81no—
298C2×C61#327611yescharacteristic
299C2×C61D4×SD1611yescharacteristic
300C2×C61#329011yescharacteristic
301D61—42no—
302C2×#137—1221no—
303Q82C2×D6111yescharacteristic
304#554—611no—
305D4×Q8C2×D6111yescharacteristic
306#547—611no—
307#501—1221no—
308#500—611no—
309#498—21no—
310#498—611no—
311#498C2×D6111yescharacteristic
312#494—611no—
313#487—611no—
314#487—611no—
315#483—1221no—
316#481—1221no—
317#479—1221no—
318#477—611no—
319#476—21no—
320#476—611no—
321#476—611no—
322#476C2×D6111yescharacteristic
323#475—21no—
324#475—611no—
325#475C2×D6111yescharacteristic
326#474—1221no—
327#470—1221no—
328#465—1221no—
329#462—611no—
330#461—611no—
331#452—1221no—
332#451—1221no—
333#446—611no—
334C8×Q8C2×D6111yescharacteristic
335#440—1221no—
336C4×SD16—21no—
337C4×SD16—611no—
338C4×SD16C2×D6111yescharacteristic
339#436—611no—
340C8×D4—611no—
341C2×#106—1221no—
342#334—611no—
343#334—611no—
344#333—611no—
345#333—611no—
346#332—611no—
347#332—611no—
348Q8⋊C8—611no—
349Q8⋊C8—611no—
350D4⋊C8—611no—
351D4⋊C8—611no—
352C82—611no—
353C61#1989511yescharacteristic, complemented, Sylow
354C2×#41—1221no—
355C22×Q8—21no—
356#137—2444no—
357C2×Q16—1221no—
358C2×Q16—1221no—
359C2×Q16—1221no—
360C2×SD16—611no—
361C2×SD16—1221no—
362C2×SD16C22⋊D6111yescharacteristic
363C2×M16—1221no—
364C22×C8—1221no—
365C4⋊Q8—21no—
366C4⋊Q8—21no—
367C4⋊Q8—21no—
368C4⋊Q8—41no—
369C4⋊Q8—611no—
370C4⋊Q8—611no—
371C4⋊Q8C22×D6111yescharacteristic
372C4⋊Q8C22×D6111yescharacteristic
373#125—1221no—
374#123—2441no—
375C22⋊Q8—21no—
376C22⋊Q8—41no—
377C22⋊Q8—1221no—
378C22⋊Q8—1221no—
379C22⋊Q8—2441no—
380#121—1221no—
381C4×Q8—21no—
382C4×Q8—21no—
383C4×Q8C22×D6111yescharacteristic
384C4×Q8C22×D6111yescharacteristic
385C4×D4—21no—
386C4×D4—611no—
387C4×D4—1221no—
388C4×D4C22×D6111yescharacteristic
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391#107—1221no—
392#107—1221no—
393#107—1221no—
394#106—21no—
395#106—21no—
396#106—21no—
397#106—21no—
398#106—611no—
399#106—1221no—
400#106—1221no—
401#106—1221no—
402#106—1221no—
403#106—1221no—
404#106C22⋊D6111yescharacteristic
405C4⋊C8—21no—
406C4⋊C8—611no—
407C4⋊C8—611no—
408C4⋊C8—611no—
409C4⋊C8—1221no—
410C4⋊C8C22×D6111yescharacteristic
411#103—21no—
412#103—611no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#103—1221no—
417#103—2441no—
418#103—2441no—
419#103C22⋊D6111yescharacteristic
420#102—21no—
421#102—611no—
422#102—1221no—
423#102—1221no—
424#102C22⋊D6111yescharacteristic
425C22⋊C8—1221no—
426C22⋊C8—1221no—
427#97—1221no—
428C4×C8—21no—
429C4×C8—611no—
430C4×C8—611no—
431C4×C8C22×D6111yescharacteristic
432#41—4881no—
433#41—2442no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—41no—
437C2×Q8—41no—
438C2×Q8—42no—
439C2×Q8—81no—
440C2×Q8—1221no—
441C2×Q8—1221no—
442C2×Q8D4×D6111yescharacteristic
443C2×Q8C2×C22⋊D6111yescharacteristic
444C2×D4—611no—
445C2×D4—2441no—
446C2×D4C2×C22⋊D6111yescharacteristic
447C22×C4—21no—
448C22×C4—41no—
449C22×C4—1221no—
450C22×C4—2441no—
451C22×C4—2441no—
452Q16—4881no—
453Q16—2442no—
454Q16—2442no—
455SD16—22no—
456SD16—614no—
457SD16—2442no—
458M16—2442no—
459C2×C8—21no—
460C2×C8—611no—
461C2×C8—611no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—2442no—
467C2×C8C2×C22⋊D6111yescharacteristic
468C2×C8C2×C22⋊D6111yescharacteristic
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—611no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—2441no—
484C4⋊C4—2441no—
485C4⋊C4—2441no—
486C4⋊C4D4×D6111yescharacteristic
487C4⋊C4C2×C22⋊D6111yescharacteristic
488C4⋊C4C2×C22⋊D6111yescharacteristic
489C22⋊C4—21no—
490C22⋊C4—41no—
491C22⋊C4—1221no—
492C22⋊C4—2441no—
493C22⋊C4—2441no—
494C42—21no—
495C42—21no—
496C42—41no—
497C42C23×D6111yescharacteristic
498C23—21no—
499C23—1221no—
500Q8—21no—
501Q8—21no—
502Q8—21no—
503Q8—41no—
504Q8—41no—
505Q8—82no—
506Q8—81no—
507Q8—161no—
508Q8—4881no—
509Q8—2441no—
510Q8—2441no—
511Q8SD16×D6112yes—
512D4—21no—
513D4—21no—
514D4—612no—
515D4—1221no—
516D4—4882no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—82no—
527C2×C4—1221no—
528C2×C4—9761no—
529C2×C4—4882no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4C2×D4×D6111yescharacteristic
535C2×C4C22×C22⋊D6111yescharacteristic
536C2×C4#1451982211yescharacteristic
537C8—21no—
538C8—41no—
539C8—612no—
540C8—1221no—
541C8—2441no—
542C8—2441no—
543C8—2441no—
544C8—2441no—
545C8#1451980812yescomplemented
546C22—41no—
547C22—41no—
548C22—41no—
549C22—1222no—
550C22—2441no—
551C22—2441no—
552C22D4×C22⋊D6111yescharacteristic
553C4—22no—
554C4—21no—
555C4—41no—
556C4—41no—
557C4—41no—
558C4—41no—
559C4—81no—
560C4—81no—
561C4—4881no—
562C4—4881no—
563C4—2441no—
564C4C2×SD16×D6111yescharacteristic
565C4C2×#1451980811yescharacteristic
566C4#1684056811yescharacteristic
567C4#1684057711yescharacteristic
568C2—81no—
569C2—2442no—
570C2#2145746511yescharacteristic
571C2SD16×C22⋊D6111yescharacteristic
572C2#2145759911yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingQ8×SD16×C61C2
socleC22×C61D42
prime radical#16840417C22
hypercenterQ8×SD16D61
hypocenterC61#19895
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ Q8×SD16

Complements

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

NH ≅ G/Ntype
1. #21458725C2semidirect
4. #21445414C2semidirect
5. #21445340C2semidirect
6. #21445311C2semidirect
7. #21445201C2semidirect
8. #21445191C2semidirect
9. #21445040C2semidirect
10. #21445025C2semidirect
11. #21444984C2semidirect
13. #21444710C2semidirect
14. C8×#11758201C2semidirect
15. Q8×SD16×C61C2semidirect
192. #11758201SD16semidirect
252. C8×C61#137semidirect
283. Q8×SD16D61semidirect
353. C61#19895semidirect
545. C8#14519808semidirect

Maximal subgroups

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

subgroupindex
#214587252
#214581782
#214574652
#214454142
#214453402
#214453112
#214452012
#214451912
#214450402
#214450252
#214449842
#214449692
#214447102
C8×#117582012
Q8×SD16×C612
#1989561