Finite Group Databasebeta

← #29670695

Subgroups — #29670695

Order 15616.

590 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11758202C211yesmaximal, characteristic, complemented
2#21458557C211yesmaximal, characteristic, complemented
3#21457758C211yesmaximal, characteristic
4#21445469C211yesmaximal, characteristic, complemented
5#21445356C211yesmaximal, characteristic, complemented
6#21445277C211yesmaximal, characteristic, complemented
7#21445261C211yesmaximal, characteristic, complemented
8#21445216C211yesmaximal, characteristic, complemented
9#21445040C211yesmaximal, characteristic, complemented
10#21444989C211yesmaximal, characteristic, complemented
11#21444946C211yesmaximal, characteristic
12#21444896C211yesmaximal, characteristic, complemented
13#21444669C211yesmaximal, characteristic, complemented
14SD16⋊#92418C211yesmaximal, characteristic, complemented
15C61×#3362C211yesmaximal, characteristic
16Q8×Dic122C2211yescharacteristic
17#16840482C2211yescharacteristic
18#16840165—21no—
19#16840165C2211yescharacteristic
20#16840134C2211yescharacteristic
21#16840120—21no—
22#16840113C2211yescharacteristic
23#16840112C2211yescharacteristic
24#16840108C2211yescharacteristic
25#16840083—21no—
26#16840057—21no—
27#16840057C2211yescharacteristic
28#16840054—21no—
29#16840054C2211yescharacteristic
30#16840052C2211yescharacteristic
31#16840047C2211yescharacteristic
32#16840046—21no—
33#16839989—21no—
34#16839982—21no—
35#16839972C2211yescharacteristic
36#16839959C2211yescharacteristic
37#16839924—21no—
38#16839892C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40#16839866—21no—
41C4×#11758202—21no—
42C4×#11758202C2211yescharacteristic
43D4×#92418C2211yescharacteristic
44#16839758—21no—
45#16839515C2211yescharacteristic
46#16839514C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839509C2211yescharacteristic
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53#16839492C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839483C2211yescharacteristic
56Q82×C61C2211yescharacteristic
57C61×#550C2211yescharacteristic
58C61×#486—21no—
59C61×#486C2211yescharacteristic
60C61×#476—21no—
61C61×#476C2211yescharacteristic
62C61×#460—21no—
63C61×#460C2211yescharacteristic
64C8×Q8×C61C2211yescharacteristic
65C4×SD16×C61—21no—
66C4×SD16×C61C2211yescharacteristic
67C2×#11758202—21no—
68C2×#11758202D411yescharacteristic
69C2×#11758200—21no—
70C2×#11758200—41no—
71C22×#92418—21no—
72#14519777—21no—
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519776—21no—
76#14519776—41no—
77#14519770—41no—
78#14519764—21no—
79#14519756—21no—
80Q8×Dic61—21no—
81C4×Dic122—21no—
82C4×Dic122C2311yescharacteristic
83#14519742—21no—
84#14519736—21no—
85#14519710—21no—
86#14519710—21no—
87#14519710—21no—
88#14519710—21no—
89#14519710D411yescharacteristic
90#14519708—21no—
91#14519708—21no—
92C4⋊#92418—21no—
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96#14519706—21no—
97#14519702—21no—
98#14519702—21no—
99#14519701—21no—
100#14519701—21no—
101#14519701—41no—
102#14519701D411yescharacteristic
103#14519700—21no—
104#14519700—21no—
105#14519700D411yescharacteristic
106#14519698—21no—
107#14519698—21no—
108#14519698—21no—
109#14519698—21no—
110C22⋊#92418—21no—
111#14519687—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×#41×C61—21no—
116C2×SD16×C61—21no—
117C2×SD16×C61D411yescharacteristic
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61—41no—
121C4⋊Q8×C61C2311yescharacteristic
122#127×C61—21no—
123#127×C61C2311yescharacteristic
124#121×C61—21no—
125#121×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×C61D411yescharacteristic
134C4⋊C8×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136#103×C61D411yescharacteristic
137#102×C61—21no—
138#102×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61D411yescharacteristic
142C4×C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144#41×C61—42no—
145#41×C61—81no—
146C2×Q8×C61—21no—
147C2×Q8×C61—41no—
148C2×Q8×C61C2×D411yescharacteristic
149C2×Q8×C61C2×D411yescharacteristic
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—41no—
154C2×D4×C61—41no—
155C2×D4×C61C2×D411yescharacteristic
156C22×C4×C61—21no—
157C22×C4×C61—41no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—41no—
161C2×Dic122C2×D411yescharacteristic
162SD16×C61—22no—
163SD16×C61—42no—
164C2×C8×C61—21no—
165C2×C8×C61C2×D411yescharacteristic
166C2×C8×C61C2×D411yescharacteristic
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×C61C2×D411yescharacteristic
174C4⋊C4×C61C2×D411yescharacteristic
175C4⋊C4×C61C2×D411yescharacteristic
176C22⋊C4×C61—21no—
177C22⋊C4×C61—41no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61—41no—
181C42×C61C2411yescharacteristic
182C22⋊Dic61—41no—
183C22⋊Dic61—41no—
184#11758202—22no—
185#11758202—42no—
186#11758200—81no—
187#11758200—82no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—41no—
193C4⋊Dic61C2×D411yescharacteristic
194#11758196—21no—
195#11758196—41no—
196#11758196—41no—
197C4×Dic61—21no—
198C4×Dic61—41no—
199C4×Dic61—41no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—81no—
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207C23×C61—21no—
208C23×C61—41no—
209Q8×C61—21no—
210Q8×C61—21no—
211Q8×C61—81no—
212Q8×C61—161no—
213Q8×C61#13512yes—
214Q8×C61#13712yes—
215D4×C61—21no—
216D4×C61—21no—
217D4×C61—41no—
218D4×C61—41no—
219D4×C61—41no—
220D4×C61—41no—
221D4×C61—82no—
222D4×C61—81no—
223D4×C61—161no—
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—82no—
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61H(5,𝔽2)11yescharacteristic
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240Dic122—21no—
241Dic122—21no—
242Dic122—41no—
243Dic122—41no—
244Dic122—81no—
245Dic122—161no—
246C8×C61—21no—
247C8×C61—21no—
248C8×C61—41no—
249#92418—21no—
250#92418—21no—
251#92418—41no—
252#92418—41no—
253#92418—41no—
254#92418—41no—
255#19844—611nomaximal, Sylow
256C22×C61—41no—
257C22×C61—41no—
258C22×C61—41no—
259C22×C61—81no—
260C22×C61—81no—
261C22×C61—81no—
262C22×C61D4211yescharacteristic
263C4×C61—22no—
264C4×C61—22no—
265C4×C61—21no—
266C4×C61—41no—
267C4×C61—41no—
268C4×C61—41no—
269C4×C61—81no—
270C4×C61C2×#13511yescharacteristic
271C4×C61C2×#13711yescharacteristic
272C4×C61#57711yescharacteristic
273C4×C61#57811yescharacteristic
274Dic61—41no—
275Dic61—41no—
276Dic61—81no—
277Dic61—81no—
278Q8×Q16—611no—
279#3369—611no—
280#3362D6111yescharacteristic
281#3304—611no—
282#1719—611no—
283#1698—611no—
284#1684—611no—
285#1682—611no—
286#1675—611no—
287#1645—611no—
288#1638—611no—
289#1631—611no—
290#1623—611no—
291#1584—611no—
292SD16⋊C8—611no—
293C2×C61—81no—
294C2×C61—161no—
295C2×C61#326611yescharacteristic
296C2×C61#328511yescharacteristic
297C2×C61#328811yescharacteristic
298Q82—611no—
299Q82C2×D6111yescharacteristic
300#556—611no—
301#550C2×D6111yescharacteristic
302#500—611no—
303#500—1221no—
304#492—611no—
305#489—1221no—
306#487—611no—
307#487—611no—
308#486—21no—
309#486—611no—
310#486C2×D6111yescharacteristic
311#482—1221no—
312#477—611no—
313#477—611no—
314#477—1221no—
315#477—1221no—
316#476—21no—
317#476—611no—
318#476C2×D6111yescharacteristic
319#475—611no—
320#475—1221no—
321#465—1221no—
322#464—1221no—
323#462—611no—
324#460—21no—
325#460—611no—
326#460C2×D6111yescharacteristic
327#452—1221no—
328#446—611no—
329C8×Q8—611no—
330C8×Q8C2×D6111yescharacteristic
331#441—1221no—
332C4×Q16—611no—
333C4×Q16—1221no—
334C4×SD16—21no—
335C4×SD16C2×D6111yescharacteristic
336C8×D4—611no—
337#416—1221no—
338#335—611no—
339#335—611no—
340#333—611no—
341#333—611no—
342#332—611no—
343#331—611no—
344Q8⋊C8—611no—
345Q8⋊C8—611no—
346Q8⋊C8—611no—
347D4⋊C8—611no—
348#322—611no—
349C61#1984411yescharacteristic, complemented, Sylow
350C2×#41—21no—
351C2×Q16—611no—
352C2×Q16—1221no—
353C2×SD16—21no—
354C2×SD16—1221no—
355C2×SD16—2441no—
356C2×SD16C22⋊D6111yescharacteristic
357C22×C8—1221no—
358C4⋊Q8—21no—
359C4⋊Q8—21no—
360C4⋊Q8—41no—
361C4⋊Q8—611no—
362C4⋊Q8—611no—
363C4⋊Q8—1221no—
364C4⋊Q8—1221no—
365C4⋊Q8—2441no—
366C4⋊Q8C22×D6111yescharacteristic
367#127—21no—
368#127C22×D6111yescharacteristic
369#126—2441no—
370#124—1221no—
371C22⋊Q8—1221no—
372#121—21no—
373#121—41no—
374C4×Q8—21no—
375C4×Q8—21no—
376C4×Q8—611no—
377C4×Q8—1221no—
378C4×Q8—1221no—
379C4×Q8—1221no—
380C4×Q8C22×D6111yescharacteristic
381C4×Q8C22×D6111yescharacteristic
382C4×D4—21no—
383C4×D4C22×D6111yescharacteristic
384#117—1221no—
385#107—611no—
386#107—1221no—
387#107—1221no—
388#107—1221no—
389#107—1221no—
390#106—21no—
391#106—1221no—
392#106—1221no—
393#106C22⋊D6111yescharacteristic
394C4⋊C8—21no—
395C4⋊C8—611no—
396C4⋊C8—611no—
397C4⋊C8—611no—
398C4⋊C8—1221no—
399C4⋊C8—1221no—
400C4⋊C8C22×D6111yescharacteristic
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—2441no—
410#103C22⋊D6111yescharacteristic
411#102—21no—
412#102—21no—
413#102—21no—
414#102—21no—
415#102—1221no—
416#102—1221no—
417#102—1221no—
418#102—1221no—
419#102C22⋊D6111yescharacteristic
420C22⋊C8—1221no—
421#97—1221no—
422C4×C8—21no—
423C4×C8—611no—
424C4×C8—611no—
425C4×C8—1221no—
426C4×C8C22×D6111yescharacteristic
427#41—42no—
428#41—81no—
429C2×Q8—21no—
430C2×Q8—41no—
431C2×Q8—611no—
432C2×Q8—1221no—
433C2×Q8—1221no—
434C2×Q8—2441no—
435C2×Q8D4×D6111yescharacteristic
436C2×Q8C2×C22⋊D6111yescharacteristic
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—41no—
441C2×D4—41no—
442C2×D4C2×C22⋊D6111yescharacteristic
443C22×C4—21no—
444C22×C4—41no—
445Q16—1222no—
446Q16—2442no—
447SD16—22no—
448SD16—42no—
449SD16—4881no—
450SD16—4882no—
451C2×C8—21no—
452C2×C8—611no—
453C2×C8—611no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—4881no—
459C2×C8C2×C22⋊D6111yescharacteristic
460C2×C8C2×C22⋊D6111yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—611no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—2441no—
474C4⋊C4—2441no—
475C4⋊C4—2441no—
476C4⋊C4D4×D6111yescharacteristic
477C4⋊C4C2×C22⋊D6111yescharacteristic
478C4⋊C4C2×C22⋊D6111yescharacteristic
479C22⋊C4—21no—
480C22⋊C4—41no—
481C22⋊C4—2441no—
482C22⋊C4—2441no—
483C42—21no—
484C42—21no—
485C42—41no—
486C42—1221no—
487C42—2441no—
488C42—2441no—
489C42C23×D6111yescharacteristic
490C23—21no—
491C23—41no—
492Q8—21no—
493Q8—21no—
494Q8—81no—
495Q8—161no—
496Q8—1221no—
497Q8—1221no—
498Q8—9761no—
499Q8—4881no—
500Q8—2441no—
501Q8—2441no—
502Q8#1451979712yes—
503Q8#1451980612yes—
504D4—21no—
505D4—21no—
506D4—41no—
507D4—41no—
508D4—41no—
509D4—41no—
510D4—82no—
511D4—81no—
512D4—161no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—82no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4C2×D4×D6111yescharacteristic
527C2×C4C22×C22⋊D6111yescharacteristic
528C2×C4#1451982211yescharacteristic
529C8—21no—
530C8—21no—
531C8—41no—
532C8—1221no—
533C8—1221no—
534C8—2441no—
535C8—2441no—
536C8—2441no—
537C8—2441no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22D4×C22⋊D6111yescharacteristic
545C4—22no—
546C4—21no—
547C4—22no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—81no—
552C4—4881no—
553C4—4881no—
554C4—2441no—
555C4—2441no—
556C4C2×#1451979711yescharacteristic
557C4C2×#1451980611yescharacteristic
558C4#1684056811yescharacteristic
559C4#1684057711yescharacteristic
560C2—81no—
561C2—161no—
562C2#2145732911yescharacteristic
563C2#2145755611yescharacteristic
564C2#2145758311yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3362C2
socleC22×C61D42
prime radicalC61×#550C22
hypercenter#3362D61
hypocenterC61#19844
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3362

Complements

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

NH ≅ G/Ntype
1. Q8×#11758202C2semidirect
2. #21458557C2semidirect
4. #21445469C2semidirect
5. #21445356C2semidirect
6. #21445277C2semidirect
7. #21445261C2semidirect
8. #21445216C2semidirect
9. #21445040C2semidirect
10. #21444989C2semidirect
12. #21444896C2semidirect
13. #21444669C2semidirect
14. SD16⋊#92418C2semidirect
349. C61#19844semidirect

Maximal subgroups

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

subgroupindex
Q8×#117582022
#214585572
#214577582
#214454692
#214453562
#214452772
#214452612
#214452162
#214450402
#214449892
#214449462
#214448962
#214446692
SD16⋊#924182
C61×#33622
#1984461