Finite Group Databasebeta

← #29670719

Subgroups — #29670719

Order 15616.

566 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11758202C211yesmaximal, characteristic, complemented
2#21458511C211yesmaximal, characteristic
3#21458495C211yesmaximal, characteristic, complemented
4#21445469C211yesmaximal, characteristic
5#21445401C211yesmaximal, characteristic, complemented
6#21445261C211yesmaximal, characteristic
7#21445232C211yesmaximal, characteristic
8#21445221C211yesmaximal, characteristic, complemented
9#21445040C211yesmaximal, characteristic, complemented
10#21444975C211yesmaximal, characteristic, complemented
11#21444940C211yesmaximal, characteristic
12#21444896C211yesmaximal, characteristic
13#21444667C211yesmaximal, characteristic, complemented
14#21444638C211yesmaximal, characteristic, complemented
15C61×#3375C211yesmaximal, characteristic
16Q8×Dic122C2211yescharacteristic
17#16840490C2211yescharacteristic
18#16840165—21no—
19#16840165C2211yescharacteristic
20#16840134C2211yescharacteristic
21#16840116—21no—
22#16840113C2211yescharacteristic
23#16840112C2211yescharacteristic
24#16840111—21no—
25#16840108C2211yescharacteristic
26#16840062—21no—
27#16840057—21no—
28#16840057C2211yescharacteristic
29#16840057C2211yescharacteristic
30#16840056—21no—
31#16840054—21no—
32#16840054C2211yescharacteristic
33#16840047C2211yescharacteristic
34#16839981—21no—
35#16839972C2211yescharacteristic
36#16839970—21no—
37#16839959C2211yescharacteristic
38Q8×#92418C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40C8×Dic122C2211yescharacteristic
41C4×#11758202—21no—
42C4×#11758202C2211yescharacteristic
43#16839854—21no—
44C4×#11758200—21no—
45#16839514C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839509C2211yescharacteristic
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53Q8⋊#92418C2211yescharacteristic
54#16839492C2211yescharacteristic
55#16839483C2211yescharacteristic
56C61×#559C2211yescharacteristic
57Q82×C61C2211yescharacteristic
58C61×#489—21no—
59C61×#486C2211yescharacteristic
60C61×#477C2211yescharacteristic
61C61×#474—21no—
62C61×#464—21no—
63C61×#460C2211yescharacteristic
64C61×#446C2211yescharacteristic
65C61×#440—21no—
66C4×Q16×C61C2211yescharacteristic
67C2×#11758202—21no—
68C2×#11758202D411yescharacteristic
69C2×#11758200—21no—
70C2×#11758200—41no—
71#14519777—21no—
72#14519777C2311yescharacteristic
73#14519777C2311yescharacteristic
74#14519776—21no—
75#14519776—41no—
76#14519775—21no—
77#14519769—21no—
78#14519767—41no—
79Q8×Dic61—21no—
80C4×Dic122—21no—
81C4×Dic122—21no—
82C4×Dic122C2311yescharacteristic
83#14519742—21no—
84#14519710—21no—
85#14519710—21no—
86#14519710—21no—
87#14519710—21no—
88#14519710—21no—
89#14519710D411yescharacteristic
90#14519708—21no—
91C4⋊#92418—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—21no—
102#14519701—41no—
103#14519701D411yescharacteristic
104#14519700—21no—
105#14519700—21no—
106#14519700D411yescharacteristic
107#14519698—21no—
108#14519698—21no—
109#14519698—21no—
110C4×#92418—21no—
111C4×#92418—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
115C2×Q16×C61D411yescharacteristic
116C2×SD16×C61—21no—
117C4⋊Q8×C61—21no—
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61—41no—
120C4⋊Q8×C61C2311yescharacteristic
121#127×C61C2311yescharacteristic
122#124×C61—21no—
123#124×C61—21no—
124#123×C61—41no—
125C4×Q8×C61—21no—
126C4×Q8×C61—21no—
127C4×Q8×C61C2311yescharacteristic
128C4×Q8×C61C2311yescharacteristic
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61—21no—
132#107×C61—21no—
133#107×C61D411yescharacteristic
134C4⋊C8×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136#103×C61—21no—
137#103×C61D411yescharacteristic
138#103×C61D411yescharacteristic
139#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142#97×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—41no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×Q8×C61C2×D411yescharacteristic
148C2×Q8×C61C2×D411yescharacteristic
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C22×C4×C61—41no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—41no—
156C2×Dic122C2×D411yescharacteristic
157Q16×C61—22no—
158SD16×C61—42no—
159C2×C8×C61—21no—
160C2×C8×C61C2×D411yescharacteristic
161C2×C8×C61C2×D411yescharacteristic
162C4⋊C4×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—
169C4⋊C4×C61C2×D411yescharacteristic
170C4⋊C4×C61C2×D411yescharacteristic
171C4⋊C4×C61C2×D411yescharacteristic
172C22⋊C4×C61—41no—
173C22⋊C4×C61—41no—
174C22⋊C4×C61—41no—
175C42×C61—21no—
176C42×C61—21no—
177C42×C61—21no—
178C42×C61—41no—
179C42×C61C2411yescharacteristic
180#11758202—22no—
181#11758202—42no—
182#11758200—42no—
183#11758200—82no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—41no—
189C4⋊Dic61C2×D411yescharacteristic
190#11758196—21no—
191#11758196—41no—
192#11758196—41no—
193#11758196—41no—
194#11758196—41no—
195C4×Dic61—21no—
196C4×Dic61—41no—
197C4×Dic61—41no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418C2×D411yescharacteristic
203C2×#92418C2×D411yescharacteristic
204C23×C61—41no—
205Q8×C61—21no—
206Q8×C61—21no—
207Q8×C61—21no—
208Q8×C61—21no—
209Q8×C61—21no—
210Q8×C61—81no—
211Q8×C61—161no—
212Q8×C61#13512yes—
213D4×C61—41no—
214D4×C61—41no—
215D4×C61—41no—
216D4×C61—41no—
217D4×C61—81no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—41no—
225C2×C4×C61—161no—
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61H(5,𝔽2)11yescharacteristic
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233Dic122—21no—
234Dic122—41no—
235Dic122—41no—
236Dic122—81no—
237Dic122—161no—
238Dic122#13712yes—
239C8×C61—21no—
240C8×C61—21no—
241C8×C61—41no—
242#92418—21no—
243#92418—21no—
244#92418—41no—
245#92418—41no—
246#92418—41no—
247#92418—41no—
248#19846—611nomaximal, Sylow
249C22×C61—81no—
250C22×C61—81no—
251C22×C61—81no—
252C22×C61D4211yescharacteristic
253C4×C61—22no—
254C4×C61—21no—
255C4×C61—41no—
256C4×C61—41no—
257C4×C61—41no—
258C4×C61—41no—
259C4×C61—41no—
260C4×C61—81no—
261C4×C61C2×#13511yescharacteristic
262C4×C61C2×#13711yescharacteristic
263C4×C61#57811yescharacteristic
264C4×C61#57811yescharacteristic
265Dic61—22no—
266Dic61—41no—
267Dic61—81no—
268Dic61—81no—
269Q8×Q16—611no—
270#3375D6111yescharacteristic
271#3365—611no—
272#3363—611no—
273#1719—611no—
274#1704—611no—
275#1682—611no—
276#1678—611no—
277#1676—611no—
278#1645—611no—
279#1636—611no—
280#1630—611no—
281#1623—611no—
282#1584—611no—
283#1579—611no—
284C2×C61—161no—
285C2×C61#327011yescharacteristic
286C2×C61#328511yescharacteristic
287C2×C61#328811yescharacteristic
288#559C2×D6111yescharacteristic
289Q82—611no—
290Q82C2×D6111yescharacteristic
291#557—611no—
292#500—611no—
293#500—1221no—
294#492—611no—
295#489—21no—
296#488—1221no—
297#487—611no—
298#487—611no—
299#487—1221no—
300#486—611no—
301#486C2×D6111yescharacteristic
302#478—1221no—
303#477—611no—
304#477—611no—
305#477—611no—
306#477—1221no—
307#477—1221no—
308#477—1221no—
309#477C2×D6111yescharacteristic
310#475—611no—
311#474—21no—
312#464—21no—
313#464—1221no—
314#462—611no—
315#462—1221no—
316#460—611no—
317#460C2×D6111yescharacteristic
318#446C2×D6111yescharacteristic
319C8×Q8—611no—
320C8×Q8—611no—
321C8×Q8—611no—
322#440—21no—
323C4×Q16—611no—
324C4×Q16—1221no—
325C4×Q16—1221no—
326C4×Q16C2×D6111yescharacteristic
327C4×SD16—1221no—
328#335—611no—
329#334—611no—
330#333—611no—
331#333—611no—
332#332—611no—
333#331—611no—
334Q8⋊C8—611no—
335Q8⋊C8—611no—
336Q8⋊C8—611no—
337Q8⋊C8—611no—
338#322—611no—
339C61#1984611yescharacteristic, complemented, Sylow
340C2×Q16—611no—
341C2×Q16—1221no—
342C2×Q16C22⋊D6111yescharacteristic
343C2×SD16—21no—
344C2×SD16—1221no—
345C2×SD16—2441no—
346C4⋊Q8—21no—
347C4⋊Q8—21no—
348C4⋊Q8—41no—
349C4⋊Q8—611no—
350C4⋊Q8—611no—
351C4⋊Q8—1221no—
352C4⋊Q8—1221no—
353C4⋊Q8—1221no—
354C4⋊Q8—2441no—
355C4⋊Q8C22×D6111yescharacteristic
356#127C22×D6111yescharacteristic
357#125—1221no—
358#125—2441no—
359#124—21no—
360#124—21no—
361#123—41no—
362C4×Q8—21no—
363C4×Q8—21no—
364C4×Q8—611no—
365C4×Q8—1221no—
366C4×Q8—1221no—
367C4×Q8—1221no—
368C4×Q8—1221no—
369C4×Q8C22×D6111yescharacteristic
370C4×Q8C22×D6111yescharacteristic
371C4×Q8C22×D6111yescharacteristic
372C4×D4—21no—
373C4×D4—21no—
374#107—21no—
375#107—611no—
376#107—1221no—
377#107—1221no—
378#107—1221no—
379#107—1221no—
380#107—1221no—
381#107C22⋊D6111yescharacteristic
382#106—1221no—
383C4⋊C8—21no—
384C4⋊C8—611no—
385C4⋊C8—611no—
386C4⋊C8—611no—
387C4⋊C8—1221no—
388C4⋊C8—1221no—
389C4⋊C8—1221no—
390C4⋊C8C22×D6111yescharacteristic
391#103—21no—
392#103—611no—
393#103—611no—
394#103—1221no—
395#103—1221no—
396#103—1221no—
397#103—1221no—
398#103—1221no—
399#103—1221no—
400#103—1221no—
401#103—2441no—
402#103C22⋊D6111yescharacteristic
403#103C22⋊D6111yescharacteristic
404#102—21no—
405#102—21no—
406#102—21no—
407#102—1221no—
408#102—1221no—
409#102—1221no—
410#97—21no—
411C4×C8—611no—
412C4×C8—611no—
413C4×C8—1221no—
414C4×C8—1221no—
415C4×C8—1221no—
416C4×C8C22×D6111yescharacteristic
417C2×Q8—21no—
418C2×Q8—41no—
419C2×Q8—611no—
420C2×Q8—1221no—
421C2×Q8—1221no—
422C2×Q8—2441no—
423C2×Q8D4×D6111yescharacteristic
424C2×Q8C2×C22⋊D6111yescharacteristic
425C2×Q8C2×C22⋊D6111yescharacteristic
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C22×C4—41no—
430Q16—22no—
431Q16—1222no—
432Q16—2442no—
433SD16—42no—
434SD16—4882no—
435SD16—2442no—
436C2×C8—21no—
437C2×C8—611no—
438C2×C8—611no—
439C2×C8—1221no—
440C2×C8—1221no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8C2×C22⋊D6111yescharacteristic
444C2×C8C2×C22⋊D6111yescharacteristic
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—611no—
453C4⋊C4—1221no—
454C4⋊C4—1221no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—1221no—
458C4⋊C4—2441no—
459C4⋊C4—2441no—
460C4⋊C4—2441no—
461C4⋊C4—2441no—
462C4⋊C4—2441no—
463C4⋊C4D4×D6111yescharacteristic
464C4⋊C4C2×C22⋊D6111yescharacteristic
465C4⋊C4C2×C22⋊D6111yescharacteristic
466C22⋊C4—41no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—41no—
473C42—1221no—
474C42—2441no—
475C42—2441no—
476C42C23×D6111yescharacteristic
477C23—41no—
478Q8—21no—
479Q8—21no—
480Q8—21no—
481Q8—21no—
482Q8—21no—
483Q8—81no—
484Q8—161no—
485Q8—612no—
486Q8—1221no—
487Q8—9761no—
488Q8—4881no—
489Q8—2441no—
490Q8—2441no—
491Q8#1451979712yes—
492D4—41no—
493D4—41no—
494D4—41no—
495D4—41no—
496D4—81no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—41no—
504C2×C4—161no—
505C2×C4—1221no—
506C2×C4—1221no—
507C2×C4—2441no—
508C2×C4—2441no—
509C2×C4C2×D4×D6111yescharacteristic
510C2×C4C22×C22⋊D6111yescharacteristic
511C2×C4#1451982211yescharacteristic
512C8—21no—
513C8—21no—
514C8—41no—
515C8—1221no—
516C8—1221no—
517C8—2441no—
518C8—2441no—
519C8—2441no—
520C8—2441no—
521C22—81no—
522C22—81no—
523C22—81no—
524C22D4×C22⋊D6111yescharacteristic
525C4—22no—
526C4—21no—
527C4—41no—
528C4—41no—
529C4—41no—
530C4—41no—
531C4—41no—
532C4—81no—
533C4—1222no—
534C4—4881no—
535C4—4881no—
536C4—2441no—
537C4C2×#1451979711yescharacteristic
538C4#137×D6111yescharacteristic
539C4#1684057711yescharacteristic
540C4#1684057811yescharacteristic
541C2—161no—
542C2#2145738311yescharacteristic
543C2#2145755611yescharacteristic
544C2#2145758311yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3375C2
socleC22×C61D42
prime radical#127×C61C23
hypercenter#3375D61
hypocenterC61#19846
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3375

Complements

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

NH ≅ G/Ntype
1. Q8×#11758202C2semidirect
3. #21458495C2semidirect
5. #21445401C2semidirect
8. #21445221C2semidirect
9. #21445040C2semidirect
10. #21444975C2semidirect
13. #21444667C2semidirect
14. #21444638C2semidirect
339. C61#19846semidirect

Maximal subgroups

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

subgroupindex
Q8×#117582022
#214585112
#214584952
#214454692
#214454012
#214452612
#214452322
#214452212
#214450402
#214449752
#214449402
#214448962
#214446672
#214446382
C61×#33752
#1984661