Finite Group Databasebeta

← #29670235

Subgroups — #29670235

Order 15616.

580 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458731C211yesmaximal, characteristic
2Q16×Dic122C211yesmaximal, characteristic, complemented
3Q8×#11758190C211yesmaximal, characteristic
4#21457697C211yesmaximal, characteristic
5#21445496C211yesmaximal, characteristic, complemented
6#21445325C211yesmaximal, characteristic, complemented
7#21445278C211yesmaximal, characteristic, complemented
8#21445223C211yesmaximal, characteristic, complemented
9#21445219C211yesmaximal, characteristic
10#21445042C211yesmaximal, characteristic, complemented
11#21444986C211yesmaximal, characteristic
12#21444961C211yesmaximal, characteristic, complemented
13#21444956C211yesmaximal, characteristic
14#21444633C211yesmaximal, characteristic
15C61×#1584C211yesmaximal, characteristic, complemented
16Q8×Dic122C2211yescharacteristic
17Q8×Dic122C2211yescharacteristic
18#16840472C2211yescharacteristic
19Q8×D244C2211yescharacteristic
20#16840171—21no—
21#16840167C2211yescharacteristic
22#16840167C2211yescharacteristic
23#16840166—21no—
24#16840160—21no—
25#16840160C2211yescharacteristic
26#16840142C2211yescharacteristic
27#16840113C2211yescharacteristic
28#16840113C2211yescharacteristic
29#16840099—21no—
30#16840070—21no—
31#16840058C2211yescharacteristic
32#16840057C2211yescharacteristic
33#16840057C2211yescharacteristic
34#16840057C2211yescharacteristic
35#16840055—21no—
36#16840055—21no—
37#16840053—21no—
38#16840053C2211yescharacteristic
39#16840053C2211yescharacteristic
40#16840048—21no—
41#16840048C2211yescharacteristic
42#16840043—21no—
43#16840002—21no—
44#16839972C2211yescharacteristic
45#16839916—21no—
46C8×Dic122C2211yescharacteristic
47#16839864—21no—
48Q16×Dic61—21no—
49C4×#11758190—21no—
50C4×#11758190C2211yescharacteristic
51C8×D244C2211yescharacteristic
52#103×D61—21no—
53#16839515C2211yescharacteristic
54#16839510C2211yescharacteristic
55#16839510C2211yescharacteristic
56#16839510C2211yescharacteristic
57#16839507C2211yescharacteristic
58#16839494C2211yescharacteristic
59#16839491C2211yescharacteristic
60C61×#446C2211yescharacteristic
61C8×Q8×C61C2211yescharacteristic
62C4×Q16×C61C2211yescharacteristic
63C61×#335C2211yescharacteristic
64C61×Q8⋊C8C2211yescharacteristic
65C61×Q8⋊C8C2211yescharacteristic
66C61×#322C2211yescharacteristic
67C2×Q8×D61—21no—
68C2×Dic244—41no—
69C2×#11758202—21no—
70C2×#11758190D411yescharacteristic
71C2×C8×D61—21no—
72#14519777—21no—
73#14519777—21no—
74#14519777—21no—
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519777C2311yescharacteristic
78#14519777C2311yescharacteristic
79#14519776—21no—
80#14519776—21no—
81#14519776—41no—
82#14519776—41no—
83#14519769—21no—
84#14519757—21no—
85#14519757—41no—
86#14519755—41no—
87#14519753—21no—
88Q8×Dic61—21no—
89Q8×Dic61—21no—
90C4×Dic122—21no—
91C4×Dic122—21no—
92C4×Dic122C2311yescharacteristic
93C4×D244—21no—
94C4×D244—21no—
95C4×D244C2311yescharacteristic
96C4⋊C4×D61—21no—
97#14519711—21no—
98#14519711—21no—
99#14519711—21no—
100#14519711—21no—
101#14519711—21no—
102#14519710—21no—
103#14519710—21no—
104#14519710—21no—
105#14519709—21no—
106#14519709—21no—
107#14519709—21no—
108#14519709—21no—
109#14519709—21no—
110#14519709D411yescharacteristic
111C4⋊#92418C2311yescharacteristic
112C4⋊#92418C2311yescharacteristic
113#14519706—21no—
114#14519702—21no—
115#14519702—21no—
116#14519702—21no—
117#14519702—21no—
118#14519702—41no—
119#14519702D411yescharacteristic
120#14519701—21no—
121#14519701—21no—
122#14519701—21no—
123#14519700—21no—
124#14519699—21no—
125#14519699—21no—
126#14519699D411yescharacteristic
127#14519689—21no—
128C8×Dic61—21no—
129C2×Q16×C61D411yescharacteristic
130C4×Q8×C61C2311yescharacteristic
131C4×Q8×C61C2311yescharacteristic
132#107×C61D411yescharacteristic
133C4⋊C8×C61—21no—
134C4⋊C8×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137#103×C61D411yescharacteristic
138#103×C61D411yescharacteristic
139#97×C61—21no—
140C4×C8×C61—21no—
141C4×C8×C61C2311yescharacteristic
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61C2×D411yescharacteristic
145C2×Q8×C61C2×D411yescharacteristic
146Q8×D61—42no—
147Q8×D61—81no—
148C2×D244C2×D411yescharacteristic
149C2×C4×D61—21no—
150C2×C4×D61—41no—
151C2×C4×D61—41no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—41no—
157C2×Dic122—41no—
158C2×Dic122—41no—
159C2×Dic122C2×D411yescharacteristic
160Q16×C61—22no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61C2×D411yescharacteristic
165C2×C8×C61C2×D411yescharacteristic
166C2×C8×C61C2×D411yescharacteristic
167C2×C8×C61C2×D411yescharacteristic
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61C2×D411yescharacteristic
171C4⋊C4×C61C2×D411yescharacteristic
172C42×C61—21no—
173C42×C61—21no—
174C42×C61C2411yescharacteristic
175#11758202—81no—
176#11758198—21no—
177#11758198—41no—
178#11758198—41no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—41no—
188C4⋊Dic61—41no—
189C4⋊Dic61—41no—
190C4⋊Dic61C2×D411yescharacteristic
191C4⋊Dic61C2×D411yescharacteristic
192#11758196—21no—
193#11758196—41no—
194#11758196—41no—
195#11758196—41no—
196#11758196—41no—
197C4×Dic61—21no—
198C4×Dic61—41no—
199C4×Dic61—41no—
200C2×#92418—21no—
201C2×#92418—21no—
202Dic244—82no—
203#11758190Q1614yescomplemented
204C8×D61—81no—
205C22×D61—21no—
206Q8×C61—21no—
207Q8×C61—21no—
208Q8×C61—21no—
209Q8×C61#13712yes—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61H(5,𝔽2)11yescharacteristic
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—21no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224D244—21no—
225D244C2×Q1612yes—
226C4×D61—41no—
227C4×D61—41no—
228C4×D61—82no—
229C4×D61—161no—
230Dic122—21no—
231Dic122—41no—
232Dic122—41no—
233Dic122—82no—
234Dic122—81no—
235Dic122—81no—
236Dic122—81no—
237Dic122—161no—
238Dic122—161no—
239Dic122C2×Q1612yes—
240C8×C61—21no—
241C8×C61—21no—
242C8×C61—21no—
243C8×C61—41no—
244C8×C61—41no—
245C8×C61—41no—
246C8×C61C2×Q1612yes—
247#92418—41no—
248#92418—41no—
249#19813—611nomaximal, Sylow
250C22×C61D4211yescharacteristic
251C2×D61—22no—
252C2×D61—41no—
253C2×D61—41no—
254C4×C61—22no—
255C4×C61—21no—
256C4×C61—41no—
257C4×C61—41no—
258C4×C61—41no—
259C4×C61C22×Q1611yescharacteristic
260C4×C61C2×#13711yescharacteristic
261C4×C61#57711yescharacteristic
262C4×C61#57811yescharacteristic
263Dic61—22no—
264Dic61—41no—
265Dic61—41no—
266Dic61—81no—
267Dic61—81no—
268Dic61—81no—
269Dic61—81no—
270#3385—611no—
271Q8×Q16—611no—
272Q8×SD16—611no—
273#3299—611no—
274#1723—611no—
275#1692—611no—
276#1684—611no—
277#1676—611no—
278#1675—611no—
279#1645—611no—
280#1637—611no—
281#1633—611no—
282#1632—611no—
283#1584D6111yescharacteristic, complemented
284SD16⋊C8—611no—
285C2×C61#326611yescharacteristic
286C2×C61D4×Q1611yescharacteristic
287C2×C61#328811yescharacteristic
288D61—42no—
289Q82—611no—
290Q82—611no—
291#554—611no—
292D4×Q8—611no—
293#501—1221no—
294#500—611no—
295#500—611no—
296#500—1221no—
297#498—611no—
298#498—1221no—
299#494—611no—
300#487—611no—
301#487—611no—
302#484—1221no—
303#479—1221no—
304#477—611no—
305#477—611no—
306#477—611no—
307#477—611no—
308#477—1221no—
309#477—1221no—
310#476—611no—
311#476—611no—
312#476—1221no—
313#475—611no—
314#475—1221no—
315#474—1221no—
316#467—1221no—
317#462—611no—
318#451—1221no—
319#446C2×D6111yescharacteristic
320C8×Q8—611no—
321C8×Q8C2×D6111yescharacteristic
322#440—1221no—
323C4×Q16—1221no—
324C4×Q16C2×D6111yescharacteristic
325C4×SD16—611no—
326C4×SD16—1221no—
327C8×D4—611no—
328C2×#103—1221no—
329#335—611no—
330#335C2×D6111yescharacteristic
331#333—611no—
332#333—611no—
333#333—611no—
334#332—611no—
335Q8⋊C8—611no—
336Q8⋊C8C2×D6111yescharacteristic
337Q8⋊C8C2×D6111yescharacteristic
338D4⋊C8—611no—
339#322C2×D6111yescharacteristic
340C61#1981311yescharacteristic, complemented, Sylow
341C22×Q8—1221no—
342C2×Q16—1221no—
343C2×Q16—2441no—
344C2×Q16D24411yescharacteristic
345C2×SD16—611no—
346C22×C8—1221no—
347C4⋊Q8—611no—
348C4⋊Q8—611no—
349C4⋊Q8—611no—
350C4⋊Q8—611no—
351C4⋊Q8—1221no—
352C4⋊Q8—1221no—
353C4⋊Q8—1221no—
354C4⋊Q8—1221no—
355C4⋊Q8—1221no—
356C4⋊Q8—2441no—
357C4⋊Q8—2441no—
358#125—1221no—
359C22⋊Q8—1221no—
360C22⋊Q8—1221no—
361C22⋊Q8—2441no—
362C22⋊Q8—2441no—
363C4×Q8—611no—
364C4×Q8—1221no—
365C4×Q8—1221no—
366C4×Q8—1221no—
367C4×Q8—1221no—
368C4×Q8C22×D6111yescharacteristic
369C4×Q8C22×D6111yescharacteristic
370C4×D4—611no—
371C4×D4—1221no—
372C4×D4—1221no—
373C2×C4⋊C4—1221no—
374#107—1221no—
375#107—1221no—
376#107—1221no—
377#107—1221no—
378#107—1221no—
379#107—1221no—
380#107—1221no—
381#107—1221no—
382#107D24411yescharacteristic
383#106—611no—
384#106—1221no—
385#106—1221no—
386#106—1221no—
387#106—1221no—
388#106—1221no—
389C4⋊C8—21no—
390C4⋊C8—21no—
391C4⋊C8—611no—
392C4⋊C8—611no—
393C4⋊C8—1221no—
394C4⋊C8C22×D6111yescharacteristic
395C4⋊C8C22×D6111yescharacteristic
396#103—611no—
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—2441no—
406#103D24411yescharacteristic
407#103D24411yescharacteristic
408#102—611no—
409#102—1221no—
410#102—1221no—
411C22⋊C8—1221no—
412#97—21no—
413C4×C8—21no—
414C4×C8—1221no—
415C4×C8C22×D6111yescharacteristic
416C4×C8C22×D6111yescharacteristic
417C4×C8C22×D6111yescharacteristic
418C2×Q8—611no—
419C2×Q8—1221no—
420C2×Q8—1221no—
421C2×Q8—1221no—
422C2×Q8—1221no—
423C2×Q8—4881no—
424C2×Q8—2441no—
425C2×Q8—2441no—
426C2×Q8—2441no—
427C2×Q8—2442no—
428C2×Q8C2×D24411yescharacteristic
429C2×Q8C2×D24411yescharacteristic
430C2×D4—611no—
431C22×C4—1221no—
432C22×C4—2441no—
433C22×C4—2441no—
434Q16—22no—
435Q16—4882no—
436Q16—4881no—
437SD16—614no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8—4881no—
444C2×C8C2×D24411yescharacteristic
445C2×C8C2×D24411yescharacteristic
446C2×C8D4×D6111yescharacteristic
447C2×C8D4×D6111yescharacteristic
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—611no—
451C4⋊C4—611no—
452C4⋊C4—1221no—
453C4⋊C4—1221no—
454C4⋊C4—1221no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—2441no—
462C4⋊C4—2441no—
463C4⋊C4—2441no—
464C4⋊C4—2441no—
465C4⋊C4—2441no—
466C4⋊C4—2441no—
467C4⋊C4—2441no—
468C4⋊C4C2×D24411yescharacteristic
469C4⋊C4C2×D24411yescharacteristic
470C22⋊C4—1221no—
471C22⋊C4—2441no—
472C22⋊C4—2441no—
473C42—21no—
474C42—21no—
475C42—1221no—
476C42—2441no—
477C42—2441no—
478C42C23×D6111yescharacteristic
479C23—1221no—
480Q8—21no—
481Q8—21no—
482Q8—21no—
483Q8—612no—
484Q8—1221no—
485Q8—9761no—
486Q8—9761no—
487Q8—4881no—
488Q8—4881no—
489Q8—4881no—
490Q8—4882no—
491Q8—2441no—
492Q8—2441no—
493Q8#1451981012yes—
494D4—612no—
495D4—1221no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—1221no—
501C2×C4—1221no—
502C2×C4—1221no—
503C2×C4—9761no—
504C2×C4—4882no—
505C2×C4—2441no—
506C2×C4—2441no—
507C2×C4—2441no—
508C2×C4—2441no—
509C2×C4—2441no—
510C2×C4—2441no—
511C2×C4C2×D4×D6111yescharacteristic
512C2×C4C22×D24411yescharacteristic
513C2×C4#1451982311yescharacteristic
514C8—21no—
515C8—21no—
516C8—21no—
517C8—41no—
518C8—41no—
519C8—41no—
520C8—2441no—
521C8—2441no—
522C8Q16×D6112yes—
523C22—1222no—
524C22—2441no—
525C22—2441no—
526C22D4×D24411yescharacteristic
527C4—21no—
528C4—22no—
529C4—41no—
530C4—41no—
531C4—41no—
532C4—1222no—
533C4—4881no—
534C4—4881no—
535C4—4881no—
536C4—4881no—
537C4—2441no—
538C4—2441no—
539C4C2×Q16×D6111yescharacteristic
540C4C2×#1451981011yescharacteristic
541C4#1684057411yescharacteristic
542C4#1684057611yescharacteristic
543C2—2442no—
544C2#2145733411yescharacteristic
545C2Q16×D24411yescharacteristic
546C2#2145758611yescharacteristic
subgroupG/H
centerC22D4×D244
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1584C2
socleC22×C61D42
prime radicalC2×D244C2×D4
hypercenter#1584D61
hypocenterC61#19813
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1584

Complements

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

NH ≅ G/Ntype
2. Q16×Dic122C2semidirect
5. #21445496C2semidirect
6. #21445325C2semidirect
7. #21445278C2semidirect
8. #21445223C2semidirect
10. #21445042C2semidirect
12. #21444961C2semidirect
15. C61×#1584C2semidirect
203. #11758190Q16semidirect
283. #1584D61semidirect
340. C61#19813semidirect

Maximal subgroups

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

subgroupindex
#214587312
Q16×Dic1222
Q8×#117581902
#214576972
#214454962
#214453252
#214452782
#214452232
#214452192
#214450422
#214449862
#214449612
#214449562
#214446332
C61×#15842
#1981361