Finite Group Databasebeta

← #27366948

Subgroups — #27366948

Order 13568.

766 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285167C211yesmaximal, characteristic
2#20284746C211yesmaximal, characteristic, complemented
3C2×#16810823C211yesmaximal, characteristic, complemented
4#20274199C212yesmaximal, complemented
5#20274187C212yesmaximal, complemented
6#20274075C212yesmaximal, complemented
7#20274063C212yesmaximal, complemented
8#20273467C211yesmaximal, characteristic, complemented
9#20273393C211yesmaximal, characteristic, complemented
10#20273384C211yesmaximal, characteristic, complemented
11Q8×M16×C53C211yesmaximal, characteristic, complemented
12C2×#13133245—21no—
13C2×#13133245—21no—
14C22×#11719924—21no—
15#16811173C2211yescharacteristic
16C2×#13133214C2211yescharacteristic
17#16811012—22no—
18#16811000C2212yes—
19#16810996C2212yes—
20#16810941—21no—
21#16810908C2211yescharacteristic, complemented
22#16810908C2211yescharacteristic, complemented
23#16810823—21no—
24#16810823—21no—
25#16810823C2212yes—
26#16810823C2212yes—
27#16810823C2211yescharacteristic, complemented
28#16810823C2211yescharacteristic, complemented
29#16810762—21no—
30C4×#11719924—21no—
31C4×#11719924—21no—
32#16810676C2211yescharacteristic
33C2×C4⋊#88322C2211yescharacteristic
34C2×#13133137—21no—
35#16810376C2212yescomplemented
36#16810374C2212yescomplemented
37#16810370C2212yescomplemented
38#16810366C2212yes—
39#16810366C2212yes—
40#16810366C2212yescomplemented
41C2×C4×Q8×C53C2211yescharacteristic
42C53×#446—22no—
43C53×#446C2212yescomplemented
44C8×Q8×C53C2212yescomplemented
45C53×#423—21no—
46C53×#423C2211yescharacteristic
47C4×M16×C53—21no—
48C4×M16×C53C2211yescharacteristic
49C2×#11719945—21no—
50C22×D212—21no—
51C22×D212—21no—
52C22×D212—21no—
53#13133245—44no—
54#13133245—44no—
55C2×#11719913—22no—
56C2×#11719913—22no—
57C2×#11719924—21no—
58C2×#11719924—21no—
59C2×#11719924—22no—
60C2×#11719924—22no—
61C2×#11719924—44no—
62C2×D424—22no—
63C2×D424—22no—
64C22×#88322—21no—
65#13133217C2311yescharacteristic
66#13133214—21no—
67#13133214—21no—
68#13133214—21no—
69#13133214C2312yes—
70#13133214C2311yescharacteristic
71#13133205—21no—
72#13133191—21no—
73#13133191—21no—
74C4×D212—21no—
75C4×D212—21no—
76#13133148—21no—
77#13133148—21no—
78C4⋊#88322—21no—
79C4⋊#88322C2312yes—
80C4⋊#88322C2311yescharacteristic
81C4⋊#88322C2311yescharacteristic
82#13133141—21no—
83#13133141—21no—
84#13133137—22no—
85#13133137—22no—
86#13133137—21no—
87#13133137—21no—
88C22⋊#88322—21no—
89C4×#88322—21no—
90C22×Q8×C53D411yescharacteristic
91C2×M16×C53—21no—
92C2×M16×C53D411yescharacteristic
93C2×M16×C53D411yescharacteristic
94C4×Q8×C53—21no—
95C4×Q8×C53—21no—
96C4×Q8×C53D411yescharacteristic
97C4×Q8×C53D411yescharacteristic
98C4×Q8×C53C2311yescharacteristic
99C4×Q8×C53C2311yescharacteristic
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53D411yescharacteristic
102C2×C42×C53—21no—
103C2×C42×C53C2311yescharacteristic
104C4⋊C8×C53—22no—
105C4⋊C8×C53—22no—
106C4⋊C8×C53—21no—
107C4⋊C8×C53C2312yes—
108#97×C53—22no—
109#97×C53D412yes—
110C4×C8×C53—22no—
111C4×C8×C53C2312yes—
112C23×D53—41no—
113C2×Q8×C53—21no—
114C2×Q8×C53—22no—
115C2×Q8×C53—41no—
116C2×Q8×C53C2×D411yescharacteristic
117C2×Q8×C53C2×D411yescharacteristic
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53#4111yescharacteristic
123C2×C22⋊D53—41no—
124#11719945—42no—
125#11719945—42no—
126C2×D212—21no—
127C2×D212—22no—
128C2×D212—21no—
129C2×D212—22no—
130C2×D212—21no—
131C2×D212—22no—
132C2×D212—42no—
133C2×D212—42no—
134C2×D212—42no—
135C2×D212—42no—
136C2×D212—41no—
137C2×D212—41no—
138C2×D212—41no—
139C2×D212—82no—
140C2×C4×D53—41no—
141C2×Dic106—21no—
142M16×C53—22no—
143M16×C53—42no—
144M16×C53SD1614yescomplemented
145C2×C8×C53—22no—
146C2×C8×C53C2×D412yes—
147C2×C8×C53C2×D412yes—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53C2×D411yescharacteristic
154C4⋊C4×C53C2×D411yescharacteristic
155C42×C53—21no—
156C42×C53—21no—
157C42×C53—21no—
158C42×C53—21no—
159C42×C53C2×D411yescharacteristic
160C42×C53C2×D411yescharacteristic
161C42×C53#4111yescharacteristic
162C42×C53C2411yescharacteristic
163#11719924—42no—
164#11719924—44no—
165#11719924—42no—
166#11719924—44no—
167#11719921—41no—
168#11719921—41no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C2×#88322—21no—
172C2×#88322—21no—
173C2×#88322—22no—
174C2×#88322—42no—
175D424—44no—
176D424—44no—
177#11719913—44no—
178#11719913—44no—
179C23×C53#12111yescharacteristic
180C22×D53—41no—
181C22×D53—42no—
182C22×D53—82no—
183C22×D53—82no—
184C22×D53—82no—
185Q8×C53—21no—
186Q8×C53—21no—
187Q8×C53—21no—
188Q8×C53—42no—
189Q8×C53AGL(1,ℤ/8)12yescomplemented
190C2×C4×C53—21no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—41no—
197C2×C4×C53—42no—
198C2×C4×C53—81no—
199C2×C4×C53#12111yescharacteristic
200C2×C4×C53#12111yescharacteristic
201C2×C4×C53#12111yescharacteristic
202C2×C4×C53C2×SD1612yes—
203C2×C4×C53AGL(1,ℤ/8)12yes—
204C2×C4×C53C22×D411yescharacteristic
205C2×C4×C53C22×D411yescharacteristic
206C2×C4×C53C2×#4111yescharacteristic
207C22⋊D53—82no—
208C2×Dic53—41no—
209D212—42no—
210D212—41no—
211D212—42no—
212D212—42no—
213D212—41no—
214D212—41no—
215D212—42no—
216D212—41no—
217D212—81no—
218D212—82no—
219D212—82no—
220D212—82no—
221D212—82no—
222C4×D53—82no—
223Dic106—41no—
224Dic106—41no—
225C8×C53—22no—
226C8×C53—42no—
227C8×C53C2×SD1614yescomplemented
228#88322—42no—
229#88322—41no—
230#88322—41no—
231#19170—531nomaximal, Sylow
232C22×C53—21no—
233C22×C53—21no—
234C22×C53#46012yes—
235C22×C53C2×#12111yescharacteristic
236C2×D53—81no—
237C2×D53—82no—
238C2×D53—82no—
239C2×D53—81no—
240C2×D53—82no—
241C2×D53—81no—
242C2×D53—164no—
243C4×C53—21no—
244C4×C53—22no—
245C4×C53—21no—
246C4×C53—21no—
247C4×C53—41no—
248C4×C53—41no—
249C4×C53—41no—
250C4×C53#46012yes—
251C4×C53C22×SD1611yescharacteristic
252C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
253C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
254C4×C53#57511yescharacteristic
255Dic53—81no—
256#3151—531no—
257#3103—531no—
258C2×#460—531no—
259Q8×M16D5311yescharacteristic, complemented
260#1688—532no—
261#1686—532no—
262#1668—532no—
263#1666—532no—
264#1565—531no—
265#1547—531no—
266#1545—531no—
267C2×C53—22no—
268C2×C53C2×#46011yescharacteristic
269C2×C53#305511yescharacteristic
270C2×C53#306811yescharacteristic
271D53—161no—
272D53—162no—
273C2×AGL(1,ℤ/8)—1061no—
274C2×AGL(1,ℤ/8)—1061no—
275C22×SD16—1061no—
276#532—531no—
277C2×#127—531no—
278C2×C4×Q8C2×D5311yescharacteristic
279#496—1062no—
280#493—532no—
281#492—532no—
282#481—1061no—
283#475—531no—
284#475—531no—
285#460—531no—
286#460—531no—
287#460—532no—
288#460—532no—
289#460—1061no—
290#460—1061no—
291#448—1061no—
292#446—22no—
293#446C2×D5312yescomplemented
294C8×Q8C2×D5312yescomplemented
295C4×SD16—1061no—
296C4×SD16—1061no—
297#431—531no—
298#423—21no—
299#423C2×D5311yescharacteristic
300C2×C4⋊C8—531no—
301C2×#102—1061no—
302C4×M16—21no—
303C4×M16C2×D5311yescharacteristic
304#335—532no—
305#334—532no—
306#332—532no—
307#331—532no—
308#331—532no—
309#331—532no—
310C53#1917011yescharacteristic, complemented, Sylow
311C2×#41—1061no—
312C22×Q8D21211yescharacteristic
313C22×D4—1061no—
314C22×D4—1061no—
315C22×D4—1061no—
316AGL(1,ℤ/8)—2124no—
317AGL(1,ℤ/8)—2124no—
318C2×SD16—1062no—
319C2×SD16—1062no—
320C2×SD16—1061no—
321C2×SD16—1061no—
322C2×SD16—1062no—
323C2×SD16—1062no—
324C2×SD16—2124no—
325C2×D8—1062no—
326C2×D8—1062no—
327C2×M16—21no—
328C2×M16C22⋊D5311yescharacteristic
329C2×M16C22⋊D5311yescharacteristic
330C22×C8—1061no—
331C4⋊Q8—531no—
332#127—532no—
333#127—531no—
334#127—1061no—
335#127—1061no—
336#127—1061no—
337#124—1061no—
338#121—1061no—
339#121—1061no—
340C4×Q8—21no—
341C4×Q8—21no—
342C4×Q8D21211yescharacteristic
343C4×Q8D21211yescharacteristic
344C4×Q8C22×D5311yescharacteristic
345C4×Q8C22×D5311yescharacteristic
346C4×D4—1061no—
347C4×D4—1061no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4D21211yescharacteristic
350C2×C42—21no—
351C2×C42C22×D5311yescharacteristic
352#106—1061no—
353#106—1061no—
354C4⋊C8—22no—
355C4⋊C8—22no—
356C4⋊C8—21no—
357C4⋊C8—531no—
358C4⋊C8—531no—
359C4⋊C8—532no—
360C4⋊C8—1061no—
361C4⋊C8C22×D5312yes—
362#103—1061no—
363#103—1061no—
364#102—1062no—
365#102—1062no—
366#102—1061no—
367#102—1061no—
368C22⋊C8—1061no—
369#97—22no—
370#97C22⋊D5312yes—
371C4×C8—22no—
372C4×C8—1061no—
373C4×C8C22×D5312yes—
374C24—2121no—
375#41—2122no—
376#41—2122no—
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—41no—
380C2×Q8—1061no—
381C2×Q8C2×D21211yescharacteristic
382C2×Q8C2×D21211yescharacteristic
383C2×D4—1061no—
384C2×D4—1062no—
385C2×D4—1061no—
386C2×D4—1062no—
387C2×D4—1061no—
388C2×D4—1062no—
389C2×D4—4242no—
390C2×D4—2122no—
391C2×D4—2122no—
392C2×D4—2121no—
393C2×D4—2121no—
394C2×D4—2121no—
395C2×D4—2121no—
396C2×D4—2122no—
397C2×D4—2122no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—2121no—
401C22×C4C2×D21211yescharacteristic
402C22×C4#1171994511yescharacteristic
403C22×C4C2×C22⋊D5311yescharacteristic
404SD16—2124no—
405SD16—2124no—
406SD16—2122no—
407SD16—2124no—
408SD16—2122no—
409SD16—2124no—
410D8—2124no—
411D8—2124no—
412M16—22no—
413M16—42no—
414M16#1171992414yescomplemented
415C2×C8—22no—
416C2×C8—1061no—
417C2×C8—1061no—
418C2×C8—1062no—
419C2×C8—2122no—
420C2×C8C2×C22⋊D5312yes—
421C2×C8C2×C22⋊D5312yes—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—1061no—
428C4⋊C4—1061no—
429C4⋊C4C2×D21211yescharacteristic
430C4⋊C4C2×D21211yescharacteristic
431C22⋊C4—2121no—
432C22⋊C4—2121no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42C2×D21211yescharacteristic
438C42#1171994511yescharacteristic
439C42C2×C22⋊D5311yescharacteristic
440C42C23×D5311yescharacteristic
441C23—4242no—
442C23—4242no—
443C23—4242no—
444C23—2121no—
445C23—2122no—
446C23#1313319111yescharacteristic
447Q8—21no—
448Q8—21no—
449Q8—21no—
450Q8—42no—
451Q8—2121no—
452Q8—2121no—
453Q8#1313324512yescomplemented
454D4—4241no—
455D4—4242no—
456D4—4242no—
457D4—4242no—
458D4—4242no—
459D4—4242no—
460D4—2121no—
461D4—2122no—
462D4—2121no—
463D4—2122no—
464D4—2122no—
465D4—2121no—
466D4—2121no—
467D4—2122no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—81no—
477C2×C4—4242no—
478C2×C4—2121no—
479C2×C4#1313319111yescharacteristic
480C2×C4#1313319111yescharacteristic
481C2×C4#1313319111yescharacteristic
482C2×C4C2×#1171992412yes—
483C2×C4#1313324212yes—
484C2×C4C22×C22⋊D5311yescharacteristic
485C2×C4C22×D21211yescharacteristic
486C2×C4C2×#1171994511yescharacteristic
487C8—22no—
488C8—42no—
489C8—2122no—
490C8—2121no—
491C8—2121no—
492C8C2×#1171992414yescomplemented
493C22—21no—
494C22—21no—
495C22—8484no—
496C22—4242no—
497C22—4241no—
498C22—4242no—
499C22—4241no—
500C22—4241no—
501C22—4242no—
502C22#1681082312yes—
503C22C2×#1313319111yescharacteristic
504C4—21no—
505C4—21no—
506C4—21no—
507C4—22no—
508C4—41no—
509C4—41no—
510C4—41no—
511C4—4241no—
512C4#1681082312yes—
513C4C22×#1171992411yescharacteristic
514C4C2×#1313324211yescharacteristic
515C4C2×#1313324511yescharacteristic
516C4#1681142311yescharacteristic
517C2—22no—
518C2—8482no—
519C2—8481no—
520C2C2×#1681082311yescharacteristic
521C2#2028433411yescharacteristic
522C2#2028444311yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingQ8×M16×C53C2
socleC22×C53C2×#121
prime radical#20285167C2
hypercenterQ8×M16D53
hypocenterC53#19170
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
2. #20284746C2semidirect
3. C2×#16810823C2semidirect
4. #20274199C2semidirect
5. #20274187C2semidirect
6. #20274075C2semidirect
7. #20274063C2semidirect
8. #20273467C2semidirect
9. #20273393C2semidirect
10. #20273384C2semidirect
11. Q8×M16×C53C2semidirect
21. #16810908C22semidirect
22. #16810908C22semidirect
27. #16810823C22semidirect
28. #16810823C22semidirect
35. #16810376C22semidirect
36. #16810374C22semidirect
37. #16810370C22semidirect
40. #16810366C22semidirect
43. C53×#446C22semidirect
44. C8×Q8×C53C22semidirect
144. M16×C53SD16semidirect
189. Q8×C53AGL(1,ℤ/8)semidirect
227. C8×C53C2×SD16semidirect
259. Q8×M16D53semidirect
293. #446C2×D53semidirect
294. C8×Q8C2×D53semidirect
310. C53#19170semidirect
414. M16#11719924semidirect
453. Q8#13133245semidirect
492. C8C2×#11719924semidirect

Maximal subgroups

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

subgroupindex
#202851672
#202847462
C2×#168108232
#202741992
#202741872
#202740752
#202740632
#202734672
#202733932
#202733842
Q8×M16×C532
#1917053