Finite Group Databasebeta

← #27366860

Subgroups — #27366860

Order 13568.

844 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285167C211yesmaximal, characteristic, complemented
2#20284715C211yesmaximal, characteristic, complemented
3C2×#16810818C211yesmaximal, characteristic, complemented
4#20274194C212yesmaximal, complemented
5#20274180C212yesmaximal, complemented
6#20274070C212yesmaximal, complemented
7#20274056C212yesmaximal, complemented
8#20273467C211yesmaximal, characteristic, complemented
9#20273393C211yesmaximal, characteristic, complemented
10#20273384C211yesmaximal, characteristic, complemented
11D4×M16×C53C211yesmaximal, characteristic, complemented
12C2×#13133245—21no—
13C2×#13133245—21no—
14C22×#11719922—21no—
15#16811173C2211yescharacteristic
16C2×#13133214C2211yescharacteristic
17#16811012—22no—
18#16811000C2212yescomplemented
19#16810996C2212yescomplemented
20#16810936—21no—
21#16810903C2211yescharacteristic, complemented
22#16810903C2211yescharacteristic, complemented
23#16810818—21no—
24#16810818—21no—
25#16810818C2212yes—
26#16810818C2212yes—
27#16810818C2211yescharacteristic, complemented
28#16810818C2211yescharacteristic, complemented
29#16810757—21no—
30C4×#11719922—21no—
31C4×#11719922—21no—
32#16810676C2211yescharacteristic
33C2×C4⋊#88322C2211yescharacteristic
34C2×#13133137—21no—
35#16810376C2212yescomplemented
36#16810374C2212yescomplemented
37#16810370C2212yescomplemented
38#16810366C2212yescomplemented
39#16810366C2212yescomplemented
40#16810366C2212yescomplemented
41C22×M16×C53—21no—
42C2×C4×D4×C53C2211yescharacteristic
43C53×#436C2212yescomplemented
44C53×#435—22no—
45C8×D4×C53C2212yescomplemented
46C53×#423C2211yescharacteristic
47C53×#407—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×D424—22no—
58C2×D424—22no—
59C2×#11719922—21no—
60C2×#11719922—21no—
61C2×#11719922—22no—
62C2×#11719922—22no—
63C2×#11719922—44no—
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#13133150—21no—
77#13133150—21no—
78C4⋊#88322—21no—
79C4⋊#88322C2312yes—
80C4⋊#88322C2311yescharacteristic
81C4⋊#88322C2311yescharacteristic
82#13133138—21no—
83#13133138—21no—
84#13133137—22no—
85#13133137—22no—
86#13133137—21no—
87#13133137—21no—
88C22⋊#88322—21no—
89C4×#88322—21no—
90C22×D4×C53D411yescharacteristic
91C23×C4×C53—21no—
92C2×M16×C53—22no—
93C2×M16×C53—21no—
94C2×M16×C53—44no—
95C2×M16×C53D411yescharacteristic
96C2×M16×C53D411yescharacteristic
97C22×C8×C53—22no—
98C4×D4×C53—21no—
99C4×D4×C53—21no—
100C4×D4×C53D411yescharacteristic
101C4×D4×C53D411yescharacteristic
102C4×D4×C53C2311yescharacteristic
103C4×D4×C53C2311yescharacteristic
104C2×C4⋊C4×C53D411yescharacteristic
105C2×C22⋊C4×C53—21no—
106C2×C42×C53C2311yescharacteristic
107C4⋊C8×C53—21no—
108C4⋊C8×C53C2312yes—
109C22⋊C8×C53—22no—
110C22⋊C8×C53—22no—
111#97×C53D412yes—
112C4×C8×C53C2312yes—
113C24×C53—21no—
114C23×D53—41no—
115C2×D4×C53—22no—
116C2×D4×C53—21no—
117C2×D4×C53—41no—
118C2×D4×C53C2×D411yescharacteristic
119C2×D4×C53C2×D411yescharacteristic
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—42no—
126C22×C4×C53—42no—
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53#4111yescharacteristic
130C2×C22⋊D53—41no—
131#11719945—42no—
132#11719945—42no—
133C2×D212—21no—
134C2×D212—22no—
135C2×D212—21no—
136C2×D212—22no—
137C2×D212—21no—
138C2×D212—22no—
139C2×D212—42no—
140C2×D212—42no—
141C2×D212—42no—
142C2×D212—42no—
143C2×D212—41no—
144C2×D212—41no—
145C2×D212—41no—
146C2×D212—82no—
147C2×C4×D53—41no—
148C2×Dic106—21no—
149M16×C53—22no—
150M16×C53—42no—
151M16×C53—44no—
152M16×C53D814yescomplemented
153C2×C8×C53—22no—
154C2×C8×C53—44no—
155C2×C8×C53C2×D412yes—
156C2×C8×C53C2×D412yes—
157C4⋊C4×C53—21no—
158C4⋊C4×C53C2×D411yescharacteristic
159C4⋊C4×C53C2×D411yescharacteristic
160C22⋊C4×C53—21no—
161C22⋊C4×C53—21no—
162C22⋊C4×C53—21no—
163C22⋊C4×C53—21no—
164C42×C53C2×D411yescharacteristic
165C42×C53C2×D411yescharacteristic
166C42×C53#4111yescharacteristic
167C42×C53C2411yescharacteristic
168#11719922—42no—
169#11719922—44no—
170#11719922—42no—
171#11719922—44no—
172#11719921—41no—
173#11719921—41no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C2×#88322—21no—
177C2×#88322—21no—
178C2×#88322—22no—
179C2×#88322—42no—
180D424—44no—
181D424—44no—
182#11719913—44no—
183#11719913—44no—
184C23×C53—21no—
185C23×C53—21no—
186C23×C53—42no—
187C23×C53—42no—
188C23×C53—81no—
189C23×C53#12111yescharacteristic
190C22×D53—41no—
191C22×D53—42no—
192C22×D53—82no—
193C22×D53—82no—
194C22×D53—82no—
195D4×C53—21no—
196D4×C53—21no—
197D4×C53—21no—
198D4×C53—42no—
199D4×C53AGL(1,ℤ/8)12yescomplemented
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—41no—
205C2×C4×C53—42no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53—41no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53#12111yescharacteristic
213C2×C4×C53#12111yescharacteristic
214C2×C4×C53#12111yescharacteristic
215C2×C4×C53C2×D812yes—
216C2×C4×C53AGL(1,ℤ/8)12yes—
217C2×C4×C53C22×D411yescharacteristic
218C2×C4×C53C22×D411yescharacteristic
219C2×C4×C53C2×#4111yescharacteristic
220C22⋊D53—82no—
221C2×Dic53—41no—
222D212—42no—
223D212—41no—
224D212—42no—
225D212—42no—
226D212—41no—
227D212—41no—
228D212—42no—
229D212—41no—
230D212—81no—
231D212—82no—
232D212—82no—
233D212—82no—
234D212—82no—
235C4×D53—82no—
236Dic106—41no—
237Dic106—41no—
238C8×C53—22no—
239C8×C53—42no—
240C8×C53C2×D814yescomplemented
241#88322—42no—
242#88322—41no—
243#88322—41no—
244#19162—531nomaximal, Sylow
245C22×C53—21no—
246C22×C53—21no—
247C22×C53—22no—
248C22×C53—41no—
249C22×C53—41no—
250C22×C53—41no—
251C22×C53—41no—
252C22×C53—41no—
253C22×C53—84no—
254C22×C53#45912yes—
255C22×C53C2×#12111yescharacteristic
256C2×D53—81no—
257C2×D53—82no—
258C2×D53—82no—
259C2×D53—81no—
260C2×D53—82no—
261C2×D53—81no—
262C2×D53—164no—
263C4×C53—21no—
264C4×C53—21no—
265C4×C53—21no—
266C4×C53—41no—
267C4×C53—41no—
268C4×C53#45912yes—
269C4×C53C22×D811yescharacteristic
270C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
271C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
272C4×C53#57511yescharacteristic
273Dic53—81no—
274#3151—531no—
275#3100—531no—
276C2×#459—531no—
277D4×M16D5311yescharacteristic, complemented
278#1687—532no—
279#1685—532no—
280#1667—532no—
281#1665—532no—
282#1565—531no—
283#1547—531no—
284#1545—531no—
285C2×C53—22no—
286C2×C53—42no—
287C2×C53—81no—
288C2×C53C2×#45911yescharacteristic
289C2×C53#305511yescharacteristic
290C2×C53#306811yescharacteristic
291D53—161no—
292D53—162no—
293C2×AGL(1,ℤ/8)—1061no—
294C2×AGL(1,ℤ/8)—1061no—
295C22×D8—1061no—
296C22×M16—21no—
297#532—531no—
298C2×#127—531no—
299C2×C4×D4C2×D5311yescharacteristic
300#496—1062no—
301#493—532no—
302#492—532no—
303#480—1061no—
304#474—531no—
305#474—531no—
306#459—531no—
307#459—531no—
308#459—532no—
309#459—532no—
310#459—1061no—
311#459—1061no—
312#447—1061no—
313C4×D8—1061no—
314C4×D8—1061no—
315#436C2×D5312yescomplemented
316#435—22no—
317C8×D4C2×D5312yescomplemented
318#431—531no—
319#423C2×D5311yescharacteristic
320C2×C4⋊C8—531no—
321C2×#102—1061no—
322#407—21no—
323C4×M16C2×D5311yescharacteristic
324#335—532no—
325#334—532no—
326#332—532no—
327#331—532no—
328#331—532no—
329#331—532no—
330C53#1916211yescharacteristic, complemented, Sylow
331C2×#41—1061no—
332C22×D4—1061no—
333C22×D4—1061no—
334C22×D4—1061no—
335C22×D4D21211yescharacteristic
336C23×C4—21no—
337AGL(1,ℤ/8)—2124no—
338AGL(1,ℤ/8)—2124no—
339C2×SD16—1062no—
340C2×SD16—1062no—
341C2×D8—1062no—
342C2×D8—1062no—
343C2×D8—1061no—
344C2×D8—1061no—
345C2×D8—1062no—
346C2×D8—1062no—
347C2×D8—2124no—
348C2×M16—21no—
349C2×M16—22no—
350C2×M16—44no—
351C2×M16C22⋊D5311yescharacteristic
352C2×M16C22⋊D5311yescharacteristic
353C22×C8—22no—
354C22×C8—1061no—
355C4⋊Q8—531no—
356#127—532no—
357#127—531no—
358#127—1061no—
359#127—1061no—
360#127—1061no—
361#124—1061no—
362#121—1061no—
363#121—1061no—
364C4×D4—21no—
365C4×D4—21no—
366C4×D4—1061no—
367C4×D4—1061no—
368C4×D4D21211yescharacteristic
369C4×D4D21211yescharacteristic
370C4×D4C22×D5311yescharacteristic
371C4×D4C22×D5311yescharacteristic
372C2×C4⋊C4D21211yescharacteristic
373C2×C22⋊C4—21no—
374C2×C42C22×D5311yescharacteristic
375#107—1061no—
376#107—1061no—
377C4⋊C8—21no—
378C4⋊C8—531no—
379C4⋊C8—531no—
380C4⋊C8—532no—
381C4⋊C8—1061no—
382C4⋊C8C22×D5312yes—
383#102—1061no—
384#102—1061no—
385#102—1062no—
386#102—1062no—
387#102—1061no—
388#102—1061no—
389C22⋊C8—22no—
390C22⋊C8—22no—
391C22⋊C8—1061no—
392#97C22⋊D5312yes—
393C4×C8—1061no—
394C4×C8C22×D5312yes—
395C24—21no—
396C24—2121no—
397#41—2122no—
398#41—2122no—
399C2×Q8—1061no—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—41no—
403C2×D4—1061no—
404C2×D4—1062no—
405C2×D4—1061no—
406C2×D4—1062no—
407C2×D4—1061no—
408C2×D4—1062no—
409C2×D4—4242no—
410C2×D4—2122no—
411C2×D4—2122no—
412C2×D4—2121no—
413C2×D4—2121no—
414C2×D4—2121no—
415C2×D4—2121no—
416C2×D4—2122no—
417C2×D4—2122no—
418C2×D4C2×D21211yescharacteristic
419C2×D4C2×D21211yescharacteristic
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—2121no—
428C22×C4C2×D21211yescharacteristic
429C22×C4#1171994511yescharacteristic
430C22×C4C2×C22⋊D5311yescharacteristic
431SD16—2124no—
432SD16—2124no—
433D8—2124no—
434D8—2124no—
435D8—2122no—
436D8—2124no—
437D8—2122no—
438D8—2124no—
439M16—22no—
440M16—42no—
441M16—44no—
442M16#1171992214yescomplemented
443C2×C8—22no—
444C2×C8—44no—
445C2×C8—1061no—
446C2×C8—1061no—
447C2×C8—1062no—
448C2×C8—2122no—
449C2×C8C2×C22⋊D5312yes—
450C2×C8C2×C22⋊D5312yes—
451C4⋊C4—21no—
452C4⋊C4—1061no—
453C4⋊C4—1061no—
454C4⋊C4C2×D21211yescharacteristic
455C4⋊C4C2×D21211yescharacteristic
456C22⋊C4—21no—
457C22⋊C4—21no—
458C22⋊C4—21no—
459C22⋊C4—21no—
460C22⋊C4—2121no—
461C22⋊C4—2121no—
462C42C2×D21211yescharacteristic
463C42#1171994511yescharacteristic
464C42C2×C22⋊D5311yescharacteristic
465C42C23×D5311yescharacteristic
466C23—21no—
467C23—21no—
468C23—42no—
469C23—42no—
470C23—81no—
471C23—4242no—
472C23—4242no—
473C23—4242no—
474C23—2121no—
475C23—2122no—
476C23#1313319111yescharacteristic
477Q8—2121no—
478Q8—2121no—
479D4—21no—
480D4—21no—
481D4—21no—
482D4—42no—
483D4—4241no—
484D4—4242no—
485D4—4242no—
486D4—4242no—
487D4—4242no—
488D4—4242no—
489D4—2121no—
490D4—2122no—
491D4—2121no—
492D4—2122no—
493D4—2122no—
494D4—2121no—
495D4—2121no—
496D4—2122no—
497D4#1313324512yescomplemented
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—4242no—
511C2×C4—2121no—
512C2×C4#1313319111yescharacteristic
513C2×C4#1313319111yescharacteristic
514C2×C4#1313319111yescharacteristic
515C2×C4C2×#1171992212yes—
516C2×C4#1313324212yes—
517C2×C4C22×C22⋊D5311yescharacteristic
518C2×C4C22×D21211yescharacteristic
519C2×C4C2×#1171994511yescharacteristic
520C8—22no—
521C8—42no—
522C8—2122no—
523C8—2121no—
524C8—2121no—
525C8C2×#1171992214yescomplemented
526C22—21no—
527C22—21no—
528C22—22no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—84no—
535C22—8484no—
536C22—4242no—
537C22—4241no—
538C22—4242no—
539C22—4241no—
540C22—4241no—
541C22—4242no—
542C22#1681081812yes—
543C22C2×#1313319111yescharacteristic
544C4—21no—
545C4—21no—
546C4—21no—
547C4—41no—
548C4—41no—
549C4—4241no—
550C4#1681081812yes—
551C4C22×#1171992211yescharacteristic
552C4C2×#1313324211yescharacteristic
553C4C2×#1313324511yescharacteristic
554C4#1681142311yescharacteristic
555C2—22no—
556C2—42no—
557C2—81no—
558C2—8482no—
559C2—8481no—
560C2C2×#1681081811yescharacteristic
561C2#2028433411yescharacteristic
562C2#2028444311yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingD4×M16×C53C2
socleC22×C53C2×#121
hypercenterD4×M16D53
hypocenterC53#19162
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ D4×M16

Complements

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

NH ≅ G/Ntype
1. #20285167C2semidirect
2. #20284715C2semidirect
3. C2×#16810818C2semidirect
4. #20274194C2semidirect
5. #20274180C2semidirect
6. #20274070C2semidirect
7. #20274056C2semidirect
8. #20273467C2semidirect
9. #20273393C2semidirect
10. #20273384C2semidirect
11. D4×M16×C53C2semidirect
18. #16811000C22semidirect
19. #16810996C22semidirect
21. #16810903C22semidirect
22. #16810903C22semidirect
27. #16810818C22semidirect
28. #16810818C22semidirect
35. #16810376C22semidirect
36. #16810374C22semidirect
37. #16810370C22semidirect
38. #16810366C22semidirect
39. #16810366C22semidirect
40. #16810366C22semidirect
43. C53×#436C22semidirect
45. C8×D4×C53C22semidirect
152. M16×C53D8semidirect
199. D4×C53AGL(1,ℤ/8)semidirect
240. C8×C53C2×D8semidirect
277. D4×M16D53semidirect
315. #436C2×D53semidirect
317. C8×D4C2×D53semidirect
330. C53#19162semidirect
442. M16#11719922semidirect
497. D4#13133245semidirect
525. C8C2×#11719922semidirect

Maximal subgroups

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

subgroupindex
#202851672
#202847152
C2×#168108182
#202741942
#202741802
#202740702
#202740562
#202734672
#202733932
#202733842
D4×M16×C532
#1916253