Finite Group Databasebeta

← #27366904

Subgroups — #27366904

Order 13568.

758 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285090C211yesmaximal, characteristic, complemented
2#20284723C211yesmaximal, characteristic, complemented
3C2×#16810818C211yesmaximal, characteristic
4#20274276C212yesmaximal, complemented
5#20274248C212yesmaximal, complemented
6#20274152C212yesmaximal, complemented
7#20274124C212yesmaximal, complemented
8#20273467C211yesmaximal, characteristic, complemented
9#20273406C211yesmaximal, characteristic, complemented
10#20273384C211yesmaximal, characteristic, complemented
11D4×M16×C53C211yesmaximal, characteristic, complemented
12C22×#11719922—21no—
13#16811180C2211yescharacteristic
14C2×#13133214C2211yescharacteristic
15#16810988—22no—
16#16810970C2212yescomplemented
17#16810970C2212yescomplemented
18#16810953—21no—
19#16810913C2211yescharacteristic, complemented
20#16810913C2211yescharacteristic, complemented
21#16810828C2211yescharacteristic, complemented
22#16810828C2211yescharacteristic, complemented
23#16810818—21no—
24#16810818—21no—
25#16810818C2212yes—
26#16810818C2212yes—
27#16810774—21no—
28C4×#11719923—21no—
29C4×#11719923—21no—
30#16810676C2211yescharacteristic
31C2×C4⋊#88322C2211yescharacteristic
32#16810629—21no—
33#16810629—21no—
34C2×#13133137—21no—
35#16810376C2212yescomplemented
36#16810374C2212yescomplemented
37#16810371C2212yescomplemented
38#16810368C2212yescomplemented
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
49C22×D212—21no—
50C22×D212—21no—
51C22×D212—21no—
52C2×#11719923—21no—
53C2×#11719923—21no—
54C2×#11719922—22no—
55C2×#11719922—22no—
56C2×#11719922—44no—
57C22×#88322—21no—
58#13133217C2311yescharacteristic
59#13133217C2311yescharacteristic
60#13133214—21no—
61#13133214—21no—
62#13133214—21no—
63#13133214C2312yes—
64#13133209—21no—
65#13133196—21no—
66#13133196—21no—
67C4×Dic106—21no—
68C4×Dic106—21no—
69#13133176—21no—
70#13133148—21no—
71#13133148—21no—
72C4⋊#88322—21no—
73C4⋊#88322C2312yes—
74C4⋊#88322C2311yescharacteristic
75C4⋊#88322C2311yescharacteristic
76#13133140—21no—
77#13133140—21no—
78#13133139—22no—
79#13133139—22no—
80#13133139—22no—
81#13133139—22no—
82#13133138—21no—
83#13133138—21no—
84#13133137—22no—
85#13133137—22no—
86C22⋊#88322—21no—
87C4×#88322—21no—
88C22×D4×C53D411yescharacteristic
89C23×C4×C53—21no—
90C2×M16×C53—22no—
91C2×M16×C53—21no—
92C2×M16×C53—44no—
93C2×M16×C53D411yescharacteristic
94C2×M16×C53D411yescharacteristic
95C22×C8×C53—22no—
96C4×D4×C53—21no—
97C4×D4×C53—21no—
98C4×D4×C53D411yescharacteristic
99C4×D4×C53D411yescharacteristic
100C4×D4×C53C2311yescharacteristic
101C4×D4×C53C2311yescharacteristic
102C2×C4⋊C4×C53D411yescharacteristic
103C2×C22⋊C4×C53—21no—
104C2×C42×C53C2311yescharacteristic
105C4⋊C8×C53—21no—
106C4⋊C8×C53C2312yes—
107C22⋊C8×C53—22no—
108C22⋊C8×C53—22no—
109#97×C53D412yes—
110C4×C8×C53C2312yes—
111C24×C53—21no—
112C23×D53—41no—
113C2×D4×C53—22no—
114C2×D4×C53—21no—
115C2×D4×C53—41no—
116C2×D4×C53C2×D411yescharacteristic
117C2×D4×C53C2×D411yescharacteristic
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—42no—
124C22×C4×C53—42no—
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53#4111yescharacteristic
128C2×D212—22no—
129C2×D212—22no—
130C2×D212—22no—
131C2×D212—42no—
132C2×D212—42no—
133C2×D212—42no—
134C2×D212—42no—
135C2×D212—41no—
136C2×D212—41no—
137C2×D212—41no—
138C2×D212—82no—
139C2×Dic106—21no—
140C2×Dic106—21no—
141M16×C53—22no—
142M16×C53—22no—
143M16×C53—44no—
144M16×C53—42no—
145C2×C8×C53—22no—
146C2×C8×C53—44no—
147C2×C8×C53C2×D412yes—
148C2×C8×C53C2×D412yes—
149C4⋊C4×C53—21no—
150C4⋊C4×C53C2×D411yescharacteristic
151C4⋊C4×C53C2×D411yescharacteristic
152C22⋊C4×C53—21no—
153C22⋊C4×C53—21no—
154C22⋊C4×C53—21no—
155C22⋊C4×C53—21no—
156C42×C53C2×D411yescharacteristic
157C42×C53C2×D411yescharacteristic
158C42×C53#4111yescharacteristic
159C42×C53C2411yescharacteristic
160C22⋊Dic53—41no—
161#11719923—42no—
162#11719923—42no—
163#11719922—44no—
164#11719922—44no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169#11719919—41no—
170#11719919—41no—
171C4×Dic53—41no—
172C2×#88322—21no—
173C2×#88322—21no—
174C2×#88322—22no—
175C2×#88322—42no—
176C23×C53—21no—
177C23×C53—21no—
178C23×C53—42no—
179C23×C53—42no—
180C23×C53—81no—
181C23×C53#12111yescharacteristic
182C22×D53—42no—
183C22×D53—82no—
184C22×D53—82no—
185C22×D53—82no—
186D4×C53—21no—
187D4×C53—21no—
188D4×C53—21no—
189D4×C53—42no—
190D4×C53AGL(1,ℤ/8)12yes—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—41no—
196C2×C4×C53—42no—
197C2×C4×C53—41no—
198C2×C4×C53—41no—
199C2×C4×C53—41no—
200C2×C4×C53—41no—
201C2×C4×C53—42no—
202C2×C4×C53—42no—
203C2×C4×C53#12111yescharacteristic
204C2×C4×C53#12111yescharacteristic
205C2×C4×C53#12111yescharacteristic
206C2×C4×C53#13512yes—
207C2×C4×C53AGL(1,ℤ/8)12yes—
208C2×C4×C53C22×D411yescharacteristic
209C2×C4×C53C22×D411yescharacteristic
210C2×C4×C53C2×#4111yescharacteristic
211C2×Dic53—41no—
212C2×Dic53—41no—
213D212—42no—
214D212—42no—
215D212—42no—
216D212—42no—
217D212—82no—
218D212—82no—
219D212—82no—
220D212—82no—
221Dic106—41no—
222Dic106—41no—
223Dic106—81no—
224C8×C53—22no—
225C8×C53—22no—
226C8×C53—42no—
227#88322—42no—
228#88322—41no—
229#88322—41no—
230#19166—531nomaximal, Sylow
231C22×C53—21no—
232C22×C53—21no—
233C22×C53—22no—
234C22×C53—41no—
235C22×C53—41no—
236C22×C53—41no—
237C22×C53—41no—
238C22×C53—41no—
239C22×C53—84no—
240C22×C53#46312yes—
241C22×C53C2×#12111yescharacteristic
242C2×D53—82no—
243C2×D53—82no—
244C2×D53—82no—
245C2×D53—164no—
246C4×C53—21no—
247C4×C53—21no—
248C4×C53—21no—
249C4×C53—41no—
250C4×C53—41no—
251C4×C53#46312yes—
252C4×C53C2×#13511yescharacteristic
253C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
254C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
255C4×C53#57511yescharacteristic
256Dic53—81no—
257Dic53—81no—
258#3138—531no—
259#3101—531no—
260C2×#459—531no—
261D4×M16D5311yescharacteristic, complemented
262#1701—532no—
263#1697—532no—
264#1681—532no—
265#1677—532no—
266#1565—531no—
267#1550—531no—
268#1545—531no—
269C2×C53—22no—
270C2×C53—42no—
271C2×C53—81no—
272C2×C53C2×#46311yescharacteristic
273C2×C53#305511yescharacteristic
274C2×C53#306811yescharacteristic
275D53—162no—
276C22×D8—1061no—
277C22×M16—21no—
278#533—531no—
279C2×#127—531no—
280C2×C4×D4C2×D5311yescharacteristic
281#490—1062no—
282#486—532no—
283#486—532no—
284#483—1061no—
285#476—531no—
286#476—531no—
287#461—531no—
288#461—531no—
289#459—532no—
290#459—532no—
291#459—1061no—
292#459—1061no—
293#450—1061no—
294C4×SD16—1061no—
295C4×SD16—1061no—
296#436C2×D5312yescomplemented
297#435—22no—
298C8×D4C2×D5312yescomplemented
299#431—531no—
300#423C2×D5311yescharacteristic
301C2×C4⋊C8—531no—
302#418—1061no—
303#418—1061no—
304C2×#102—1061no—
305#407—21no—
306C4×M16C2×D5311yescharacteristic
307#335—532no—
308#334—532no—
309#333—532no—
310#332—532no—
311#331—532no—
312#331—532no—
313C53#1916611yescharacteristic, complemented, Sylow
314C22×D4—1061no—
315C22×D4—1061no—
316C22×D4—1061no—
317C22×D4D21211yescharacteristic
318C23×C4—21no—
319C2×SD16—1061no—
320C2×SD16—1061no—
321C2×D8—1062no—
322C2×D8—1062no—
323C2×D8—2124no—
324C2×M16—21no—
325C2×M16—22no—
326C2×M16—44no—
327C2×M16C22⋊D5311yescharacteristic
328C2×M16C22⋊D5311yescharacteristic
329C22×C8—22no—
330C22×C8—1061no—
331C4⋊Q8—531no—
332C4⋊Q8—531no—
333#127—532no—
334#127—1061no—
335#127—1061no—
336#127—1061no—
337#125—1061no—
338C22⋊Q8—1061no—
339C22⋊Q8—1061no—
340C4×Q8—1061no—
341C4×Q8—1061no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4D21211yescharacteristic
345C4×D4D21211yescharacteristic
346C4×D4C22×D5311yescharacteristic
347C4×D4C22×D5311yescharacteristic
348#117—1061no—
349C2×C4⋊C4D21211yescharacteristic
350C2×C22⋊C4—21no—
351C2×C42C22×D5311yescharacteristic
352#106—1061no—
353#106—1061no—
354C4⋊C8—21no—
355C4⋊C8—531no—
356C4⋊C8—531no—
357C4⋊C8—532no—
358C4⋊C8—1061no—
359C4⋊C8C22×D5312yes—
360#103—1061no—
361#103—1061no—
362#102—1061no—
363#102—1061no—
364#102—1062no—
365#102—1062no—
366#102—1062no—
367#102—1062no—
368#102—1062no—
369#102—1062no—
370C22⋊C8—22no—
371C22⋊C8—22no—
372C22⋊C8—1061no—
373#97C22⋊D5312yes—
374C4×C8—1061no—
375C4×C8C22×D5312yes—
376C24—21no—
377C24—2121no—
378C2×Q8—1061no—
379C2×Q8—1061no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—41no—
383C2×D4—1062no—
384C2×D4—1062no—
385C2×D4—1062no—
386C2×D4—4242no—
387C2×D4—2122no—
388C2×D4—2122no—
389C2×D4—2121no—
390C2×D4—2121no—
391C2×D4—2121no—
392C2×D4—2122no—
393C2×D4—2122no—
394C2×D4C2×D21211yescharacteristic
395C2×D4C2×D21211yescharacteristic
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4C2×D21211yescharacteristic
404C22×C4#1171994511yescharacteristic
405C22×C4C2×C22⋊D5311yescharacteristic
406SD16—2122no—
407SD16—2122no—
408D8—2124no—
409D8—2124no—
410M16—22no—
411M16—22no—
412M16—44no—
413M16—42no—
414C2×C8—22no—
415C2×C8—44no—
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—1061no—
424C4⋊C4—1061no—
425C4⋊C4—1061no—
426C4⋊C4—1061no—
427C4⋊C4—2121no—
428C4⋊C4—2121no—
429C4⋊C4C2×D21211yescharacteristic
430C4⋊C4C2×D21211yescharacteristic
431C22⋊C4—21no—
432C22⋊C4—21no—
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—2121no—
436C42—2121no—
437C42C2×D21211yescharacteristic
438C42#1171994511yescharacteristic
439C42C2×C22⋊D5311yescharacteristic
440C42C23×D5311yescharacteristic
441C23—21no—
442C23—21no—
443C23—42no—
444C23—42no—
445C23—81no—
446C23—4242no—
447C23—4242no—
448C23—4242no—
449C23—2122no—
450C23#1313319111yescharacteristic
451Q8—4241no—
452Q8—2121no—
453Q8—2121no—
454D4—21no—
455D4—21no—
456D4—21no—
457D4—42no—
458D4—4242no—
459D4—4242no—
460D4—4242no—
461D4—4242no—
462D4—2122no—
463D4—2122no—
464D4—2122no—
465D4—2122no—
466D4#1313324512yes—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—2121no—
480C2×C4—2121no—
481C2×C4#1313319111yescharacteristic
482C2×C4#1313319111yescharacteristic
483C2×C4#1313319111yescharacteristic
484C2×C4#1313323712yes—
485C2×C4#1313324312yes—
486C2×C4C22×C22⋊D5311yescharacteristic
487C2×C4C22×D21211yescharacteristic
488C2×C4C2×#1171994511yescharacteristic
489C8—22no—
490C8—22no—
491C8—42no—
492C8—2122no—
493C8—2121no—
494C8—2121no—
495C22—21no—
496C22—21no—
497C22—22no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—84no—
504C22—8484no—
505C22—4242no—
506C22—4242no—
507C22—4242no—
508C22#1681083812yes—
509C22C2×#1313319111yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—41no—
514C4—41no—
515C4—4241no—
516C4—4241no—
517C4#1681083812yes—
518C4C2×#1313323711yescharacteristic
519C4C2×#1313324311yescharacteristic
520C4C2×#1313324511yescharacteristic
521C4#1681142311yescharacteristic
522C2—22no—
523C2—42no—
524C2—81no—
525C2—8482no—
526C2C2×#1681083811yescharacteristic
527C2#2028433411yescharacteristic
528C2#2028444311yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingD4×M16×C53C2
socleC22×C53C2×#121
prime radicalC2×#16810818C2
hypercenterD4×M16D53
hypocenterC53#19166
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. #20285090C2semidirect
2. #20284723C2semidirect
4. #20274276C2semidirect
5. #20274248C2semidirect
6. #20274152C2semidirect
7. #20274124C2semidirect
8. #20273467C2semidirect
9. #20273406C2semidirect
10. #20273384C2semidirect
11. D4×M16×C53C2semidirect
16. #16810970C22semidirect
17. #16810970C22semidirect
19. #16810913C22semidirect
20. #16810913C22semidirect
21. #16810828C22semidirect
22. #16810828C22semidirect
35. #16810376C22semidirect
36. #16810374C22semidirect
37. #16810371C22semidirect
38. #16810368C22semidirect
39. #16810366C22semidirect
40. #16810366C22semidirect
43. C53×#436C22semidirect
45. C8×D4×C53C22semidirect
261. D4×M16D53semidirect
296. #436C2×D53semidirect
298. C8×D4C2×D53semidirect
313. C53#19166semidirect

Maximal subgroups

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

subgroupindex
#202850902
#202847232
C2×#168108182
#202742762
#202742482
#202741522
#202741242
#202734672
#202734062
#202733842
D4×M16×C532
#1916653