Finite Group Databasebeta

← #14736629

Subgroups — #14736629

Order 2816.

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#13079068C211yesmaximal, characteristic
2#13078647C211yesmaximal, characteristic, complemented
3C2×#10622640C211yesmaximal, characteristic, complemented
4#13068100C212yesmaximal, complemented
5#13068088C212yesmaximal, complemented
6#13067976C212yesmaximal, complemented
7#13067964C212yesmaximal, complemented
8#13067368C211yesmaximal, characteristic, complemented
9#13067294C211yesmaximal, characteristic, complemented
10#13067285C211yesmaximal, characteristic, complemented
11Q8×C11×M16C211yesmaximal, characteristic, complemented
12C2×C4×Q8×C11C2211yescharacteristic
13C22×#4325—21no—
14C2×#66961—21no—
15C2×#66961—21no—
16#10623008C2211yescharacteristic
17C2×#66927C2211yescharacteristic
18C11×#446—22no—
19C11×#446C2212yescomplemented
20C8×Q8×C11C2212yescomplemented
21C11×#423—21no—
22C11×#423C2211yescharacteristic
23C4×C11×M16—21no—
24C4×C11×M16C2211yescharacteristic
25#10622840—21no—
26#10622640—21no—
27#10622640—21no—
28#10622640C2212yes—
29#10622640C2212yes—
30#10622640C2211yescharacteristic, complemented
31#10622640C2211yescharacteristic, complemented
32C4×#4325—21no—
33C4×#4325—21no—
34#10622633C2211yescharacteristic, complemented
35#10622633C2211yescharacteristic, complemented
36#10622581—21no—
37C2×#66873—21no—
38#10622538C2211yescharacteristic
39C2×C4⋊(C11⋊C8)C2211yescharacteristic
40#10622325—22no—
41#10622308C2212yes—
42#10622306C2212yes—
43#10622123C2212yes—
44#10622123C2212yes—
45#10622123C2212yescomplemented
46#10622120C2212yescomplemented
47#10622096C2212yescomplemented
48#10622095C2212yescomplemented
49C22×Q8×C11D411yescharacteristic
50C2×#4339—21no—
51C22×D44—21no—
52C22×D44—21no—
53C22×D44—21no—
54C2×C11×M16—21no—
55C2×C11×M16D411yescharacteristic
56C2×C11×M16D411yescharacteristic
57C4×Q8×C11—21no—
58C4×Q8×C11—21no—
59C4×Q8×C11D411yescharacteristic
60C4×Q8×C11D411yescharacteristic
61C4×Q8×C11C2311yescharacteristic
62C4×Q8×C11C2311yescharacteristic
63C2×C11×C4⋊C4—21no—
64C2×C11×C4⋊C4D411yescharacteristic
65C2×C42×C11—21no—
66C2×C42×C11C2311yescharacteristic
67C2×#4325—22no—
68C2×#4325—22no—
69C2×#4325—21no—
70C2×#4325—21no—
71C2×#4325—44no—
72#66983—21no—
73#66983—21no—
74C22×C11⋊C8—21no—
75#66961—44no—
76#66961—44no—
77C2×D88—22no—
78C2×D88—22no—
79C2×#4314—22no—
80C2×#4314—22no—
81#66928—21no—
82#66927—21no—
83#66927—21no—
84#66927—21no—
85#66927C2312yes—
86#66927C2311yescharacteristic
87C4×D44—21no—
88C4×D44—21no—
89#66922C2311yescharacteristic
90C11×C4⋊C8—22no—
91C11×C4⋊C8—22no—
92C11×C4⋊C8—21no—
93C11×C4⋊C8C2312yes—
94C11×#97—22no—
95C11×#97D412yes—
96C4×C8×C11—22no—
97C4×C8×C11C2312yes—
98#66899—21no—
99#66899—21no—
100C22⋊(C11⋊C8)—21no—
101#66873—22no—
102#66873—22no—
103#66873—21no—
104#66873—21no—
105#66872—21no—
106#66872—21no—
107C4⋊(C11⋊C8)—21no—
108C4⋊(C11⋊C8)C2312yes—
109C4⋊(C11⋊C8)C2311yescharacteristic
110C4⋊(C11⋊C8)C2311yescharacteristic
111C4×C11⋊C8—21no—
112#19170—111nomaximal, Sylow
113C23×D11—41no—
114C2×Q8×C11—21no—
115C2×Q8×C11—22no—
116C2×Q8×C11—41no—
117C2×Q8×C11C2×D411yescharacteristic
118C2×Q8×C11C2×D411yescharacteristic
119C22×C4×C11—21no—
120C22×C4×C11—21no—
121C22×C4×C11C2×D411yescharacteristic
122C22×C4×C11C2×D411yescharacteristic
123C22×C4×C11#4111yescharacteristic
124C2×C22⋊D11—41no—
125#4339—42no—
126#4339—42no—
127C2×D44—21no—
128C2×D44—22no—
129C2×D44—22no—
130C2×D44—21no—
131C2×D44—21no—
132C2×D44—22no—
133C2×D44—42no—
134C2×D44—42no—
135C2×D44—41no—
136C2×D44—41no—
137C2×D44—41no—
138C2×D44—42no—
139C2×D44—42no—
140C2×D44—82no—
141C2×C4×D11—41no—
142C2×Dic22—21no—
143C11×M16—22no—
144C11×M16—42no—
145C11×M16SD1614yescomplemented
146C2×C8×C11—22no—
147C2×C8×C11C2×D412yes—
148C2×C8×C11C2×D412yes—
149C11×C4⋊C4—21no—
150C11×C4⋊C4—21no—
151C11×C4⋊C4—21no—
152C11×C4⋊C4—21no—
153C11×C4⋊C4—21no—
154C11×C4⋊C4C2×D411yescharacteristic
155C11×C4⋊C4C2×D411yescharacteristic
156C42×C11—21no—
157C42×C11—21no—
158C42×C11—21no—
159C42×C11—21no—
160C42×C11C2×D411yescharacteristic
161C42×C11C2×D411yescharacteristic
162C42×C11#4111yescharacteristic
163C42×C11C2411yescharacteristic
164#4325—42no—
165#4325—44no—
166#4325—42no—
167#4325—44no—
168(C2×D11)⋊C4—41no—
169(C2×D11)⋊C4—41no—
170C4⋊Dic11—21no—
171C4⋊Dic11—21no—
172C2×C11⋊C8—21no—
173C2×C11⋊C8—21no—
174C2×C11⋊C8—22no—
175C2×C11⋊C8—42no—
176D88—44no—
177D88—44no—
178#4314—44no—
179#4314—44no—
180#3151—111no—
181#3103—111no—
182C2×#460—111no—
183Q8×M16D1111yescharacteristic, complemented
184#1688—112no—
185#1686—112no—
186#1668—112no—
187#1666—112no—
188#1565—111no—
189#1547—111no—
190#1545—111no—
191C23×C11#12111yescharacteristic
192C22×D11—41no—
193C22×D11—42no—
194C22×D11—82no—
195C22×D11—82no—
196C22×D11—82no—
197Q8×C11—21no—
198Q8×C11—21no—
199Q8×C11—21no—
200Q8×C11—42no—
201Q8×C11AGL(1,ℤ/8)12yescomplemented
202C2×C4×C11—21no—
203C2×C4×C11—21no—
204C2×C4×C11—21no—
205C2×C4×C11—21no—
206C2×C4×C11—21no—
207C2×C4×C11—21no—
208C2×C4×C11—41no—
209C2×C4×C11—42no—
210C2×C4×C11—81no—
211C2×C4×C11#12111yescharacteristic
212C2×C4×C11#12111yescharacteristic
213C2×C4×C11#12111yescharacteristic
214C2×C4×C11C2×SD1612yes—
215C2×C4×C11AGL(1,ℤ/8)12yes—
216C2×C4×C11C22×D411yescharacteristic
217C2×C4×C11C22×D411yescharacteristic
218C2×C4×C11C2×#4111yescharacteristic
219C22⋊D11—82no—
220C2×Dic11—41no—
221D44—42no—
222D44—41no—
223D44—42no—
224D44—41no—
225D44—42no—
226D44—41no—
227D44—41no—
228D44—42no—
229D44—81no—
230D44—82no—
231D44—82no—
232D44—82no—
233D44—82no—
234C4×D11—82no—
235Dic22—41no—
236Dic22—41no—
237C8×C11—22no—
238C8×C11—42no—
239C8×C11C2×SD1614yescomplemented
240C11⋊C8—42no—
241C11⋊C8—41no—
242C11⋊C8—41no—
243C2×AGL(1,ℤ/8)—221no—
244C2×AGL(1,ℤ/8)—221no—
245C22×SD16—221no—
246#532—111no—
247C2×#127—111no—
248C2×C4×Q8C2×D1111yescharacteristic
249#496—222no—
250#493—112no—
251#492—112no—
252#481—221no—
253#475—111no—
254#475—111no—
255#460—111no—
256#460—111no—
257#460—112no—
258#460—112no—
259#460—221no—
260#460—221no—
261#448—221no—
262#446—22no—
263#446C2×D1112yescomplemented
264C8×Q8C2×D1112yescomplemented
265C4×SD16—221no—
266C4×SD16—221no—
267#431—111no—
268#423—21no—
269#423C2×D1111yescharacteristic
270C2×C4⋊C8—111no—
271C2×#102—221no—
272C4×M16—21no—
273C4×M16C2×D1111yescharacteristic
274#335—112no—
275#334—112no—
276#332—112no—
277#331—112no—
278#331—112no—
279#331—112no—
280C22×C11—21no—
281C22×C11—21no—
282C22×C11#46012yes—
283C22×C11C2×#12111yescharacteristic
284C2×D11—82no—
285C2×D11—81no—
286C2×D11—82no—
287C2×D11—81no—
288C2×D11—82no—
289C2×D11—81no—
290C2×D11—164no—
291C4×C11—21no—
292C4×C11—22no—
293C4×C11—21no—
294C4×C11—21no—
295C4×C11—41no—
296C4×C11—41no—
297C4×C11—41no—
298C4×C11#46012yes—
299C4×C11C22×SD1611yescharacteristic
300C4×C11C2×AGL(1,ℤ/8)11yescharacteristic
301C4×C11C2×AGL(1,ℤ/8)11yescharacteristic
302C4×C11#57511yescharacteristic
303Dic11—81no—
304C2×#41—221no—
305C22×Q8D4411yescharacteristic
306C22×D4—221no—
307C22×D4—221no—
308C22×D4—221no—
309AGL(1,ℤ/8)—444no—
310AGL(1,ℤ/8)—444no—
311C2×SD16—222no—
312C2×SD16—222no—
313C2×SD16—221no—
314C2×SD16—221no—
315C2×SD16—222no—
316C2×SD16—222no—
317C2×SD16—444no—
318C2×D8—222no—
319C2×D8—222no—
320C2×M16—21no—
321C2×M16C22⋊D1111yescharacteristic
322C2×M16C22⋊D1111yescharacteristic
323C22×C8—221no—
324C4⋊Q8—111no—
325#127—112no—
326#127—111no—
327#127—221no—
328#127—221no—
329#127—221no—
330#124—221no—
331#121—221no—
332#121—221no—
333C4×Q8—21no—
334C4×Q8—21no—
335C4×Q8D4411yescharacteristic
336C4×Q8D4411yescharacteristic
337C4×Q8C22×D1111yescharacteristic
338C4×Q8C22×D1111yescharacteristic
339C4×D4—221no—
340C4×D4—221no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4D4411yescharacteristic
343C2×C42—21no—
344C2×C42C22×D1111yescharacteristic
345#106—221no—
346#106—221no—
347C4⋊C8—22no—
348C4⋊C8—22no—
349C4⋊C8—21no—
350C4⋊C8—111no—
351C4⋊C8—111no—
352C4⋊C8—112no—
353C4⋊C8—221no—
354C4⋊C8C22×D1112yes—
355#103—221no—
356#103—221no—
357#102—222no—
358#102—222no—
359#102—221no—
360#102—221no—
361C22⋊C8—221no—
362#97—22no—
363#97C22⋊D1112yes—
364C4×C8—22no—
365C4×C8—221no—
366C4×C8C22×D1112yes—
367C2×C11—22no—
368C2×C11C2×#46011yescharacteristic
369C2×C11#305511yescharacteristic
370C2×C11#306811yescharacteristic
371D11—161no—
372D11—162no—
373C24—441no—
374#41—442no—
375#41—442no—
376C2×Q8—21no—
377C2×Q8—22no—
378C2×Q8—41no—
379C2×Q8—221no—
380C2×Q8C2×D4411yescharacteristic
381C2×Q8C2×D4411yescharacteristic
382C2×D4—221no—
383C2×D4—222no—
384C2×D4—221no—
385C2×D4—222no—
386C2×D4—221no—
387C2×D4—222no—
388C2×D4—442no—
389C2×D4—442no—
390C2×D4—441no—
391C2×D4—441no—
392C2×D4—441no—
393C2×D4—441no—
394C2×D4—442no—
395C2×D4—442no—
396C2×D4—882no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—441no—
400C22×C4C2×D4411yescharacteristic
401C22×C4#433911yescharacteristic
402C22×C4C2×C22⋊D1111yescharacteristic
403SD16—444no—
404SD16—444no—
405SD16—442no—
406SD16—444no—
407SD16—442no—
408SD16—444no—
409D8—444no—
410D8—444no—
411M16—22no—
412M16—42no—
413M16#432514yescomplemented
414C2×C8—22no—
415C2×C8—221no—
416C2×C8—221no—
417C2×C8—222no—
418C2×C8—442no—
419C2×C8C2×C22⋊D1112yes—
420C2×C8C2×C22⋊D1112yes—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—221no—
427C4⋊C4—221no—
428C4⋊C4C2×D4411yescharacteristic
429C4⋊C4C2×D4411yescharacteristic
430C22⋊C4—441no—
431C22⋊C4—441no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42C2×D4411yescharacteristic
437C42#433911yescharacteristic
438C42C2×C22⋊D1111yescharacteristic
439C42C23×D1111yescharacteristic
440C11#1917011yescharacteristic, complemented, Sylow
441C23—441no—
442C23—442no—
443C23—882no—
444C23—882no—
445C23—882no—
446C23#6698311yescharacteristic
447Q8—21no—
448Q8—21no—
449Q8—21no—
450Q8—42no—
451Q8—441no—
452Q8—441no—
453Q8#6696112yescomplemented
454D4—442no—
455D4—441no—
456D4—441no—
457D4—442no—
458D4—441no—
459D4—442no—
460D4—441no—
461D4—442no—
462D4—882no—
463D4—881no—
464D4—882no—
465D4—882no—
466D4—882no—
467D4—882no—
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—441no—
478C2×C4—882no—
479C2×C4#6698311yescharacteristic
480C2×C4#6698311yescharacteristic
481C2×C4#6698311yescharacteristic
482C2×C4C2×#432512yes—
483C2×C4#6700212yes—
484C2×C4C22×D4411yescharacteristic
485C2×C4C2×#433911yescharacteristic
486C2×C4C22×C22⋊D1111yescharacteristic
487C8—22no—
488C8—42no—
489C8—442no—
490C8—441no—
491C8—441no—
492C8C2×#432514yescomplemented
493C22—21no—
494C22—21no—
495C22—881no—
496C22—882no—
497C22—882no—
498C22—881no—
499C22—882no—
500C22—881no—
501C22—1764no—
502C22#1062264012yes—
503C22C2×#6698311yescharacteristic
504C4—21no—
505C4—21no—
506C4—21no—
507C4—22no—
508C4—41no—
509C4—41no—
510C4—41no—
511C4—881no—
512C4#1062264012yes—
513C4C2×#6696111yescharacteristic
514C4#1062327911yescharacteristic
515C4C22×#432511yescharacteristic
516C4C2×#6700211yescharacteristic
517C2—22no—
518C2—1762no—
519C2—1761no—
520C2C2×#1062264011yescharacteristic
521C2#1307823511yescharacteristic
522C2#1307834411yescharacteristic
subgroupG/H
centerC22C2×#66983
commutator, cosocleC42×C11C24
FrattiniC42C23×D11
FittingQ8×C11×M16C2
socleC22×C11C2×#121
prime radical#13079068C2
hypercenterQ8×M16D11
hypocenterC11#19170
LCSG ▷ C42×C11 ▷ C22×C11 ▷ C11
UCSC1 ◁ C22 ◁ C2×C42 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
2. #13078647C2semidirect
3. C2×#10622640C2semidirect
4. #13068100C2semidirect
5. #13068088C2semidirect
6. #13067976C2semidirect
7. #13067964C2semidirect
8. #13067368C2semidirect
9. #13067294C2semidirect
10. #13067285C2semidirect
11. Q8×C11×M16C2semidirect
19. C11×#446C22semidirect
20. C8×Q8×C11C22semidirect
30. #10622640C22semidirect
31. #10622640C22semidirect
34. #10622633C22semidirect
35. #10622633C22semidirect
45. #10622123C22semidirect
46. #10622120C22semidirect
47. #10622096C22semidirect
48. #10622095C22semidirect
145. C11×M16SD16semidirect
183. Q8×M16D11semidirect
201. Q8×C11AGL(1,ℤ/8)semidirect
239. C8×C11C2×SD16semidirect
263. #446C2×D11semidirect
264. C8×Q8C2×D11semidirect
413. M16#4325semidirect
440. C11#19170semidirect
453. Q8#66961semidirect
492. C8C2×#4325semidirect

Maximal subgroups

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

subgroupindex
#130790682
#130786472
C2×#106226402
#130681002
#130680882
#130679762
#130679642
#130673682
#130672942
#130672852
Q8×C11×M162
#1917011