Finite Group Databasebeta

← #14736616

Subgroups — #14736616

Order 2816.

728 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#13079010C211yesmaximal, characteristic, complemented
2#13078616C211yesmaximal, characteristic
3C2×#10622632C211yesmaximal, characteristic, complemented
4#13068177C212yesmaximal, complemented
5#13068163C212yesmaximal, complemented
6#13068053C212yesmaximal, complemented
7#13068039C212yesmaximal, complemented
8#13067368C211yesmaximal, characteristic, complemented
9#13067302C211yesmaximal, characteristic, complemented
10#13067294C211yesmaximal, characteristic, complemented
11D4×C11×M16C211yesmaximal, characteristic, complemented
12C22×C11×M16—21no—
13C2×C4×D4×C11C2211yescharacteristic
14C22×#4324—21no—
15#10623008C2211yescharacteristic
16C2×#66922C2211yescharacteristic
17C11×#436C2212yes—
18C11×#435—22no—
19C8×D4×C11C2212yes—
20C11×#423C2211yescharacteristic
21C11×#407—21no—
22C4×C11×M16C2211yescharacteristic
23#10622830—21no—
24#10622746—21no—
25#10622746—21no—
26#10622632—21no—
27#10622632—21no—
28#10622632C2212yes—
29#10622632C2212yes—
30#10622628C2211yescharacteristic
31#10622628C2211yescharacteristic
32C4×#4323—21no—
33C4×#4323—21no—
34#10622620C2211yescharacteristic
35#10622620C2211yescharacteristic
36C2×#66874—21no—
37#10622580—21no—
38#10622538C2211yescharacteristic
39C2×C4⋊(C11⋊C8)C2211yescharacteristic
40#10622327—22no—
41#10622307C2212yescomplemented
42#10622294C2212yescomplemented
43#10622124C2212yescomplemented
44#10622124C2212yescomplemented
45#10622123C2212yescomplemented
46#10622120C2212yescomplemented
47#10622096C2212yescomplemented
48#10622095C2212yescomplemented
49C22×D4×C11D411yescharacteristic
50C23×C4×C11—21no—
51C2×#4339—21no—
52C22×Dic22—21no—
53C2×C11×M16—21no—
54C2×C11×M16—22no—
55C2×C11×M16—44no—
56C2×C11×M16D411yescharacteristic
57C2×C11×M16D411yescharacteristic
58C22×C8×C11—22no—
59C4×D4×C11—21no—
60C4×D4×C11—21no—
61C4×D4×C11D411yescharacteristic
62C4×D4×C11D411yescharacteristic
63C4×D4×C11C2311yescharacteristic
64C4×D4×C11C2311yescharacteristic
65C2×C11×C4⋊C4D411yescharacteristic
66C2×C11×C22⋊C4—21no—
67C2×C42×C11C2311yescharacteristic
68C2×#4324—22no—
69C2×#4324—22no—
70C2×#4324—44no—
71C2×#4323—21no—
72C2×#4323—21no—
73#66983—21no—
74#66983—21no—
75C2×C4⋊Dic11—21no—
76C2×C4⋊Dic11—21no—
77C22×C11⋊C8—21no—
78#66928—21no—
79#66927C2311yescharacteristic
80C4×D44—21no—
81C4×D44—21no—
82#66922—21no—
83#66922—21no—
84#66922—21no—
85#66922C2312yes—
86#66922C2311yescharacteristic
87C11×C4⋊C8—21no—
88C11×C4⋊C8C2312yes—
89C11×C22⋊C8—22no—
90C11×C22⋊C8—22no—
91C11×#97D412yes—
92C4×C8×C11C2312yes—
93#66896—21no—
94#66896—21no—
95C22⋊(C11⋊C8)—21no—
96#66885—22no—
97#66885—22no—
98#66880—22no—
99#66880—22no—
100#66874—22no—
101#66874—22no—
102#66873—21no—
103#66873—21no—
104#66871—21no—
105#66871—21no—
106C4⋊(C11⋊C8)—21no—
107C4⋊(C11⋊C8)C2312yes—
108C4⋊(C11⋊C8)C2311yescharacteristic
109C4⋊(C11⋊C8)C2311yescharacteristic
110C4×C11⋊C8—21no—
111#19169—111nomaximal, Sylow
112C24×C11—21no—
113C2×D4×C11—22no—
114C2×D4×C11—21no—
115C2×D4×C11—41no—
116C2×D4×C11C2×D411yescharacteristic
117C2×D4×C11C2×D411yescharacteristic
118C22×C4×C11—21no—
119C22×C4×C11—21no—
120C22×C4×C11—21no—
121C22×C4×C11—21no—
122C22×C4×C11—21no—
123C22×C4×C11—42no—
124C22×C4×C11—42no—
125C22×C4×C11C2×D411yescharacteristic
126C22×C4×C11C2×D411yescharacteristic
127C22×C4×C11#4111yescharacteristic
128C2×C22⋊D11—41no—
129C22×Dic11—41no—
130#4339—42no—
131#4339—42no—
132C2×D44—21no—
133C2×D44—21no—
134C2×D44—21no—
135C2×C4×D11—41no—
136C2×Dic22—22no—
137C2×Dic22—21no—
138C2×Dic22—42no—
139C2×Dic22—42no—
140C2×Dic22—41no—
141C11×M16—22no—
142C11×M16—22no—
143C11×M16—44no—
144C11×M16—42no—
145C2×C8×C11—22no—
146C2×C8×C11—44no—
147C2×C8×C11C2×D412yes—
148C2×C8×C11C2×D412yes—
149C11×C4⋊C4—21no—
150C11×C4⋊C4C2×D411yescharacteristic
151C11×C4⋊C4C2×D411yescharacteristic
152C11×C22⋊C4—21no—
153C11×C22⋊C4—21no—
154C11×C22⋊C4—21no—
155C11×C22⋊C4—21no—
156C42×C11C2×D411yescharacteristic
157C42×C11C2×D411yescharacteristic
158C42×C11#4111yescharacteristic
159C42×C11C2411yescharacteristic
160#4324—44no—
161#4324—44no—
162#4323—42no—
163#4323—42no—
164(C2×D11)⋊C4—41no—
165(C2×D11)⋊C4—41no—
166C4⋊Dic11—21no—
167C4⋊Dic11—22no—
168C4⋊Dic11—22no—
169C4⋊Dic11—21no—
170C4⋊Dic11—41no—
171C4⋊Dic11—41no—
172C2×C11⋊C8—21no—
173C2×C11⋊C8—21no—
174C2×C11⋊C8—22no—
175C2×C11⋊C8—42no—
176#3141—111no—
177#3100—111no—
178C2×#461—111no—
179D4×M16D1111yescharacteristic, complemented
180#1701—112no—
181#1699—112no—
182#1681—112no—
183#1679—112no—
184#1565—111no—
185#1549—111no—
186#1547—111no—
187C23×C11—21no—
188C23×C11—21no—
189C23×C11—42no—
190C23×C11—42no—
191C23×C11—81no—
192C23×C11#12111yescharacteristic
193C22×D11—41no—
194D4×C11—21no—
195D4×C11—21no—
196D4×C11—21no—
197D4×C11—42no—
198D4×C11AGL(1,ℤ/8)12yes—
199C2×C4×C11—21no—
200C2×C4×C11—21no—
201C2×C4×C11—21no—
202C2×C4×C11—21no—
203C2×C4×C11—41no—
204C2×C4×C11—42no—
205C2×C4×C11—41no—
206C2×C4×C11—41no—
207C2×C4×C11—41no—
208C2×C4×C11—41no—
209C2×C4×C11—42no—
210C2×C4×C11—42no—
211C2×C4×C11#12111yescharacteristic
212C2×C4×C11#12111yescharacteristic
213C2×C4×C11#12111yescharacteristic
214C2×C4×C11#13512yes—
215C2×C4×C11AGL(1,ℤ/8)12yes—
216C2×C4×C11C22×D411yescharacteristic
217C2×C4×C11C22×D411yescharacteristic
218C2×C4×C11C2×#4111yescharacteristic
219C22⋊D11—82no—
220C2×Dic11—42no—
221C2×Dic11—41no—
222C2×Dic11—82no—
223D44—41no—
224D44—41no—
225D44—41no—
226D44—41no—
227D44—81no—
228C4×D11—82no—
229Dic22—41no—
230Dic22—42no—
231Dic22—41no—
232Dic22—42no—
233Dic22—82no—
234C8×C11—22no—
235C8×C11—22no—
236C8×C11—42no—
237C11⋊C8—42no—
238C11⋊C8—41no—
239C11⋊C8—41no—
240C22×SD16—221no—
241C22×M16—21no—
242#532—111no—
243C2×C4⋊Q8—111no—
244C2×C4×D4C2×D1111yescharacteristic
245#489—222no—
246#487—112no—
247#486—112no—
248#480—221no—
249#474—111no—
250#474—111no—
251#461—112no—
252#461—112no—
253#461—221no—
254#461—221no—
255#459—111no—
256#459—111no—
257#447—221no—
258C4×D8—221no—
259C4×D8—221no—
260#436C2×D1112yes—
261#435—22no—
262C8×D4C2×D1112yes—
263#431—111no—
264#423C2×D1111yescharacteristic
265C2×C4⋊C8—111no—
266#417—221no—
267#417—221no—
268C2×#103—221no—
269#407—21no—
270C4×M16C2×D1111yescharacteristic
271#335—112no—
272#334—112no—
273#332—112no—
274#332—112no—
275#332—112no—
276#331—112no—
277C22×C11—21no—
278C22×C11—21no—
279C22×C11—22no—
280C22×C11—41no—
281C22×C11—41no—
282C22×C11—41no—
283C22×C11—41no—
284C22×C11—41no—
285C22×C11—84no—
286C22×C11#46312yes—
287C22×C11C2×#12111yescharacteristic
288C2×D11—81no—
289C2×D11—81no—
290C2×D11—81no—
291C4×C11—21no—
292C4×C11—21no—
293C4×C11—21no—
294C4×C11—41no—
295C4×C11—41no—
296C4×C11#46312yes—
297C4×C11C2×#13511yescharacteristic
298C4×C11C2×AGL(1,ℤ/8)11yescharacteristic
299C4×C11C2×AGL(1,ℤ/8)11yescharacteristic
300C4×C11#57511yescharacteristic
301Dic11—81no—
302Dic11—82no—
303C2×#41—221no—
304C22×Q8—221no—
305C22×D4D4411yescharacteristic
306C23×C4—21no—
307C2×SD16—222no—
308C2×SD16—222no—
309C2×SD16—444no—
310C2×D8—221no—
311C2×D8—221no—
312C2×M16—21no—
313C2×M16—22no—
314C2×M16—44no—
315C2×M16C22⋊D1111yescharacteristic
316C2×M16C22⋊D1111yescharacteristic
317C22×C8—22no—
318C22×C8—221no—
319C4⋊Q8—112no—
320C4⋊Q8—111no—
321C4⋊Q8—221no—
322C4⋊Q8—221no—
323C4⋊Q8—221no—
324#127—111no—
325#124—221no—
326#121—221no—
327#121—221no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—221no—
331C4×D4—221no—
332C4×D4D4411yescharacteristic
333C4×D4D4411yescharacteristic
334C4×D4C22×D1111yescharacteristic
335C4×D4C22×D1111yescharacteristic
336C2×C4⋊C4—221no—
337C2×C4⋊C4—221no—
338C2×C4⋊C4D4411yescharacteristic
339C2×C22⋊C4—21no—
340C2×C42C22×D1111yescharacteristic
341#107—221no—
342#107—221no—
343C4⋊C8—21no—
344C4⋊C8—112no—
345C4⋊C8—111no—
346C4⋊C8—111no—
347C4⋊C8—221no—
348C4⋊C8C22×D1112yes—
349#103—222no—
350#103—222no—
351#103—222no—
352#103—222no—
353#102—221no—
354#102—221no—
355#102—222no—
356#102—222no—
357#102—221no—
358#102—221no—
359C22⋊C8—22no—
360C22⋊C8—22no—
361C22⋊C8—221no—
362#97C22⋊D1112yes—
363C4×C8—221no—
364C4×C8C22×D1112yes—
365C2×C11—22no—
366C2×C11—42no—
367C2×C11—81no—
368C2×C11C2×#46311yescharacteristic
369C2×C11#305511yescharacteristic
370C2×C11#306911yescharacteristic
371D11—161no—
372C24—21no—
373#41—442no—
374#41—442no—
375C2×Q8—222no—
376C2×Q8—221no—
377C2×Q8—442no—
378C2×Q8—442no—
379C2×Q8—441no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—41no—
383C2×D4—221no—
384C2×D4—221no—
385C2×D4—221no—
386C2×D4—441no—
387C2×D4C2×D4411yescharacteristic
388C2×D4C2×D4411yescharacteristic
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—441no—
397C22×C4—441no—
398C22×C4C2×D4411yescharacteristic
399C22×C4#433911yescharacteristic
400C22×C4C2×C22⋊D1111yescharacteristic
401SD16—444no—
402SD16—444no—
403D8—442no—
404D8—442no—
405M16—22no—
406M16—22no—
407M16—44no—
408M16—42no—
409C2×C8—22no—
410C2×C8—44no—
411C2×C8—221no—
412C2×C8—221no—
413C2×C8—222no—
414C2×C8—442no—
415C2×C8C2×C22⋊D1112yes—
416C2×C8C2×C22⋊D1112yes—
417C4⋊C4—21no—
418C4⋊C4—221no—
419C4⋊C4—222no—
420C4⋊C4—221no—
421C4⋊C4—222no—
422C4⋊C4—441no—
423C4⋊C4—441no—
424C4⋊C4C2×D4411yescharacteristic
425C4⋊C4C2×D4411yescharacteristic
426C22⋊C4—21no—
427C22⋊C4—21no—
428C22⋊C4—21no—
429C22⋊C4—21no—
430C22⋊C4—441no—
431C22⋊C4—441no—
432C42C2×D4411yescharacteristic
433C42#433911yescharacteristic
434C42C2×C22⋊D1111yescharacteristic
435C42C23×D1111yescharacteristic
436C11#1916911yescharacteristic, complemented, Sylow
437C23—21no—
438C23—21no—
439C23—42no—
440C23—42no—
441C23—81no—
442C23—441no—
443C23#6698311yescharacteristic
444Q8—442no—
445Q8—442no—
446Q8—441no—
447Q8—441no—
448Q8—882no—
449D4—21no—
450D4—21no—
451D4—21no—
452D4—42no—
453D4—441no—
454D4—441no—
455D4—441no—
456D4—441no—
457D4—881no—
458D4—882no—
459D4#6696112yes—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—441no—
473C2×C4—442no—
474C2×C4—882no—
475C2×C4—882no—
476C2×C4#6698311yescharacteristic
477C2×C4#6698311yescharacteristic
478C2×C4#6698311yescharacteristic
479C2×C4#6698512yes—
480C2×C4#6700312yes—
481C2×C4C22×D4411yescharacteristic
482C2×C4C2×#433911yescharacteristic
483C2×C4C22×C22⋊D1111yescharacteristic
484C8—22no—
485C8—22no—
486C8—42no—
487C8—442no—
488C8—441no—
489C8—441no—
490C22—21no—
491C22—21no—
492C22—22no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—84no—
499C22—881no—
500C22—881no—
501C22—881no—
502C22#1062262912yes—
503C22C2×#6698311yescharacteristic
504C4—21no—
505C4—21no—
506C4—21no—
507C4—41no—
508C4—41no—
509C4—881no—
510C4—882no—
511C4#1062262912yes—
512C4C2×#6696111yescharacteristic
513C4#1062327911yescharacteristic
514C4C2×#6698511yescharacteristic
515C4C2×#6700311yescharacteristic
516C2—22no—
517C2—42no—
518C2—81no—
519C2—1761no—
520C2C2×#1062262911yescharacteristic
521C2#1307823511yescharacteristic
522C2#1307835311yescharacteristic
subgroupG/H
centerC22C2×#66983
commutator, cosocleC42×C11C24
FrattiniC42C23×D11
FittingD4×C11×M16C2
socleC22×C11C2×#121
prime radical#13078616C2
hypercenterD4×M16D11
hypocenterC11#19169
LCSG ▷ C42×C11 ▷ C22×C11 ▷ C11
UCSC1 ◁ C22 ◁ C2×C42 ◁ D4×M16

Complements

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

NH ≅ G/Ntype
1. #13079010C2semidirect
3. C2×#10622632C2semidirect
4. #13068177C2semidirect
5. #13068163C2semidirect
6. #13068053C2semidirect
7. #13068039C2semidirect
8. #13067368C2semidirect
9. #13067302C2semidirect
10. #13067294C2semidirect
11. D4×C11×M16C2semidirect
41. #10622307C22semidirect
42. #10622294C22semidirect
43. #10622124C22semidirect
44. #10622124C22semidirect
45. #10622123C22semidirect
46. #10622120C22semidirect
47. #10622096C22semidirect
48. #10622095C22semidirect
179. D4×M16D11semidirect
436. C11#19169semidirect

Maximal subgroups

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

subgroupindex
#130790102
#130786162
C2×#106226322
#130681772
#130681632
#130680532
#130680392
#130673682
#130673022
#130672942
D4×C11×M162
#1916911