Finite Group Databasebeta

← #14736612

Subgroups — #14736612

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#13078612C211yesmaximal, characteristic
2C2×#10622696C211yesmaximal, characteristic, complemented
3M16×D44C211yesmaximal, characteristic, complemented
4#13068174C212yesmaximal, complemented
5#13068160C212yesmaximal, complemented
6#13068050C212yesmaximal, complemented
7#13068036C212yesmaximal, complemented
8#13067368C211yesmaximal, characteristic, complemented
9#13067301C211yesmaximal, characteristic, complemented
10#13067293C211yesmaximal, characteristic, complemented
11C11×#3141C211yesmaximal, characteristic, complemented
12C11×#532C2211yescharacteristic
13C2×C11×C4⋊Q8C2211yescharacteristic
14C22×#4325—21no—
15C2×M16×D11—21no—
16C2×C4×D44C2211yescharacteristic
17C11×#489—22no—
18C11×#487C2212yescomplemented
19C11×#486C2212yescomplemented
20C11×#417—21no—
21C11×#417—21no—
22C4×C11×M16C2211yescharacteristic
23C2×#66899—21no—
24#10622739—21no—
25#10622700C2211yescharacteristic
26#10622700C2211yescharacteristic
27#10622696—21no—
28#10622696—21no—
29#10622696C2212yes—
30#10622696C2212yes—
31#10622685C2211yescharacteristic
32#10622685C2211yescharacteristic
33#10622649—21no—
34#10622646—21no—
35C4×#4323—21no—
36C4×#4323—21no—
37#10622538C2211yescharacteristic
38#10622537C2211yescharacteristic
39C2×C4⋊(C11⋊C8)C2211yescharacteristic
40#10622319—22no—
41#10622301C2212yes—
42C8×D44C2212yes—
43#10622185C2212yescomplemented
44#10622185C2212yescomplemented
45#10622185C2212yescomplemented
46#10622182C2212yescomplemented
47#10622096C2212yescomplemented
48#10622095C2212yescomplemented
49C2×C11×#41—21no—
50C22×Q8×C11—21no—
51C22×D44D411yescharacteristic
52C22×C4×D11—21no—
53C2×C11×M16D411yescharacteristic
54C2×C11×M16D411yescharacteristic
55C11×C4⋊Q8—21no—
56C11×C4⋊Q8—21no—
57C11×C4⋊Q8—21no—
58C11×C4⋊Q8C2312yes—
59C11×C4⋊Q8C2311yescharacteristic
60C11×#127C2311yescharacteristic
61C11×#124—21no—
62C11×#121—21no—
63C11×#121—21no—
64C4×D4×C11—21no—
65C4×D4×C11—21no—
66C2×C11×C4⋊C4—21no—
67C2×C11×C4⋊C4—21no—
68C2×C42×C11C2311yescharacteristic
69C2×#4325—22no—
70C2×#4325—22no—
71C2×#4325—44no—
72C2×#4323—21no—
73C2×#4323—21no—
74C2×(C2×D11)⋊C4—21no—
75C2×C4⋊Dic11D411yescharacteristic
76C2×#4318—21no—
77C22×C11⋊C8—21no—
78M16×D11—44no—
79C2×#4313—22no—
80C2×C8×D11—22no—
81C4×D44—21no—
82C4×D44—21no—
83C4×D44D411yescharacteristic
84C4×D44D411yescharacteristic
85C4×D44C2311yescharacteristic
86C4×D44C2311yescharacteristic
87C11×#103—22no—
88C11×#103—22no—
89C11×#102—22no—
90C11×#102—22no—
91C11×#97D412yes—
92C4×C8×C11C2312yes—
93#66899—22no—
94#66899—22no—
95#66896—21no—
96#66896—21no—
97C22⋊(C11⋊C8)—21no—
98(C2×D11)⋊C8—22no—
99(C2×D11)⋊C8—22no—
100#66873—21no—
101#66873—21no—
102#66871—21no—
103#66871—21no—
104C4⋊(C11⋊C8)—21no—
105C4⋊(C11⋊C8)—21no—
106C4⋊(C11⋊C8)C2312yes—
107C4⋊(C11⋊C8)C2312yes—
108C4⋊(C11⋊C8)C2311yescharacteristic
109C4⋊(C11⋊C8)C2311yescharacteristic
110C4×C11⋊C8—21no—
111#19169—111nomaximal, Sylow
112C23×D11—21no—
113C11×#41—42no—
114C11×#41—42no—
115C2×Q8×C11—22no—
116C2×Q8×C11—21no—
117C2×Q8×C11—42no—
118C2×Q8×C11—42no—
119C2×Q8×C11—41no—
120C2×D4×C11—21no—
121C2×D4×C11—21no—
122C2×D4×C11—21no—
123C2×D4×C11—41no—
124C22×C4×C11—41no—
125C22×C4×C11—41no—
126C22×C4×C11C2×D411yescharacteristic
127C22×C4×C11C2×D411yescharacteristic
128C22×C4×C11#4111yescharacteristic
129C22×Dic11—21no—
130C2×D44—22no—
131C2×D44—21no—
132C2×D44—41no—
133C2×D44C2×D411yescharacteristic
134C2×D44C2×D411yescharacteristic
135C2×C4×D11—21no—
136C2×C4×D11—21no—
137C2×C4×D11—21no—
138C2×C4×D11—21no—
139C2×C4×D11—42no—
140C2×C4×D11—42no—
141C11×M16—22no—
142C11×M16—22no—
143C2×C8×C11C2×D412yes—
144C2×C8×C11C2×D412yes—
145C11×C4⋊C4—21no—
146C11×C4⋊C4—22no—
147C11×C4⋊C4—22no—
148C11×C4⋊C4—21no—
149C11×C4⋊C4—41no—
150C11×C4⋊C4—41no—
151C11×C22⋊C4—41no—
152C11×C22⋊C4—41no—
153C42×C11C2×D411yescharacteristic
154C42×C11C2×D411yescharacteristic
155C42×C11#4111yescharacteristic
156C42×C11C2411yescharacteristic
157#4325—44no—
158#4325—44no—
159#4323—42no—
160#4323—42no—
161(C2×D11)⋊C4—21no—
162(C2×D11)⋊C4—21no—
163(C2×D11)⋊C4—21no—
164(C2×D11)⋊C4—21no—
165C4⋊Dic11—21no—
166C4⋊Dic11C2×D411yescharacteristic
167C4⋊Dic11C2×D411yescharacteristic
168#4318—42no—
169C2×C11⋊C8—21no—
170C2×C11⋊C8—21no—
171C2×C11⋊C8—22no—
172C2×C11⋊C8—22no—
173C2×C11⋊C8—42no—
174#4313—44no—
175C8×D11—44no—
176#3141D1111yescharacteristic, complemented
177#3100—111no—
178C2×#461—111no—
179D4×M16—111no—
180#1701—112no—
181#1699—112no—
182#1681—112no—
183#1679—112no—
184#1565—111no—
185#1549—111no—
186#1547—111no—
187C23×C11—41no—
188C23×C11#12111yescharacteristic
189C22×D11—21no—
190C22×D11—21no—
191C22×D11—42no—
192C22×D11—42no—
193C22×D11—81no—
194Q8×C11—41no—
195Q8×C11—42no—
196Q8×C11—41no—
197Q8×C11—42no—
198Q8×C11—82no—
199D4×C11—41no—
200D4×C11—41no—
201D4×C11—41no—
202D4×C11—41no—
203D4×C11—81no—
204D4×C11—82no—
205C2×C4×C11—21no—
206C2×C4×C11—21no—
207C2×C4×C11—41no—
208C2×C4×C11—41no—
209C2×C4×C11—42no—
210C2×C4×C11—82no—
211C2×C4×C11—82no—
212C2×C4×C11#12111yescharacteristic
213C2×C4×C11#12111yescharacteristic
214C2×C4×C11#12111yescharacteristic
215C2×C4×C11#13512yes—
216C2×C4×C11AGL(1,ℤ/8)12yes—
217C2×C4×C11C22×D411yescharacteristic
218C2×C4×C11C22×D411yescharacteristic
219C2×C4×C11C2×#4111yescharacteristic
220C2×Dic11—21no—
221C2×Dic11—21no—
222C2×Dic11—42no—
223D44—21no—
224D44—21no—
225D44—21no—
226D44—42no—
227D44AGL(1,ℤ/8)12yes—
228C4×D11—41no—
229C4×D11—41no—
230C4×D11—41no—
231C4×D11—41no—
232C4×D11—42no—
233C4×D11—42no—
234C8×C11—22no—
235C8×C11—22no—
236C11⋊C8—42no—
237C11⋊C8—42no—
238C11⋊C8—41no—
239C11⋊C8—41no—
240C22×SD16—221no—
241C22×M16—221no—
242#532C2×D1111yescharacteristic
243C2×C4⋊Q8C2×D1111yescharacteristic
244C2×C4×D4—111no—
245#489—22no—
246#487C2×D1112yescomplemented
247#486C2×D1112yescomplemented
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#436—112no—
261#435—222no—
262C8×D4—112no—
263#431—111no—
264#423—111no—
265C2×C4⋊C8—111no—
266#417—21no—
267#417—21no—
268C2×#103—221no—
269#407—221no—
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—81no—
280C22×C11—81no—
281C22×C11—81no—
282C22×C11#46312yes—
283C22×C11C2×#12111yescharacteristic
284C2×D11—22no—
285C2×D11—41no—
286C2×D11—41no—
287C2×D11—41no—
288C2×D11—41no—
289C2×D11—41no—
290C2×D11—84no—
291C4×C11—21no—
292C4×C11—21no—
293C4×C11—21no—
294C4×C11—81no—
295C4×C11—82no—
296C4×C11#46312yes—
297C4×C11C2×#13511yescharacteristic
298C4×C11C2×AGL(1,ℤ/8)11yescharacteristic
299C4×C11C2×AGL(1,ℤ/8)11yescharacteristic
300C4×C11#57511yescharacteristic
301Dic11—41no—
302Dic11—41no—
303C2×#41—21no—
304C22×Q8—21no—
305C22×D4—111no—
306C23×C4—221no—
307C2×SD16—222no—
308C2×SD16—222no—
309C2×SD16—444no—
310C2×D8—221no—
311C2×D8—221no—
312C2×M16—221no—
313C2×M16—222no—
314C2×M16—444no—
315C2×M16C22⋊D1111yescharacteristic
316C2×M16C22⋊D1111yescharacteristic
317C22×C8—221no—
318C22×C8—222no—
319C4⋊Q8—21no—
320C4⋊Q8—21no—
321C4⋊Q8—21no—
322C4⋊Q8C22×D1112yes—
323C4⋊Q8C22×D1111yescharacteristic
324#127C22×D1111yescharacteristic
325#124—21no—
326#121—21no—
327#121—21no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—111no—
331C4×D4—111no—
332C4×D4—111no—
333C4×D4—111no—
334C4×D4—221no—
335C4×D4—221no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—111no—
339C2×C22⋊C4—221no—
340C2×C42C22×D1111yescharacteristic
341#107—221no—
342#107—221no—
343C4⋊C8—112no—
344C4⋊C8—112no—
345C4⋊C8—111no—
346C4⋊C8—111no—
347C4⋊C8—221no—
348C4⋊C8—221no—
349#103—22no—
350#103—22no—
351#103—222no—
352#103—222no—
353#102—22no—
354#102—22no—
355#102—221no—
356#102—221no—
357#102—221no—
358#102—221no—
359C22⋊C8—221no—
360C22⋊C8—222no—
361C22⋊C8—222no—
362#97C22⋊D1112yes—
363C4×C8—221no—
364C4×C8C22×D1112yes—
365C2×C11—22no—
366C2×C11—161no—
367C2×C11C2×#46311yescharacteristic
368C2×C11#305511yescharacteristic
369C2×C11#306911yescharacteristic
370D11—42no—
371D11—81no—
372C24—221no—
373#41—42no—
374#41—42no—
375C2×Q8—22no—
376C2×Q8—21no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8—41no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—41no—
384C2×D4—111no—
385C2×D4—111no—
386C2×D4—222no—
387C2×D4—221no—
388C2×D4—441no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—221no—
392C22×C4—221no—
393C22×C4—221no—
394C22×C4—221no—
395C22×C4—221no—
396C22×C4—442no—
397C22×C4—442no—
398C22×C4D4×D1111yescharacteristic
399C22×C4#434111yescharacteristic
400C22×C4C2×C22⋊D1111yescharacteristic
401SD16—444no—
402SD16—444no—
403D8—442no—
404D8—442no—
405M16—22no—
406M16—22no—
407M16—444no—
408M16—442no—
409C2×C8—221no—
410C2×C8—221no—
411C2×C8—222no—
412C2×C8—222no—
413C2×C8—444no—
414C2×C8—442no—
415C2×C8C2×C22⋊D1112yes—
416C2×C8C2×C22⋊D1112yes—
417C4⋊C4—21no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—22no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—111no—
424C4⋊C4—111no—
425C4⋊C4—221no—
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—221no—
429C22⋊C4—221no—
430C22⋊C4—221no—
431C22⋊C4—221no—
432C42D4×D1111yescharacteristic
433C42#434111yescharacteristic
434C42C2×C22⋊D1111yescharacteristic
435C42C23×D1111yescharacteristic
436C11#1916911yescharacteristic, complemented, Sylow
437C23—41no—
438C23—221no—
439C23—221no—
440C23—442no—
441C23—442no—
442C23—881no—
443C23#6699111yescharacteristic
444Q8—42no—
445Q8—42no—
446Q8—41no—
447Q8—41no—
448Q8—82no—
449D4—41no—
450D4—41no—
451D4—41no—
452D4—41no—
453D4—81no—
454D4—82no—
455D4—112no—
456D4—221no—
457D4—221no—
458D4—221no—
459D4—442no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—221no—
468C2×C4—221no—
469C2×C4—442no—
470C2×C4—441no—
471C2×C4—441no—
472C2×C4—441no—
473C2×C4—441no—
474C2×C4—442no—
475C2×C4—442no—
476C2×C4#6699111yescharacteristic
477C2×C4#6699111yescharacteristic
478C2×C4#6699111yescharacteristic
479C2×C4#6700212yes—
480C2×C4#6700312yes—
481C2×C4C2×D4×D1111yescharacteristic
482C2×C4C2×#434111yescharacteristic
483C2×C4C22×C22⋊D1111yescharacteristic
484C8—22no—
485C8—22no—
486C8—442no—
487C8—441no—
488C8—441no—
489C8—442no—
490C22—21no—
491C22—21no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—222no—
496C22—441no—
497C22—441no—
498C22—441no—
499C22—441no—
500C22—441no—
501C22—884no—
502C22#1062267012yes—
503C22C2×#6699111yescharacteristic
504C4—21no—
505C4—21no—
506C4—21no—
507C4—81no—
508C4—82no—
509C4—441no—
510C4—441no—
511C4#1062267012yes—
512C4D11×AGL(1,ℤ/8)11yescharacteristic
513C4#1062330411yescharacteristic
514C4C2×#6700211yescharacteristic
515C4C2×#6700311yescharacteristic
516C2—22no—
517C2—161no—
518C2—442no—
519C2—881no—
520C2C2×#1062267011yescharacteristic
521C2#1307822911yescharacteristic
522C2#1307834911yescharacteristic
subgroupG/H
centerC22C2×#66991
commutator, cosocleC42×C11C24
FrattiniC42C23×D11
FittingC11×#3141C2
socleC22×C11C2×#121
prime radical#13078612C2
hypercenter#3141D11
hypocenterC11#19169
LCSG ▷ C42×C11 ▷ C22×C11 ▷ C11
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3141

Complements

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

NH ≅ G/Ntype
2. C2×#10622696C2semidirect
3. M16×D44C2semidirect
4. #13068174C2semidirect
5. #13068160C2semidirect
6. #13068050C2semidirect
7. #13068036C2semidirect
8. #13067368C2semidirect
9. #13067301C2semidirect
10. #13067293C2semidirect
11. C11×#3141C2semidirect
18. C11×#487C22semidirect
19. C11×#486C22semidirect
43. #10622185C22semidirect
44. #10622185C22semidirect
45. #10622185C22semidirect
46. #10622182C22semidirect
47. #10622096C22semidirect
48. #10622095C22semidirect
176. #3141D11semidirect
246. #487C2×D11semidirect
247. #486C2×D11semidirect
436. C11#19169semidirect

Maximal subgroups

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

subgroupindex
#130786122
C2×#106226962
M16×D442
#130681742
#130681602
#130680502
#130680362
#130673682
#130673012
#130672932
C11×#31412
#1916911