Finite Group Databasebeta

← #14780591

Subgroups — #14780591

Order 2816.

2142 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#4324C211yesmaximal, characteristic, complemented
2C2×#10623356C211yesmaximal, characteristic, complemented
3C2×#10622617C211yesmaximal, characteristic, complemented
4C2×#10622828C211yesmaximal, characteristic, complemented
5C2×#10622847C211yesmaximal, characteristic, complemented
6#13069619C218yesmaximal, complemented
7#13068332C211yesmaximal, characteristic, complemented
8C2×C11×#535C211yesmaximal, characteristic
9C22×C11×#41C2211yescharacteristic
10C23×D4×C11C2211yescharacteristic
11C23×Dic22C2211yescharacteristic
12C11×#535—24no—
13C11×#535—24no—
14C2×C11×#127—21no—
15C2×C11×#124—21no—
16C2×C11×#121C2214yes—
17C2×C11×#120—22no—
18C2×C11×#117C2211yescharacteristic
19#10623356—24no—
20#10623356—24no—
21C2×#66997—21no—
22C2×#66989—21no—
23C2×#66988—22no—
24C22×#4324—24no—
25C22×#4324—24no—
26C22×#4324—24no—
27C22×#4324C2214yes—
28C2×#66979C2211yescharacteristic
29C2×#66977C2214yescomplemented
30C22×#4318C2211yescharacteristic
31C23×C11⋊C8C2211yescharacteristic
32#10622847—24no—
33#10622828—24no—
34C2×#66896C2214yes—
35C2×C22⋊(C11⋊C8)C2214yescomplemented
36#10622617—24no—
37C2×#66874C2214yescomplemented
38#10622167C2214yescomplemented
39C25×C11—21no—
40C2×C11×#41—22no—
41C2×C11×#41—22no—
42C2×C11×#41—22no—
43C2×C11×#41—22no—
44C2×C11×#41—22no—
45C2×C11×#41—22no—
46C22×Q8×C11D411yescharacteristic
47C22×D4×C11—24no—
48C22×D4×C11—22no—
49C22×D4×C11—22no—
50C22×D4×C11—22no—
51C22×D4×C11—21no—
52C22×D4×C11—22no—
53C22×D4×C11—22no—
54C22×D4×C11D412yes—
55C22×D4×C11D411yescharacteristic
56C22×D4×C11D414yes—
57C22×D4×C11C2312yes—
58C23×C4×C11C2312yes—
59C23×C4×C11C2311yescharacteristic
60C23×Dic11—21no—
61C22×Dic22—22no—
62C22×Dic22—22no—
63C22×Dic22—22no—
64C22×Dic22—22no—
65C22×Dic22—21no—
66C22×Dic22—22no—
67C22×Dic22D414yescomplemented
68C22×Dic22C2312yes—
69C11×#127—28no—
70C11×#124—28no—
71C11×#121—28no—
72C11×#121—28no—
73C11×#120—216no—
74C11×#117—22no—
75C11×#117D414yes—
76C2×C11×C4⋊C4C2312yes—
77C2×C11×C22⋊C4—21no—
78C2×C11×C22⋊C4—24no—
79C2×C42×C11—21no—
80C2×C22⋊Dic11—24no—
81C2×C22⋊Dic11—21no—
82#66997—28no—
83#66989—28no—
84#66988—216no—
85C2×#4324—232no—
86C2×#4324—28no—
87C2×#4324—28no—
88C2×#4324—28no—
89#66979—22no—
90#66979D414yescomplemented
91C2×C4⋊Dic11C2312yes—
92#66977—28no—
93#66977—28no—
94C2×Dic11⋊C4—24no—
95C2×C4×Dic11—21no—
96C2×#4318—22no—
97C2×#4318—22no—
98C2×#4318—22no—
99C22×C11⋊C8—22no—
100C22×C11⋊C8—22no—
101C22×C11⋊C8D414yescomplemented
102C22×C11⋊C8C2312yes—
103C22×C11⋊C8C2312yes—
104#66896—28no—
105C22⋊(C11⋊C8)—28no—
106#66874—28no—
107#23388—111nomaximal, Sylow
108C24×C11—22no—
109C24×C11—24no—
110C24×C11—42no—
111C24×C11—42no—
112C24×C11—84no—
113C24×C11C2×D412yes—
114C24×C11C2×D411yescharacteristic
115C11×#41—44no—
116C11×#41—44no—
117C11×#41—44no—
118C11×#41—44no—
119C2×Q8×C11—22no—
120C2×Q8×C11—22no—
121C2×Q8×C11—22no—
122C2×D4×C11—24no—
123C2×D4×C11—24no—
124C2×D4×C11—24no—
125C2×D4×C11—216no—
126C2×D4×C11—28no—
127C2×D4×C11—22no—
128C2×D4×C11—24no—
129C2×D4×C11—22no—
130C2×D4×C11—28no—
131C2×D4×C11—22no—
132C2×D4×C11—22no—
133C2×D4×C11—22no—
134C2×D4×C11—22no—
135C2×D4×C11—22no—
136C2×D4×C11—22no—
137C2×D4×C11—42no—
138C2×D4×C11—44no—
139C2×D4×C11—42no—
140C2×D4×C11—44no—
141C2×D4×C11—88no—
142C2×D4×C11C2×D414yes—
143C22×C4×C11—22no—
144C22×C4×C11—22no—
145C22×C4×C11—22no—
146C22×C4×C11—22no—
147C22×C4×C11—22no—
148C22×C4×C11—22no—
149C22×C4×C11—22no—
150C22×C4×C11C2×D412yes—
151C22×C4×C11C2×D411yescharacteristic
152C22×C4×C11C2×D412yes—
153C22×C4×C11C2×D412yes—
154C22×C4×C11C2×D412yes—
155C22×C4×C11C2×D412yes—
156C22×C4×C11C2×D412yes—
157C22×C4×C11#4112yes—
158C22×C4×C11C2411yescharacteristic
159C22×Dic11—24no—
160C22×Dic11—22no—
161C22×Dic11—22no—
162C22×Dic11—42no—
163C22×Dic11—42no—
164C2×Dic22—28no—
165C2×Dic22—24no—
166C2×Dic22—28no—
167C2×Dic22—22no—
168C2×Dic22—22no—
169C2×Dic22—22no—
170C2×Dic22—22no—
171C2×Dic22—22no—
172C2×Dic22—42no—
173C2×Dic22—44no—
174C2×Dic22—42no—
175C2×Dic22—44no—
176C2×Dic22C2×D414yes—
177C11×C4⋊C4—22no—
178C11×C4⋊C4C2×D414yes—
179C11×C22⋊C4—216no—
180C11×C22⋊C4—24no—
181C42×C11—24no—
182C22⋊Dic11—216no—
183C22⋊Dic11—24no—
184#4324—232no—
185C4⋊Dic11—22no—
186C4⋊Dic11C2×D414yes—
187Dic11⋊C4—216no—
188C4×Dic11—24no—
189#4318—44no—
190C2×C11⋊C8—28no—
191C2×C11⋊C8—22no—
192C2×C11⋊C8—22no—
193C2×C11⋊C8—22no—
194C2×C11⋊C8—22no—
195C2×C11⋊C8—22no—
196C2×C11⋊C8C2×D414yes—
197C23×SD16—111no—
198C2×#536—111no—
199C2×#535D1111yescharacteristic
200C2×#419—111no—
201C2×#418—111no—
202C2×#408—111no—
203#2000—118no—
204#1735—111no—
205C23×C11—24no—
206C23×C11—22no—
207C23×C11—24no—
208C23×C11—22no—
209C23×C11—22no—
210C23×C11—28no—
211C23×C11—42no—
212C23×C11—42no—
213C23×C11—48no—
214C23×C11—48no—
215C23×C11—88no—
216C23×C11—88no—
217C23×C11—88no—
218C23×C11#12012yes—
219C23×C11#12012yes—
220C23×C11#12112yes—
221C23×C11C22×D411yescharacteristic
222Q8×C11—44no—
223D4×C11—24no—
224D4×C11—24no—
225D4×C11—48no—
226D4×C11—42no—
227D4×C11—42no—
228D4×C11—48no—
229D4×C11—44no—
230D4×C11—816no—
231C2×C4×C11—22no—
232C2×C4×C11—24no—
233C2×C4×C11—22no—
234C2×C4×C11—22no—
235C2×C4×C11—44no—
236C2×C4×C11—42no—
237C2×C4×C11—42no—
238C2×C4×C11—42no—
239C2×C4×C11—42no—
240C2×C4×C11#12014yes—
241C2×C4×C11#12012yes—
242C2×C4×C11#12012yes—
243C2×C4×C11#12114yes—
244C2×C4×C11#12114yes—
245C2×C4×C11#12112yes—
246C2×C4×C11#12714yes—
247C2×C4×C11C22×D411yescharacteristic
248C2×C4×C11C22×D411yescharacteristic
249C2×C4×C11C22×D411yescharacteristic
250C2×C4×C11C22×D411yescharacteristic
251C2×C4×C11C22×D411yescharacteristic
252C2×C4×C11C2×#4111yescharacteristic
253C2×Dic11—24no—
254C2×Dic11—24no—
255C2×Dic11—42no—
256C2×Dic11—42no—
257C2×Dic11—44no—
258C2×Dic11—48no—
259Dic22—24no—
260Dic22—24no—
261Dic22—42no—
262Dic22—42no—
263Dic22—48no—
264C11⋊C8—24no—
265C11⋊C8—42no—
266C22×#41C2×D1111yescharacteristic
267C23×Q8—111no—
268C23×D4C2×D1111yescharacteristic
269C22×SD16—114no—
270C22×SD16—224no—
271C22×SD16—224no—
272C22×SD16—224no—
273C22×M16—111no—
274C23×C8—111no—
275#536—224no—
276#536—224no—
277#535—24no—
278#535—24no—
279C2×C4⋊Q8—221no—
280C2×#127—21no—
281C2×#124—21no—
282C2×#124—221no—
283C2×#123—222no—
284C2×C22⋊Q8—114no—
285C2×#121C2×D1114yes—
286C2×#120—22no—
287C2×#117—111no—
288C2×#117C2×D1111yescharacteristic
289#419—224no—
290#418—224no—
291C2×#103—114no—
292C2×#102—114no—
293#408—224no—
294C2×C22⋊C8—114no—
295#340—114no—
296C22×C11—24no—
297C22×C11—24no—
298C22×C11—24no—
299C22×C11—42no—
300C22×C11—44no—
301C22×C11—44no—
302C22×C11—48no—
303C22×C11—44no—
304C22×C11—44no—
305C22×C11—816no—
306C22×C11—84no—
307C22×C11—84no—
308C22×C11#39214yes—
309C22×C11C2×#12011yescharacteristic
310C22×C11C2×#12011yescharacteristic
311C22×C11C2×#12111yescharacteristic
312C22×C11#57714yes—
313C4×C11—42no—
314C4×C11—44no—
315C4×C11#39214yes—
316C4×C11C2×#12011yescharacteristic
317C4×C11C2×#12111yescharacteristic
318C4×C11C2×#12111yescharacteristic
319C4×C11C2×#12711yescharacteristic
320Dic11—44no—
321Dic11—44no—
322C25—21no—
323C2×#41—22no—
324C2×#41—22no—
325C2×#41—22no—
326C2×#41—22no—
327C2×#41—22no—
328C2×#41—22no—
329C22×Q8—114no—
330C22×Q8—112no—
331C22×Q8—222no—
332C22×Q8—221no—
333C22×Q8—222no—
334C22×Q8—222no—
335C22×Q8—222no—
336C22×Q8—222no—
337C22×Q8C22⋊D1111yescharacteristic
338C22×D4—24no—
339C22×D4—22no—
340C22×D4—22no—
341C22×D4—22no—
342C22×D4—22no—
343C22×D4—21no—
344C22×D4—22no—
345C22×D4C22⋊D1112yes—
346C22×D4C22⋊D1111yescharacteristic
347C22×D4C22⋊D1114yes—
348C22×D4C22×D1112yes—
349C23×C4—221no—
350C23×C4C22×D1112yes—
351C23×C4C22×D1111yescharacteristic
352C2×SD16—228no—
353C2×SD16—228no—
354C2×SD16—2232no—
355C2×SD16—228no—
356C2×M16—222no—
357C2×M16—222no—
358C2×M16—222no—
359C22×C8—114no—
360C22×C8—112no—
361C22×C8—112no—
362C22×C8—222no—
363C22×C8—222no—
364C4⋊Q8—228no—
365#127—28no—
366#124—28no—
367#124—228no—
368#123—2216no—
369C22⋊Q8—228no—
370C22⋊Q8—228no—
371#121—28no—
372#121—28no—
373#120—216no—
374#117—22no—
375#117—114no—
376#117—222no—
377#117C22⋊D1114yes—
378C2×C4⋊C4—112no—
379C2×C4⋊C4—224no—
380C2×C4⋊C4C22×D1112yes—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—221no—
384C2×C22⋊C4—224no—
385C2×C42—21no—
386C2×C42—221no—
387#103—228no—
388#102—228no—
389C22⋊C8—228no—
390C2×C11—24no—
391C2×C11—82no—
392C2×C11—84no—
393C2×C11C2×#39211yescharacteristic
394C2×C11#300912yes—
395C2×C11#301012yes—
396C2×C11#305712yes—
397C24—22no—
398C24—24no—
399C24—42no—
400C24—42no—
401C24—84no—
402C24C2×C22⋊D1112yes—
403C24C2×C22⋊D1111yescharacteristic
404#41—44no—
405#41—44no—
406#41—44no—
407#41—44no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—114no—
412C2×Q8—224no—
413C2×Q8—222no—
414C2×Q8—228no—
415C2×Q8—228no—
416C2×Q8—222no—
417C2×Q8—222no—
418C2×Q8—222no—
419C2×Q8—222no—
420C2×Q8—442no—
421C2×Q8—444no—
422C2×Q8—444no—
423C2×Q8—442no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—216no—
428C2×D4—28no—
429C2×D4—22no—
430C2×D4—28no—
431C2×D4—22no—
432C2×D4—24no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—42no—
440C2×D4—44no—
441C2×D4—42no—
442C2×D4—44no—
443C2×D4—88no—
444C2×D4C2×C22⋊D1114yes—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—224no—
453C22×C4—222no—
454C22×C4—222no—
455C22×C4—442no—
456C22×C4—442no—
457C22×C4D4×D1112yes—
458C22×C4D4×D1112yes—
459C22×C4D4×D1112yes—
460C22×C4#434112yes—
461C22×C4C2×C22⋊D1112yes—
462C22×C4C2×C22⋊D1111yescharacteristic
463C22×C4C2×C22⋊D1112yes—
464C22×C4C2×C22⋊D1112yes—
465C22×C4C23×D1111yescharacteristic
466SD16—2232no—
467M16—444no—
468C2×C8—114no—
469C2×C8—228no—
470C2×C8—222no—
471C2×C8—222no—
472C2×C8—222no—
473C2×C8—222no—
474C2×C8—222no—
475C4⋊C4—22no—
476C4⋊C4—114no—
477C4⋊C4—2216no—
478C4⋊C4—222no—
479C4⋊C4C2×C22⋊D1114yes—
480C22⋊C4—216no—
481C22⋊C4—24no—
482C22⋊C4—2216no—
483C22⋊C4—224no—
484C42—24no—
485C42—224no—
486C11#2338811yescharacteristic, complemented, Sylow
487C23—24no—
488C23—22no—
489C23—22no—
490C23—28no—
491C23—24no—
492C23—22no—
493C23—42no—
494C23—42no—
495C23—48no—
496C23—48no—
497C23—88no—
498C23—88no—
499C23—88no—
500C23#6699012yes—
501C23#6699212yes—
502C23#6700612yes—
503C23C22×C22⋊D1111yescharacteristic
504Q8—44no—
505Q8—224no—
506Q8—224no—
507Q8—442no—
508Q8—442no—
509Q8—448no—
510D4—24no—
511D4—24no—
512D4—48no—
513D4—42no—
514D4—42no—
515D4—48no—
516D4—44no—
517D4—816no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—224no—
528C2×C4—224no—
529C2×C4—442no—
530C2×C4—448no—
531C2×C4—444no—
532C2×C4—442no—
533C2×C4#6699014yes—
534C2×C4#6699012yes—
535C2×C4#6699114yes—
536C2×C4#6699214yes—
537C2×C4#6699212yes—
538C2×C4#6699314yes—
539C2×C4#6700612yes—
540C2×C4C2×D4×D1111yescharacteristic
541C2×C4C2×D4×D1111yescharacteristic
542C2×C4C2×D4×D1111yescharacteristic
543C2×C4C2×#434111yescharacteristic
544C2×C4C22×C22⋊D1111yescharacteristic
545C2×C4C22×C22⋊D1111yescharacteristic
546C8—224no—
547C8—442no—
548C22—24no—
549C22—24no—
550C22—24no—
551C22—44no—
552C22—48no—
553C22—44no—
554C22—42no—
555C22—44no—
556C22—44no—
557C22—816no—
558C22—84no—
559C22—84no—
560C22#1062283614yes—
561C22C2×#6699011yescharacteristic
562C22C2×#6699211yescharacteristic
563C22#1062336714yes—
564C22C2×#6700611yescharacteristic
565C4—44no—
566C4—42no—
567C4—444no—
568C4—444no—
569C4#1062283614yes—
570C4C2×#6699011yescharacteristic
571C4C2×#6699111yescharacteristic
572C4C2×#6699211yescharacteristic
573C4C2×#6699311yescharacteristic
574C2—24no—
575C2—82no—
576C2—84no—
577C2C2×#1062283611yescharacteristic
578C2#1307789312yes—
579C2#1307790012yes—
580C2#1307825212yes—
subgroupG/H
centerC23C22×C22⋊D11
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
Fitting, prime radicalC2×C11×#535C2
socleC23×C11C22×D4
hypercenterC2×#535D11
hypocenterC11#23388
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
1. C23×#4324C2semidirect
2. C2×#10623356C2semidirect
3. C2×#10622617C2semidirect
4. C2×#10622828C2semidirect
5. C2×#10622847C2semidirect
6. #13069619C2semidirect
7. #13068332C2semidirect
29. C2×#66977C22semidirect
35. C2×C22⋊(C11⋊C8)C22semidirect
37. C2×#66874C22semidirect
38. #10622167C22semidirect
67. C22×Dic22D4semidirect
90. #66979D4semidirect
101. C22×C11⋊C8D4semidirect
486. C11#23388semidirect

Maximal subgroups

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

subgroupindex
C23×#43242
C2×#106233562
C2×#106226172
C2×#106228282
C2×#106228472
#130696192
#130683322
C2×C11×#5352
#2338811