Finite Group Databasebeta

← #14754814

Subgroups — #14754814

Order 2816.

1458 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623054C211yesmaximal, characteristic
2C22×#66872C211yesmaximal, characteristic, complemented
3C2×#10622579C211yesmaximal, characteristic
4C2×#10622615C211yesmaximal, characteristic
5#13068883C214yesmaximal, complemented
6#13068876C214yesmaximal, complemented
7#13068337C211yesmaximal, characteristic, complemented
8#13068313C211yesmaximal, characteristic, complemented
9C11×#2317C211yesmaximal, characteristic, complemented
10C22×#4339C2211yescharacteristic
11C2×C11×#117C2211yescharacteristic
12C2×C11×#117C2211yescharacteristic
13C22×C11×C4⋊C4C2211yescharacteristic
14C22×C4⋊Dic11C2211yescharacteristic
15C22×#4318C2211yescharacteristic
16C23×C11⋊C8C2211yescharacteristic
17#10623054—24no—
18#10623054—24no—
19C2×#66945—21no—
20C2×C4⋊C4×D11—21no—
21C2×(C22⋊D11)⋊C4—22no—
22C2×C4×D44C2212yes—
23C2×C4×Dic22C2212yes—
24C11×#389C2214yes—
25C11×#385—22no—
26C11×#384—21no—
27C11×#382—21no—
28#10622615C418yescomplemented
29C2×#66874C2212yes—
30C2×#66874C2212yes—
31#10622579—24no—
32C2×#66873C2212yes—
33C2×#66873C2212yes—
34C2×#66872—24no—
35C2×#66872C2214yes—
36#10622167C2214yes—
37#10622162C2214yes—
38C23×C4×C11—21no—
39C23×C4×C11C2311yescharacteristic
40C22×C22⋊D11—21no—
41C23×Dic11—21no—
42C2×#4339—22no—
43C2×#4339—22no—
44C2×#4339—22no—
45C2×#4339C2×C414yes—
46C2×#4339D414yes—
47C2×#4339Q814yes—
48C22×D44C2311yescharacteristic
49C22×C4×D11—21no—
50C22×C4×D11—22no—
51C22×Dic22C2311yescharacteristic
52C11×#117—22no—
53C11×#117—22no—
54C11×#117—22no—
55C11×#117C2×C414yescomplemented
56C2×C11×C4⋊C4—22no—
57C2×C11×C4⋊C4—22no—
58C2×C11×C4⋊C4—21no—
59C2×C11×C4⋊C4—24no—
60C2×C11×C4⋊C4D411yescharacteristic
61C2×C11×C4⋊C4D411yescharacteristic
62C2×C11×C4⋊C4C2312yes—
63C2×C11×C4⋊C4C2312yes—
64C2×C11×C22⋊C4—21no—
65C2×C11×C22⋊C4D412yes—
66C2×C42×C11—21no—
67C2×C42×C11C2312yes—
68C2×(C2×D11)⋊C4—22no—
69C2×C4⋊Dic11—21no—
70C2×C4⋊Dic11—22no—
71C2×C4⋊Dic11D414yes—
72C2×C4⋊Dic11C2312yes—
73C2×Dic11⋊C4—22no—
74C2×C4×Dic11—22no—
75C2×#4318—22no—
76C2×#4318—22no—
77C2×#4318—22no—
78C22×C11⋊C8—22no—
79C22×C11⋊C8—22no—
80C22×C11⋊C8C2×C414yescomplemented
81C22×C11⋊C8C2312yes—
82C22×C11⋊C8C2312yes—
83#66945—28no—
84C4⋊C4×D11—28no—
85(C22⋊D11)⋊C4—216no—
86C4×D44—24no—
87C4×D44—24no—
88C4×Dic22—24no—
89C4×Dic22—24no—
90C11×#95—24no—
91#66874—24no—
92#66874C2×C418yescomplemented
93#66873—24no—
94#66873C2×C418yescomplemented
95#66872—28no—
96#20882—111nomaximal, Sylow
97C24×C11C2×D411yescharacteristic
98C23×D11—21no—
99C22×C4×C11—22no—
100C22×C4×C11—22no—
101C22×C4×C11—42no—
102C22×C4×C11—82no—
103C22×C4×C11C22×C412yes—
104C22×C4×C11C2×D412yes—
105C22×C4×C11C2×D412yes—
106C22×C4×C11C2×D412yes—
107C22×C4×C11C2×D411yescharacteristic
108C22×C4×C11C2×D412yes—
109C22×C4×C11C2×Q812yes—
110C22×C4×C11#4112yes—
111C22×C4×C11#4112yes—
112C22×C4×C11C2411yescharacteristic
113C2×C22⋊D11—24no—
114C2×C22⋊D11—24no—
115C2×C22⋊D11—24no—
116C22×Dic11—24no—
117C22×Dic11—21no—
118C22×Dic11—22no—
119C22×Dic11—82no—
120#4339—28no—
121#4339—28no—
122#4339—28no—
123#4339C4⋊C4116yescomplemented
124C2×D44—21no—
125C2×D44—21no—
126C2×D44—21no—
127C2×D44C22×C412yes—
128C2×D44C2×D412yes—
129C2×D44C2×Q812yes—
130C2×C4×D11—28no—
131C2×C4×D11—24no—
132C2×C4×D11—24no—
133C2×C4×D11—24no—
134C2×C4×D11—84no—
135C2×Dic22—21no—
136C2×Dic22—21no—
137C2×Dic22—21no—
138C2×Dic22C22×C412yes—
139C2×Dic22C2×D412yes—
140C2×Dic22C2×Q812yes—
141C11×C4⋊C4—22no—
142C11×C4⋊C4—22no—
143C11×C4⋊C4—22no—
144C11×C4⋊C4—22no—
145C11×C4⋊C4—22no—
146C11×C4⋊C4—42no—
147C11×C4⋊C4—84no—
148C11×C4⋊C4C22×C414yes—
149C11×C22⋊C4—24no—
150C11×C22⋊C4—24no—
151C42×C11—24no—
152C42×C11—22no—
153C42×C11—22no—
154(C2×D11)⋊C4—28no—
155C4⋊Dic11—24no—
156C4⋊Dic11—22no—
157C4⋊Dic11C2×D414yes—
158Dic11⋊C4—28no—
159C4×Dic11—28no—
160#4318—44no—
161C2×C11⋊C8—28no—
162C2×C11⋊C8—22no—
163C2×C11⋊C8—22no—
164C2×C11⋊C8—22no—
165C2×C11⋊C8—22no—
166C2×C11⋊C8—22no—
167C2×C11⋊C8C22×C414yes—
168C2×#520—111no—
169C22×#106—111no—
170C2×#417—111no—
171C2×#416—111no—
172#2317D1111yescharacteristic, complemented
173#1865—114no—
174#1864—114no—
175#1736—111no—
176#1729—111no—
177C23×C11—22no—
178C23×C11—22no—
179C23×C11C4×D412yes—
180C23×C11#12012yes—
181C23×C11C22⋊Q812yes—
182C23×C11C22×D411yescharacteristic
183C22×D11—22no—
184C22×D11—22no—
185C22×D11—22no—
186C22×D11—42no—
187C22×D11—42no—
188C2×C4×C11—24no—
189C2×C4×C11—22no—
190C2×C4×C11—22no—
191C2×C4×C11—24no—
192C2×C4×C11—24no—
193C2×C4×C11—42no—
194C2×C4×C11—42no—
195C2×C4×C11—42no—
196C2×C4×C11—44no—
197C2×C4×C11—84no—
198C2×C4×C11C2×C4⋊C414yes—
199C2×C4×C11C4×D414yes—
200C2×C4×C11C4×D412yes—
201C2×C4×C11#12012yes—
202C2×C4×C11C22⋊Q814yes—
203C2×C4×C11C22⋊Q812yes—
204C2×C4×C11#12314yes—
205C2×C4×C11C23×C411yescharacteristic
206C2×C4×C11C22×D411yescharacteristic
207C2×C4×C11C22×D411yescharacteristic
208C2×C4×C11C22×Q811yescharacteristic
209C2×C4×C11C2×#4111yescharacteristic
210C2×C4×C11C2×#4111yescharacteristic
211C22⋊D11—216no—
212C2×Dic11—22no—
213C2×Dic11—22no—
214C2×Dic11—22no—
215C2×Dic11—24no—
216C2×Dic11—44no—
217C2×Dic11—84no—
218D44—22no—
219D44—22no—
220D44—22no—
221D44C2×C4⋊C414yes—
222C4×D11—216no—
223C4×D11—88no—
224Dic22—22no—
225Dic22—22no—
226Dic22—22no—
227Dic22C2×C4⋊C414yes—
228C11⋊C8—24no—
229C11⋊C8—42no—
230C22×#41—111no—
231C22×M16—111no—
232C23×C8—111no—
233#520—224no—
234#520—224no—
235C2×C4×Q8—112no—
236C2×C4×D4—112no—
237C2×C4×D4—222no—
238C2×#117—221no—
239C2×#117C2×D1111yescharacteristic
240C2×#117C2×D1111yescharacteristic
241C22×C4⋊C4—111no—
242C22×C4⋊C4—221no—
243C22×C4⋊C4C2×D1111yescharacteristic
244C2×#106—114no—
245C2×#106—224no—
246#417—224no—
247#416—118no—
248C2×#103—112no—
249C2×#103—112no—
250C2×#102—112no—
251C2×#102—112no—
252#389C2×D1114yes—
253#385—22no—
254#384—21no—
255#382—21no—
256#340—114no—
257#336—114no—
258C22×C11—24no—
259C22×C11—24no—
260C22×C11—22no—
261C22×C11—22no—
262C22×C11#38514yes—
263C22×C11C2×C4×D411yescharacteristic
264C22×C11C2×#12011yescharacteristic
265C22×C11C2×C22⋊Q811yescharacteristic
266C22×C11#57712yes—
267C22×C11#57712yes—
268C2×D11—24no—
269C2×D11—44no—
270C2×D11—44no—
271C2×D11—42no—
272C2×D11—42no—
273C4×C11—44no—
274C4×C11—42no—
275C4×C11—42no—
276C4×C11—82no—
277C4×C11#38514yes—
278C4×C11C22×C4⋊C411yescharacteristic
279C4×C11C2×C4×D411yescharacteristic
280C4×C11C2×C22⋊Q811yescharacteristic
281C4×C11C2×#12311yescharacteristic
282Dic11—24no—
283Dic11—44no—
284C2×#41—114no—
285C2×#41—114no—
286C2×#41—114no—
287C2×#41—222no—
288C2×#41—222no—
289C2×#41—222no—
290C22×Q8—111no—
291C22×D4—111no—
292C22×D4—221no—
293C23×C4—21no—
294C23×C4—221no—
295C23×C4—221no—
296C23×C4—222no—
297C23×C4C22×D1111yescharacteristic
298C2×M16—222no—
299C2×M16—222no—
300C2×M16—222no—
301C22×C8—114no—
302C22×C8—112no—
303C22×C8—112no—
304C22×C8—222no—
305C22×C8—222no—
306C4×Q8—224no—
307C4×Q8—224no—
308C4×D4—224no—
309C4×D4—224no—
310C4×D4—2216no—
311#117—22no—
312#117—22no—
313#117—22no—
314#117—228no—
315#117C4×D1114yescomplemented
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—114no—
321C2×C4⋊C4—112no—
322C2×C4⋊C4—228no—
323C2×C4⋊C4—222no—
324C2×C4⋊C4—221no—
325C2×C4⋊C4—222no—
326C2×C4⋊C4C22⋊D1111yescharacteristic
327C2×C4⋊C4C22⋊D1111yescharacteristic
328C2×C4⋊C4C22×D1112yes—
329C2×C4⋊C4C22×D1112yes—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—222no—
332C2×C22⋊C4C22⋊D1112yes—
333C2×C42—21no—
334C2×C42—222no—
335C2×C42C22×D1112yes—
336#106—228no—
337#103—118no—
338#103—224no—
339#102—118no—
340#102—224no—
341#95—24no—
342C2×C11—24no—
343C2×C11C2×#38511yescharacteristic
344C2×C11#295212yes—
345C2×C11#300911yescharacteristic
346C2×C11#301011yescharacteristic
347C2×C11#308412yes—
348D11—44no—
349C24—221no—
350C24C2×C22⋊D1111yescharacteristic
351#41—1116no—
352#41—228no—
353#41—228no—
354#41—228no—
355C2×Q8—112no—
356C2×Q8—112no—
357C2×Q8—112no—
358C2×Q8—221no—
359C2×Q8—221no—
360C2×Q8—221no—
361C2×D4—112no—
362C2×D4—112no—
363C2×D4—112no—
364C2×D4—224no—
365C2×D4—224no—
366C2×D4—224no—
367C2×D4—221no—
368C2×D4—221no—
369C2×D4—221no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—42no—
373C22×C4—82no—
374C22×C4—228no—
375C22×C4—221no—
376C22×C4—224no—
377C22×C4—224no—
378C22×C4—224no—
379C22×C4—222no—
380C22×C4—224no—
381C22×C4—884no—
382C22×C4—882no—
383C22×C4C2×C4×D1112yes—
384C22×C4#433912yes—
385C22×C4D4×D1112yes—
386C22×C4D4×D1112yes—
387C22×C4Q8×D1112yes—
388C22×C4#434312yes—
389C22×C4C2×C22⋊D1112yes—
390C22×C4C2×C22⋊D1112yes—
391C22×C4C2×C22⋊D1111yescharacteristic
392C22×C4C23×D1111yescharacteristic
393M16—444no—
394C2×C8—114no—
395C2×C8—228no—
396C2×C8—222no—
397C2×C8—222no—
398C2×C8—222no—
399C2×C8—222no—
400C2×C8—222no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—42no—
407C4⋊C4—84no—
408C4⋊C4—114no—
409C4⋊C4—228no—
410C4⋊C4—224no—
411C4⋊C4—222no—
412C4⋊C4C2×C4×D1114yes—
413C22⋊C4—24no—
414C22⋊C4—24no—
415C22⋊C4—228no—
416C42—24no—
417C42—22no—
418C42—22no—
419C42—228no—
420C11#2088211yescharacteristic, complemented, Sylow
421C23—22no—
422C23—22no—
423C23—222no—
424C23—222no—
425C23—222no—
426C23—442no—
427C23—442no—
428C23C4×C22⋊D1112yes—
429C23#6699012yes—
430C23#6699912yes—
431C23C22×C22⋊D1111yescharacteristic
432Q8—114no—
433Q8—222no—
434Q8—222no—
435Q8—222no—
436D4—114no—
437D4—2216no—
438D4—222no—
439D4—222no—
440D4—222no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—84no—
451C2×C4—222no—
452C2×C4—222no—
453C2×C4—222no—
454C2×C4—2216no—
455C2×C4—224no—
456C2×C4—444no—
457C2×C4—888no—
458C2×C4—884no—
459C2×C4C4⋊C4×D1114yes—
460C2×C4#6694614yes—
461C2×C4#6694714yes—
462C2×C4#6694914yes—
463C2×C4C4×C22⋊D1112yes—
464C2×C4#6699012yes—
465C2×C4#6699912yes—
466C2×C4C22×C4×D1111yescharacteristic
467C2×C4C2×#433911yescharacteristic
468C2×C4C2×D4×D1111yescharacteristic
469C2×C4C2×D4×D1111yescharacteristic
470C2×C4C2×Q8×D1111yescharacteristic
471C2×C4C2×#434311yescharacteristic
472C8—224no—
473C8—442no—
474C22—22no—
475C22—22no—
476C22—24no—
477C22—24no—
478C22—224no—
479C22—442no—
480C22—442no—
481C22—444no—
482C22—444no—
483C22#1062260214yes—
484C22C2×C4×C22⋊D1111yescharacteristic
485C22C2×#6699011yescharacteristic
486C22C2×#6699911yescharacteristic
487C22#1062336712yes—
488C22#1062336812yes—
489C4—42no—
490C4—42no—
491C4—44no—
492C4—82no—
493C4—224no—
494C4—444no—
495C4#1062260214yes—
496C4C2×C4⋊C4×D1111yescharacteristic
497C4C2×#6694611yescharacteristic
498C4C2×#6694711yescharacteristic
499C4C2×#6694911yescharacteristic
500C2—24no—
501C2—444no—
502C2C2×#1062260211yescharacteristic
503C2#1307749712yes—
504C2#1307789111yescharacteristic
505C2#1307790011yescharacteristic
506C2#1307847412yes—
subgroupG/H
centerC23C22×C22⋊D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#2317C2
socleC23×C11C22×D4
prime radicalC22×#4339C22
cosocleC22×C4×C11C24
hypercenter#2317D11
hypocenterC11#20882
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
2. C22×#66872C2semidirect
5. #13068883C2semidirect
6. #13068876C2semidirect
7. #13068337C2semidirect
8. #13068313C2semidirect
9. C11×#2317C2semidirect
28. #10622615C4semidirect
55. C11×#117C2×C4semidirect
80. C22×C11⋊C8C2×C4semidirect
92. #66874C2×C4semidirect
94. #66873C2×C4semidirect
123. #4339C4⋊C4semidirect
172. #2317D11semidirect
315. #117C4×D11semidirect
420. C11#20882semidirect

Maximal subgroups

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

subgroupindex
C2×#106230542
C22×#668722
C2×#106225792
C2×#106226152
#130688832
#130688762
#130683372
#130683132
C11×#23172
#2088211