Finite Group Databasebeta

← #14755185

Subgroups — #14755185

Order 2816.

1766 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#67004C211yesmaximal, characteristic
2C2×#10623358C211yesmaximal, characteristic
3C22×#66880C211yesmaximal, characteristic, complemented
4#13071200C211yesmaximal, characteristic, complemented
5#13068955C214yesmaximal, complemented
6#13068948C214yesmaximal, complemented
7#13068337C211yesmaximal, characteristic, complemented
8#13068312C211yesmaximal, characteristic, complemented
9C2×C11×#416C211yesmaximal, characteristic
10C22×C11×#41C2211yescharacteristic
11C23×Dic22C2211yescharacteristic
12C23×C8×C11C2211yescharacteristic
13C2×C11×#117C2211yescharacteristic
14#10623358—24no—
15#10623358—24no—
16C2×#67004—24no—
17C2×#67004—24no—
18C2×#67004C418yescomplemented
19C2×Q8×Dic11C2212yes—
20C22×#4326C2212yes—
21C2×D4×Dic11—22no—
22C2×D4×Dic11C2212yes—
23C22×#4324C2212yes—
24C2×#66979—21no—
25C2×#66979C2211yescharacteristic
26C22×C4⋊Dic11—21no—
27C22×C4⋊Dic11C2211yescharacteristic
28C22×#4318C2211yescharacteristic
29C11×#416—24no—
30C2×C11×#103C2212yes—
31C2×C11×#102C2212yes—
32C2×#66880—24no—
33C2×#66880C2214yes—
34#10622587C2214yes—
35#10622556—22no—
36#10622542—21no—
37#10622542—21no—
38#10622188C2214yes—
39#10622167C2214yes—
40C2×C11×#41—22no—
41C2×C11×#41—22no—
42C2×C11×#41—22no—
43C2×C11×#41C2×C414yes—
44C2×C11×#41D414yes—
45C2×C11×#41D414yes—
46C22×Q8×C11C2311yescharacteristic
47C22×D4×C11—21no—
48C22×D4×C11C2311yescharacteristic
49C23×C4×C11—21no—
50C23×C4×C11C2311yescharacteristic
51C23×Dic11—21no—
52C23×Dic11—21no—
53C23×Dic11—22no—
54C22×Dic22—22no—
55C22×Dic22—22no—
56C22×Dic22—22no—
57C22×Dic22—22no—
58C22×Dic22—21no—
59C22×Dic22—22no—
60C22×Dic22C2×C414yescomplemented
61C22×Dic22C2312yes—
62C22×C8×C11—22no—
63C22×C8×C11—22no—
64C22×C8×C11—22no—
65C22×C8×C11C2312yes—
66C11×#117—22no—
67C11×#117—22no—
68C2×C11×C4⋊C4C2312yes—
69C2×C11×C22⋊C4—21no—
70C2×C42×C11—21no—
71C2×C22⋊Dic11—22no—
72C2×C22⋊Dic11D412yes—
73#67004—232no—
74Q8×Dic11—24no—
75Q8×Dic11—24no—
76C2×#4326—24no—
77C2×#4326—24no—
78C2×#4326C2×C418yescomplemented
79D4×Dic11—216no—
80D4×Dic11—24no—
81D4×Dic11—24no—
82C2×#4324—24no—
83C2×#4324—24no—
84C2×#4324C2×C418yescomplemented
85#66979—28no—
86#66979—22no—
87#66979—22no—
88C2×C4⋊Dic11—28no—
89C2×C4⋊Dic11—21no—
90C2×C4⋊Dic11—22no—
91C2×C4⋊Dic11—22no—
92C2×C4⋊Dic11D411yescharacteristic
93C2×C4⋊Dic11D411yescharacteristic
94C2×C4⋊Dic11D414yes—
95C2×C4⋊Dic11C2312yes—
96C2×C4×Dic11—22no—
97C2×C4×Dic11C2312yes—
98C2×#4318—22no—
99C2×#4318—22no—
100C2×#4318C2×C414yescomplemented
101C22×C11⋊C8C2312yes—
102C11×#103—24no—
103C11×#102—24no—
104C11⋊#95—24no—
105C11⋊#95—24no—
106#66880—28no—
107#20931—111nomaximal, Sylow
108C24×C11—21no—
109C24×C11C2×D411yescharacteristic
110C11×#41—28no—
111C11×#41—28no—
112C11×#41—28no—
113C11×#41C22⋊C4116yes—
114C2×Q8×C11—21no—
115C2×Q8×C11—21no—
116C2×Q8×C11—21no—
117C2×Q8×C11C22×C412yes—
118C2×Q8×C11C2×D412yes—
119C2×Q8×C11C2×D412yes—
120C2×D4×C11—24no—
121C2×D4×C11—24no—
122C2×D4×C11—24no—
123C2×D4×C11—21no—
124C2×D4×C11—21no—
125C2×D4×C11—21no—
126C2×D4×C11C22×C412yes—
127C2×D4×C11C2×D412yes—
128C2×D4×C11C2×D412yes—
129C22×C4×C11—21no—
130C22×C4×C11—24no—
131C22×C4×C11—24no—
132C22×C4×C11—24no—
133C22×C4×C11—22no—
134C22×C4×C11C22×C412yes—
135C22×C4×C11C2×D412yes—
136C22×C4×C11C2×D412yes—
137C22×C4×C11C2×D411yescharacteristic
138C22×C4×C11C2×D412yes—
139C22×C4×C11#4112yes—
140C22×C4×C11#4112yes—
141C22×C4×C11C2411yescharacteristic
142C22×Dic11—28no—
143C22×Dic11—24no—
144C22×Dic11—24no—
145C22×Dic11—22no—
146C22×Dic11—22no—
147C22×Dic11—42no—
148C22×Dic11—42no—
149C22×Dic11—82no—
150C22×Dic11—84no—
151C22×Dic11C2×D412yes—
152C22×Dic11C2×D412yes—
153C2×Dic22—28no—
154C2×Dic22—28no—
155C2×Dic22—24no—
156C2×Dic22—22no—
157C2×Dic22—22no—
158C2×Dic22—22no—
159C2×Dic22—22no—
160C2×Dic22—22no—
161C2×Dic22—42no—
162C2×Dic22—44no—
163C2×Dic22—42no—
164C2×Dic22—44no—
165C2×Dic22C22×C414yes—
166C2×C8×C11—22no—
167C2×C8×C11—22no—
168C2×C8×C11—22no—
169C2×C8×C11—44no—
170C11×C4⋊C4—22no—
171C11×C4⋊C4—22no—
172C11×C22⋊C4—42no—
173C42×C11—42no—
174C22⋊Dic11—28no—
175C22⋊Dic11—24no—
176#4326—216no—
177#4324—216no—
178C4⋊Dic11—28no—
179C4⋊Dic11—22no—
180C4⋊Dic11—24no—
181C4⋊Dic11—22no—
182C4⋊Dic11—22no—
183C4⋊Dic11C2×D414yes—
184C4×Dic11—28no—
185C4×Dic11—22no—
186C4×Dic11—22no—
187#4318—28no—
188C2×C11⋊C8—22no—
189C2×C11⋊C8—22no—
190C2×C11⋊C8C22×C414yes—
191C22×#137—111no—
192C2×#520—111no—
193C2×#416D1111yescharacteristic
194C22×#103—111no—
195#2310—111no—
196#1878—114no—
197#1877—114no—
198#1736—111no—
199#1729—111no—
200C23×C11—22no—
201C23×C11—22no—
202C23×C11—22no—
203C23×C11—22no—
204C23×C11—22no—
205C23×C11—42no—
206C23×C11—42no—
207C23×C11C4×D412yes—
208C23×C11#12012yes—
209C23×C11#12112yes—
210C23×C11C22×D411yescharacteristic
211Q8×C11—22no—
212Q8×C11—22no—
213Q8×C11—22no—
214Q8×C11C2×C22⋊C414yes—
215D4×C11—216no—
216D4×C11—22no—
217D4×C11—22no—
218D4×C11—22no—
219D4×C11C2×C22⋊C414yes—
220C2×C4×C11—22no—
221C2×C4×C11—22no—
222C2×C4×C11—22no—
223C2×C4×C11—216no—
224C2×C4×C11—42no—
225C2×C4×C11—42no—
226C2×C4×C11—42no—
227C2×C4×C11C2×C22⋊C414yes—
228C2×C4×C11C4×D414yes—
229C2×C4×C11C4×D412yes—
230C2×C4×C11#12012yes—
231C2×C4×C11#12114yes—
232C2×C4×C11#12112yes—
233C2×C4×C11#12314yes—
234C2×C4×C11C23×C411yescharacteristic
235C2×C4×C11C22×D411yescharacteristic
236C2×C4×C11C22×D411yescharacteristic
237C2×C4×C11C22×D411yescharacteristic
238C2×C4×C11C2×#4111yescharacteristic
239C2×C4×C11C2×#4111yescharacteristic
240C2×Dic11—24no—
241C2×Dic11—22no—
242C2×Dic11—22no—
243C2×Dic11—24no—
244C2×Dic11—24no—
245C2×Dic11—24no—
246C2×Dic11—42no—
247C2×Dic11—42no—
248C2×Dic11—44no—
249C2×Dic11—48no—
250C2×Dic11—84no—
251C2×Dic11—88no—
252Dic22—24no—
253Dic22—24no—
254Dic22—42no—
255Dic22—42no—
256Dic22—48no—
257C8×C11—42no—
258C11⋊C8—24no—
259C22×#41C2×D1111yescharacteristic
260C23×Q8—111no—
261C2×#137—118no—
262C2×#137—224no—
263C2×#137—224no—
264C22×Q16—112no—
265C22×SD16—112no—
266C22×M16—111no—
267C23×C8C2×D1111yescharacteristic
268#520—224no—
269#520—224no—
270C2×C4×Q8—112no—
271C2×C4×D4—112no—
272C2×C4×D4—222no—
273C2×#117—111no—
274C2×#117—221no—
275C2×#117C2×D1111yescharacteristic
276C22×C4⋊C4—111no—
277C22×C4⋊C4—221no—
278#416—24no—
279C2×#103—114no—
280C2×#103—224no—
281C2×#103C2×D1112yes—
282C2×#102C2×D1112yes—
283#391—114no—
284#391—221no—
285#391—221no—
286#381—222no—
287#340—114no—
288#336—114no—
289C22×C11—24no—
290C22×C11—22no—
291C22×C11—22no—
292C22×C11—24no—
293C22×C11—24no—
294C22×C11—44no—
295C22×C11—42no—
296C22×C11—42no—
297C22×C11—44no—
298C22×C11#38614yes—
299C22×C11C2×C4×D411yescharacteristic
300C22×C11C2×#12011yescharacteristic
301C22×C11C2×#12111yescharacteristic
302C22×C11#57712yes—
303C22×C11#57812yes—
304C4×C11—24no—
305C4×C11—82no—
306C4×C11#38614yes—
307C4×C11C22×C22⋊C411yescharacteristic
308C4×C11C2×C4×D411yescharacteristic
309C4×C11C2×#12111yescharacteristic
310C4×C11C2×#12311yescharacteristic
311Dic11—42no—
312Dic11—44no—
313Dic11—42no—
314Dic11—44no—
315C2×#41—22no—
316C2×#41—22no—
317C2×#41—22no—
318C2×#41C4×D1114yes—
319C2×#41D4414yes—
320C2×#41C22⋊D1114yes—
321C22×Q8—114no—
322C22×Q8—112no—
323C22×Q8—221no—
324C22×Q8—222no—
325C22×Q8—222no—
326C22×Q8—222no—
327C22×Q8—222no—
328C22×Q8—222no—
329C22×Q8C22×D1111yescharacteristic
330C22×D4—21no—
331C22×D4C22×D1111yescharacteristic
332C23×C4—21no—
333C23×C4—222no—
334C23×C4—221no—
335C23×C4—221no—
336C23×C4C22×D1111yescharacteristic
337#137—2232no—
338C2×Q16—118no—
339C2×Q16—224no—
340C2×Q16—224no—
341C2×SD16—118no—
342C2×SD16—224no—
343C2×SD16—224no—
344C2×M16—114no—
345C2×M16—222no—
346C2×M16—222no—
347C22×C8—22no—
348C22×C8—22no—
349C22×C8—22no—
350C22×C8—112no—
351C22×C8C22×D1112yes—
352C4×Q8—224no—
353C4×Q8—224no—
354C4×D4—224no—
355C4×D4—2216no—
356C4×D4—224no—
357#117—22no—
358#117—22no—
359#117—228no—
360#117—222no—
361#117—222no—
362C2×C4⋊C4—114no—
363C2×C4⋊C4—112no—
364C2×C4⋊C4—111no—
365C2×C4⋊C4—111no—
366C2×C4⋊C4—222no—
367C2×C4⋊C4—228no—
368C2×C4⋊C4—222no—
369C2×C4⋊C4—221no—
370C2×C4⋊C4C22×D1112yes—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—112no—
373C2×C22⋊C4—222no—
374C2×C42—21no—
375C2×C42—112no—
376C2×C42—222no—
377#103—24no—
378#103—228no—
379#102—24no—
380#95—224no—
381#95—224no—
382C2×C11—24no—
383C2×C11—44no—
384C2×C11C2×#38611yescharacteristic
385C2×C11#294712yes—
386C2×C11#300911yescharacteristic
387C2×C11#301111yescharacteristic
388C2×C11#304612yes—
389C24—21no—
390C24D4×D1111yescharacteristic
391#41—28no—
392#41—28no—
393#41—28no—
394#41(C2×D11)⋊C4116yes—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—114no—
399C2×Q8—224no—
400C2×Q8—222no—
401C2×Q8—228no—
402C2×Q8—228no—
403C2×Q8—222no—
404C2×Q8—222no—
405C2×Q8—222no—
406C2×Q8—222no—
407C2×Q8—442no—
408C2×Q8—444no—
409C2×Q8—442no—
410C2×Q8—444no—
411C2×Q8C2×C4×D1112yes—
412C2×Q8C2×D4412yes—
413C2×Q8C2×C22⋊D1112yes—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4C2×C4×D1112yes—
421C2×D4C2×D4412yes—
422C2×D4C2×C22⋊D1112yes—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—112no—
429C22×C4—112no—
430C22×C4—224no—
431C22×C4—222no—
432C22×C4—222no—
433C22×C4—228no—
434C22×C4—224no—
435C22×C4—442no—
436C22×C4—442no—
437C22×C4—884no—
438C22×C4—882no—
439C22×C4C2×C4×D1112yes—
440C22×C4C2×D4412yes—
441C22×C4D4×D1111yescharacteristic
442C22×C4D4×D1112yes—
443C22×C4#434112yes—
444C22×C4#434112yes—
445C22×C4C2×C22⋊D1112yes—
446C22×C4C23×D1111yescharacteristic
447Q16—2216no—
448SD16—2216no—
449M16—228no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—44no—
454C2×C8—114no—
455C2×C8—222no—
456C2×C8—222no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—114no—
460C4⋊C4—228no—
461C4⋊C4—224no—
462C4⋊C4—222no—
463C4⋊C4—222no—
464C4⋊C4—222no—
465C22⋊C4—42no—
466C22⋊C4—228no—
467C22⋊C4—224no—
468C42—42no—
469C42—228no—
470C42—222no—
471C42—222no—
472C11#2093111yescharacteristic, complemented, Sylow
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—42no—
479C23—42no—
480C23(C22⋊D11)⋊C412yes—
481C23#6693512yes—
482C23#6699212yes—
483C23C2×D4×D1111yescharacteristic
484Q8—22no—
485Q8—22no—
486Q8—22no—
487Q8—224no—
488Q8—224no—
489Q8—442no—
490Q8—442no—
491Q8—448no—
492Q8C2×(C2×D11)⋊C414yes—
493D4—216no—
494D4—22no—
495D4—22no—
496D4—22no—
497D4C2×(C2×D11)⋊C414yes—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—216no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—224no—
506C2×C4—222no—
507C2×C4—224no—
508C2×C4—222no—
509C2×C4—224no—
510C2×C4—224no—
511C2×C4—442no—
512C2×C4—444no—
513C2×C4—442no—
514C2×C4—448no—
515C2×C4—884no—
516C2×C4—888no—
517C2×C4(C22⋊D11)⋊C414yes—
518C2×C4(C22⋊D11)⋊C412yes—
519C2×C4#6693512yes—
520C2×C4#6693914yes—
521C2×C4C2×(C2×D11)⋊C414yes—
522C2×C4#6699214yes—
523C2×C4#6699212yes—
524C2×C4C22×C4×D1111yescharacteristic
525C2×C4C22×D4411yescharacteristic
526C2×C4C2×D4×D1111yescharacteristic
527C2×C4C2×#434111yescharacteristic
528C2×C4C2×#434111yescharacteristic
529C2×C4C22×C22⋊D1111yescharacteristic
530C8—42no—
531C8—224no—
532C22—22no—
533C22—24no—
534C22—22no—
535C22—24no—
536C22—24no—
537C22—42no—
538C22—44no—
539C22—42no—
540C22—44no—
541C22#1062257114yes—
542C22C2×(C22⋊D11)⋊C411yescharacteristic
543C22C2×#6693511yescharacteristic
544C22#1062330112yes—
545C22#1062330212yes—
546C22C2×#6699211yescharacteristic
547C4—24no—
548C4—82no—
549C4—442no—
550C4—444no—
551C4—442no—
552C4—444no—
553C4#1062257114yes—
554C4C2×(C22⋊D11)⋊C411yescharacteristic
555C4C2×#6693911yescharacteristic
556C4C22×(C2×D11)⋊C411yescharacteristic
557C4C2×#6699211yescharacteristic
558C2—24no—
559C2—44no—
560C2C2×#1062257111yescharacteristic
561C2#1307746212yes—
562C2#1307789611yescharacteristic
563C2#1307791411yescharacteristic
564C2#1307816412yes—
subgroupG/H
centerC23C2×D4×D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC2×C11×#416C2
socleC23×C11C22×D4
prime radicalC22×C11×#41C22
cosocleC22×C4×C11C24
hypercenterC2×#416D11
hypocenterC11#20931
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
3. C22×#66880C2semidirect
4. #13071200C2semidirect
5. #13068955C2semidirect
6. #13068948C2semidirect
7. #13068337C2semidirect
8. #13068312C2semidirect
18. C2×#67004C4semidirect
60. C22×Dic22C2×C4semidirect
78. C2×#4326C2×C4semidirect
84. C2×#4324C2×C4semidirect
100. C2×#4318C2×C4semidirect
472. C11#20931semidirect

Maximal subgroups

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

subgroupindex
C22×#670042
C2×#106233582
C22×#668802
#130712002
#130689552
#130689482
#130683372
#130683122
C2×C11×#4162
#2093111