Finite Group Databasebeta

← #14753300

Subgroups — #14753300

Order 2816.

1462 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622612C211yesmaximal, characteristic, complemented
2C2×#10622845C211yesmaximal, characteristic
3C2×#10622846C211yesmaximal, characteristic
4#13071239C211yesmaximal, characteristic, complemented
5#13068631C212yesmaximal, complemented
6#13068626C212yesmaximal, complemented
7#13068621C212yesmaximal, complemented
8#13068616C212yesmaximal, complemented
9#13068335C211yesmaximal, characteristic, complemented
10C2×#10622167C211yesmaximal, characteristic, complemented
11C2×C11×#520C211yesmaximal, characteristic
12C22×C11×#41C2211yescharacteristic
13C11×#520—24no—
14C11×#520C418yescomplemented
15C2×C4×Q8×C11C2212yes—
16C2×C4×D4×C11—22no—
17C2×C4×D4×C11C2212yes—
18C2×C11×#117—21no—
19C2×C11×#117C2211yescharacteristic
20C22×C11×C4⋊C4—21no—
21C22×C11×C4⋊C4C2211yescharacteristic
22C2×#66979C2211yescharacteristic
23C22×C4⋊Dic11C2211yescharacteristic
24C22×#4318C2211yescharacteristic
25C23×C11⋊C8C2211yescharacteristic
26#10622846—24no—
27#10622845—24no—
28C2×#66899C2211yescharacteristic
29C2×#66899C2212yes—
30C2×#66899C2211yescharacteristic
31C2×#66896C2211yescharacteristic
32C2×#66896C2212yes—
33C2×#66896C2211yescharacteristic
34#10622612—24no—
35#10622598—21no—
36#10622598—21no—
37#10622563—22no—
38#10622549C2212yes—
39#10622548C2212yes—
40C2×C4⋊(C11⋊C8)C2212yes—
41C2×C4⋊(C11⋊C8)C2212yes—
42#10622167C2212yes—
43#10622167C2212yes—
44#10622167C2212yes—
45#10622167C2212yes—
46C2×C11×#41—22no—
47C2×C11×#41—22no—
48C2×C11×#41—22no—
49C2×C11×#41C2×C414yes—
50C2×C11×#41D414yes—
51C2×C11×#41Q814yes—
52C22×Q8×C11C2311yescharacteristic
53C22×D4×C11—21no—
54C22×D4×C11C2311yescharacteristic
55C23×C4×C11—21no—
56C23×C4×C11—21no—
57C23×C4×C11—22no—
58C23×C4×C11C2311yescharacteristic
59C23×Dic11—21no—
60C4×Q8×C11—24no—
61C4×Q8×C11C2×C418yescomplemented
62C4×D4×C11—216no—
63C4×D4×C11—24no—
64C4×D4×C11C2×C418yescomplemented
65C11×#117—28no—
66C11×#117C2×C414yes—
67C11×#117D414yes—
68C2×C11×C4⋊C4—21no—
69C2×C11×C4⋊C4—22no—
70C2×C11×C4⋊C4—22no—
71C2×C11×C4⋊C4—28no—
72C2×C11×C4⋊C4C2×C414yes—
73C2×C11×C4⋊C4D411yescharacteristic
74C2×C11×C4⋊C4D411yescharacteristic
75C2×C11×C4⋊C4C2312yes—
76C2×C11×C22⋊C4—22no—
77C2×C11×C22⋊C4D412yes—
78C2×C42×C11—22no—
79C2×C42×C11C2312yes—
80C2×C22⋊Dic11—21no—
81#66979—22no—
82#66979—22no—
83C2×C4⋊Dic11—22no—
84C2×C4⋊Dic11—22no—
85C2×C4⋊Dic11—21no—
86C2×C4⋊Dic11—22no—
87C2×C4⋊Dic11—22no—
88C2×C4⋊Dic11C2311yescharacteristic
89C2×C4⋊Dic11C2312yes—
90C2×C4⋊Dic11C2311yescharacteristic
91C2×C4×Dic11—21no—
92C2×#4318—22no—
93C2×#4318—22no—
94C2×#4318—22no—
95C22×C11⋊C8—22no—
96C22×C11⋊C8—22no—
97C22×C11⋊C8—22no—
98C22×C11⋊C8C2311yescharacteristic
99C22×C11⋊C8C2311yescharacteristic
100C22×C11⋊C8C2312yes—
101#66899—24no—
102#66899—22no—
103#66899—22no—
104#66896—24no—
105#66896—22no—
106#66896—22no—
107C11⋊#95—22no—
108C11⋊#95—22no—
109C4⋊(C11⋊C8)—24no—
110C4⋊(C11⋊C8)—24no—
111#20652—111nomaximal, Sylow
112C24×C11—21no—
113C24×C11C2×D411yescharacteristic
114C11×#41—28no—
115C11×#41—28no—
116C11×#41—28no—
117C11×#41C4⋊C4116yescomplemented
118C2×Q8×C11—21no—
119C2×Q8×C11—21no—
120C2×Q8×C11—21no—
121C2×Q8×C11C22×C412yes—
122C2×Q8×C11C2×D412yes—
123C2×Q8×C11C2×Q812yes—
124C2×D4×C11—24no—
125C2×D4×C11—24no—
126C2×D4×C11—24no—
127C2×D4×C11—21no—
128C2×D4×C11—21no—
129C2×D4×C11—21no—
130C2×D4×C11C22×C412yes—
131C2×D4×C11C2×D412yes—
132C2×D4×C11C2×Q812yes—
133C22×C4×C11—21no—
134C22×C4×C11—24no—
135C22×C4×C11—24no—
136C22×C4×C11—24no—
137C22×C4×C11—22no—
138C22×C4×C11—24no—
139C22×C4×C11—28no—
140C22×C4×C11—82no—
141C22×C4×C11—82no—
142C22×C4×C11—81no—
143C22×C4×C11—81no—
144C22×C4×C11C22×C412yes—
145C22×C4×C11C2×D412yes—
146C22×C4×C11C2×D412yes—
147C22×C4×C11C2×D411yescharacteristic
148C22×C4×C11C2×D412yes—
149C22×C4×C11C2×D412yes—
150C22×C4×C11C2×Q812yes—
151C22×C4×C11#4112yes—
152C22×C4×C11#4112yes—
153C22×C4×C11C2411yescharacteristic
154C22×Dic11—21no—
155C22×Dic11—22no—
156C22×Dic11—21no—
157C22×Dic11—42no—
158C22×Dic11—81no—
159C22×Dic11—81no—
160C11×C4⋊C4—24no—
161C11×C4⋊C4—28no—
162C11×C4⋊C4—22no—
163C11×C4⋊C4C22⋊C414yes—
164C11×C4⋊C4C22⋊C414yes—
165C11×C4⋊C4C22×C414yes—
166C11×C22⋊C4—28no—
167C11×C22⋊C4C22⋊C418yes—
168C42×C11—28no—
169C42×C11C22×C414yes—
170C42×C11C2×D414yes—
171C22⋊Dic11—42no—
172C4⋊Dic11—22no—
173C4⋊Dic11—21no—
174C4⋊Dic11—21no—
175C4⋊Dic11—22no—
176C4⋊Dic11—21no—
177C4⋊Dic11—21no—
178C4⋊Dic11—42no—
179C4⋊Dic11—82no—
180C4⋊Dic11—82no—
181C4×Dic11—42no—
182#4318—44no—
183C2×C11⋊C8—21no—
184C2×C11⋊C8—21no—
185C2×C11⋊C8—22no—
186C2×C11⋊C8—21no—
187C2×C11⋊C8—21no—
188C2×C11⋊C8—22no—
189C2×C11⋊C8—21no—
190C2×C11⋊C8—21no—
191C2×C11⋊C8—22no—
192C2×C11⋊C8—44no—
193C2×#520D1111yescharacteristic
194C2×#424—111no—
195C2×#417—111no—
196C2×#416—111no—
197#2317—111no—
198#1805—112no—
199#1804—112no—
200#1803—112no—
201#1802—112no—
202#1736—111no—
203C2×#340—111no—
204C23×C11—21no—
205C23×C11—21no—
206C23×C11—21no—
207C23×C11—21no—
208C23×C11—22no—
209C23×C11—22no—
210C23×C11—22no—
211C23×C11—41no—
212C23×C11—41no—
213C23×C11—41no—
214C23×C11—41no—
215C23×C11C2×C22⋊C412yes—
216C23×C11#12112yes—
217C23×C11C22⋊Q812yes—
218C23×C11C22×D411yescharacteristic
219Q8×C11—22no—
220Q8×C11—22no—
221Q8×C11—22no—
222Q8×C11C2×C4⋊C414yes—
223D4×C11—216no—
224D4×C11—22no—
225D4×C11—22no—
226D4×C11—22no—
227D4×C11C2×C4⋊C414yes—
228C2×C4×C11—22no—
229C2×C4×C11—22no—
230C2×C4×C11—22no—
231C2×C4×C11—22no—
232C2×C4×C11—22no—
233C2×C4×C11—22no—
234C2×C4×C11—22no—
235C2×C4×C11—216no—
236C2×C4×C11—24no—
237C2×C4×C11—42no—
238C2×C4×C11—42no—
239C2×C4×C11—88no—
240C2×C4×C11—84no—
241C2×C4×C11C2×C22⋊C414yes—
242C2×C4×C11C2×C22⋊C414yes—
243C2×C4×C11C2×C22⋊C412yes—
244C2×C4×C11C2×C4⋊C414yes—
245C2×C4×C11#11714yes—
246C2×C4×C11#12114yes—
247C2×C4×C11#12112yes—
248C2×C4×C11C22⋊Q814yes—
249C2×C4×C11C22⋊Q812yes—
250C2×C4×C11C23×C411yescharacteristic
251C2×C4×C11C22×D411yescharacteristic
252C2×C4×C11C22×D411yescharacteristic
253C2×C4×C11C22×Q811yescharacteristic
254C2×C4×C11C2×#4111yescharacteristic
255C2×C4×C11C2×#4111yescharacteristic
256C2×Dic11—41no—
257C2×Dic11—41no—
258C2×Dic11—42no—
259C2×Dic11—41no—
260C2×Dic11—42no—
261C2×Dic11—41no—
262C2×Dic11—42no—
263C2×Dic11—41no—
264C2×Dic11—41no—
265C2×Dic11—84no—
266C11⋊C8—41no—
267C11⋊C8—41no—
268C11⋊C8—42no—
269C22×#41C2×D1111yescharacteristic
270C22×M16—111no—
271C23×C8—111no—
272#520—24no—
273#520Dic1118yescomplemented
274C2×C4×Q8C2×D1112yes—
275C2×C4×D4—22no—
276C2×C4×D4C2×D1112yes—
277C2×#117—21no—
278C2×#117—111no—
279C2×#117C2×D1111yescharacteristic
280C22×C4⋊C4—21no—
281C22×C4⋊C4—111no—
282C22×C4⋊C4C2×D1111yescharacteristic
283#424—224no—
284C2×C4⋊C8—112no—
285C2×C4⋊C8—112no—
286#417—224no—
287#416—224no—
288C2×#103—111no—
289C2×#103—112no—
290C2×#103—111no—
291C2×#102—111no—
292C2×#102—112no—
293C2×#102—111no—
294#389—221no—
295#389—221no—
296#385—222no—
297#384—112no—
298#382—112no—
299#340—112no—
300#340—112no—
301#340—112no—
302#340—112no—
303C22×C11—22no—
304C22×C11—22no—
305C22×C11—22no—
306C22×C11—22no—
307C22×C11—22no—
308C22×C11—22no—
309C22×C11—24no—
310C22×C11—42no—
311C22×C11—42no—
312C22×C11—42no—
313C22×C11—42no—
314C22×C11—42no—
315C22×C11—42no—
316C22×C11#38014yes—
317C22×C11C22×C22⋊C411yescharacteristic
318C22×C11C2×#12111yescharacteristic
319C22×C11C2×C22⋊Q811yescharacteristic
320C22×C11#57611yescharacteristic
321C22×C11#57711yescharacteristic
322C22×C11#57711yescharacteristic
323C22×C11#57811yescharacteristic
324C4×C11—24no—
325C4×C11—24no—
326C4×C11—24no—
327C4×C11—44no—
328C4×C11#38014yes—
329C4×C11C22×C4⋊C411yescharacteristic
330C4×C11C2×#11711yescharacteristic
331C4×C11C2×#12111yescharacteristic
332C4×C11C2×C22⋊Q811yescharacteristic
333Dic11—81no—
334Dic11—82no—
335Dic11—81no—
336C2×#41—22no—
337C2×#41—22no—
338C2×#41—22no—
339C2×#41Dic2214yes—
340C2×#41D4414yes—
341C2×#41C2×Dic1114yes—
342C22×Q8C22×D1111yescharacteristic
343C22×D4—21no—
344C22×D4C22×D1111yescharacteristic
345C23×C4—21no—
346C23×C4—21no—
347C23×C4—22no—
348C23×C4—221no—
349C23×C4C22×D1111yescharacteristic
350C2×M16—222no—
351C2×M16—222no—
352C2×M16—222no—
353C22×C8—111no—
354C22×C8—111no—
355C22×C8—112no—
356C22×C8—222no—
357C22×C8—222no—
358C22×C8—222no—
359C4×Q8—24no—
360C4×Q8C2×Dic1118yescomplemented
361C4×D4—216no—
362C4×D4—24no—
363C4×D4C2×Dic1118yescomplemented
364#117—28no—
365#117—222no—
366#117—222no—
367#117C2×Dic1114yes—
368#117C22⋊D1114yes—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—28no—
373C2×C4⋊C4—112no—
374C2×C4⋊C4—111no—
375C2×C4⋊C4—111no—
376C2×C4⋊C4—222no—
377C2×C4⋊C4—222no—
378C2×C4⋊C4—221no—
379C2×C4⋊C4—222no—
380C2×C4⋊C4—222no—
381C2×C4⋊C4C2×Dic1114yes—
382C2×C4⋊C4C22⋊D1111yescharacteristic
383C2×C4⋊C4C22⋊D1111yescharacteristic
384C2×C4⋊C4C22×D1112yes—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—221no—
387C2×C22⋊C4C22⋊D1112yes—
388C2×C42—22no—
389C2×C42—221no—
390C2×C42C22×D1112yes—
391C4⋊C8—224no—
392C4⋊C8—224no—
393#103—222no—
394#103—222no—
395#103—224no—
396#102—222no—
397#102—222no—
398#102—224no—
399#95—222no—
400#95—222no—
401C2×C11—24no—
402C2×C11—44no—
403C2×C11C2×#38011yescharacteristic
404C2×C11#289711yescharacteristic
405C2×C11#289811yescharacteristic
406C2×C11#304511yescharacteristic
407C2×C11#304611yescharacteristic
408C2×C11#308311yescharacteristic
409C2×C11#308411yescharacteristic
410C24—21no—
411C24C2×C22⋊D1111yescharacteristic
412#41—28no—
413#41—28no—
414#41—28no—
415#41C4⋊Dic11116yescomplemented
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8C2×Dic2212yes—
420C2×Q8C2×D4412yes—
421C2×Q8C22×Dic1112yes—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4C2×Dic2212yes—
429C2×D4C2×D4412yes—
430C2×D4C22×Dic1112yes—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—28no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4—81no—
441C22×C4—81no—
442C22×C4—221no—
443C22×C4—222no—
444C22×C4—221no—
445C22×C4—442no—
446C22×C4—881no—
447C22×C4—881no—
448C22×C4C2×Dic2212yes—
449C22×C4C2×D4412yes—
450C22×C4#433912yes—
451C22×C4#433912yes—
452C22×C4C22×Dic1112yes—
453C22×C4C2×C22⋊D1112yes—
454C22×C4C2×C22⋊D1112yes—
455C22×C4C2×C22⋊D1111yescharacteristic
456C22×C4C2×C22⋊D1112yes—
457C22×C4C23×D1111yescharacteristic
458M16—444no—
459C2×C8—221no—
460C2×C8—221no—
461C2×C8—222no—
462C2×C8—221no—
463C2×C8—221no—
464C2×C8—222no—
465C2×C8—221no—
466C2×C8—221no—
467C2×C8—222no—
468C2×C8—444no—
469C4⋊C4—24no—
470C4⋊C4—28no—
471C4⋊C4—22no—
472C4⋊C4—222no—
473C4⋊C4—221no—
474C4⋊C4—221no—
475C4⋊C4—222no—
476C4⋊C4—221no—
477C4⋊C4—221no—
478C4⋊C4—442no—
479C4⋊C4—882no—
480C4⋊C4—882no—
481C4⋊C4C22⋊Dic1114yes—
482C4⋊C4C22⋊Dic1114yes—
483C4⋊C4C22×Dic1114yes—
484C22⋊C4—28no—
485C22⋊C4—442no—
486C22⋊C4C22⋊Dic1118yes—
487C42—28no—
488C42—442no—
489C42C22×Dic1114yes—
490C42C2×C22⋊D1114yes—
491C11#2065211yescharacteristic, complemented, Sylow
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23#6697712yes—
504C23#6698312yes—
505C23C2×C22⋊Dic1112yes—
506C23C22×C22⋊D1111yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—22no—
510Q8C2×C4⋊Dic1114yes—
511D4—216no—
512D4—22no—
513D4—22no—
514D4—22no—
515D4C2×C4⋊Dic1114yes—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—216no—
524C2×C4—24no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—88no—
528C2×C4—84no—
529C2×C4—441no—
530C2×C4—441no—
531C2×C4—442no—
532C2×C4—441no—
533C2×C4—442no—
534C2×C4—441no—
535C2×C4—441no—
536C2×C4—442no—
537C2×C4—441no—
538C2×C4—884no—
539C2×C4#6697714yes—
540C2×C4#6697712yes—
541C2×C4C2×C4⋊Dic1114yes—
542C2×C4#6697914yes—
543C2×C4#6698314yes—
544C2×C4#6698312yes—
545C2×C4C2×C22⋊Dic1114yes—
546C2×C4C2×C22⋊Dic1114yes—
547C2×C4C2×C22⋊Dic1112yes—
548C2×C4C22×Dic2211yescharacteristic
549C2×C4C22×D4411yescharacteristic
550C2×C4C2×#433911yescharacteristic
551C2×C4C2×#433911yescharacteristic
552C2×C4C23×Dic1111yescharacteristic
553C2×C4C22×C22⋊D1111yescharacteristic
554C8—441no—
555C8—441no—
556C8—442no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—24no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22#1062282514yes—
571C22C2×#6697711yescharacteristic
572C22C2×#6698311yescharacteristic
573C22#1062336611yescharacteristic
574C22#1062336711yescharacteristic
575C22#1062336811yescharacteristic
576C22#1062336911yescharacteristic
577C22C22×C22⋊Dic1111yescharacteristic
578C4—24no—
579C4—24no—
580C4—24no—
581C4—44no—
582C4—882no—
583C4—881no—
584C4—881no—
585C4#1062282514yes—
586C4C2×#6697711yescharacteristic
587C4C22×C4⋊Dic1111yescharacteristic
588C4C2×#6697911yescharacteristic
589C4C2×#6698311yescharacteristic
590C2—24no—
591C2—44no—
592C2C2×#1062282511yescharacteristic
593C2#1307714211yescharacteristic
594C2#1307714911yescharacteristic
595C2#1307815511yescharacteristic
596C2#1307816611yescharacteristic
597C2#1307846711yescharacteristic
598C2#1307847811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC2×C11×#520C2
socleC23×C11C22×D4
prime radicalC22×C11×#41C22
cosocleC22×C4×C11C24
hypercenterC2×#520D11
hypocenterC11#20652
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
1. C2×#10622612C2semidirect
4. #13071239C2semidirect
5. #13068631C2semidirect
6. #13068626C2semidirect
7. #13068621C2semidirect
8. #13068616C2semidirect
9. #13068335C2semidirect
10. C2×#10622167C2semidirect
14. C11×#520C4semidirect
61. C4×Q8×C11C2×C4semidirect
64. C4×D4×C11C2×C4semidirect
117. C11×#41C4⋊C4semidirect
273. #520Dic11semidirect
360. C4×Q8C2×Dic11semidirect
363. C4×D4C2×Dic11semidirect
415. #41C4⋊Dic11semidirect
491. C11#20652semidirect

Maximal subgroups

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

subgroupindex
C2×#106226122
C2×#106228452
C2×#106228462
#130712392
#130686312
#130686262
#130686212
#130686162
#130683352
C2×#106221672
C2×C11×#5202
#2065211