Finite Group Databasebeta

← #14916616

Subgroups — #14916616

Order 2816.

2510 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623256C211yesmaximal, characteristic
2#13071771C214yesmaximal, complemented
3#13071656C212yesmaximal, complemented
4#13071653C212yesmaximal, complemented
5#13071561C212yesmaximal, complemented
6#13071540C214yesmaximal, complemented
7#13071517C212yesmaximal, complemented
8#13071428C214yesmaximal, complemented
9#13071365C211yesmaximal, characteristic, complemented
10#13071341C211yesmaximal, characteristic, complemented
11#13071339C211yesmaximal, characteristic, complemented
12#13071207C214yesmaximal, complemented
13#13071022C211yesmaximal, characteristic, complemented
14#13071008C211yesmaximal, characteristic, complemented
15C11×#2331C211yesmaximal, characteristic, complemented
16C22×#4343C2212yes—
17C2×C11×C4⋊Q8C2211yescharacteristic
18C2×#67000C2212yes—
19#10623256C4116yescomplemented
20C2×#66948C2212yes—
21C2×#66948C2212yes—
22C2×#66946C2212yes—
23C2×#66946C2212yes—
24C2×#66946C2212yes—
25C2×#66945C2212yes—
26C2×C4⋊C4×D11C2214yes—
27C2×#66927C2211yescharacteristic
28C2×C4×D44C2212yes—
29C2×#66925C2214yes—
30C2×C42×D11C2211yescharacteristic
31C11×#391C2214yes—
32C11×#389C2212yes—
33C11×#387C2214yes—
34C11×#382C2211yescharacteristic
35C4×C11×C4⋊C4C2212yes—
36C11×#376C2211yescharacteristic
37#10622843C2214yes—
38#10622603C2214yes—
39#10622602C2214yes—
40#10622601C2214yes—
41#10622601C2212yes—
42#10622598C2212yes—
43#10622594C2214yes—
44#10622589C2212yes—
45C4⋊C4×Dic11C2212yes—
46#10622553C2214yes—
47#10622552C2214yes—
48#10622552C2212yes—
49C4×(C2×D11)⋊C4C2211yescharacteristic
50C4×(C2×D11)⋊C4C2211yescharacteristic
51C4×(C2×D11)⋊C4C2214yes—
52#10622549C2211yescharacteristic
53#10622545C2211yescharacteristic
54C4×Dic11⋊C4C2211yescharacteristic
55C4×Dic11⋊C4C2211yescharacteristic
56#10622283C2214yes—
57#10622282C2214yes—
58#10622281C2214yes—
59#10622280C2212yes—
60#10622280C2212yes—
61#10622280C2214yes—
62#10622280C2214yes—
63#10622280C2212yes—
64#10622279C2214yes—
65#10622278C2214yes—
66#10622278C2214yes—
67#10622278C2214yes—
68#10622278C2214yes—
69#10622278C2214yes—
70#10622278C2214yes—
71#10622277C2212yes—
72#10622264C2212yes—
73#10622263C2212yes—
74#10622262C2212yes—
75#10622262C2212yes—
76C22×Q8×C11C2312yes—
77C2×#4343—48no—
78C2×#4343C2×C4116yescomplemented
79C22×D44C2312yes—
80C22×D44C2312yes—
81C22×D44C2312yes—
82C22×D44C2314yes—
83C22×C4×D11C2311yescharacteristic
84C22×C4×D11C2311yescharacteristic
85C22×C4×D11C2314yes—
86C22×C4×D11C2314yes—
87C22×C4×D11C2314yes—
88C22×C4×D11C2311yescharacteristic
89C11×C4⋊Q8C2×C418yescomplemented
90C2×C11×C4⋊C4C2312yes—
91C2×C11×C4⋊C4C2314yes—
92C2×C11×C4⋊C4C2312yes—
93C2×C11×C4⋊C4C2312yes—
94C2×C42×C11C2311yescharacteristic
95C2×C42×C11C2311yescharacteristic
96C2×C42×C11C2314yes—
97C2×C42×C11C2311yescharacteristic
98#67000C2×C4116yescomplemented
99C2×(C2×D11)⋊C4C2314yes—
100C2×(C2×D11)⋊C4C2314yes—
101C2×(C2×D11)⋊C4C2314yes—
102C2×(C2×D11)⋊C4C2314yes—
103C2×(C2×D11)⋊C4C2314yes—
104C2×(C2×D11)⋊C4C2312yes—
105C2×(C2×D11)⋊C4C2312yes—
106C2×(C2×D11)⋊C4C2314yes—
107C2×(C2×D11)⋊C4C2314yes—
108C2×(C2×D11)⋊C4C2314yes—
109C2×(C2×D11)⋊C4C2314yes—
110C2×C4⋊Dic11C2312yes—
111C2×C4⋊Dic11C2314yes—
112C2×Dic11⋊C4C2312yes—
113C2×Dic11⋊C4C2314yes—
114C2×Dic11⋊C4C2312yes—
115C2×Dic11⋊C4C2314yes—
116C2×C4×Dic11C2311yescharacteristic
117C2×C4×Dic11C2311yescharacteristic
118C2×C4×Dic11C2314yes—
119C2×C4×Dic11C2314yes—
120C2×C4×Dic11C2311yescharacteristic
121C2×C4×Dic11C2311yescharacteristic
122C2×C4×Dic11C2311yescharacteristic
123#66948C2×C4116yescomplemented
124#66948C2×C4116yescomplemented
125#66946—44no—
126#66946C2×C4116yescomplemented
127#66946C2×C4116yescomplemented
128#66945—44no—
129C4⋊C4×D11—48no—
130#66927C2×C418yes—
131C4×D44—44no—
132#66925—48no—
133C42×D11C2×C418yescomplemented
134C11×#95C2312yes—
135C11×#95C2312yes—
136C11×#95C2312yes—
137C11×#95C2312yes—
138C11×#95C2314yes—
139C11×#95C2314yes—
140C11⋊#95C2312yes—
141C11⋊#95C2312yes—
142C11⋊#95C2312yes—
143C11⋊#95C2312yes—
144C11⋊#95C2312yes—
145C11⋊#95C2312yes—
146C11⋊#95C2312yes—
147C11⋊#95C2312yes—
148C11⋊#95C2314yes—
149C11⋊#95C2314yes—
150#34367—111nomaximal, Sylow
151C23×D11C2411yescharacteristic
152C23×D11C2414yes—
153C2×Q8×C11—44no—
154C2×Q8×C11C22×C418yes—
155C22×C4×C11C2412yes—
156C22×C4×C11C2412yes—
157C22×C4×C11C2414yes—
158C22×C4×C11C2411yescharacteristic
159C22×C4×C11C2411yescharacteristic
160C22×C4×C11C2414yes—
161C22×C4×C11C2411yescharacteristic
162C22×Dic11C2412yes—
163C22×Dic11C2412yes—
164C22×Dic11C2414yes—
165C22×Dic11C2411yescharacteristic
166C22×Dic11C2411yescharacteristic
167C22×Dic11C2411yescharacteristic
168#4343—432no—
169C2×D44—24no—
170C2×D44—24no—
171C2×D44—44no—
172C2×D44—44no—
173C2×D44—44no—
174C2×D44—44no—
175C2×D44C22×C418yes—
176C2×D44C22×C418yes—
177C2×D44C22×C418yes—
178C2×D44C22×C4116yes—
179C2×C4×D11—28no—
180C2×C4×D11—28no—
181C2×C4×D11—24no—
182C2×C4×D11—22no—
183C2×C4×D11—22no—
184C2×C4×D11—42no—
185C2×C4×D11—44no—
186C2×C4×D11—44no—
187C2×C4×D11—44no—
188C2×C4×D11—44no—
189C2×C4×D11—44no—
190C2×C4×D11—44no—
191C2×C4×D11C22×C414yes—
192C2×C4×D11C22×C414yes—
193C2×C4×D11C22×C4116yes—
194C2×C4×D11C22×C414yes—
195C11×C4⋊C4—24no—
196C11×C4⋊C4—44no—
197C11×C4⋊C4C22×C418yes—
198C11×C4⋊C4C22×C418yes—
199C42×C11—22no—
200C42×C11—22no—
201C42×C11—44no—
202C42×C11C22×C414yes—
203(C2×D11)⋊C4—44no—
204(C2×D11)⋊C4—44no—
205(C2×D11)⋊C4—44no—
206(C2×D11)⋊C4—44no—
207(C2×D11)⋊C4—44no—
208(C2×D11)⋊C4—44no—
209(C2×D11)⋊C4—42no—
210(C2×D11)⋊C4—44no—
211(C2×D11)⋊C4—42no—
212(C2×D11)⋊C4C22×C4116yes—
213(C2×D11)⋊C4C22×C4116yes—
214C4⋊Dic11—24no—
215C4⋊Dic11—44no—
216Dic11⋊C4—24no—
217Dic11⋊C4—28no—
218Dic11⋊C4—24no—
219Dic11⋊C4—44no—
220C4×Dic11—22no—
221C4×Dic11—22no—
222C4×Dic11—44no—
223C4×Dic11—44no—
224C4×Dic11C22×C414yes—
225C4×Dic11C22×C414yes—
226C4×Dic11C22×C414yes—
227C2×#532—111no—
228#2377—114no—
229#2363—112no—
230#2363—112no—
231#2354—112no—
232#2352—114no—
233D4×C4⋊C4—112no—
234#2337—114no—
235#2331—111no—
236#2331D1111yescharacteristic, complemented
237#2328—111no—
238#2328—111no—
239#2311—114no—
240#2285—111no—
241#2283—111no—
242C23×C11C2511yescharacteristic
243C22×D11—21no—
244C22×D11—24no—
245C22×D11—24no—
246C22×D11—24no—
247C22×D11—22no—
248C22×D11—21no—
249C22×D11—24no—
250C22×D11—22no—
251C22×D11—24no—
252C22×D11—24no—
253C22×D11C23×C418yes—
254C22×D11C23×C412yes—
255Q8×C11—48no—
256C2×C4×C11—24no—
257C2×C4×C11—24no—
258C2×C4×C11—22no—
259C2×C4×C11—22no—
260C2×C4×C11—24no—
261C2×C4×C11—22no—
262C2×C4×C11—24no—
263C2×C4×C11—22no—
264C2×C4×C11—22no—
265C2×C4×C11—22no—
266C2×C4×C11—24no—
267C2×C4×C11—22no—
268C2×C4×C11—22no—
269C2×C4×C11C23×C412yes—
270C2×C4×C11C23×C412yes—
271C2×C4×C11C23×C418yes—
272C2×C4×C11C23×C412yes—
273C2×C4×C11H(5,𝔽2)12yes—
274C2×C4×C112−1+412yes—
275C2×Dic11—21no—
276C2×Dic11—21no—
277C2×Dic11—22no—
278C2×Dic11—22no—
279C2×Dic11—22no—
280C2×Dic11—24no—
281C2×Dic11—24no—
282C2×Dic11—22no—
283C2×Dic11—22no—
284C2×Dic11—24no—
285C2×Dic11—22no—
286C2×Dic11—22no—
287C2×Dic11C23×C412yes—
288C2×Dic11C23×C412yes—
289C2×Dic11C23×C412yes—
290C2×Dic11H(5,𝔽2)14yes—
291D44—216no—
292D44—48no—
293D44—48no—
294D44—416no—
295C4×D11—28no—
296C4×D11—232no—
297C4×D11—28no—
298C4×D11—44no—
299C4×D11—88no—
300C4×D11—88no—
301C22×#41—112no—
302#532—1116no—
303C2×C4⋊Q8C2×D1111yescharacteristic
304C2×#127—111no—
305C2×#124—112no—
306C2×#121—112no—
307C2×#121—112no—
308C2×C4×D4—112no—
309C2×C4×D4—112no—
310C2×C4×D4—112no—
311C2×C4×D4—112no—
312C2×#117—112no—
313C2×#117—114no—
314C22×C4⋊C4—114no—
315C22×C42—111no—
316#391—114no—
317#391C2×D1114yes—
318#390—114no—
319#390—112no—
320#390—114no—
321#390—112no—
322#389—112no—
323#389C2×D1112yes—
324#388—114no—
325#388—114no—
326#388—114no—
327#388—114no—
328#387—114no—
329#387C2×D1114yes—
330#386—114no—
331#386—114no—
332#386—114no—
333#386—114no—
334#386—114no—
335#386—114no—
336#385—114no—
337#385—114no—
338#384—112no—
339#383—112no—
340#382—111no—
341#382—112no—
342#382C2×D1111yescharacteristic
343#381—112no—
344#380—112no—
345#380—112no—
346#380—114no—
347#380—114no—
348#380—112no—
349C4×C4⋊C4—112no—
350C4×C4⋊C4—111no—
351C4×C4⋊C4—111no—
352C4×C4⋊C4—112no—
353C4×C4⋊C4—112no—
354C4×C4⋊C4C2×D1112yes—
355C4×C22⋊C4—114no—
356C4×C22⋊C4—111no—
357C4×C22⋊C4—111no—
358C4×C22⋊C4—114no—
359#376—111no—
360#376C2×D1111yescharacteristic
361C22×C11C24×C411yescharacteristic
362C22×C11C2×H(5,𝔽2)11yescharacteristic
363C22×C11C2×H(5,𝔽2)12yes—
364C22×C11C2×2−1+411yescharacteristic
365C22×C11#58512yes—
366C2×D11—28no—
367C2×D11—22no—
368C2×D11—22no—
369C2×D11—28no—
370C2×D11—22no—
371C2×D11—28no—
372C2×D11—42no—
373C2×D11—44no—
374C2×D11—44no—
375C2×D11—44no—
376C2×D11—44no—
377C2×D11—42no—
378C4×C11—22no—
379C4×C11—28no—
380C4×C11—22no—
381C4×C11—44no—
382C4×C11—42no—
383C4×C11—42no—
384C4×C11#52014yes—
385Dic11—22no—
386Dic11—22no—
387Dic11—42no—
388Dic11—44no—
389Dic11—42no—
390Dic11#51814yes—
391C2×#41—1116no—
392C2×#41—448no—
393C22×Q8C22×D1112yes—
394C22×D4—112no—
395C22×D4—112no—
396C22×D4—114no—
397C22×D4—112no—
398C23×C4—114no—
399C23×C4—111no—
400C23×C4—111no—
401C23×C4—111no—
402C23×C4—114no—
403C23×C4—114no—
404C4⋊Q8C4×D1118yescomplemented
405#127—118no—
406#124—1116no—
407#121—1116no—
408#121—1116no—
409C4×D4—1116no—
410C4×D4—1116no—
411C4×D4—444no—
412C4×D4—444no—
413#117—448no—
414#117—444no—
415C2×C4⋊C4—112no—
416C2×C4⋊C4—114no—
417C2×C4⋊C4—112no—
418C2×C4⋊C4—112no—
419C2×C4⋊C4—114no—
420C2×C4⋊C4—114no—
421C2×C4⋊C4—448no—
422C2×C4⋊C4C22×D1112yes—
423C2×C4⋊C4C22×D1114yes—
424C2×C4⋊C4C22×D1112yes—
425C2×C4⋊C4C22×D1112yes—
426C2×C22⋊C4—114no—
427C2×C22⋊C4—114no—
428C2×C22⋊C4—114no—
429C2×C22⋊C4—114no—
430C2×C22⋊C4—114no—
431C2×C22⋊C4—114no—
432C2×C22⋊C4—114no—
433C2×C22⋊C4—112no—
434C2×C22⋊C4—112no—
435C2×C22⋊C4—114no—
436C2×C22⋊C4—114no—
437C2×C42—111no—
438C2×C42—111no—
439C2×C42—111no—
440C2×C42—118no—
441C2×C42—111no—
442C2×C42—111no—
443C2×C42—114no—
444C2×C42—114no—
445C2×C42C22×D1111yescharacteristic
446C2×C42C22×D1111yescharacteristic
447C2×C42C22×D1114yes—
448C2×C42C22×D1111yescharacteristic
449#95—112no—
450#95—112no—
451#95—112no—
452#95—112no—
453#95—114no—
454#95—112no—
455#95—112no—
456#95—114no—
457#95—112no—
458#95—112no—
459#95C22×D1112yes—
460#95C22×D1112yes—
461#95C22×D1114yes—
462#95C22×D1112yes—
463#95C22×D1112yes—
464#95C22×D1114yes—
465C2×C11C2×#51811yescharacteristic
466C2×C11C2×#52011yescharacteristic
467C2×C11#342811yescharacteristic
468C2×C11#354612yes—
469C2×C11#354712yes—
470D11—48no—
471D11—42no—
472C24—114no—
473C24—111no—
474#41—4432no—
475C2×Q8—44no—
476C2×Q8C2×C4×D1118yes—
477C2×D4—118no—
478C2×D4—118no—
479C2×D4—1116no—
480C2×D4—118no—
481C2×D4—224no—
482C2×D4—224no—
483C2×D4—444no—
484C2×D4—444no—
485C2×D4—444no—
486C2×D4—444no—
487C22×C4—111no—
488C22×C4—111no—
489C22×C4—111no—
490C22×C4—114no—
491C22×C4—114no—
492C22×C4—1116no—
493C22×C4—114no—
494C22×C4—112no—
495C22×C4—112no—
496C22×C4—114no—
497C22×C4—228no—
498C22×C4—228no—
499C22×C4—224no—
500C22×C4—222no—
501C22×C4—222no—
502C22×C4—442no—
503C22×C4—444no—
504C22×C4—444no—
505C22×C4—444no—
506C22×C4—444no—
507C22×C4—444no—
508C22×C4—444no—
509C22×C4C23×D1112yes—
510C22×C4C23×D1112yes—
511C22×C4C23×D1114yes—
512C22×C4C23×D1111yescharacteristic
513C22×C4C23×D1111yescharacteristic
514C22×C4C23×D1114yes—
515C22×C4C23×D1111yescharacteristic
516C4⋊C4—24no—
517C4⋊C4—44no—
518C4⋊C4—224no—
519C4⋊C4—228no—
520C4⋊C4—224no—
521C4⋊C4—224no—
522C4⋊C4—444no—
523C4⋊C4—444no—
524C4⋊C4C2×C4×D1118yes—
525C4⋊C4C2×C4×D1118yes—
526C22⋊C4—1116no—
527C22⋊C4—1116no—
528C22⋊C4—444no—
529C22⋊C4—444no—
530C22⋊C4—444no—
531C22⋊C4—444no—
532C22⋊C4—444no—
533C22⋊C4—444no—
534C22⋊C4—442no—
535C22⋊C4—444no—
536C22⋊C4—442no—
537C42—22no—
538C42—22no—
539C42—44no—
540C42—114no—
541C42—114no—
542C42—114no—
543C42—222no—
544C42—222no—
545C42—444no—
546C42—444no—
547C42C2×C4×D1114yes—
548C11#3436711yescharacteristic, complemented, Sylow
549C23—118no—
550C23—112no—
551C23—221no—
552C23—224no—
553C23—221no—
554C23—224no—
555C23—224no—
556C23—224no—
557C23—222no—
558C23—222no—
559C23—224no—
560C23—224no—
561C23C24×D1111yescharacteristic
562Q8—48no—
563D4—2216no—
564D4—448no—
565D4—4416no—
566D4—448no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—112no—
581C2×C4—114no—
582C2×C4—112no—
583C2×C4—112no—
584C2×C4—228no—
585C2×C4—2232no—
586C2×C4—221no—
587C2×C4—221no—
588C2×C4—222no—
589C2×C4—228no—
590C2×C4—222no—
591C2×C4—222no—
592C2×C4—224no—
593C2×C4—222no—
594C2×C4—224no—
595C2×C4—222no—
596C2×C4—222no—
597C2×C4—224no—
598C2×C4—222no—
599C2×C4—444no—
600C2×C4—888no—
601C2×C4—888no—
602C2×C4C22×C4×D1112yes—
603C2×C4C22×C4×D1112yes—
604C2×C4C22×C4×D1118yes—
605C2×C4C22×C4×D1112yes—
606C2×C4#6704012yes—
607C2×C4#6704212yes—
608C22—222no—
609C22—228no—
610C22—222no—
611C22—228no—
612C22—222no—
613C22—228no—
614C22—444no—
615C22—442no—
616C22—444no—
617C22—442no—
618C22—444no—
619C22—444no—
620C22C23×C4×D1111yescharacteristic
621C22C2×#6704011yescharacteristic
622C22C2×#6704211yescharacteristic
623C22D11×H(5,𝔽2)12yes—
624C22#1062345712yes—
625C4—22no—
626C4—28no—
627C4—22no—
628C4—42no—
629C4—44no—
630C4—42no—
631C4—114no—
632C4—222no—
633C4—222no—
634C4—442no—
635C4—444no—
636C4—442no—
637C4#1062310514yes—
638C2—442no—
639C2—448no—
640C2D11×#51811yescharacteristic
641C2C2×#1062310511yescharacteristic
642C2#1308181811yescharacteristic
643C2#1308298512yes—
644C2#1308300212yes—
subgroupG/H
center, FrattiniC23C24×D11
commutatorC22×C11C24×C4
FittingC11×#2331C2
socle, cosocleC23×C11C25
prime radicalC2×#10623256C2
hypercenter#2331D11
hypocenterC11#34367
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C23 ◁ #2331

Complements

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

NH ≅ G/Ntype
2. #13071771C2semidirect
3. #13071656C2semidirect
4. #13071653C2semidirect
5. #13071561C2semidirect
6. #13071540C2semidirect
7. #13071517C2semidirect
8. #13071428C2semidirect
9. #13071365C2semidirect
10. #13071341C2semidirect
11. #13071339C2semidirect
12. #13071207C2semidirect
13. #13071022C2semidirect
14. #13071008C2semidirect
15. C11×#2331C2semidirect
19. #10623256C4semidirect
78. C2×#4343C2×C4semidirect
89. C11×C4⋊Q8C2×C4semidirect
98. #67000C2×C4semidirect
123. #66948C2×C4semidirect
124. #66948C2×C4semidirect
126. #66946C2×C4semidirect
127. #66946C2×C4semidirect
133. C42×D11C2×C4semidirect
236. #2331D11semidirect
404. C4⋊Q8C4×D11semidirect
548. C11#34367semidirect

Maximal subgroups

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

subgroupindex
C2×#106232562
#130717712
#130716562
#130716532
#130715612
#130715402
#130715172
#130714282
#130713652
#130713412
#130713392
#130712072
#130710222
#130710082
C11×#23312
#3436711