Finite Group Databasebeta

← #14788886

Subgroups — #14788886

Order 2816.

1482 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623008C211yesmaximal, characteristic
2C2×#10622641C212yesmaximal, complemented
3C2×#10622660C212yesmaximal, complemented
4#13069862C211yesmaximal, characteristic, complemented
5#13069850C211yesmaximal, characteristic, complemented
6#13069715C211yesmaximal, characteristic, complemented
7#13069703C211yesmaximal, characteristic, complemented
8#13069548C212yesmaximal, complemented
9#13068596C211yesmaximal, characteristic, complemented
10#13068554C212yesmaximal, complemented
11C4×C11×C22⋊Q8C211yesmaximal, characteristic, complemented
12C22×#4339C2212yes—
13C2×C11×C22⋊Q8C2211yescharacteristic
14C2×C4×Q8×C11C2211yescharacteristic
15C22×C42×C11C2211yescharacteristic
16C22×#4326C2212yes—
17C22×#4325C2212yes—
18C2×#66983—21no—
19C2×#66983—21no—
20#10623008—28no—
21C2×#66928C2212yes—
22C2×#66927C2211yescharacteristic
23C2×C4×D44—21no—
24C2×C4×D44—21no—
25C2×#66922C2211yescharacteristic
26C11×#391C2211yescharacteristic
27C11×#389C2211yescharacteristic
28C11×#387—21no—
29C11×#385—21no—
30C11×#380C2211yescharacteristic
31C4×C11×C4⋊C4—21no—
32C4×C11×C4⋊C4C2211yescharacteristic
33C4×C11×C22⋊C4—21no—
34C2×C22⋊(C11⋊C8)C2212yes—
35#10622660—28no—
36#10622641—28no—
37C2×#66874C2212yes—
38C2×#66874C2212yes—
39C2×#66874C2212yes—
40C2×#66873C2212yes—
41C2×#66873C2212yes—
42C2×#66873C2212yes—
43C2×C4⋊(C11⋊C8)C2212yes—
44#10622162C2211yescharacteristic
45#10622162C2211yescharacteristic
46C22×Q8×C11C2311yescharacteristic
47C23×C4×C11D411yescharacteristic
48C23×C4×C11D411yescharacteristic
49C23×C4×C11C2311yescharacteristic
50C22×C22⋊D11—22no—
51C2×#4339—24no—
52C2×#4339—24no—
53C2×#4339—24no—
54C2×#4339—24no—
55C2×#4339—42no—
56C2×#4339—42no—
57C22×D44—21no—
58C22×D44—21no—
59C22×D44C2312yes—
60C22×C4×D11—22no—
61C22×Dic22C2312yes—
62C11×C22⋊Q8—22no—
63C11×C22⋊Q8D414yescomplemented
64C4×Q8×C11—24no—
65C2×C11×C4⋊C4—21no—
66C2×C11×C4⋊C4—21no—
67C2×C11×C4⋊C4—21no—
68C2×C11×C4⋊C4C2311yescharacteristic
69C2×C11×C4⋊C4C2311yescharacteristic
70C2×C11×C4⋊C4C2311yescharacteristic
71C2×C11×C22⋊C4—21no—
72C2×C11×C22⋊C4—21no—
73C2×C42×C11—24no—
74C2×C42×C11—21no—
75C2×C42×C11—21no—
76C2×C42×C11D411yescharacteristic
77C2×C42×C11D411yescharacteristic
78C2×C42×C11C2311yescharacteristic
79C2×C42×C11C2311yescharacteristic
80C2×#4326—24no—
81C2×#4326—24no—
82C2×#4326—42no—
83C2×#4325—24no—
84C2×#4325—24no—
85C2×#4325—42no—
86#66983—28no—
87#66983—28no—
88C2×(C2×D11)⋊C4—22no—
89C2×(C2×D11)⋊C4—22no—
90C2×C4⋊Dic11—21no—
91C2×C4⋊Dic11—21no—
92C22×C11⋊C8C2312yes—
93C22×C11⋊C8C2312yes—
94#66928—24no—
95#66928D418yes—
96#66927—24no—
97C4×D44—28no—
98C4×D44—28no—
99#66922—24no—
100C11×#95—21no—
101C11×#95—21no—
102C11×#95—21no—
103C11×#95—21no—
104C11×#95—21no—
105C22⋊(C11⋊C8)—24no—
106#66874—24no—
107#66874—24no—
108#66874—42no—
109#66873—24no—
110#66873—24no—
111#66873—42no—
112C4⋊(C11⋊C8)—24no—
113#24216—111nomaximal, Sylow
114C24×C11C2×D411yescharacteristic
115C23×D11—22no—
116C2×Q8×C11—21no—
117C2×Q8×C11—22no—
118C2×Q8×C11—22no—
119C2×Q8×C11C2×D412yes—
120C22×C4×C11—22no—
121C22×C4×C11—22no—
122C22×C4×C11—21no—
123C22×C4×C11—21no—
124C22×C4×C11—22no—
125C22×C4×C11—22no—
126C22×C4×C11—22no—
127C22×C4×C11—22no—
128C22×C4×C11—21no—
129C22×C4×C11—21no—
130C22×C4×C11—41no—
131C22×C4×C11—41no—
132C22×C4×C11C2×D411yescharacteristic
133C22×C4×C11C2×D411yescharacteristic
134C22×C4×C11C2×D411yescharacteristic
135C22×C4×C11C2×D412yes—
136C22×C4×C11C2×D411yescharacteristic
137C22×C4×C11C2×D411yescharacteristic
138C22×C4×C11#4112yes—
139C22×C4×C11C2411yescharacteristic
140C2×C22⋊D11—28no—
141C2×C22⋊D11—44no—
142C2×C22⋊D11—44no—
143C22×Dic11—22no—
144#4339—48no—
145#4339—48no—
146#4339—48no—
147#4339—48no—
148C2×D44—24no—
149C2×D44—24no—
150C2×D44—22no—
151C2×D44—22no—
152C2×D44—22no—
153C2×D44—22no—
154C2×D44—22no—
155C2×D44—42no—
156C2×D44—42no—
157C2×D44—42no—
158C2×D44—42no—
159C2×D44C2×D414yes—
160C2×C4×D11—28no—
161C2×C4×D11—44no—
162C2×C4×D11—44no—
163C2×Dic22—22no—
164C2×Dic22—22no—
165C2×Dic22—22no—
166C2×Dic22—22no—
167C2×Dic22—22no—
168C2×Dic22C2×D414yes—
169C11×C4⋊C4—24no—
170C11×C4⋊C4—21no—
171C11×C4⋊C4—22no—
172C11×C4⋊C4—22no—
173C11×C4⋊C4—42no—
174C11×C4⋊C4—42no—
175C11×C4⋊C4C2×D412yes—
176C11×C22⋊C4—24no—
177C11×C22⋊C4—42no—
178C42×C11—22no—
179C42×C11—22no—
180C42×C11—22no—
181C42×C11—42no—
182C42×C11—42no—
183C42×C11C2×D414yes—
184#4326—48no—
185#4325—48no—
186(C2×D11)⋊C4—28no—
187(C2×D11)⋊C4—28no—
188C4⋊Dic11—24no—
189C4⋊Dic11—24no—
190C2×C11⋊C8—22no—
191C2×C11⋊C8—22no—
192C2×C11⋊C8—22no—
193C2×C11⋊C8—22no—
194C2×C11⋊C8—22no—
195C2×C11⋊C8—22no—
196C2×#532—111no—
197C2×#464—112no—
198C2×#452—112no—
199C4×C22⋊Q8D1111yescharacteristic, complemented
200#2043—111no—
201#2041—111no—
202#2016—111no—
203#2014—111no—
204#1983—112no—
205#1797—111no—
206#1788—112no—
207C23×C11—21no—
208C23×C11—21no—
209C23×C11—21no—
210C23×C11—21no—
211C23×C11—21no—
212C23×C11—21no—
213C23×C11#12112yes—
214C23×C11C22×D411yescharacteristic
215C22×D11—24no—
216C22×D11—42no—
217C22×D11—42no—
218C22×D11—42no—
219C22×D11—42no—
220C22×D11—42no—
221C22×D11—42no—
222Q8×C11—24no—
223Q8×C11—42no—
224C2×C4×C11—21no—
225C2×C4×C11—21no—
226C2×C4×C11—21no—
227C2×C4×C11—21no—
228C2×C4×C11—21no—
229C2×C4×C11—21no—
230C2×C4×C11—21no—
231C2×C4×C11—21no—
232C2×C4×C11—21no—
233C2×C4×C11—21no—
234C2×C4×C11—24no—
235C2×C4×C11—24no—
236C2×C4×C11—22no—
237C2×C4×C11—22no—
238C2×C4×C11—44no—
239C2×C4×C11—44no—
240C2×C4×C11—41no—
241C2×C4×C11—41no—
242C2×C4×C11—41no—
243C2×C4×C11—41no—
244C2×C4×C11—41no—
245C2×C4×C11—41no—
246C2×C4×C11—41no—
247C2×C4×C11—41no—
248C2×C4×C11—41no—
249C2×C4×C11—41no—
250C2×C4×C11#12012yes—
251C2×C4×C11#12012yes—
252C2×C4×C11#12112yes—
253C2×C4×C11#13512yes—
254C2×C4×C11#13712yes—
255C2×C4×C11C22×D411yescharacteristic
256C2×C4×C11C22×D412yes—
257C2×C4×C11C2×#4111yescharacteristic
258C2×C4×C11H(5,𝔽2)12yes—
259C22⋊D11—416no—
260C2×Dic11—24no—
261C2×Dic11—42no—
262C2×Dic11—42no—
263D44—24no—
264D44—24no—
265D44—48no—
266D44—48no—
267D44—42no—
268D44—42no—
269C4×D11—416no—
270Dic22—24no—
271Dic22—24no—
272Dic22—42no—
273Dic22—42no—
274C11⋊C8—42no—
275C11⋊C8—42no—
276C22×#41—112no—
277C22×Q16—112no—
278C22×SD16—112no—
279#532—228no—
280C2×C4⋊Q8—111no—
281C2×#127—111no—
282C2×#124—112no—
283C2×C22⋊Q8C2×D1111yescharacteristic
284C2×#121—221no—
285C2×#121—221no—
286C2×C4×Q8C2×D1111yescharacteristic
287C2×C4×D4—221no—
288C2×C4×D4—221no—
289C22×C42C2×D1111yescharacteristic
290#464—228no—
291#452—228no—
292C2×C4⋊C8—112no—
293C2×#103—112no—
294C2×#103—112no—
295C2×#103—112no—
296C2×#102—112no—
297C2×#102—112no—
298C2×#102—112no—
299C2×C22⋊C8—112no—
300#391C2×D1111yescharacteristic
301#389C2×D1111yescharacteristic
302#387—21no—
303#385—21no—
304#380C2×D1111yescharacteristic
305C4×C4⋊C4—21no—
306C4×C4⋊C4C2×D1111yescharacteristic
307C4×C22⋊C4—21no—
308#336—111no—
309#336—111no—
310C22×C11—22no—
311C22×C11—22no—
312C22×C11—22no—
313C22×C11—41no—
314C22×C11—41no—
315C22×C11—41no—
316C22×C11—41no—
317C22×C11C2×#12011yescharacteristic
318C22×C11C2×#12111yescharacteristic
319C22×C11#53511yescharacteristic
320C22×C11C2×#13511yescharacteristic
321C22×C11C2×#13711yescharacteristic
322C22×C11#57511yescharacteristic
323C22×C11#57511yescharacteristic
324C2×D11—44no—
325C2×D11—44no—
326C2×D11—44no—
327C2×D11—82no—
328C2×D11—82no—
329C2×D11—82no—
330C2×D11—82no—
331C4×C11—22no—
332C4×C11—22no—
333C4×C11—22no—
334C4×C11—22no—
335C4×C11—22no—
336C4×C11—42no—
337C4×C11—42no—
338C4×C11—81no—
339C4×C11—81no—
340C4×C11D4212yes—
341C4×C11#54612yes—
342Dic11—44no—
343C2×#41—224no—
344C2×#41—224no—
345C2×#41—224no—
346C2×#41—224no—
347C2×#41—442no—
348C2×#41—442no—
349C22×Q8—112no—
350C22×Q8C22×D1111yescharacteristic
351C22×D4—112no—
352C22×D4—222no—
353C22×D4—221no—
354C22×D4—221no—
355C23×C4—222no—
356C23×C4C22⋊D1111yescharacteristic
357C23×C4C22⋊D1111yescharacteristic
358C23×C4C22×D1111yescharacteristic
359C2×Q16—224no—
360C2×Q16—224no—
361C2×Q16—442no—
362C2×SD16—224no—
363C2×SD16—224no—
364C2×SD16—442no—
365C22×C8—112no—
366C22×C8—112no—
367C4⋊Q8—224no—
368#127—224no—
369#124—118no—
370#124—224no—
371C22⋊Q8—22no—
372C22⋊Q8D4414yescomplemented
373#121—228no—
374#121—228no—
375C4×Q8—24no—
376C4×D4—228no—
377C4×D4—228no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—221no—
382C2×C4⋊C4—221no—
383C2×C4⋊C4C22×D1111yescharacteristic
384C2×C4⋊C4C22×D1111yescharacteristic
385C2×C4⋊C4C22×D1111yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—222no—
389C2×C22⋊C4—222no—
390C2×C42—24no—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42C22⋊D1111yescharacteristic
394C2×C42C22⋊D1111yescharacteristic
395C2×C42C22×D1111yescharacteristic
396C2×C42C22×D1111yescharacteristic
397C4⋊C8—224no—
398#103—224no—
399#103—224no—
400#103—442no—
401#102—224no—
402#102—224no—
403#102—442no—
404C22⋊C8—224no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—21no—
410C2×C11—42no—
411C2×C11#240811yescharacteristic
412C2×C11#300211yescharacteristic
413C2×C11#300711yescharacteristic
414C2×C11#303511yescharacteristic
415C2×C11#303711yescharacteristic
416C2×C11#321011yescharacteristic
417C2×C11#321911yescharacteristic
418D11—84no—
419C24—222no—
420C24C2×C22⋊D1111yescharacteristic
421#41—448no—
422#41—448no—
423#41—448no—
424#41—448no—
425C2×Q8—21no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—114no—
429C2×Q8—222no—
430C2×Q8—222no—
431C2×Q8—222no—
432C2×Q8—222no—
433C2×Q8—222no—
434C2×Q8C2×D4412yes—
435C2×D4—114no—
436C2×D4—224no—
437C2×D4—224no—
438C2×D4—228no—
439C2×D4—222no—
440C2×D4—222no—
441C2×D4—222no—
442C2×D4—222no—
443C2×D4—222no—
444C2×D4—442no—
445C2×D4—442no—
446C2×D4—444no—
447C2×D4—444no—
448C2×D4—442no—
449C2×D4—442no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—228no—
463C22×C4—222no—
464C22×C4—444no—
465C22×C4—444no—
466C22×C4C2×D4412yes—
467C22×C4#433912yes—
468C22×C4C2×C22⋊D1111yescharacteristic
469C22×C4C2×C22⋊D1111yescharacteristic
470C22×C4C2×C22⋊D1111yescharacteristic
471C22×C4C2×C22⋊D1111yescharacteristic
472C22×C4C2×C22⋊D1111yescharacteristic
473C22×C4C23×D1111yescharacteristic
474Q16—448no—
475SD16—448no—
476C2×C8—222no—
477C2×C8—222no—
478C2×C8—222no—
479C2×C8—222no—
480C2×C8—222no—
481C2×C8—222no—
482C4⋊C4—24no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—224no—
489C4⋊C4—224no—
490C4⋊C4C2×D4412yes—
491C22⋊C4—24no—
492C22⋊C4—42no—
493C22⋊C4—228no—
494C22⋊C4—228no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—42no—
499C42—42no—
500C42D4×D1114yes—
501C11#2421611yescharacteristic, complemented, Sylow
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—224no—
509C23—442no—
510C23—442no—
511C23—442no—
512C23—442no—
513C23—442no—
514C23—442no—
515C23#6698312yes—
516C23C22×C22⋊D1111yescharacteristic
517Q8—24no—
518Q8—42no—
519Q8—224no—
520Q8—224no—
521Q8—442no—
522Q8—442no—
523D4—224no—
524D4—224no—
525D4—448no—
526D4—448no—
527D4—4416no—
528D4—442no—
529D4—442no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—224no—
557C2×C4—4416no—
558C2×C4—442no—
559C2×C4—442no—
560C2×C4#6698312yes—
561C2×C4#6699012yes—
562C2×C4#6699012yes—
563C2×C4#6699512yes—
564C2×C4#6700312yes—
565C2×C4C22×D4411yescharacteristic
566C2×C4C2×#433911yescharacteristic
567C2×C4C2×D4×D1112yes—
568C2×C4#6704212yes—
569C8—442no—
570C8—442no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—444no—
579C22—444no—
580C22—444no—
581C22—882no—
582C22—882no—
583C22—882no—
584C22—882no—
585C22C2×#6698311yescharacteristic
586C22C2×#6699011yescharacteristic
587C22C2×#6699511yescharacteristic
588C22C2×#6700311yescharacteristic
589C22#1062335311yescharacteristic
590C22#1062335311yescharacteristic
591C22#1062336111yescharacteristic
592C4—22no—
593C4—22no—
594C4—22no—
595C4—22no—
596C4—22no—
597C4—42no—
598C4—42no—
599C4—81no—
600C4—81no—
601C4—444no—
602C4D4×D4412yes—
603C4#1062308112yes—
604C2—42no—
605C2—884no—
606C2#1307197211yescharacteristic
607C2#1307783211yescharacteristic
608C2#1307787411yescharacteristic
609C2#1307807111yescharacteristic
610C2#1307808811yescharacteristic
611C2#1307956811yescharacteristic
612C2#1307964911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D11
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC4×C11×C22⋊Q8C2
socleC23×C11C22×D4
prime radicalC2×#10623008C2
hypercenterC4×C22⋊Q8D11
hypocenterC11#24216
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#10622641C2semidirect
3. C2×#10622660C2semidirect
4. #13069862C2semidirect
5. #13069850C2semidirect
6. #13069715C2semidirect
7. #13069703C2semidirect
8. #13069548C2semidirect
9. #13068596C2semidirect
10. #13068554C2semidirect
11. C4×C11×C22⋊Q8C2semidirect
63. C11×C22⋊Q8D4semidirect
199. C4×C22⋊Q8D11semidirect
372. C22⋊Q8D44semidirect
501. C11#24216semidirect

Maximal subgroups

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

subgroupindex
C2×#106230082
C2×#106226412
C2×#106226602
#130698622
#130698502
#130697152
#130697032
#130695482
#130685962
#130685542
C4×C11×C22⋊Q82
#2421611