Finite Group Databasebeta

← #14781755

Subgroups — #14781755

Order 2816.

1570 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622732C211yesmaximal, characteristic, complemented
2C2×#10622846C211yesmaximal, characteristic, complemented
3C22×#66896C211yesmaximal, characteristic, complemented
4#13071208C211yesmaximal, characteristic, complemented
5#13071198C211yesmaximal, characteristic, complemented
6#13069929C214yesmaximal, complemented
7#13069924C214yesmaximal, complemented
8#13068336C211yesmaximal, characteristic, complemented
9C22×C11×AGL(1,ℤ/8)C211yesmaximal, characteristic
10C22×C11×#41C2211yescharacteristic
11C23×D4×C11C2211yescharacteristic
12C2×C11×AGL(1,ℤ/8)—24no—
13C2×C11×AGL(1,ℤ/8)—24no—
14C2×C11×AGL(1,ℤ/8)—24no—
15C22×C11×SD16C2212yes—
16C22×C11×D8C2212yes—
17C22×C11×M16C2211yescharacteristic
18C2×#66979C2211yescharacteristic
19C2×#66979C2211yescharacteristic
20C22×C4⋊Dic11C2211yescharacteristic
21C23×C11⋊C8C2211yescharacteristic
22#10622860—22no—
23#10622846—24no—
24#10622843C2212yescomplemented
25C2×#66899C2212yescomplemented
26#10622834—22no—
27#10622834C2214yes—
28#10622834C2212yescomplemented
29#10622833—22no—
30C2×#66896—24no—
31C2×#66896C2214yes—
32C2×#66896C2212yescomplemented
33#10622825—22no—
34#10622732—24no—
35C2×Dic11⋊C8C2214yescomplemented
36#10622189C2214yescomplemented
37C25×C11—21no—
38C2×C11×#41—22no—
39C2×C11×#41—22no—
40C2×C11×#41—22no—
41C2×C11×#41—22no—
42C2×C11×#41—22no—
43C2×C11×#41D414yes—
44C22×Q8×C11C2311yescharacteristic
45C22×D4×C11—21no—
46C22×D4×C11—22no—
47C22×D4×C11—22no—
48C22×D4×C11—21no—
49C22×D4×C11—22no—
50C22×D4×C11—22no—
51C22×D4×C11—22no—
52C22×D4×C11D414yes—
53C22×D4×C11C2312yes—
54C22×D4×C11C2311yescharacteristic
55C23×C4×C11—21no—
56C23×C4×C11C2311yescharacteristic
57C23×Dic11—21no—
58C11×AGL(1,ℤ/8)—232no—
59C2×C11×SD16—24no—
60C2×C11×SD16—24no—
61C2×C11×SD16—24no—
62C2×C11×D8—24no—
63C2×C11×D8—24no—
64C2×C11×D8—24no—
65C2×C11×M16—22no—
66C2×C11×M16—22no—
67C2×C11×M16D414yes—
68C22×C8×C11C2312yes—
69C2×C22⋊Dic11—21no—
70C2×C22⋊Dic11—24no—
71C2×C22⋊Dic11—24no—
72C2×C22⋊Dic11—24no—
73C2×C22⋊Dic11D412yes—
74#66979—22no—
75#66979—22no—
76#66979—22no—
77#66979—22no—
78C2×C4⋊Dic11—22no—
79C2×C4⋊Dic11—22no—
80C2×C4⋊Dic11—21no—
81C2×C4⋊Dic11D412yes—
82C2×C4⋊Dic11C2312yes—
83C2×C4⋊Dic11C2312yes—
84C2×C4×Dic11—21no—
85C2×C4×Dic11C2312yes—
86C22×C11⋊C8—22no—
87C22×C11⋊C8—22no—
88C22×C11⋊C8—22no—
89C22×C11⋊C8C2312yes—
90#66899—24no—
91#66896—28no—
92#66896—24no—
93C11⋊#95—24no—
94Dic11⋊C8—28no—
95#23544—111nomaximal, Sylow
96C24×C11—21no—
97C24×C11—22no—
98C24×C11—24no—
99C24×C11—42no—
100C24×C11—42no—
101C24×C11—82no—
102C24×C11—82no—
103C24×C11C2×D411yescharacteristic
104C11×#41—28no—
105C11×#41—28no—
106C11×#41—44no—
107C11×#41—44no—
108C2×Q8×C11—21no—
109C2×Q8×C11—21no—
110C2×Q8×C11—21no—
111C2×Q8×C11—21no—
112C2×Q8×C11—21no—
113C2×Q8×C11C2×D412yes—
114C2×D4×C11—24no—
115C2×D4×C11—28no—
116C2×D4×C11—24no—
117C2×D4×C11—28no—
118C2×D4×C11—21no—
119C2×D4×C11—21no—
120C2×D4×C11—22no—
121C2×D4×C11—22no—
122C2×D4×C11—22no—
123C2×D4×C11—22no—
124C2×D4×C11—21no—
125C2×D4×C11—21no—
126C2×D4×C11—22no—
127C2×D4×C11—21no—
128C2×D4×C11—42no—
129C2×D4×C11—42no—
130C2×D4×C11—42no—
131C2×D4×C11—42no—
132C2×D4×C11—44no—
133C2×D4×C11—44no—
134C2×D4×C11C2×D412yes—
135C2×D4×C11C2×D414yes—
136C22×C4×C11—21no—
137C22×C4×C11—24no—
138C22×C4×C11—42no—
139C22×C4×C11—42no—
140C22×C4×C11C2×D412yes—
141C22×C4×C11C2×D412yes—
142C22×C4×C11C2×D412yes—
143C22×C4×C11C2×D411yescharacteristic
144C22×C4×C11#4112yes—
145C22×C4×C11#4112yes—
146C22×C4×C11#4112yes—
147C22×C4×C11C2411yescharacteristic
148C22×Dic11—22no—
149C22×Dic11—22no—
150C22×Dic11—42no—
151C22×Dic11—82no—
152C22×Dic11C2×D414yes—
153C11×SD16—216no—
154C11×D8—216no—
155C11×M16—28no—
156C2×C8×C11—22no—
157C2×C8×C11—22no—
158C2×C8×C11C2×D414yes—
159C22⋊Dic11—24no—
160C22⋊Dic11—42no—
161C22⋊Dic11—84no—
162C22⋊Dic11—84no—
163C22⋊Dic11—84no—
164C4⋊Dic11—24no—
165C4⋊Dic11—22no—
166C4⋊Dic11—22no—
167C4⋊Dic11—22no—
168C4⋊Dic11—22no—
169C4⋊Dic11—42no—
170C4×Dic11—22no—
171C4×Dic11—22no—
172C4×Dic11—42no—
173C2×C11⋊C8—22no—
174C2×C11⋊C8—22no—
175C2×C11⋊C8—22no—
176C2×C11⋊C8—44no—
177C22×AGL(1,ℤ/8)D1111yescharacteristic
178C2×#424—111no—
179C2×#416—111no—
180C22×#102—111no—
181#2311—111no—
182#2309—111no—
183#2055—114no—
184#2054—114no—
185#1736—111no—
186C23×C11—22no—
187C23×C11—22no—
188C23×C11—22no—
189C23×C11—24no—
190C23×C11—41no—
191C23×C11—41no—
192C23×C11—42no—
193C23×C11—42no—
194C23×C11—42no—
195C23×C11—44no—
196C23×C11—44no—
197C23×C11—42no—
198C23×C11—48no—
199C23×C11—84no—
200C23×C11—84no—
201C23×C11—88no—
202C23×C11—88no—
203C23×C11#12112yes—
204C23×C11#12112yes—
205C23×C11#12112yes—
206C23×C11C22×D411yescharacteristic
207Q8×C11—22no—
208Q8×C11—22no—
209Q8×C11—41no—
210Q8×C11—41no—
211D4×C11—22no—
212D4×C11—24no—
213D4×C11—24no—
214D4×C11—22no—
215D4×C11—48no—
216D4×C11—42no—
217D4×C11—41no—
218D4×C11—42no—
219D4×C11—41no—
220D4×C11—48no—
221C2×C4×C11—22no—
222C2×C4×C11—41no—
223C2×C4×C11—41no—
224C2×C4×C11—48no—
225C2×C4×C11#12014yes—
226C2×C4×C11#12112yes—
227C2×C4×C11#12112yes—
228C2×C4×C11#12112yes—
229C2×C4×C11#12314yes—
230C2×C4×C11#12314yes—
231C2×C4×C11#12414yes—
232C2×C4×C11C22×D411yescharacteristic
233C2×C4×C11C22×D411yescharacteristic
234C2×C4×C11C22×D411yescharacteristic
235C2×C4×C11C2×#4111yescharacteristic
236C2×C4×C11C2×#4111yescharacteristic
237C2×C4×C11C2×#4111yescharacteristic
238C2×Dic11—24no—
239C2×Dic11—24no—
240C2×Dic11—24no—
241C2×Dic11—42no—
242C2×Dic11—42no—
243C2×Dic11—42no—
244C2×Dic11—44no—
245C2×Dic11—42no—
246C2×Dic11—84no—
247C8×C11—24no—
248C11⋊C8—42no—
249C22×#41C2×D1111yescharacteristic
250C23×D4C2×D1111yescharacteristic
251C2×AGL(1,ℤ/8)—24no—
252C2×AGL(1,ℤ/8)—24no—
253C2×AGL(1,ℤ/8)—24no—
254C22×SD16C2×D1112yes—
255C22×D8C2×D1112yes—
256C22×M16C2×D1111yescharacteristic
257C23×C8—111no—
258C2×#117—111no—
259C2×#117—111no—
260C22×C4⋊C4—111no—
261#424—224no—
262C2×C4⋊C8—114no—
263#416—224no—
264C2×#103—112no—
265C2×#102—112no—
266C2×#102—114no—
267C2×#102—224no—
268#391—112no—
269#390—114no—
270#390—112no—
271#390—222no—
272#386—222no—
273#380—222no—
274#379—222no—
275#340—114no—
276C22×C11—22no—
277C22×C11—22no—
278C22×C11—24no—
279C22×C11—24no—
280C22×C11—42no—
281C22×C11—44no—
282C22×C11—44no—
283C22×C11—44no—
284C22×C11—44no—
285C22×C11—816no—
286C22×C11—81no—
287C22×C11—81no—
288C22×C11—84no—
289C22×C11—82no—
290C22×C11—84no—
291C22×C11#39414yes—
292C22×C11C2×#12111yescharacteristic
293C22×C11C2×#12111yescharacteristic
294C22×C11C2×#12111yescharacteristic
295C22×C11#57612yes—
296C22×C11#57712yes—
297C4×C11—42no—
298C4×C11#39414yes—
299C4×C11C2×#12011yescharacteristic
300C4×C11C2×#12311yescharacteristic
301C4×C11C2×#12311yescharacteristic
302C4×C11C2×#12411yescharacteristic
303Dic11—44no—
304Dic11—82no—
305Dic11—82no—
306C25—21no—
307C2×#41—22no—
308C2×#41—22no—
309C2×#41—22no—
310C2×#41—22no—
311C2×#41—22no—
312C2×#41C22⋊D1114yes—
313C22×Q8C22×D1111yescharacteristic
314C22×D4—21no—
315C22×D4—21no—
316C22×D4—22no—
317C22×D4—22no—
318C22×D4—22no—
319C22×D4—22no—
320C22×D4—22no—
321C22×D4C22⋊D1114yes—
322C22×D4C22×D1111yescharacteristic
323C22×D4C22×D1112yes—
324C23×C4—21no—
325C23×C4—221no—
326C23×C4C22×D1111yescharacteristic
327AGL(1,ℤ/8)—232no—
328C2×SD16—24no—
329C2×SD16—24no—
330C2×SD16—24no—
331C2×D8—24no—
332C2×D8—24no—
333C2×D8—24no—
334C2×M16—22no—
335C2×M16—22no—
336C2×M16C22⋊D1114yes—
337C22×C8—112no—
338C22×C8—222no—
339C22×C8—222no—
340C22×C8—222no—
341C22×C8C22×D1112yes—
342#117—222no—
343#117—222no—
344#117—222no—
345#117—222no—
346C2×C4⋊C4—112no—
347C2×C4⋊C4—112no—
348C2×C4⋊C4—112no—
349C2×C4⋊C4—222no—
350C2×C4⋊C4—222no—
351C2×C4⋊C4—221no—
352C2×C22⋊C4—112no—
353C2×C22⋊C4—221no—
354C2×C22⋊C4—224no—
355C2×C22⋊C4—224no—
356C2×C22⋊C4—224no—
357C2×C42—112no—
358C2×C42—221no—
359C4⋊C8—228no—
360#103—224no—
361#102—228no—
362#102—224no—
363#95—224no—
364C2×C11—24no—
365C2×C11—82no—
366C2×C11—84no—
367C2×C11C2×#39411yescharacteristic
368C2×C11#304111yescharacteristic
369C2×C11#304211yescharacteristic
370C2×C11#304512yes—
371C2×C11#306212yes—
372C24—21no—
373C24—22no—
374C24—24no—
375C24—42no—
376C24—42no—
377C24—82no—
378C24—82no—
379C24D4×D1111yescharacteristic
380#41—28no—
381#41—28no—
382#41—44no—
383#41—44no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8C2×C22⋊D1112yes—
390C2×D4—24no—
391C2×D4—28no—
392C2×D4—24no—
393C2×D4—28no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—22no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4C2×C22⋊D1112yes—
411C2×D4C2×C22⋊D1114yes—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—114no—
417C22×C4—222no—
418C22×C4—222no—
419C22×C4—442no—
420C22×C4—882no—
421C22×C4D4×D1111yescharacteristic
422C22×C4#434112yes—
423C22×C4#434112yes—
424C22×C4#434112yes—
425C22×C4C2×C22⋊D1112yes—
426C22×C4C2×C22⋊D1112yes—
427C22×C4C2×C22⋊D1112yes—
428C22×C4C23×D1111yescharacteristic
429SD16—216no—
430D8—216no—
431M16—28no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—222no—
435C2×C8—222no—
436C2×C8—222no—
437C2×C8—444no—
438C2×C8C2×C22⋊D1114yes—
439C4⋊C4—224no—
440C4⋊C4—222no—
441C4⋊C4—222no—
442C4⋊C4—222no—
443C4⋊C4—222no—
444C4⋊C4—442no—
445C22⋊C4—224no—
446C22⋊C4—442no—
447C22⋊C4—884no—
448C22⋊C4—884no—
449C22⋊C4—884no—
450C42—222no—
451C42—222no—
452C42—442no—
453C11#2354411yescharacteristic, complemented, Sylow
454C23—22no—
455C23—22no—
456C23—22no—
457C23—24no—
458C23—42no—
459C23—44no—
460C23—44no—
461C23—42no—
462C23—41no—
463C23—42no—
464C23—41no—
465C23—42no—
466C23—48no—
467C23—84no—
468C23—84no—
469C23—88no—
470C23—88no—
471C23#6699112yes—
472C23#6699212yes—
473C23#6699212yes—
474C23C2×D4×D1111yescharacteristic
475Q8—22no—
476Q8—22no—
477Q8—41no—
478Q8—41no—
479D4—24no—
480D4—22no—
481D4—22no—
482D4—24no—
483D4—48no—
484D4—42no—
485D4—41no—
486D4—42no—
487D4—41no—
488D4—48no—
489C2×C4—22no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—48no—
493C2×C4—224no—
494C2×C4—224no—
495C2×C4—224no—
496C2×C4—442no—
497C2×C4—444no—
498C2×C4—442no—
499C2×C4—442no—
500C2×C4—442no—
501C2×C4—884no—
502C2×C4#6698814yes—
503C2×C4#6698814yes—
504C2×C4#6698914yes—
505C2×C4#6699112yes—
506C2×C4#6699212yes—
507C2×C4#6699212yes—
508C2×C4#6700614yes—
509C2×C4C2×#434111yescharacteristic
510C2×C4C2×#434111yescharacteristic
511C2×C4C2×#434111yescharacteristic
512C2×C4C22×C22⋊D1111yescharacteristic
513C2×C4C22×C22⋊D1111yescharacteristic
514C2×C4C22×C22⋊D1111yescharacteristic
515C8—24no—
516C8—442no—
517C22—24no—
518C22—22no—
519C22—22no—
520C22—24no—
521C22—44no—
522C22—44no—
523C22—44no—
524C22—42no—
525C22—44no—
526C22—816no—
527C22—84no—
528C22—82no—
529C22—81no—
530C22—81no—
531C22—84no—
532C22#1062283514yes—
533C22#1062330512yes—
534C22#1062330612yes—
535C22C2×#6699111yescharacteristic
536C22C2×#6699211yescharacteristic
537C22C2×#6699211yescharacteristic
538C4—42no—
539C4—444no—
540C4—882no—
541C4—882no—
542C4#1062283514yes—
543C4C2×#6698811yescharacteristic
544C4C2×#6698811yescharacteristic
545C4C2×#6698911yescharacteristic
546C4C2×#6700611yescharacteristic
547C2—24no—
548C2—84no—
549C2—82no—
550C2C2×#1062283511yescharacteristic
551C2#1307812111yescharacteristic
552C2#1307813011yescharacteristic
553C2#1307815912yes—
554C2#1307829012yes—
subgroupG/H
centerC23C2×D4×D11
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
Fitting, prime radicalC22×C11×AGL(1,ℤ/8)C2
socleC23×C11C22×D4
hypercenterC22×AGL(1,ℤ/8)D11
hypocenterC11#23544
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#10622732C2semidirect
2. C2×#10622846C2semidirect
3. C22×#66896C2semidirect
4. #13071208C2semidirect
5. #13071198C2semidirect
6. #13069929C2semidirect
7. #13069924C2semidirect
8. #13068336C2semidirect
24. #10622843C22semidirect
25. C2×#66899C22semidirect
28. #10622834C22semidirect
32. C2×#66896C22semidirect
35. C2×Dic11⋊C8C22semidirect
36. #10622189C22semidirect
453. C11#23544semidirect

Maximal subgroups

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

subgroupindex
C2×#106227322
C2×#106228462
C22×#668962
#130712082
#130711982
#130699292
#130699242
#130683362
C22×C11×AGL(1,ℤ/8)2
#2354411