Finite Group Databasebeta

← #14763726

Subgroups — #14763726

Order 2816.

1130 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623109C211yesmaximal, characteristic
2C2×#10622450C211yesmaximal, characteristic
3#13071021C211yesmaximal, characteristic
4#13069566C212yesmaximal, complemented
5#13069537C212yesmaximal, complemented
6#13069048C212yesmaximal
7#13069046C212yesmaximal
8#13068622C211yesmaximal, characteristic, complemented
9#13068621C211yesmaximal, characteristic, complemented
10#13068618C211yesmaximal, characteristic, complemented
11C11×#1803C211yesmaximal, characteristic, complemented
12C2×C4×Q8×C11C2211yescharacteristic
13C2×#66999C2211yescharacteristic
14C2×#66997C2211yescharacteristic
15C22×#4313C2212yes—
16#10623109—24no—
17#10623109C418yescomplemented
18C2×#66950C2211yescharacteristic
19C2×#66949—21no—
20C2×#66945—21no—
21C2×#66942—21no—
22C2×#66941C2211yescharacteristic
23C2×#66925—21no—
24C2×C42×D11C2211yescharacteristic
25C2×C4×Dic22—21no—
26C2×C4×Dic22C2211yescharacteristic
27C2×C11×C4⋊C8C2211yescharacteristic
28C2×C11×#103C2212yes—
29C11×#384C2211yescharacteristic
30C2×#66899C2212yes—
31C2×#66880C2212yes—
32C2×Dic11⋊C8C2212yes—
33#10622600—21no—
34#10622600C2212yes—
35#10622590—21no—
36C4⋊C4×Dic11—21no—
37C4⋊C4×Dic11C2212yes—
38C2×#66874C2212yes—
39#10622548C2211yescharacteristic
40C2×C4⋊(C11⋊C8)C2211yescharacteristic
41#10622450—24no—
42C11×#340C2212yes—
43#10622189C2212yes—
44#10622167C2212yes—
45#10622167C2212yes—
46C22×Q8×C11C2311yescharacteristic
47C22×C4×D11D411yescharacteristic
48C22×C4×D11D411yescharacteristic
49C22×C4×D11C2311yescharacteristic
50C22×Dic22—21no—
51C22×Dic22C2311yescharacteristic
52C22×C8×C11C2312yes—
53C4×Q8×C11—22no—
54C4×Q8×C11C2×C414yes—
55C2×C11×C4⋊C4—21no—
56C2×C11×C4⋊C4—21no—
57C2×C11×C4⋊C4—21no—
58C2×C11×C4⋊C4C2312yes—
59C2×C11×C4⋊C4C2311yescharacteristic
60C2×C42×C11—21no—
61C2×C42×C11C2311yescharacteristic
62#66999—22no—
63#66999C2×C414yes—
64#66997—22no—
65#66997C2×C414yes—
66C2×(C2×D11)⋊C4—21no—
67C2×(C2×D11)⋊C4—21no—
68C2×(C2×D11)⋊C4—22no—
69C2×C4⋊Dic11—21no—
70C2×C4⋊Dic11—21no—
71C2×C4⋊Dic11—21no—
72C2×C4⋊Dic11C2312yes—
73C2×C4⋊Dic11C2311yescharacteristic
74C2×Dic11⋊C4—21no—
75C2×Dic11⋊C4—21no—
76C2×Dic11⋊C4—21no—
77C2×Dic11⋊C4—21no—
78C2×Dic11⋊C4—21no—
79C2×C4×Dic11—21no—
80C2×C4×Dic11—22no—
81C2×C4×Dic11D411yescharacteristic
82C2×C4×Dic11D411yescharacteristic
83C2×C4×Dic11C2311yescharacteristic
84C22×C11⋊C8C2312yes—
85C2×#4313—24no—
86C2×#4313—42no—
87C2×#4313—42no—
88#66950—22no—
89#66950C2×C414yes—
90#66949—28no—
91#66945—44no—
92#66942—28no—
93#66941—22no—
94#66941C2×C414yes—
95#66925—28no—
96C42×D11C2×C414yes—
97C42×D11D414yes—
98C4×Dic22—22no—
99C4×Dic22—28no—
100C4×Dic22C2×C414yes—
101C11×C4⋊C8—22no—
102C11×#103—42no—
103#66899—42no—
104C11⋊#95—22no—
105C11⋊#95—22no—
106#66880—42no—
107Dic11⋊C8—24no—
108#66874—42no—
109C4⋊(C11⋊C8)—22no—
110#21894—111nomaximal, Sylow
111C23×D11C2×D411yescharacteristic
112C2×Q8×C11—21no—
113C2×Q8×C11—21no—
114C2×Q8×C11—21no—
115C2×Q8×C11—21no—
116C2×Q8×C11—21no—
117C2×Q8×C11C22×C412yes—
118C22×C4×C11—21no—
119C22×C4×C11—21no—
120C22×C4×C11—22no—
121C22×C4×C11C2×D411yescharacteristic
122C22×C4×C11C2×D411yescharacteristic
123C22×C4×C11C2411yescharacteristic
124C22×Dic11—21no—
125C22×Dic11—21no—
126C22×Dic11—22no—
127C22×Dic11C2×D411yescharacteristic
128C22×Dic11C2×D411yescharacteristic
129C22×Dic11C2×D411yescharacteristic
130C2×C4×D11—22no—
131C2×C4×D11—22no—
132C2×C4×D11—22no—
133C2×C4×D11—22no—
134C2×C4×D11—21no—
135C2×C4×D11—21no—
136C2×C4×D11—21no—
137C2×C4×D11—21no—
138C2×C4×D11C22⋊C414yes—
139C2×C4×D11C22⋊C414yes—
140C2×C4×D11C22×C412yes—
141C2×C4×D11C2×D412yes—
142C2×Dic22—24no—
143C2×Dic22—21no—
144C2×Dic22—21no—
145C2×Dic22—21no—
146C2×Dic22—21no—
147C2×Dic22—21no—
148C2×Dic22—42no—
149C2×Dic22—42no—
150C2×Dic22C22×C412yes—
151C2×C8×C11—22no—
152C2×C8×C11—22no—
153C2×C8×C11—22no—
154C11×C4⋊C4—24no—
155C11×C4⋊C4—21no—
156C11×C4⋊C4—22no—
157C11×C4⋊C4—22no—
158C11×C4⋊C4—42no—
159C11×C4⋊C4—42no—
160C11×C4⋊C4C22×C412yes—
161C42×C11—24no—
162C42×C11C22×C412yes—
163C42×C11C2×D412yes—
164(C2×D11)⋊C4—24no—
165(C2×D11)⋊C4—24no—
166(C2×D11)⋊C4—44no—
167C4⋊Dic11—24no—
168C4⋊Dic11—21no—
169C4⋊Dic11—22no—
170C4⋊Dic11—22no—
171C4⋊Dic11—42no—
172C4⋊Dic11—42no—
173C4⋊Dic11C22×C412yes—
174Dic11⋊C4—24no—
175Dic11⋊C4—24no—
176Dic11⋊C4—24no—
177Dic11⋊C4—24no—
178Dic11⋊C4—24no—
179C4×Dic11—24no—
180C4×Dic11—44no—
181C4×Dic11C22⋊C414yes—
182C4×Dic11C22⋊C414yes—
183C4×Dic11C22×C412yes—
184C4×Dic11C2×D412yes—
185C2×C11⋊C8—22no—
186C2×C11⋊C8—22no—
187C2×C11⋊C8—22no—
188#4313—48no—
189C2×#533—111no—
190C2×#423—111no—
191#2285—111no—
192#1988—112no—
193#1981—112no—
194#1894—112no—
195#1894—112no—
196#1803—111no—
197#1803—111no—
198#1803—111no—
199#1803D1111yescharacteristic, complemented
200C23×C11C22×D411yescharacteristic
201C22×D11—21no—
202C22×D11—21no—
203C22×D11—21no—
204C22×D11—21no—
205C22×D11—21no—
206C22×D11—21no—
207C22×D11C2×C22⋊C412yes—
208Q8×C11—22no—
209Q8×C11—22no—
210Q8×C11—41no—
211Q8×C11—41no—
212C2×C4×C11—21no—
213C2×C4×C11—21no—
214C2×C4×C11—22no—
215C2×C4×C11—22no—
216C2×C4×C11—41no—
217C2×C4×C11—41no—
218C2×C4×C11—41no—
219C2×C4×C11—41no—
220C2×C4×C11—42no—
221C2×C4×C11—42no—
222C2×C4×C11—42no—
223C2×C4×C11C2×C22⋊C412yes—
224C2×C4×C11C2×C22⋊C412yes—
225C2×C4×C11#13712yes—
226C2×C4×C11#13712yes—
227C2×C4×C11C23×C411yescharacteristic
228C2×C4×C11C22×D411yescharacteristic
229C2×C4×C11H(5,𝔽2)11yescharacteristic
230C2×C4×C11H(5,𝔽2)11yescharacteristic
231C2×C4×C112−1+411yescharacteristic
232C2×C4×C112−1+411yescharacteristic
233C2×Dic11—21no—
234C2×Dic11—21no—
235C2×Dic11—21no—
236C2×Dic11—21no—
237C2×Dic11—22no—
238C2×Dic11—22no—
239C2×Dic11—41no—
240C2×Dic11—41no—
241C2×Dic11—41no—
242C2×Dic11—41no—
243C2×Dic11—42no—
244C2×Dic11—42no—
245C2×Dic11—42no—
246C2×Dic11C2×C22⋊C412yes—
247C2×Dic11C2×C22⋊C412yes—
248C2×Dic11C2×C22⋊C412yes—
249C2×Dic11AGL(1,ℤ/8)12yes—
250C2×Dic11#13712yes—
251C4×D11—22no—
252C4×D11—22no—
253C4×D11—22no—
254C4×D11—22no—
255C4×D11—28no—
256C4×D11—28no—
257Dic22—22no—
258Dic22—22no—
259Dic22—41no—
260Dic22—41no—
261Dic22—48no—
262C8×C11—42no—
263C11⋊C8—42no—
264C22×M16—112no—
265#533—118no—
266#533—224no—
267C2×C4⋊Q8—111no—
268C2×C4⋊Q8—111no—
269C2×#125—221no—
270C2×C22⋊Q8—111no—
271C2×C22⋊Q8—111no—
272C2×C22⋊Q8—221no—
273C2×C4×Q8—111no—
274C2×C4×Q8—221no—
275C2×C4×Q8C2×D1111yescharacteristic
276C2×#117—221no—
277C2×#117—221no—
278C22×C42—111no—
279#423—224no—
280C2×C4⋊C8—112no—
281C2×C4⋊C8—111no—
282C2×C4⋊C8C2×D1111yescharacteristic
283C2×#103—112no—
284C2×#103—112no—
285C2×#103—112no—
286C2×#103C2×D1112yes—
287#384—111no—
288#384C2×D1111yescharacteristic
289#382—221no—
290#380—112no—
291#380—221no—
292C4×C4⋊C4—112no—
293C4×C4⋊C4—221no—
294#340—112no—
295#340—112no—
296#340—112no—
297#340C2×D1112yes—
298C22×C11C22×C22⋊C411yescharacteristic
299C22×C11#53711yescharacteristic
300C22×C11#53711yescharacteristic
301C22×C11C2×AGL(1,ℤ/8)11yescharacteristic
302C22×C11C2×#13711yescharacteristic
303C22×C11C2×#13711yescharacteristic
304C22×C11C2×#13711yescharacteristic
305C2×D11—22no—
306C2×D11—22no—
307C2×D11—22no—
308C2×D11—41no—
309C2×D11—41no—
310C2×D11—41no—
311C2×D11—41no—
312C4×C11—22no—
313C4×C11—42no—
314C4×C11—42no—
315C4×C11—82no—
316C4×C11#41914yes—
317C4×C11#52011yescharacteristic
318C4×C11#52011yescharacteristic
319C4×C11#53511yescharacteristic
320C4×C11#53611yescharacteristic
321Dic11—22no—
322Dic11—22no—
323Dic11—42no—
324Dic11—42no—
325Dic11—82no—
326Dic11#41714yes—
327C22×Q8—111no—
328C22×Q8—221no—
329C22×Q8C22×D1111yescharacteristic
330C23×C4—111no—
331C23×C4—111no—
332C23×C4—111no—
333C2×M16—224no—
334C2×M16—442no—
335C2×M16—442no—
336C22×C8—112no—
337C22×C8C22×D1112yes—
338C4⋊Q8—114no—
339C4⋊Q8—114no—
340C4⋊Q8—222no—
341C4⋊Q8—222no—
342#125—228no—
343C22⋊Q8—114no—
344C22⋊Q8—114no—
345C22⋊Q8—222no—
346C22⋊Q8—228no—
347C22⋊Q8—222no—
348C4×Q8—22no—
349C4×Q8—114no—
350C4×Q8—228no—
351C4×Q8—222no—
352C4×Q8C4×D1114yes—
353#117—228no—
354#117—444no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—111no—
359C2×C4⋊C4—112no—
360C2×C4⋊C4—221no—
361C2×C4⋊C4—221no—
362C2×C4⋊C4—221no—
363C2×C4⋊C4—221no—
364C2×C4⋊C4—221no—
365C2×C4⋊C4—221no—
366C2×C4⋊C4—221no—
367C2×C4⋊C4—221no—
368C2×C4⋊C4C22×D1112yes—
369C2×C4⋊C4C22×D1111yescharacteristic
370C2×C22⋊C4—221no—
371C2×C22⋊C4—221no—
372C2×C22⋊C4—222no—
373C2×C42—21no—
374C2×C42—111no—
375C2×C42—111no—
376C2×C42—111no—
377C2×C42—114no—
378C2×C42—114no—
379C2×C42—221no—
380C2×C42—222no—
381C2×C42C22×D1111yescharacteristic
382C4⋊C8—22no—
383C4⋊C8—224no—
384C4⋊C8—222no—
385#103—42no—
386#103—442no—
387#103—442no—
388#103—442no—
389#95—222no—
390#95—222no—
391C2×C11#231111yescharacteristic
392C2×C11C2×#41911yescharacteristic
393C2×C11C2×#41711yescharacteristic
394C2×C11#325411yescharacteristic
395C2×C11#325411yescharacteristic
396C2×C11#325511yescharacteristic
397C2×C11#325511yescharacteristic
398D11—42no—
399C24—111no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—112no—
406C2×Q8—224no—
407C2×Q8—221no—
408C2×Q8—221no—
409C2×Q8—221no—
410C2×Q8—221no—
411C2×Q8—221no—
412C2×Q8—442no—
413C2×Q8—442no—
414C2×Q8C2×C4×D1112yes—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—111no—
419C22×C4—111no—
420C22×C4—111no—
421C22×C4—114no—
422C22×C4—114no—
423C22×C4—112no—
424C22×C4—112no—
425C22×C4—222no—
426C22×C4—222no—
427C22×C4—222no—
428C22×C4—222no—
429C22×C4—221no—
430C22×C4—221no—
431C22×C4—221no—
432C22×C4—221no—
433C22×C4—221no—
434C22×C4—221no—
435C22×C4—222no—
436C22×C4D4×D1111yescharacteristic
437C22×C4D4×D1111yescharacteristic
438C22×C4C23×D1111yescharacteristic
439M16—448no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—222no—
444C2×C8—222no—
445C2×C8—222no—
446C4⋊C4—24no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—112no—
453C4⋊C4—224no—
454C4⋊C4—224no—
455C4⋊C4—224no—
456C4⋊C4—224no—
457C4⋊C4—224no—
458C4⋊C4—224no—
459C4⋊C4—221no—
460C4⋊C4—222no—
461C4⋊C4—222no—
462C4⋊C4—442no—
463C4⋊C4—442no—
464C4⋊C4C2×C4×D1112yes—
465C22⋊C4—224no—
466C22⋊C4—224no—
467C22⋊C4—444no—
468C42—24no—
469C42—114no—
470C42—114no—
471C42—112no—
472C42—112no—
473C42—224no—
474C42—444no—
475C42C2×C4×D1112yes—
476C42D4×D1112yes—
477C11#2189411yescharacteristic, complemented, Sylow
478C23—112no—
479C23—221no—
480C23—221no—
481C23—221no—
482C23—221no—
483C23—221no—
484C23—221no—
485C23C2×D4×D1111yescharacteristic
486Q8—22no—
487Q8—22no—
488Q8—41no—
489Q8—41no—
490Q8—222no—
491Q8—222no—
492Q8—448no—
493Q8—441no—
494Q8—441no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—112no—
507C2×C4—112no—
508C2×C4—112no—
509C2×C4—112no—
510C2×C4—112no—
511C2×C4—228no—
512C2×C4—221no—
513C2×C4—221no—
514C2×C4—221no—
515C2×C4—221no—
516C2×C4—222no—
517C2×C4—222no—
518C2×C4—222no—
519C2×C4—222no—
520C2×C4—228no—
521C2×C4—222no—
522C2×C4—222no—
523C2×C4—441no—
524C2×C4—441no—
525C2×C4—441no—
526C2×C4—441no—
527C2×C4—442no—
528C2×C4—442no—
529C2×C4—442no—
530C2×C4C22⋊C4×D1112yes—
531C2×C4C22⋊C4×D1112yes—
532C2×C4#6696812yes—
533C2×C4#6697112yes—
534C2×C4C22×C4×D1111yescharacteristic
535C2×C4C2×D4×D1111yescharacteristic
536C2×C4#6703711yescharacteristic
537C2×C4#6704011yescharacteristic
538C2×C4#6704211yescharacteristic
539C2×C4#6704311yescharacteristic
540C8—42no—
541C8—442no—
542C22—222no—
543C22—222no—
544C22—222no—
545C22—441no—
546C22—441no—
547C22—441no—
548C22—441no—
549C22C2×C22⋊C4×D1111yescharacteristic
550C22#1062313911yescharacteristic
551C22#1062316011yescharacteristic
552C22C2×#6696811yescharacteristic
553C22C2×#6697111yescharacteristic
554C22D11×AGL(1,ℤ/8)11yescharacteristic
555C22D11×#13711yescharacteristic
556C4—22no—
557C4—42no—
558C4—42no—
559C4—82no—
560C4—114no—
561C4—222no—
562C4—222no—
563C4—442no—
564C4—442no—
565C4—882no—
566C4#1062241514yes—
567C4#1062303511yescharacteristic
568C4#1062305411yescharacteristic
569C4#1062313811yescharacteristic
570C4#1062315911yescharacteristic
571C2—442no—
572C2#1307120511yescharacteristic
573C2C2×#1062241511yescharacteristic
574C2D11×#41711yescharacteristic
575C2#1308000111yescharacteristic
576C2#1308000211yescharacteristic
577C2#1308000811yescharacteristic
578C2#1308000911yescharacteristic
subgroupG/H
centerC23C2×D4×D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1803C2
socleC23×C11C22×D4
prime radicalC23×D11C2×D4
cosocleC22×C4×C11C24
hypercenter#1803D11
hypocenterC11#21894
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
4. #13069566C2semidirect
5. #13069537C2semidirect
8. #13068622C2semidirect
9. #13068621C2semidirect
10. #13068618C2semidirect
11. C11×#1803C2semidirect
17. #10623109C4semidirect
199. #1803D11semidirect
477. C11#21894semidirect

Maximal subgroups

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

subgroupindex
C2×#106231092
C2×#106224502
#130710212
#130695662
#130695372
#130690482
#130690462
#130686222
#130686212
#130686182
C11×#18032
#2189411