Finite Group Databasebeta

← #14763712

Subgroups — #14763712

Order 2816.

1086 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623109C211yesmaximal, characteristic
2C2×#10622450C211yesmaximal, characteristic
3#13071009C211yesmaximal, characteristic
4#13069562C212yesmaximal, complemented
5#13069537C212yesmaximal, complemented
6#13069048C212yesmaximal
7#13069046C212yesmaximal
8#13068632C211yesmaximal, characteristic, complemented
9#13068631C211yesmaximal, characteristic, complemented
10#13068628C211yesmaximal, characteristic, complemented
11C11×#1805C211yesmaximal, characteristic, complemented
12C2×C4×Q8×C11C2211yescharacteristic
13C2×#66999C2211yescharacteristic
14C2×#66997C2211yescharacteristic
15C22×#4313C2212yes—
16#10623109—24no—
17#10623109C418yescomplemented
18C2×#66950C2211yescharacteristic
19C2×#66949—21no—
20C2×#66942—21no—
21C2×#66941C2211yescharacteristic
22C2×#66925—21no—
23C2×C42×D11C2211yescharacteristic
24C2×C4×Dic22—21no—
25C2×C4×Dic22C2211yescharacteristic
26C2×C11×C4⋊C8C2211yescharacteristic
27C2×C11×#103C2212yes—
28C11×#382C2211yescharacteristic
29C2×#66899C2212yes—
30C2×#66880C2212yes—
31C2×Dic11⋊C8C2212yes—
32#10622600C2212yes—
33C4⋊C4×Dic11C2212yes—
34C2×#66874C2212yes—
35#10622549C2211yescharacteristic
36C2×C4⋊(C11⋊C8)C2211yescharacteristic
37#10622450—24no—
38#10622279—21no—
39#10622277—21no—
40#10622264—21no—
41#10622261—21no—
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⋊C4C2312yes—
57C2×C11×C4⋊C4C2311yescharacteristic
58C2×C42×C11—21no—
59C2×C42×C11C2311yescharacteristic
60#66999—22no—
61#66999C2×C414yes—
62#66997—22no—
63#66997C2×C414yes—
64C2×(C2×D11)⋊C4—21no—
65C2×(C2×D11)⋊C4—21no—
66C2×(C2×D11)⋊C4—22no—
67C2×C4⋊Dic11—21no—
68C2×C4⋊Dic11C2312yes—
69C2×C4⋊Dic11C2311yescharacteristic
70C2×Dic11⋊C4—21no—
71C2×Dic11⋊C4—21no—
72C2×Dic11⋊C4—21no—
73C2×Dic11⋊C4—21no—
74C2×Dic11⋊C4—21no—
75C2×C4×Dic11—21no—
76C2×C4×Dic11—22no—
77C2×C4×Dic11D411yescharacteristic
78C2×C4×Dic11D411yescharacteristic
79C2×C4×Dic11C2311yescharacteristic
80C22×C11⋊C8C2312yes—
81C2×#4313—24no—
82C2×#4313—42no—
83C2×#4313—42no—
84#66950—22no—
85#66950C2×C414yes—
86#66949—28no—
87#66942—28no—
88#66941—22no—
89#66941C2×C414yes—
90#66925—28no—
91C42×D11C2×C414yes—
92C42×D11D414yes—
93C4×Dic22—22no—
94C4×Dic22—28no—
95C4×Dic22C2×C414yes—
96C11×C4⋊C8—22no—
97C11×#103—42no—
98C11×#95—21no—
99C11×#95—21no—
100#66899—42no—
101C11⋊#95—21no—
102C11⋊#95—21no—
103C11⋊#95—22no—
104C11⋊#95—22no—
105#66880—42no—
106Dic11⋊C8—24no—
107#66874—42no—
108C4⋊(C11⋊C8)—22no—
109#21892—111nomaximal, Sylow
110C23×D11C2×D411yescharacteristic
111C2×Q8×C11—21no—
112C2×Q8×C11—21no—
113C2×Q8×C11—21no—
114C2×Q8×C11—21no—
115C2×Q8×C11—21no—
116C2×Q8×C11C22×C412yes—
117C22×C4×C11—21no—
118C22×C4×C11—21no—
119C22×C4×C11—22no—
120C22×C4×C11C2×D411yescharacteristic
121C22×C4×C11C2×D411yescharacteristic
122C22×C4×C11C2411yescharacteristic
123C22×Dic11—21no—
124C22×Dic11—21no—
125C22×Dic11—22no—
126C22×Dic11C2×D411yescharacteristic
127C22×Dic11C2×D411yescharacteristic
128C22×Dic11C2×D411yescharacteristic
129C2×C4×D11—24no—
130C2×C4×D11—21no—
131C2×C4×D11—21no—
132C2×C4×D11—21no—
133C2×C4×D11—21no—
134C2×C4×D11—42no—
135C2×C4×D11C22⋊C414yes—
136C2×C4×D11C22⋊C414yes—
137C2×C4×D11C22×C412yes—
138C2×C4×D11C2×D412yes—
139C2×Dic22—24no—
140C2×Dic22—21no—
141C2×Dic22—21no—
142C2×Dic22—21no—
143C2×Dic22—21no—
144C2×Dic22—21no—
145C2×Dic22—44no—
146C2×Dic22C22×C412yes—
147C2×C8×C11—22no—
148C2×C8×C11—22no—
149C2×C8×C11—22no—
150C11×C4⋊C4—24no—
151C11×C4⋊C4—21no—
152C11×C4⋊C4—22no—
153C11×C4⋊C4—22no—
154C11×C4⋊C4C22×C412yes—
155C42×C11—24no—
156C42×C11C22×C412yes—
157C42×C11C2×D412yes—
158(C2×D11)⋊C4—24no—
159(C2×D11)⋊C4—24no—
160(C2×D11)⋊C4—44no—
161C4⋊Dic11—24no—
162C4⋊Dic11—21no—
163C4⋊Dic11—22no—
164C4⋊Dic11—22no—
165C4⋊Dic11C22×C412yes—
166Dic11⋊C4—24no—
167Dic11⋊C4—24no—
168Dic11⋊C4—24no—
169Dic11⋊C4—24no—
170Dic11⋊C4—24no—
171C4×Dic11—24no—
172C4×Dic11—44no—
173C4×Dic11C22⋊C414yes—
174C4×Dic11C22⋊C414yes—
175C4×Dic11C22×C412yes—
176C4×Dic11C2×D412yes—
177C2×C11⋊C8—22no—
178C2×C11⋊C8—22no—
179C2×C11⋊C8—22no—
180#4313—48no—
181C2×#533—111no—
182C2×#423—111no—
183#2283—111no—
184#1987—112no—
185#1981—112no—
186#1894—112no—
187#1894—112no—
188#1805—111no—
189#1805—111no—
190#1805—111no—
191#1805D1111yescharacteristic, complemented
192C23×C11C22×D411yescharacteristic
193C22×D11—21no—
194C22×D11—21no—
195C22×D11—22no—
196C22×D11—22no—
197C22×D11C2×C22⋊C412yes—
198Q8×C11—22no—
199Q8×C11—22no—
200Q8×C11—41no—
201Q8×C11—41no—
202C2×C4×C11—22no—
203C2×C4×C11—22no—
204C2×C4×C11—22no—
205C2×C4×C11—22no—
206C2×C4×C11—41no—
207C2×C4×C11—41no—
208C2×C4×C11—42no—
209C2×C4×C11—42no—
210C2×C4×C11—42no—
211C2×C4×C11—42no—
212C2×C4×C11C2×C22⋊C412yes—
213C2×C4×C11C2×C22⋊C412yes—
214C2×C4×C11C23×C411yescharacteristic
215C2×C4×C11C22×D411yescharacteristic
216C2×C4×C11H(5,𝔽2)11yescharacteristic
217C2×C4×C11H(5,𝔽2)11yescharacteristic
218C2×C4×C112−1+411yescharacteristic
219C2×C4×C112−1+411yescharacteristic
220C2×Dic11—21no—
221C2×Dic11—21no—
222C2×Dic11—22no—
223C2×Dic11—22no—
224C2×Dic11—22no—
225C2×Dic11—41no—
226C2×Dic11—41no—
227C2×Dic11—42no—
228C2×Dic11—42no—
229C2×Dic11—42no—
230C2×Dic11—42no—
231C2×Dic11C2×C22⋊C412yes—
232C2×Dic11C2×C22⋊C412yes—
233C2×Dic11C2×C22⋊C412yes—
234C2×Dic11#13714yes—
235C4×D11—22no—
236C4×D11—22no—
237C4×D11—22no—
238C4×D11—22no—
239C4×D11—28no—
240C4×D11—44no—
241Dic22—22no—
242Dic22—22no—
243Dic22—48no—
244Dic22—41no—
245Dic22—41no—
246C8×C11—42no—
247C11⋊C8—42no—
248C22×M16—112no—
249#533—118no—
250#533—224no—
251C2×C4⋊Q8—111no—
252C2×C4⋊Q8—111no—
253C2×#125—221no—
254C2×C22⋊Q8—111no—
255C2×C22⋊Q8—111no—
256C2×C22⋊Q8—221no—
257C2×C4×Q8—111no—
258C2×C4×Q8—221no—
259C2×C4×Q8C2×D1111yescharacteristic
260C2×#117—221no—
261C22×C42—111no—
262#423—224no—
263C2×C4⋊C8—112no—
264C2×C4⋊C8—111no—
265C2×C4⋊C8C2×D1111yescharacteristic
266C2×#103—112no—
267C2×#103—112no—
268C2×#103—112no—
269C2×#103C2×D1112yes—
270#383—221no—
271#382—111no—
272#382C2×D1111yescharacteristic
273#381—221no—
274#380—112no—
275C4×C4⋊C4—112no—
276C4×C22⋊C4—221no—
277#376—221no—
278#340—112no—
279#340—112no—
280#340—112no—
281#340C2×D1112yes—
282C22×C11C22×C22⋊C411yescharacteristic
283C22×C11#53711yescharacteristic
284C22×C11#53711yescharacteristic
285C22×C11C2×#13712yes—
286C22×C11#57512yes—
287C2×D11—22no—
288C2×D11—22no—
289C2×D11—22no—
290C2×D11—42no—
291C2×D11—42no—
292C4×C11—22no—
293C4×C11—22no—
294C4×C11—42no—
295C4×C11—42no—
296C4×C11—82no—
297C4×C11#52011yescharacteristic
298C4×C11#52011yescharacteristic
299C4×C11#53511yescharacteristic
300C4×C11#53611yescharacteristic
301Dic11—22no—
302Dic11—22no—
303Dic11—42no—
304Dic11—42no—
305Dic11—82no—
306Dic11#41914yes—
307C22×Q8—111no—
308C22×Q8—221no—
309C22×Q8C22×D1111yescharacteristic
310C23×C4—111no—
311C23×C4—111no—
312C23×C4—111no—
313C2×M16—224no—
314C2×M16—442no—
315C2×M16—442no—
316C22×C8—112no—
317C22×C8C22×D1112yes—
318C4⋊Q8—114no—
319C4⋊Q8—114no—
320C4⋊Q8—222no—
321C4⋊Q8—222no—
322#125—228no—
323C22⋊Q8—114no—
324C22⋊Q8—114no—
325C22⋊Q8—222no—
326C22⋊Q8—228no—
327C22⋊Q8—222no—
328C4×Q8—22no—
329C4×Q8—114no—
330C4×Q8—228no—
331C4×Q8—222no—
332C4×Q8C4×D1114yes—
333#117—228no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—111no—
336C2×C4⋊C4—112no—
337C2×C4⋊C4—221no—
338C2×C4⋊C4—221no—
339C2×C4⋊C4—221no—
340C2×C4⋊C4—221no—
341C2×C4⋊C4—221no—
342C2×C4⋊C4—221no—
343C2×C4⋊C4C22×D1112yes—
344C2×C4⋊C4C22×D1111yescharacteristic
345C2×C22⋊C4—221no—
346C2×C22⋊C4—221no—
347C2×C22⋊C4—222no—
348C2×C42—21no—
349C2×C42—111no—
350C2×C42—111no—
351C2×C42—111no—
352C2×C42—114no—
353C2×C42—114no—
354C2×C42—221no—
355C2×C42—222no—
356C2×C42C22×D1111yescharacteristic
357C4⋊C8—22no—
358C4⋊C8—224no—
359C4⋊C8—222no—
360#103—42no—
361#103—442no—
362#103—442no—
363#103—442no—
364#95—21no—
365#95—21no—
366#95—222no—
367#95—222no—
368#95—221no—
369#95—221no—
370C2×C11#231111yescharacteristic
371C2×C11C2×#41911yescharacteristic
372C2×C11#289611yescharacteristic
373C2×C11#325212yes—
374C2×C11#325212yes—
375D11—42no—
376C24—111no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—112no—
383C2×Q8—224no—
384C2×Q8—221no—
385C2×Q8—221no—
386C2×Q8—221no—
387C2×Q8—221no—
388C2×Q8—221no—
389C2×Q8—444no—
390C2×Q8C2×C4×D1112yes—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—111no—
395C22×C4—111no—
396C22×C4—111no—
397C22×C4—114no—
398C22×C4—112no—
399C22×C4—112no—
400C22×C4—114no—
401C22×C4—224no—
402C22×C4—221no—
403C22×C4—221no—
404C22×C4—221no—
405C22×C4—221no—
406C22×C4—221no—
407C22×C4—221no—
408C22×C4—222no—
409C22×C4—442no—
410C22×C4D4×D1111yescharacteristic
411C22×C4D4×D1111yescharacteristic
412C22×C4C23×D1111yescharacteristic
413M16—448no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—222no—
418C2×C8—222no—
419C2×C8—222no—
420C4⋊C4—24no—
421C4⋊C4—21no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—112no—
425C4⋊C4—224no—
426C4⋊C4—224no—
427C4⋊C4—224no—
428C4⋊C4—224no—
429C4⋊C4—224no—
430C4⋊C4—224no—
431C4⋊C4—221no—
432C4⋊C4—222no—
433C4⋊C4—222no—
434C4⋊C4C2×C4×D1112yes—
435C22⋊C4—224no—
436C22⋊C4—224no—
437C22⋊C4—444no—
438C42—24no—
439C42—114no—
440C42—114no—
441C42—112no—
442C42—112no—
443C42—224no—
444C42—444no—
445C42C2×C4×D1112yes—
446C42D4×D1112yes—
447C11#2189211yescharacteristic, complemented, Sylow
448C23—112no—
449C23—222no—
450C23—222no—
451C23—221no—
452C23—221no—
453C23C2×D4×D1111yescharacteristic
454Q8—22no—
455Q8—22no—
456Q8—41no—
457Q8—41no—
458Q8—222no—
459Q8—222no—
460Q8—448no—
461Q8—441no—
462Q8—441no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—112no—
474C2×C4—114no—
475C2×C4—112no—
476C2×C4—112no—
477C2×C4—228no—
478C2×C4—222no—
479C2×C4—221no—
480C2×C4—221no—
481C2×C4—222no—
482C2×C4—222no—
483C2×C4—222no—
484C2×C4—222no—
485C2×C4—222no—
486C2×C4—222no—
487C2×C4—444no—
488C2×C4—442no—
489C2×C4—441no—
490C2×C4—441no—
491C2×C4—442no—
492C2×C4—442no—
493C2×C4—442no—
494C2×C4C22⋊C4×D1112yes—
495C2×C4C22⋊C4×D1112yes—
496C2×C4C22×C4×D1111yescharacteristic
497C2×C4C2×D4×D1111yescharacteristic
498C2×C4#6703711yescharacteristic
499C2×C4#6704011yescharacteristic
500C2×C4#6704211yescharacteristic
501C2×C4#6704311yescharacteristic
502C8—42no—
503C8—442no—
504C22—222no—
505C22—222no—
506C22—222no—
507C22—442no—
508C22—442no—
509C22C2×C22⋊C4×D1111yescharacteristic
510C22#1062313911yescharacteristic
511C22#1062316011yescharacteristic
512C22#1062329912yes—
513C22D11×#13712yes—
514C4—22no—
515C4—22no—
516C4—42no—
517C4—42no—
518C4—82no—
519C4—114no—
520C4—222no—
521C4—222no—
522C4—442no—
523C4—442no—
524C4—882no—
525C4#1062303511yescharacteristic
526C4#1062305411yescharacteristic
527C4#1062313811yescharacteristic
528C4#1062315911yescharacteristic
529C2—442no—
530C2#1307120511yescharacteristic
531C2D11×#41911yescharacteristic
532C2#1307713111yescharacteristic
533C2#1307998612yes—
534C2#1307998812yes—
subgroupG/H
centerC23C2×D4×D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1805C2
socleC23×C11C22×D4
prime radicalC23×D11C2×D4
cosocleC22×C4×C11C24
hypercenter#1805D11
hypocenterC11#21892
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
4. #13069562C2semidirect
5. #13069537C2semidirect
8. #13068632C2semidirect
9. #13068631C2semidirect
10. #13068628C2semidirect
11. C11×#1805C2semidirect
17. #10623109C4semidirect
191. #1805D11semidirect
447. C11#21892semidirect

Maximal subgroups

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

subgroupindex
C2×#106231092
C2×#106224502
#130710092
#130695622
#130695372
#130690482
#130690462
#130686322
#130686312
#130686282
C11×#18052
#2189211