Finite Group Databasebeta

← #14762807

Subgroups — #14762807

Order 2816.

1498 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66950C211yesmaximal, characteristic
2C2×D11×C22⋊C8C211yesmaximal, characteristic
3#13070998C211yesmaximal, characteristic
4#13069038C214yesmaximal, complemented
5#13068927C214yesmaximal, complemented
6#13068702C211yesmaximal, characteristic, complemented
7#13068552C211yesmaximal, characteristic, complemented
8#13068550C211yesmaximal, characteristic, complemented
9C11×#1823C211yesmaximal, characteristic, complemented
10C23×C4×D11C2211yescharacteristic
11C23×Dic22C2211yescharacteristic
12C22×C11×C4⋊C4C2211yescharacteristic
13C22×(C2×D11)⋊C4—21no—
14C22×C4⋊Dic11—21no—
15C22×C8×D11C2212yes—
16C2×#66950—21no—
17C2×#66950—21no—
18C2×#66950—24no—
19C2×#66950C418yescomplemented
20C2×#66950C2212yes—
21C2×#66950C2212yes—
22C2×#66945C2212yes—
23C2×C11×C22⋊C8C2211yescharacteristic
24C11×#380C2211yescharacteristic
25#10622825C2211yescharacteristic
26C2×C22⋊(C11⋊C8)C2211yescharacteristic
27C2×(C2×D11)⋊C8C2212yes—
28C2×#66880C2214yescomplemented
29#10622600C2212yes—
30C2×#66874C2214yescomplemented
31#10622568—21no—
32C22⋊C4×Dic11—21no—
33D11×C22⋊C8—24no—
34#10622279—21no—
35#10622277—21no—
36C11×#340C2214yescomplemented
37#10622167C2214yescomplemented
38C24×D11D411yescharacteristic
39C23×C4×C11—21no—
40C23×C4×C11C2311yescharacteristic
41C23×Dic11—21no—
42C23×Dic11D411yescharacteristic
43C22×C4×D11—44no—
44C22×C4×D11C2×C414yes—
45C22×C4×D11D411yescharacteristic
46C22×C4×D11D411yescharacteristic
47C22×C4×D11D414yes—
48C22×C4×D11C2311yescharacteristic
49C22×C4×D11C2311yescharacteristic
50C22×Dic22—21no—
51C22×Dic22—22no—
52C22×Dic22—44no—
53C22×Dic22C2×C414yes—
54C22×Dic22C2312yes—
55C22×C8×C11C2312yes—
56C2×C11×C4⋊C4—21no—
57C2×C11×C4⋊C4—22no—
58C2×C11×C4⋊C4C2×C414yes—
59C2×C11×C4⋊C4C2312yes—
60C2×C11×C4⋊C4C2312yes—
61C2×C11×C22⋊C4—21no—
62C2×C22⋊Dic11—21no—
63C2×(C2×D11)⋊C4—28no—
64C2×(C2×D11)⋊C4—22no—
65C2×(C2×D11)⋊C4—22no—
66C2×(C2×D11)⋊C4—22no—
67C2×(C2×D11)⋊C4—22no—
68C2×C4⋊Dic11—28no—
69C2×C4⋊Dic11—22no—
70C2×C4⋊Dic11—22no—
71C2×C4⋊Dic11C2312yes—
72C2×C4×Dic11—22no—
73C22×C11⋊C8C2312yes—
74C2×C8×D11—24no—
75C2×C8×D11—44no—
76#66950—28no—
77#66950—28no—
78#66950—24no—
79#66950—24no—
80#66950C2×C418yes—
81#66950C2×C418yes—
82#66945—44no—
83C11×C22⋊C8—22no—
84C11×#95—21no—
85C11⋊#95—21no—
86C11⋊#95—22no—
87C22⋊(C11⋊C8)—22no—
88(C2×D11)⋊C8—24no—
89#66880—44no—
90#66874—44no—
91#21762—111nomaximal, Sylow
92C24×C11C2×D411yescharacteristic
93C23×D11—22no—
94C23×D11—22no—
95C23×D11—24no—
96C23×D11—42no—
97C23×D11C22⋊C414yes—
98C23×D11C2×D411yescharacteristic
99C23×D11C2×D411yescharacteristic
100C22×C4×C11—24no—
101C22×C4×C11—24no—
102C22×C4×C11—22no—
103C22×C4×C11—22no—
104C22×C4×C11—42no—
105C22×C4×C11—42no—
106C22×C4×C11C22×C412yes—
107C22×C4×C11C2×D411yescharacteristic
108C22×C4×C11C2×D412yes—
109C22×C4×C11C2411yescharacteristic
110C22×Dic11—24no—
111C22×Dic11—22no—
112C22×Dic11—22no—
113C22×Dic11—42no—
114C22×Dic11—44no—
115C22×Dic11C22⋊C414yes—
116C22×Dic11C2×D411yescharacteristic
117C22×Dic11C2×D411yescharacteristic
118C2×C4×D11—24no—
119C2×C4×D11—24no—
120C2×C4×D11—42no—
121C2×C4×D11—42no—
122C2×C4×D11—48no—
123C2×C4×D11—48no—
124C2×C4×D11C22⋊C414yes—
125C2×C4×D11C22⋊C414yes—
126C2×C4×D11C22×C412yes—
127C2×C4×D11C22×C412yes—
128C2×C4×D11C2×D412yes—
129C2×C4×D11C2×D412yes—
130C2×Dic22—24no—
131C2×Dic22—22no—
132C2×Dic22—24no—
133C2×Dic22—24no—
134C2×Dic22—48no—
135C2×Dic22—44no—
136C2×Dic22—48no—
137C2×Dic22C22×C414yes—
138C2×C8×C11—22no—
139C2×C8×C11—24no—
140C11×C4⋊C4—24no—
141C11×C4⋊C4—22no—
142C11×C4⋊C4—24no—
143C11×C4⋊C4C22×C414yes—
144C11×C22⋊C4—42no—
145C22⋊Dic11—42no—
146(C2×D11)⋊C4—28no—
147(C2×D11)⋊C4—28no—
148(C2×D11)⋊C4—44no—
149(C2×D11)⋊C4—44no—
150C4⋊Dic11—28no—
151C4⋊Dic11—28no—
152C4⋊Dic11—24no—
153C4×Dic11—44no—
154C2×C11⋊C8—22no—
155C2×C11⋊C8—24no—
156C8×D11—48no—
157C22×C22⋊Q8—111no—
158C22×C22⋊C8—111no—
159#2280—111no—
160#1892—114no—
161#1873—114no—
162#1823—111no—
163#1823D1111yescharacteristic, complemented
164#1787—111no—
165#1787—111no—
166C23×C11—22no—
167C23×C11—22no—
168C23×C11C2×C22⋊C412yes—
169C23×C11#13512yes—
170C23×C11#13512yes—
171C23×C11C22×D411yescharacteristic
172C22×D11—22no—
173C22×D11—21no—
174C22×D11—21no—
175C22×D11—22no—
176C22×D11—21no—
177C22×D11—21no—
178C22×D11—22no—
179C22×D11—28no—
180C22×D11—28no—
181C22×D11—44no—
182C22×D11—48no—
183C22×D11—48no—
184C22×D11C2×C22⋊C412yes—
185C22×D11C2×C22⋊C412yes—
186C22×D11#13714yes—
187C2×C4×C11—22no—
188C2×C4×C11—24no—
189C2×C4×C11—24no—
190C2×C4×C11—24no—
191C2×C4×C11—42no—
192C2×C4×C11—42no—
193C2×C4×C11—48no—
194C2×C4×C11—42no—
195C2×C4×C11—42no—
196C2×C4×C11—42no—
197C2×C4×C11C2×C22⋊C412yes—
198C2×C4×C11C23×C411yescharacteristic
199C2×C4×C11C22×D411yescharacteristic
200C2×C4×C11H(5,𝔽2)14yes—
201C2×Dic11—22no—
202C2×Dic11—22no—
203C2×Dic11—24no—
204C2×Dic11—44no—
205C2×Dic11—48no—
206C2×Dic11—44no—
207C2×Dic11—42no—
208C2×Dic11—42no—
209C2×Dic11—42no—
210C2×Dic11C2×C22⋊C412yes—
211C2×Dic11C2×C22⋊C412yes—
212C4×D11—24no—
213C4×D11—24no—
214C4×D11—24no—
215C4×D11—24no—
216C4×D11—44no—
217C4×D11—44no—
218Dic22—24no—
219Dic22—24no—
220Dic22—48no—
221Dic22—44no—
222C8×C11—42no—
223C11⋊C8—42no—
224C23×Q8—111no—
225C24×C4—111no—
226C23×C8—112no—
227C2×C22⋊Q8—112no—
228C2×C22⋊Q8—112no—
229C2×C22⋊Q8—118no—
230C2×C22⋊Q8—221no—
231C2×C22⋊Q8—221no—
232C2×C22⋊Q8—224no—
233C2×#117—112no—
234C22×C4⋊C4—221no—
235C22×C4⋊C4C2×D1111yescharacteristic
236C22×C22⋊C4—221no—
237C2×#103—114no—
238C2×#103—114no—
239C2×C22⋊C8—111no—
240C2×C22⋊C8—112no—
241C2×C22⋊C8—224no—
242C2×C22⋊C8C2×D1111yescharacteristic
243#381—221no—
244#380—111no—
245#380—112no—
246#380C2×D1111yescharacteristic
247#379—221no—
248C4×C22⋊C4—221no—
249C4×C22⋊C4—221no—
250#340—114no—
251#340C2×D1114yescomplemented
252C22×C11—22no—
253C22×C11—22no—
254C22×C11—24no—
255C22×C11—24no—
256C22×C11#41614yes—
257C22×C11C22×C22⋊C411yescharacteristic
258C22×C11#53412yes—
259C22×C11C2×#13511yescharacteristic
260C22×C11C2×#13511yescharacteristic
261C22×C11C2×#13712yes—
262C2×D11—22no—
263C2×D11—22no—
264C2×D11—24no—
265C2×D11—24no—
266C2×D11—22no—
267C2×D11—22no—
268C2×D11—22no—
269C2×D11—416no—
270C2×D11—42no—
271C2×D11—42no—
272C2×D11#41914yes—
273C4×C11—22no—
274C4×C11—44no—
275C4×C11—82no—
276C4×C11#51812yes—
277C4×C11#53412yes—
278Dic11—22no—
279Dic11—22no—
280Dic11—44no—
281Dic11—82no—
282C25—111no—
283C22×Q8—114no—
284C22×Q8—112no—
285C22×Q8—222no—
286C22×Q8—221no—
287C22×Q8—444no—
288C23×C4—21no—
289C23×C4—111no—
290C23×C4—111no—
291C23×C4—111no—
292C23×C4—114no—
293C23×C4—111no—
294C23×C4—111no—
295C23×C4—114no—
296C23×C4—221no—
297C23×C4—444no—
298C23×C4C22×D1111yescharacteristic
299C22×C8—112no—
300C22×C8—224no—
301C22×C8—444no—
302C22×C8C22×D1112yes—
303C22⋊Q8—118no—
304C22⋊Q8—118no—
305C22⋊Q8—228no—
306C22⋊Q8—228no—
307C22⋊Q8—224no—
308C22⋊Q8—224no—
309#117—444no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—112no—
313C2×C4⋊C4—228no—
314C2×C4⋊C4—222no—
315C2×C4⋊C4—222no—
316C2×C4⋊C4C4×D1114yes—
317C2×C4⋊C4C22×D1112yes—
318C2×C4⋊C4C22×D1112yes—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—228no—
321C2×C22⋊C4—222no—
322C2×C22⋊C4—222no—
323C2×C22⋊C4—222no—
324C2×C22⋊C4—222no—
325C2×C22⋊C4—221no—
326C2×C42—222no—
327#103—444no—
328#103—444no—
329C22⋊C8—22no—
330C22⋊C8—224no—
331C22⋊C8—222no—
332#95—21no—
333#95—222no—
334#95—221no—
335C2×C11—24no—
336C2×C11#230811yescharacteristic
337C2×C11C2×#41611yescharacteristic
338C2×C11C2×#41911yescharacteristic
339C2×C11#319012yes—
340C2×C11#319012yes—
341D11—24no—
342D11—42no—
343C24—111no—
344C24—111no—
345C24—114no—
346C24—224no—
347C24—222no—
348C24—222no—
349C24—442no—
350C24D4×D1111yescharacteristic
351C2×Q8—114no—
352C2×Q8—224no—
353C2×Q8—222no—
354C2×Q8—224no—
355C2×Q8—224no—
356C2×Q8—448no—
357C2×Q8—444no—
358C2×Q8—448no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—111no—
366C22×C4—111no—
367C22×C4—114no—
368C22×C4—112no—
369C22×C4—112no—
370C22×C4—112no—
371C22×C4—112no—
372C22×C4—114no—
373C22×C4—114no—
374C22×C4—224no—
375C22×C4—224no—
376C22×C4—224no—
377C22×C4—222no—
378C22×C4—222no—
379C22×C4—448no—
380C22×C4—448no—
381C22×C4—442no—
382C22×C4—442no—
383C22×C4—442no—
384C22×C4—444no—
385C22×C4C2×C4×D1112yes—
386C22×C4D4×D1111yescharacteristic
387C22×C4D4×D1112yes—
388C22×C4C23×D1111yescharacteristic
389C2×C8—22no—
390C2×C8—24no—
391C2×C8—222no—
392C2×C8—224no—
393C2×C8—448no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—228no—
398C4⋊C4—228no—
399C4⋊C4—224no—
400C4⋊C4C2×C4×D1114yes—
401C22⋊C4—42no—
402C22⋊C4—228no—
403C22⋊C4—228no—
404C22⋊C4—444no—
405C22⋊C4—444no—
406C22⋊C4—442no—
407C42—444no—
408C11#2176211yescharacteristic, complemented, Sylow
409C23—22no—
410C23—22no—
411C23—112no—
412C23—114no—
413C23—112no—
414C23—228no—
415C23—221no—
416C23—221no—
417C23—222no—
418C23—221no—
419C23—221no—
420C23—222no—
421C23—228no—
422C23—222no—
423C23—444no—
424C23—448no—
425C23—448no—
426C23C22⋊C4×D1112yes—
427C23#6696512yes—
428C23#6696912yes—
429C23C2×D4×D1111yescharacteristic
430Q8—224no—
431Q8—224no—
432Q8—448no—
433Q8—444no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—48no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—112no—
445C2×C4—112no—
446C2×C4—222no—
447C2×C4—222no—
448C2×C4—224no—
449C2×C4—224no—
450C2×C4—224no—
451C2×C4—224no—
452C2×C4—224no—
453C2×C4—444no—
454C2×C4—444no—
455C2×C4—444no—
456C2×C4—448no—
457C2×C4—444no—
458C2×C4—442no—
459C2×C4—442no—
460C2×C4—442no—
461C2×C4C22⋊C4×D1112yes—
462C2×C4C22×C4×D1111yescharacteristic
463C2×C4C2×D4×D1111yescharacteristic
464C2×C4#6704214yes—
465C8—42no—
466C8—442no—
467C22—22no—
468C22—24no—
469C22—24no—
470C22—22no—
471C22—114no—
472C22—222no—
473C22—224no—
474C22—224no—
475C22—222no—
476C22—222no—
477C22—222no—
478C22—222no—
479C22—4416no—
480C22—442no—
481C22—442no—
482C22#1062238514yes—
483C22C2×C22⋊C4×D1111yescharacteristic
484C22#1062318412yes—
485C22C2×#6696511yescharacteristic
486C22C2×#6696911yescharacteristic
487C22D11×#13712yes—
488C4—22no—
489C4—44no—
490C4—82no—
491C4—222no—
492C4—222no—
493C4—444no—
494C4—882no—
495C4#1062305212yes—
496C4#1062318412yes—
497C2—24no—
498C2—224no—
499C2—442no—
500C2#1307119311yescharacteristic
501C2C2×#1062238511yescharacteristic
502C2D11×#41911yescharacteristic
503C2#1307939612yes—
504C2#1307940212yes—
subgroupG/H
centerC23C2×D4×D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1823C2
socleC23×C11C22×D4
prime radicalC24×D11D4
cosocleC22×C4×C11C24
hypercenter#1823D11
hypocenterC11#21762
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
4. #13069038C2semidirect
5. #13068927C2semidirect
6. #13068702C2semidirect
7. #13068552C2semidirect
8. #13068550C2semidirect
9. C11×#1823C2semidirect
19. C2×#66950C4semidirect
28. C2×#66880C22semidirect
30. C2×#66874C22semidirect
36. C11×#340C22semidirect
37. #10622167C22semidirect
163. #1823D11semidirect
251. #340C2×D11semidirect
408. C11#21762semidirect

Maximal subgroups

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

subgroupindex
C22×#669502
C2×D11×C22⋊C82
#130709982
#130690382
#130689272
#130687022
#130685522
#130685502
C11×#18232
#2176211