Finite Group Databasebeta

← #14684788

Subgroups — #14684788

Order 2816.

3274 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66935C211yesmaximal, characteristic
2C2×#10622568C212yesmaximal, complemented
3#13066865C211yesmaximal, characteristic, complemented
4#13066864C214yesmaximal, complemented
5#13066856C214yesmaximal, complemented
6#13066855C212yesmaximal, complemented
7C11×#1440C211yesmaximal, characteristic, complemented
8C25×D11C2211yescharacteristic
9C23×C22⋊D11C2211yescharacteristic
10C23×D44C2212yes—
11C22×C11×C22⋊C4C2211yescharacteristic
12C22×C11×C22⋊C4C2212yes—
13C22×C22⋊Dic11C2212yes—
14C22×(C2×D11)⋊C4C2212yes—
15C22×(C2×D11)⋊C4C2214yes—
16C22×(C2×D11)⋊C4C2214yes—
17C22×(C2×D11)⋊C4C2214yescomplemented
18C2×#66935—28no—
19C2×#66935C418yescomplemented
20C2×C11×#95C2214yescomplemented
21C2×C11⋊#95C2214yescomplemented
22C2×C11⋊#95C2214yescomplemented
23#10622568—28no—
24C25×C11C2311yescharacteristic
25C24×D11—44no—
26C24×D11—44no—
27C24×D11—44no—
28C24×D11C2×C414yes—
29C24×D11D418yes—
30C24×D11C2311yescharacteristic
31C24×D11C2312yes—
32C23×C4×C11C2314yes—
33C23×C4×C11C2312yes—
34C22×C22⋊D11—24no—
35C22×C22⋊D11—44no—
36C22×C22⋊D11C2×C414yes—
37C23×Dic11C2314yes—
38C23×Dic11C2311yescharacteristic
39C22×D44—24no—
40C22×D44—24no—
41C22×D44—44no—
42C22×D44—44no—
43C22×D44C2×C418yes—
44C2×C11×C22⋊C4—24no—
45C2×C11×C22⋊C4—24no—
46C2×C11×C22⋊C4—44no—
47C2×C11×C22⋊C4C2×C414yes—
48C2×C22⋊Dic11—24no—
49C2×C22⋊Dic11—44no—
50C2×(C2×D11)⋊C4—24no—
51C2×(C2×D11)⋊C4—24no—
52C2×(C2×D11)⋊C4—28no—
53C2×(C2×D11)⋊C4—28no—
54C2×(C2×D11)⋊C4—44no—
55C2×(C2×D11)⋊C4—44no—
56C2×(C2×D11)⋊C4—44no—
57C2×(C2×D11)⋊C4—44no—
58C2×(C2×D11)⋊C4—44no—
59C2×(C2×D11)⋊C4—44no—
60C2×(C2×D11)⋊C4—44no—
61C2×(C2×D11)⋊C4C2×C418yes—
62#66935—232no—
63C11×#95—44no—
64C11⋊#95—44no—
65C11⋊#95—44no—
66#14657—111nomaximal, Sylow
67C24×C11—24no—
68C24×C11—24no—
69C24×C11—24no—
70C24×C11C22×C412yes—
71C24×C11C2×D414yes—
72C24×C11C2411yescharacteristic
73C23×D11—24no—
74C23×D11—24no—
75C23×D11—24no—
76C23×D11—24no—
77C23×D11—24no—
78C23×D11—24no—
79C23×D11—22no—
80C23×D11—24no—
81C23×D11—24no—
82C23×D11—24no—
83C23×D11—416no—
84C23×D11—48no—
85C23×D11—416no—
86C23×D11—416no—
87C23×D11—48no—
88C23×D11—48no—
89C23×D11—416no—
90C23×D11—416no—
91C23×D11—416no—
92C23×D11—416no—
93C23×D11C22⋊C4116yescomplemented
94C23×D11C22×C412yes—
95C23×D11C22×C414yes—
96C23×D11C2×D414yes—
97C23×D11C2×D414yes—
98C22×C4×C11—22no—
99C22×C4×C11—22no—
100C22×C4×C11—24no—
101C22×C4×C11—24no—
102C22×C4×C11—24no—
103C22×C4×C11—24no—
104C22×C4×C11—24no—
105C22×C4×C11—24no—
106C22×C4×C11—24no—
107C22×C4×C11—24no—
108C22×C4×C11—24no—
109C22×C4×C11C22×C414yes—
110C2×C22⋊D11—28no—
111C2×C22⋊D11—48no—
112C2×C22⋊D11—416no—
113C22×Dic11—22no—
114C22×Dic11—24no—
115C22×Dic11—24no—
116C22×Dic11—24no—
117C22×Dic11—24no—
118C22×Dic11—24no—
119C22×Dic11—24no—
120C22×Dic11—24no—
121C22×Dic11—24no—
122C22×Dic11C22×C412yes—
123C2×D44—216no—
124C2×D44—416no—
125C2×D44—48no—
126C2×D44—416no—
127C2×D44—48no—
128C11×C22⋊C4—28no—
129C11×C22⋊C4—48no—
130C22⋊Dic11—48no—
131(C2×D11)⋊C4—216no—
132(C2×D11)⋊C4—416no—
133(C2×D11)⋊C4—416no—
134(C2×D11)⋊C4—416no—
135C22×#120—111no—
136C2×#379—112no—
137#1440—114no—
138#1440—111no—
139#1440D1111yescharacteristic, complemented
140#1438—114no—
141#1438—112no—
142C23×C11—24no—
143C23×C11—28no—
144C23×C11—24no—
145C23×C11—24no—
146C23×C11—28no—
147C23×C11—28no—
148C23×C11—44no—
149C23×C11—44no—
150C23×C11—44no—
151C23×C11—44no—
152C23×C11C2×C22⋊C414yes—
153C23×C11C23×C411yescharacteristic
154C23×C11C22×D412yes—
155C23×C11H(5,𝔽2)14yes—
156C23×C11H(5,𝔽2)14yes—
157C23×C11H(5,𝔽2)14yes—
158C22×D11—28no—
159C22×D11—24no—
160C22×D11—24no—
161C22×D11—28no—
162C22×D11—28no—
163C22×D11—28no—
164C22×D11—24no—
165C22×D11—28no—
166C22×D11—24no—
167C22×D11—24no—
168C22×D11—24no—
169C22×D11—28no—
170C22×D11—22no—
171C22×D11—28no—
172C22×D11—24no—
173C22×D11—416no—
174C22×D11—432no—
175C22×D11—464no—
176C22×D11—464no—
177C22×D11—464no—
178C22×D11—44no—
179C22×D11—44no—
180C22×D11—44no—
181C22×D11—44no—
182C22×D11—44no—
183C22×D11—44no—
184C22×D11—44no—
185C22×D11—44no—
186C22×D11—42no—
187C22×D11—44no—
188C22×D11—42no—
189C22×D11—44no—
190C22×D11—44no—
191C22×D11—44no—
192C22×D11—44no—
193C22×D11—44no—
194C22×D11C2×C22⋊C418yes—
195C2×C4×C11—24no—
196C2×C4×C11—28no—
197C2×C4×C11—42no—
198C2×C4×C11—44no—
199C2×C4×C11—42no—
200C2×C4×C11—44no—
201C2×C4×C11—44no—
202C2×C4×C11—44no—
203C2×C4×C11—44no—
204C2×C4×C11—44no—
205C2×C4×C11—44no—
206C2×C4×C11—44no—
207C22⋊D11—416no—
208C2×Dic11—22no—
209C2×Dic11—24no—
210C2×Dic11—44no—
211C2×Dic11—44no—
212C2×Dic11—44no—
213C2×Dic11—44no—
214C2×Dic11—44no—
215C2×Dic11—44no—
216C2×Dic11—44no—
217C2×Dic11—44no—
218D44—432no—
219C26—111no—
220C23×D4—111no—
221C23×D4—112no—
222C2×#120—118no—
223C2×#120—228no—
224C22×C22⋊C4—112no—
225C22×C22⋊C4—112no—
226C22×C22⋊C4—114no—
227C22×C22⋊C4—114no—
228C22×C22⋊C4—114no—
229C22×C22⋊C4C2×D1111yescharacteristic
230C22×C22⋊C4C2×D1112yes—
231#379—228no—
232C2×#95—114no—
233C2×#95—114no—
234C2×#95C2×D1114yescomplemented
235C22×C11—28no—
236C22×C11—24no—
237C22×C11—48no—
238C22×C11—48no—
239C22×C11—48no—
240C22×C11C22×C22⋊C411yescharacteristic
241C22×C11#51812yes—
242C22×C11#51814yes—
243C22×C11#53412yes—
244C22×C11#53414yes—
245C22×C11#53412yes—
246C22×C11#53414yes—
247C22×C11#56014yes—
248C22×C11#56014yes—
249C22×C11#56014yes—
250C22×C11H(3,𝔽4)14yes—
251C2×D11—28no—
252C2×D11—28no—
253C2×D11—28no—
254C2×D11—464no—
255C2×D11—48no—
256C2×D11—48no—
257C2×D11—48no—
258C2×D11—48no—
259C2×D11—48no—
260C2×D11—48no—
261C2×D11—44no—
262C2×D11—42no—
263C2×D11—48no—
264C2×D11—84no—
265C2×D11—84no—
266C4×C11—44no—
267C4×C11—84no—
268Dic11—42no—
269Dic11—84no—
270C25—112no—
271C25—114no—
272C25—118no—
273C25—111no—
274C25—444no—
275C25—444no—
276C25—444no—
277C25C22×D1111yescharacteristic
278C22×D4—114no—
279C22×D4—118no—
280C22×D4—224no—
281C22×D4—224no—
282C22×D4—224no—
283C22×D4—444no—
284C22×D4—444no—
285C22×D4—444no—
286C23×C4—111no—
287C23×C4—114no—
288C23×C4C22×D1114yes—
289C23×C4C22×D1112yes—
290#120—2232no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—118no—
295C2×C22⋊C4—224no—
296C2×C22⋊C4—224no—
297C2×C22⋊C4—228no—
298C2×C22⋊C4—228no—
299C2×C22⋊C4—224no—
300C2×C22⋊C4—444no—
301C2×C22⋊C4—444no—
302C2×C22⋊C4—444no—
303C2×C22⋊C4—444no—
304C2×C22⋊C4—444no—
305C2×C22⋊C4—444no—
306C2×C22⋊C4—444no—
307C2×C22⋊C4—444no—
308C2×C22⋊C4C4×D1114yes—
309#95—44no—
310#95—444no—
311#95—444no—
312C2×C11—44no—
313C2×C11#230812yes—
314C2×C11#230811yescharacteristic
315C2×C11#237514yes—
316C2×C11#281214yes—
317C2×C11#281214yes—
318D11—48no—
319D11—82no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—114no—
324C24—114no—
325C24—114no—
326C24—1116no—
327C24—112no—
328C24—224no—
329C24—224no—
330C24—224no—
331C24—224no—
332C24—224no—
333C24—224no—
334C24—222no—
335C24—224no—
336C24—224no—
337C24—224no—
338C24—448no—
339C24—4416no—
340C24—448no—
341C24—4416no—
342C24—448no—
343C24—4416no—
344C24—4416no—
345C24—4416no—
346C24—4416no—
347C24—4416no—
348C24C2×C4×D1112yes—
349C24D4×D1114yes—
350C24C23×D1111yescharacteristic
351C2×D4—2216no—
352C2×D4—228no—
353C2×D4—4416no—
354C2×D4—448no—
355C2×D4—4416no—
356C2×D4—448no—
357C2×D4—448no—
358C2×D4—4416no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—112no—
371C22×C4—222no—
372C22×C4—224no—
373C22×C4—224no—
374C22×C4—224no—
375C22×C4—224no—
376C22×C4—224no—
377C22×C4—224no—
378C22×C4—224no—
379C22×C4—224no—
380C22×C4C2×C4×D1114yes—
381C22⋊C4—28no—
382C22⋊C4—48no—
383C22⋊C4—2216no—
384C22⋊C4—4416no—
385C22⋊C4—4416no—
386C22⋊C4—4416no—
387C22⋊C4—448no—
388C11#1465711yescharacteristic, complemented, Sylow
389C23—24no—
390C23—28no—
391C23—24no—
392C23—28no—
393C23—24no—
394C23—28no—
395C23—44no—
396C23—44no—
397C23—44no—
398C23—44no—
399C23—118no—
400C23—224no—
401C23—224no—
402C23—228no—
403C23—228no—
404C23—224no—
405C23—224no—
406C23—228no—
407C23—228no—
408C23—228no—
409C23—224no—
410C23—224no—
411C23—228no—
412C23—222no—
413C23—224no—
414C23—228no—
415C23—4432no—
416C23—4416no—
417C23—4464no—
418C23—4464no—
419C23—4464no—
420C23—444no—
421C23—444no—
422C23—444no—
423C23—444no—
424C23—444no—
425C23—444no—
426C23—444no—
427C23—444no—
428C23—444no—
429C23—442no—
430C23—444no—
431C23—444no—
432C23—442no—
433C23—444no—
434C23—444no—
435C23—444no—
436C23C22⋊C4×D1114yes—
437C23C22×C4×D1111yescharacteristic
438C23C2×D4×D1112yes—
439C23#6703714yes—
440C23#6704214yes—
441C23#6704214yes—
442D4—4432no—
443D4—4416no—
444C2×C4—24no—
445C2×C4—28no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—222no—
457C2×C4—224no—
458C2×C4—444no—
459C2×C4—444no—
460C2×C4—444no—
461C2×C4—444no—
462C2×C4—444no—
463C2×C4—444no—
464C2×C4—444no—
465C2×C4—444no—
466C22—28no—
467C22—24no—
468C22—48no—
469C22—48no—
470C22—48no—
471C22—228no—
472C22—228no—
473C22—228no—
474C22—4464no—
475C22—448no—
476C22—448no—
477C22—448no—
478C22—448no—
479C22—448no—
480C22—448no—
481C22—444no—
482C22—442no—
483C22—448no—
484C22—884no—
485C22—884no—
486C22C2×C22⋊C4×D1111yescharacteristic
487C22#1062305212yes—
488C22#1062307614yes—
489C22#1062312012yes—
490C22#1062313614yes—
491C22#1062315114yes—
492C22#1062318412yes—
493C22#1062318414yes—
494C22#1062319714yes—
495C22#1062321414yes—
496C22#1062324214yes—
497C4—44no—
498C4—84no—
499C4—442no—
500C4—884no—
501C2—44no—
502C2—448no—
503C2—882no—
504C2#1307119012yes—
505C2#1307119311yescharacteristic
506C2#1307175614yes—
507C2#1307661214yes—
508C2#1307661414yes—
subgroupG/H
center, FrattiniC24C23×D11
commutatorC23×C11C23×C4
FittingC11×#1440C2
socle, cosocleC24×C11C24
prime radicalC22×#66935C2
hypercenter#1440D11
hypocenterC11#14657
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ #1440

Complements

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

NH ≅ G/Ntype
2. C2×#10622568C2semidirect
3. #13066865C2semidirect
4. #13066864C2semidirect
5. #13066856C2semidirect
6. #13066855C2semidirect
7. C11×#1440C2semidirect
17. C22×(C2×D11)⋊C4C22semidirect
19. C2×#66935C4semidirect
20. C2×C11×#95C22semidirect
21. C2×C11⋊#95C22semidirect
22. C2×C11⋊#95C22semidirect
93. C23×D11C22⋊C4semidirect
139. #1440D11semidirect
234. C2×#95C2×D11semidirect
388. C11#14657semidirect

Maximal subgroups

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

subgroupindex
C22×#669352
C2×#106225682
#130668652
#130668642
#130668562
#130668552
C11×#14402
#1465711