Finite Group Databasebeta

← #14763109

Subgroups — #14763109

Order 2816.

1474 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66989C211yesmaximal, characteristic
2C2×#10622332C211yesmaximal, characteristic, complemented
3#13071018C211yesmaximal, characteristic, complemented
4#13069547C214yesmaximal, complemented
5#13069431C214yesmaximal, complemented
6#13068552C211yesmaximal, characteristic, complemented
7#13068550C211yesmaximal, characteristic, complemented
8#13068547C211yesmaximal, characteristic
9C11×#1786C211yesmaximal, characteristic
10C23×D4×C11C2211yescharacteristic
11C23×Dic22C2211yescharacteristic
12C22×C22⋊Dic11—21no—
13C22×C22⋊Dic11—21no—
14C2×#66989—21no—
15C2×#66989—21no—
16C2×#66989—24no—
17C2×#66989C418yescomplemented
18C2×#66989C2212yes—
19C2×#66989C2212yes—
20C22×C4×Dic11C2211yescharacteristic
21C2×#66930—21no—
22C2×C11×#102C2214yes—
23C2×C11×C22⋊C8C2211yescharacteristic
24C11×#380C2211yescharacteristic
25C2×#66896C2214yes—
26#10622825C2211yescharacteristic
27C2×C22⋊(C11⋊C8)C2211yescharacteristic
28C2×#66880C2214yescomplemented
29C2×#66879C2212yescomplemented
30C2×Dic11⋊C8C2212yescomplemented
31#10622589C2212yescomplemented
32C4⋊C4×Dic11C2212yescomplemented
33C2×#66874C2214yescomplemented
34#10622555—21no—
35#10622332—24no—
36#10622263—21no—
37#10622262—21no—
38C25×C11—21no—
39C22×D4×C11—21no—
40C22×D4×C11—22no—
41C22×D4×C11—44no—
42C22×D4×C11C2×C414yes—
43C22×D4×C11C2312yes—
44C23×C4×C11C2311yescharacteristic
45C23×Dic11—21no—
46C23×Dic11D411yescharacteristic
47C23×Dic11D411yescharacteristic
48C22×Dic22—21no—
49C22×Dic22—22no—
50C22×Dic22—44no—
51C22×Dic22C2×C414yes—
52C22×Dic22C2312yes—
53C22×C8×C11C2312yes—
54C2×C11×C4⋊C4C2312yes—
55C2×C11×C22⋊C4—21no—
56C2×C22⋊Dic11—28no—
57C2×C22⋊Dic11—28no—
58C2×C22⋊Dic11—22no—
59C2×C22⋊Dic11—22no—
60C2×C22⋊Dic11—22no—
61C2×C22⋊Dic11—22no—
62C2×C22⋊Dic11—21no—
63#66989—28no—
64#66989—28no—
65#66989—24no—
66#66989—24no—
67#66989C2×C418yes—
68#66989C2×C418yes—
69C2×C4⋊Dic11C2312yes—
70C2×Dic11⋊C4—22no—
71C2×Dic11⋊C4—22no—
72C2×C4×Dic11—22no—
73C2×C4×Dic11C2×C414yes—
74C2×C4×Dic11D411yescharacteristic
75C2×C4×Dic11D411yescharacteristic
76C2×C4×Dic11D414yes—
77C2×C4×Dic11C2311yescharacteristic
78C2×C4×Dic11C2311yescharacteristic
79C22×C11⋊C8C2312yes—
80#66930—44no—
81C11×#102—44no—
82C11×C22⋊C8—22no—
83C11×#95—21no—
84#66896—44no—
85C11⋊#95—21no—
86C11⋊#95—22no—
87C22⋊(C11⋊C8)—22no—
88#66880—44no—
89#66879—24no—
90Dic11⋊C8—24no—
91#66874—44no—
92#21808—111nomaximal, Sylow
93C24×C11—24no—
94C24×C11—22no—
95C24×C11—44no—
96C24×C11—44no—
97C24×C11—42no—
98C24×C11—42no—
99C24×C11C2×D411yescharacteristic
100C2×D4×C11—24no—
101C2×D4×C11—22no—
102C2×D4×C11—24no—
103C2×D4×C11—24no—
104C2×D4×C11—48no—
105C2×D4×C11—44no—
106C2×D4×C11—48no—
107C2×D4×C11C22×C414yes—
108C22×C4×C11—24no—
109C22×C4×C11—22no—
110C22×C4×C11C22×C412yes—
111C22×C4×C11C2×D411yescharacteristic
112C22×C4×C11C2×D412yes—
113C22×C4×C11C2411yescharacteristic
114C22×Dic11—24no—
115C22×Dic11—22no—
116C22×Dic11—22no—
117C22×Dic11—22no—
118C22×Dic11—22no—
119C22×Dic11—42no—
120C22×Dic11—44no—
121C22×Dic11C22⋊C414yes—
122C22×Dic11C22⋊C414yes—
123C22×Dic11C2×D411yescharacteristic
124C22×Dic11C2×D411yescharacteristic
125C22×Dic11C2×D411yescharacteristic
126C22×Dic11C2×D411yescharacteristic
127C2×Dic22—24no—
128C2×Dic22—22no—
129C2×Dic22—24no—
130C2×Dic22—24no—
131C2×Dic22—48no—
132C2×Dic22—44no—
133C2×Dic22—48no—
134C2×Dic22C22×C414yes—
135C2×C8×C11—22no—
136C2×C8×C11—24no—
137C11×C4⋊C4—24no—
138C11×C22⋊C4—42no—
139C22⋊Dic11—28no—
140C22⋊Dic11—28no—
141C22⋊Dic11—28no—
142C22⋊Dic11—28no—
143C22⋊Dic11—42no—
144C4⋊Dic11—24no—
145Dic11⋊C4—44no—
146Dic11⋊C4—44no—
147C4×Dic11—44no—
148C4×Dic11C22⋊C414yes—
149C4×Dic11C22⋊C414yes—
150C4×Dic11C22×C412yes—
151C4×Dic11C22×C412yes—
152C4×Dic11C2×D412yes—
153C4×Dic11C2×D412yes—
154C2×C11⋊C8—22no—
155C2×C11⋊C8—24no—
156C22×#124—111no—
157C2×#432—111no—
158#2285—111no—
159#1983—114no—
160#1959—114no—
161#1787—111no—
162#1787—111no—
163#1786—111no—
164#1786D1111yescharacteristic
165C23×C11—22no—
166C23×C11—22no—
167C23×C11—24no—
168C23×C11—48no—
169C23×C11—48no—
170C23×C11—44no—
171C23×C11—42no—
172C23×C11—42no—
173C23×C11—44no—
174C23×C11—48no—
175C23×C11—88no—
176C23×C11—88no—
177C23×C11C2×C22⋊C412yes—
178C23×C11AGL(1,ℤ/8)12yes—
179C23×C11#13712yes—
180C23×C11C22×D411yescharacteristic
181D4×C11—24no—
182D4×C11—24no—
183D4×C11—48no—
184D4×C11—44no—
185C2×C4×C11—22no—
186C2×C4×C11—24no—
187C2×C4×C11—24no—
188C2×C4×C11—42no—
189C2×C4×C11—42no—
190C2×C4×C11—42no—
191C2×C4×C11C2×C22⋊C412yes—
192C2×C4×C11C23×C411yescharacteristic
193C2×C4×C11C22×D411yescharacteristic
194C2×C4×C11H(5,𝔽2)14yes—
195C2×Dic11—21no—
196C2×Dic11—21no—
197C2×Dic11—21no—
198C2×Dic11—21no—
199C2×Dic11—22no—
200C2×Dic11—22no—
201C2×Dic11—28no—
202C2×Dic11—24no—
203C2×Dic11—44no—
204C2×Dic11—48no—
205C2×Dic11—44no—
206C2×Dic11—42no—
207C2×Dic11—42no—
208C2×Dic11—42no—
209C2×Dic11C2×C22⋊C412yes—
210C2×Dic11C2×C22⋊C412yes—
211C2×Dic11C2×C22⋊C412yes—
212C2×Dic11C2×C22⋊C412yes—
213Dic22—24no—
214Dic22—24no—
215Dic22—48no—
216Dic22—44no—
217C8×C11—42no—
218C11⋊C8—42no—
219C23×Q8—111no—
220C23×D4C2×D1111yescharacteristic
221C2×#124—112no—
222C2×#124—112no—
223C2×#124—118no—
224C2×#124—221no—
225C2×#124—221no—
226C2×#124—224no—
227C2×#117—221no—
228C22×C22⋊C4—221no—
229C22×C22⋊C4—221no—
230C22×C42—111no—
231#432—224no—
232C2×C4⋊C8—112no—
233C2×#103—114no—
234C2×#103—114no—
235C2×#102—114no—
236C2×#102C2×D1114yes—
237C2×C22⋊C8—111no—
238C2×C22⋊C8C2×D1111yescharacteristic
239C2×#97—112no—
240#384—112no—
241#382—221no—
242#380—111no—
243#380—221no—
244#380C2×D1111yescharacteristic
245C4×C4⋊C4—112no—
246C4×C4⋊C4—221no—
247C22×C11—22no—
248C22×C11—22no—
249C22×C11—24no—
250C22×C11—24no—
251C22×C11—44no—
252C22×C11—44no—
253C22×C11—44no—
254C22×C11—816no—
255C22×C11—84no—
256C22×C11—84no—
257C22×C11#41714yes—
258C22×C11C22×C22⋊C411yescharacteristic
259C22×C11#53512yes—
260C22×C11C2×AGL(1,ℤ/8)11yescharacteristic
261C22×C11C2×#13711yescharacteristic
262C22×C11#57512yes—
263C4×C11—22no—
264C4×C11—82no—
265C4×C11#51812yes—
266C4×C11#53512yes—
267Dic11—22no—
268Dic11—22no—
269Dic11—22no—
270Dic11—22no—
271Dic11—44no—
272Dic11—82no—
273C25—21no—
274C22×Q8—114no—
275C22×Q8—112no—
276C22×Q8—222no—
277C22×Q8—221no—
278C22×Q8—444no—
279C22×D4—21no—
280C22×D4—22no—
281C22×D4—44no—
282C22×D4C4×D1114yes—
283C22×D4C22×D1112yes—
284C23×C4—111no—
285C23×C4—111no—
286C23×C4—221no—
287C23×C4C22×D1111yescharacteristic
288C22×C8—112no—
289C22×C8C22×D1112yes—
290#124—118no—
291#124—118no—
292#124—228no—
293#124—228no—
294#124—224no—
295#124—224no—
296#117—444no—
297C2×C4⋊C4—112no—
298C2×C4⋊C4—222no—
299C2×C4⋊C4—222no—
300C2×C4⋊C4C22×D1112yes—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—228no—
303C2×C22⋊C4—222no—
304C2×C22⋊C4—222no—
305C2×C22⋊C4—222no—
306C2×C22⋊C4—222no—
307C2×C22⋊C4—228no—
308C2×C22⋊C4—221no—
309C2×C42—114no—
310C2×C42—111no—
311C2×C42—111no—
312C2×C42—114no—
313C2×C42—111no—
314C2×C42—111no—
315C2×C42—222no—
316C4⋊C8—224no—
317#103—444no—
318#103—444no—
319#102—44no—
320#102—444no—
321C22⋊C8—22no—
322C22⋊C8—222no—
323#97—224no—
324#95—21no—
325#95—222no—
326#95—221no—
327C2×C11—24no—
328C2×C11—84no—
329C2×C11#230911yescharacteristic
330C2×C11C2×#41711yescharacteristic
331C2×C11#289611yescharacteristic
332C2×C11#321312yes—
333C2×C11#321512yes—
334C24—24no—
335C24—22no—
336C24—44no—
337C24—44no—
338C24—42no—
339C24—42no—
340C24D4×D1111yescharacteristic
341C2×Q8—114no—
342C2×Q8—224no—
343C2×Q8—222no—
344C2×Q8—224no—
345C2×Q8—224no—
346C2×Q8—448no—
347C2×Q8—444no—
348C2×Q8—448no—
349C2×D4—24no—
350C2×D4—22no—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—48no—
354C2×D4—44no—
355C2×D4—48no—
356C2×D4C2×C4×D1114yes—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—114no—
360C22×C4—111no—
361C22×C4—111no—
362C22×C4—111no—
363C22×C4—111no—
364C22×C4—114no—
365C22×C4—222no—
366C22×C4—222no—
367C22×C4—224no—
368C22×C4—222no—
369C22×C4—222no—
370C22×C4—442no—
371C22×C4—444no—
372C22×C4C2×C4×D1112yes—
373C22×C4D4×D1111yescharacteristic
374C22×C4D4×D1112yes—
375C22×C4C23×D1111yescharacteristic
376C2×C8—22no—
377C2×C8—24no—
378C2×C8—222no—
379C2×C8—224no—
380C4⋊C4—24no—
381C4⋊C4—224no—
382C4⋊C4—444no—
383C4⋊C4—444no—
384C22⋊C4—42no—
385C22⋊C4—228no—
386C22⋊C4—228no—
387C22⋊C4—228no—
388C22⋊C4—228no—
389C22⋊C4—442no—
390C42—114no—
391C42—114no—
392C42—112no—
393C42—112no—
394C42—112no—
395C42—112no—
396C42—444no—
397C11#2180811yescharacteristic, complemented, Sylow
398C23—22no—
399C23—22no—
400C23—24no—
401C23—48no—
402C23—48no—
403C23—42no—
404C23—42no—
405C23—44no—
406C23—44no—
407C23—48no—
408C23—88no—
409C23—88no—
410C23C22⋊C4×D1112yes—
411C23#6696412yes—
412C23#6696812yes—
413C23C2×D4×D1111yescharacteristic
414Q8—224no—
415Q8—224no—
416Q8—448no—
417Q8—444no—
418D4—24no—
419D4—24no—
420D4—48no—
421D4—44no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—112no—
429C2×C4—112no—
430C2×C4—112no—
431C2×C4—112no—
432C2×C4—221no—
433C2×C4—221no—
434C2×C4—221no—
435C2×C4—221no—
436C2×C4—222no—
437C2×C4—222no—
438C2×C4—228no—
439C2×C4—224no—
440C2×C4—444no—
441C2×C4—448no—
442C2×C4—444no—
443C2×C4—442no—
444C2×C4—442no—
445C2×C4—442no—
446C2×C4C22⋊C4×D1112yes—
447C2×C4C22×C4×D1111yescharacteristic
448C2×C4C2×D4×D1111yescharacteristic
449C2×C4#6703714yes—
450C8—42no—
451C8—442no—
452C22—24no—
453C22—24no—
454C22—22no—
455C22—22no—
456C22—44no—
457C22—44no—
458C22—44no—
459C22—816no—
460C22—84no—
461C22—84no—
462C22#1062238414yes—
463C22C2×C22⋊C4×D1111yescharacteristic
464C22#1062312312yes—
465C22C2×#6696411yescharacteristic
466C22C2×#6696811yescharacteristic
467C22#1062330412yes—
468C4—22no—
469C4—82no—
470C4—222no—
471C4—222no—
472C4—222no—
473C4—222no—
474C4—444no—
475C4—882no—
476C4#1062301712yes—
477C4#1062312312yes—
478C2—24no—
479C2—84no—
480C2#1307119711yescharacteristic
481C2C2×#1062238411yescharacteristic
482C2#1307713011yescharacteristic
483C2#1307960012yes—
484C2#1307961612yes—
subgroupG/H
centerC23C2×D4×D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1786C2
socleC23×C11C22×D4
prime radicalC23×D4×C11C22
cosocleC22×C4×C11C24
hypercenter#1786D11
hypocenterC11#21808
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1786

Complements

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

NH ≅ G/Ntype
2. C2×#10622332C2semidirect
3. #13071018C2semidirect
4. #13069547C2semidirect
5. #13069431C2semidirect
6. #13068552C2semidirect
7. #13068550C2semidirect
17. C2×#66989C4semidirect
28. C2×#66880C22semidirect
29. C2×#66879C22semidirect
30. C2×Dic11⋊C8C22semidirect
31. #10622589C22semidirect
32. C4⋊C4×Dic11C22semidirect
33. C2×#66874C22semidirect
397. C11#21808semidirect

Maximal subgroups

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

subgroupindex
C22×#669892
C2×#106223322
#130710182
#130695472
#130694312
#130685522
#130685502
#130685472
C11×#17862
#2180811