Finite Group Databasebeta

← #14762929

Subgroups — #14762929

Order 2816.

1318 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13082260C211yesmaximal, characteristic
2#13077074C212yesmaximal, complemented
3#13069155C214yesmaximal, complemented
4#13069026C214yesmaximal, complemented
5#13068722C212yesmaximal, complemented
6#13068534C211yesmaximal, characteristic, complemented
7C11×#1782C211yesmaximal, characteristic, complemented
8C22×C11×#41C2211yescharacteristic
9C2×#67037C2211yescharacteristic
10#10623360—24no—
11#10623355—24no—
12#10623347C412yescomplemented
13#10623335C412yescomplemented
14#10623335C2212yes—
15#10623329C2211yescharacteristic
16#10623322C412yescomplemented
17C2×#66982C412yescomplemented
18C2×#66982C2212yes—
19#10623017C2214yes—
20C2×C11×#99C2212yes—
21C2×C11×#99C2212yescomplemented
22C11×#381C2212yescomplemented
23C2×#66898C2212yes—
24C2×#66898C2212yescomplemented
25#10622824C2212yescomplemented
26C2×#66870C2214yes—
27D11×#99—24no—
28#10622353—24no—
29#10622165C2214yescomplemented
30#10622164C2214yescomplemented
31C2×C11×#41—24no—
32C2×C11×#41—42no—
33C2×C11×#41—42no—
34C22×Q8×C11C2×C411yescharacteristic, complemented
35C22×D4×C11C2×C411yescharacteristic, complemented
36C22×D4×C11C2311yescharacteristic
37C22×D4×C11C2311yescharacteristic
38C23×C4×C11C2×C411yescharacteristic, complemented
39C23×C4×C11C2×C411yescharacteristic, complemented
40C23×C4×C11C2311yescharacteristic
41C22×C22⋊D11C2×C412yes—
42C22×C22⋊D11C2312yes—
43#67037—42no—
44#67037—42no—
45C2×#4341D414yes—
46C2×D4×D11D414yes—
47C2×#4339C2×C412yes—
48C2×C11×C22⋊C4C2314yes—
49#67006C2×C414yescomplemented
50C2×C22⋊Dic11C2×C412yescomplemented
51C2×C22⋊Dic11C2314yes—
52C2×C22⋊Dic11C2312yes—
53#67000—22no—
54#66999—24no—
55#66997—22no—
56#66992—24no—
57#66992—24no—
58#66990—22no—
59#66990—22no—
60#66988—22no—
61#66988—22no—
62D4×Dic11—24no—
63#66983—22no—
64#66982—24no—
65#66982—24no—
66#66982C2×C414yescomplemented
67#66982C2×C414yescomplemented
68#66982C2×C414yescomplemented
69C4×C22⋊D11—24no—
70C4×C22⋊D11—44no—
71C2×(C2×D11)⋊C4—22no—
72C2×(C2×D11)⋊C4—22no—
73#66979C2×C412yescomplemented
74#66977—22no—
75C2×Dic11⋊C4—22no—
76C2×Dic11⋊C4—22no—
77(C22⋊D11)⋊C4—44no—
78C22⋊C4×D11—24no—
79#66930—24no—
80C11×#99—24no—
81C11×#99—42no—
82C11×#95—42no—
83#66898—24no—
84#66898—42no—
85C11⋊#95—42no—
86#66870—28no—
87#21777—111nomaximal, Sylow
88C24×C11C22×C411yescharacteristic
89C24×C11C22×C411yescharacteristic
90C24×C11C2411yescharacteristic
91C23×D11C22⋊C412yes—
92C23×D11C2×D412yes—
93C11×#41—416no—
94C2×Q8×C11—22no—
95C2×Q8×C11—41no—
96C2×Q8×C11—41no—
97C2×D4×C11—22no—
98C2×D4×C11—22no—
99C2×D4×C11—22no—
100C2×D4×C11—22no—
101C2×D4×C11—22no—
102C2×D4×C11—22no—
103C2×D4×C11—41no—
104C2×D4×C11—41no—
105C2×D4×C11C2×D414yes—
106C22×C4×C11—22no—
107C22×C4×C11—22no—
108C22×C4×C11—22no—
109C22×C4×C11—22no—
110C22×C4×C11—22no—
111C22×C4×C11—24no—
112C22×C4×C11—41no—
113C22×C4×C11—41no—
114C22×C4×C11—41no—
115C22×C4×C11—41no—
116C22×C4×C11C22×C411yescharacteristic
117C22×C4×C11C22×C411yescharacteristic
118C22×C4×C11C22×C411yescharacteristic
119C22×C4×C11C22×C411yescharacteristic
120C2×C22⋊D11—24no—
121C2×C22⋊D11—24no—
122C2×C22⋊D11—24no—
123C2×C22⋊D11—42no—
124C2×C22⋊D11—42no—
125C2×C22⋊D11C22⋊C414yes—
126C2×C22⋊D11C22×C414yes—
127C2×C22⋊D11C2×D414yes—
128C22×Dic11—22no—
129C22×Dic11—22no—
130C22×Dic11—24no—
131C22×Dic11C22⋊C412yes—
132C22×Dic11C2×D412yes—
133#4341—48no—
134D4×D11—48no—
135#4339—42no—
136#4339—42no—
137C2×D44C22⋊C412yes—
138C2×C4×D11—44no—
139C2×C4×D11C22⋊C414yes—
140C2×Dic22C22⋊C412yes—
141C11×C22⋊C4—24no—
142C11×C22⋊C4—24no—
143C22⋊Dic11—24no—
144C22⋊Dic11—22no—
145C22⋊Dic11—22no—
146C22⋊Dic11—24no—
147C22⋊Dic11—24no—
148C22⋊Dic11C22×C414yes—
149(C2×D11)⋊C4—24no—
150(C2×D11)⋊C4—24no—
151(C2×D11)⋊C4—24no—
152(C2×D11)⋊C4—44no—
153C4⋊Dic11—22no—
154Dic11⋊C4—24no—
155Dic11⋊C4—24no—
156Dic11⋊C4—24no—
157Dic11⋊C4—44no—
158C4×Dic11—22no—
159C4×Dic11—44no—
160#3485—111no—
161#2883—112no—
162#1913—114no—
163#1890—114no—
164#1828—112no—
165#1782—111no—
166#1782D1111yescharacteristic, complemented
167C23×C11—21no—
168C23×C11—21no—
169C23×C11—21no—
170C23×C11—21no—
171C23×C11—21no—
172C23×C11—81no—
173C23×C11—81no—
174C23×C11—81no—
175C23×C11C2×C22⋊C412yes—
176C23×C11C23×C411yescharacteristic
177C23×C11C22×D412yes—
178C23×C11H(5,𝔽2)12yes—
179C23×C11H(5,𝔽2)12yes—
180C22×D11—24no—
181C22×D11—42no—
182C22×D11—82no—
183C22×D11—82no—
184C22×D11C2×C22⋊C414yes—
185Q8×C11—44no—
186D4×C11—28no—
187D4×C11—44no—
188D4×C11—44no—
189C2×C4×C11—21no—
190C2×C4×C11—21no—
191C2×C4×C11—21no—
192C2×C4×C11—21no—
193C2×C4×C11—21no—
194C2×C4×C11—21no—
195C2×C4×C11—21no—
196C2×C4×C11—21no—
197C2×C4×C11—21no—
198C2×C4×C11—21no—
199C2×C4×C11—21no—
200C2×C4×C11—44no—
201C2×C4×C11—44no—
202C2×C4×C11—44no—
203C2×C4×C11—44no—
204C2×C4×C11—84no—
205C2×C4×C11C2×C22⋊C412yes—
206C22⋊D11—24no—
207C22⋊D11—24no—
208C22⋊D11—28no—
209C22⋊D11—44no—
210C22⋊D11—84no—
211C2×Dic11—24no—
212C2×Dic11—24no—
213C2×Dic11—42no—
214C2×Dic11—44no—
215C2×Dic11—82no—
216C2×Dic11—82no—
217C2×Dic11—84no—
218C2×Dic11C2×C22⋊C414yes—
219D44—42no—
220C4×D11—44no—
221C4×D11—164no—
222Dic22—42no—
223C2×H(5,𝔽2)—111no—
224C22×#41C2×D1111yescharacteristic
225#547—224no—
226#546—224no—
227#536—112no—
228#534—112no—
229#534—112no—
230#525—112no—
231C2×#123—112no—
232C2×#123—112no—
233#518—111no—
234#518—114no—
235#410—224no—
236C2×#99—112no—
237C2×#99—114no—
238C2×#99—112no—
239C2×#99—224no—
240C2×#99C2×D1112yes—
241C2×#99C2×D1112yescomplemented
242#381—112no—
243#381C2×D1112yescomplemented
244#342—114no—
245#342—114no—
246C22×C11—22no—
247C22×C11—42no—
248C22×C11—42no—
249C22×C11—41no—
250C22×C11—42no—
251C22×C11—42no—
252C22×C11—42no—
253C22×C11—41no—
254C22×C11—81no—
255C22×C11—81no—
256C22×C11—81no—
257C22×C11—81no—
258C22×C11C22×C22⋊C411yescharacteristic
259C22×C11#51811yescharacteristic
260C22×C11#51811yescharacteristic
261C22×C11#53414yes—
262C2×D11—24no—
263C2×D11—44no—
264C2×D11—84no—
265C2×D11—162no—
266C4×C11—22no—
267C4×C11—41no—
268C4×C11—41no—
269C4×C11—41no—
270C4×C11—84no—
271Dic11—24no—
272Dic11—44no—
273Dic11—84no—
274H(5,𝔽2)—442no—
275H(5,𝔽2)—442no—
276C2×#41—24no—
277C2×#41—42no—
278C2×#41—42no—
279C2×#41—114no—
280C2×#41—112no—
281C22×Q8C4×D1111yescharacteristic, complemented
282C22×D4—112no—
283C22×D4—112no—
284C22×D4—114no—
285C22×D4C4×D1111yescharacteristic, complemented
286C22×D4C22×D1111yescharacteristic
287C22×D4C22×D1111yescharacteristic
288C23×C4C4×D1111yescharacteristic, complemented
289C23×C4C4×D1111yescharacteristic, complemented
290C23×C4C22×D1111yescharacteristic
291C4⋊Q8—222no—
292#124—222no—
293#123—114no—
294#123—114no—
295#123—114no—
296#123—224no—
297#123—224no—
298#123—222no—
299#123—222no—
300C22⋊Q8—222no—
301C22⋊Q8—224no—
302#121—222no—
303#121—224no—
304#121—224no—
305#120—114no—
306#120—222no—
307#120—222no—
308C4×D4—224no—
309C4×D4—224no—
310C4×D4—444no—
311C4×D4—444no—
312#117—112no—
313#117—224no—
314C2×C4⋊C4—222no—
315C2×C4⋊C4—222no—
316C2×C22⋊C4—112no—
317C2×C22⋊C4—112no—
318C2×C22⋊C4—114no—
319C2×C22⋊C4—224no—
320C2×C22⋊C4—222no—
321C2×C22⋊C4—222no—
322C2×C22⋊C4C22×D1114yes—
323#99—24no—
324#99—42no—
325#99—228no—
326#99—224no—
327#99—442no—
328#95—42no—
329#95—442no—
330C2×C11—21no—
331C2×C11—24no—
332C2×C11—21no—
333C2×C11—21no—
334C2×C11—42no—
335C2×C11—81no—
336C2×C11#230811yescharacteristic
337D11—44no—
338C24—112no—
339C24—112no—
340C24C2×C4×D1111yescharacteristic
341C24C2×C4×D1111yescharacteristic
342C24C23×D1111yescharacteristic
343#41—416no—
344#41—442no—
345#41—442no—
346#41—448no—
347C2×Q8—22no—
348C2×Q8—41no—
349C2×Q8—41no—
350C2×Q8—112no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—114no—
360C2×D4—114no—
361C2×D4—114no—
362C2×D4—112no—
363C2×D4—224no—
364C2×D4—224no—
365C2×D4—224no—
366C2×D4—448no—
367C2×D4—442no—
368C2×D4—442no—
369C2×D4D4×D1114yes—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—112no—
381C22×C4—112no—
382C22×C4—114no—
383C22×C4—222no—
384C22×C4—222no—
385C22×C4—224no—
386C22×C4—444no—
387C22×C4C2×C4×D1111yescharacteristic
388C22×C4C2×C4×D1111yescharacteristic
389C22×C4C2×C4×D1111yescharacteristic
390C22×C4C2×C4×D1111yescharacteristic
391C4⋊C4—224no—
392C4⋊C4—224no—
393C4⋊C4—224no—
394C4⋊C4—222no—
395C4⋊C4—444no—
396C22⋊C4—24no—
397C22⋊C4—24no—
398C22⋊C4—114no—
399C22⋊C4—224no—
400C22⋊C4—224no—
401C22⋊C4—224no—
402C22⋊C4—224no—
403C22⋊C4—222no—
404C22⋊C4—222no—
405C22⋊C4—224no—
406C22⋊C4—224no—
407C22⋊C4—444no—
408C42—222no—
409C42—444no—
410C11#2177711yescharacteristic, complemented, Sylow
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23—114no—
420C23—224no—
421C23—442no—
422C23—882no—
423C23—882no—
424C23C22⋊C4×D1112yes—
425C23C22×C4×D1111yescharacteristic
426C23C2×D4×D1112yes—
427C23#6703712yes—
428C23#6703712yes—
429Q8—44no—
430Q8—442no—
431D4—28no—
432D4—44no—
433D4—44no—
434D4—224no—
435D4—224no—
436D4—228no—
437D4—442no—
438D4—444no—
439D4—884no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—84no—
456C2×C4—114no—
457C2×C4—224no—
458C2×C4—224no—
459C2×C4—444no—
460C2×C4—442no—
461C2×C4—444no—
462C2×C4—882no—
463C2×C4—882no—
464C2×C4—884no—
465C2×C4—1764no—
466C2×C4C22⋊C4×D1112yes—
467C22—22no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—41no—
472C22—42no—
473C22—42no—
474C22—41no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22—81no—
479C22—224no—
480C22—444no—
481C22—884no—
482C22—1762no—
483C22C2×C22⋊C4×D1111yescharacteristic
484C22#1062301711yescharacteristic
485C22#1062301711yescharacteristic
486C22#1062312014yes—
487C4—22no—
488C4—41no—
489C4—41no—
490C4—41no—
491C4—84no—
492C4—224no—
493C4—444no—
494C4—884no—
495C2—24no—
496C2—21no—
497C2—21no—
498C2—21no—
499C2—42no—
500C2—81no—
501C2—444no—
502C2#1307119211yescharacteristic
subgroupG/H
centerC2#13071192
commutatorC23×C11C23×C4
FrattiniC24C23×D11
FittingC11×#1782C2
socleC2×C11#2308
prime radical#13082260C2
cosocleC24×C11C24
hypercenter#1782D11
hypocenterC11#21777
LCSG ▷ C23×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C2 ◁ C24 ◁ #1782

Complements

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

NH ≅ G/Ntype
2. #13077074C2semidirect
3. #13069155C2semidirect
4. #13069026C2semidirect
5. #13068722C2semidirect
6. #13068534C2semidirect
7. C11×#1782C2semidirect
12. #10623347C4semidirect
13. #10623335C4semidirect
16. #10623322C4semidirect
17. C2×#66982C4semidirect
21. C2×C11×#99C22semidirect
22. C11×#381C22semidirect
24. C2×#66898C22semidirect
25. #10622824C22semidirect
29. #10622165C22semidirect
30. #10622164C22semidirect
34. C22×Q8×C11C2×C4semidirect
35. C22×D4×C11C2×C4semidirect
38. C23×C4×C11C2×C4semidirect
39. C23×C4×C11C2×C4semidirect
49. #67006C2×C4semidirect
50. C2×C22⋊Dic11C2×C4semidirect
66. #66982C2×C4semidirect
67. #66982C2×C4semidirect
68. #66982C2×C4semidirect
73. #66979C2×C4semidirect
166. #1782D11semidirect
241. C2×#99C2×D11semidirect
243. #381C2×D11semidirect
281. C22×Q8C4×D11semidirect
285. C22×D4C4×D11semidirect
288. C23×C4C4×D11semidirect
289. C23×C4C4×D11semidirect
410. C11#21777semidirect

Maximal subgroups

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

subgroupindex
#130822602
#130770742
#130691552
#130690262
#130687222
#130685342
C11×#17822
#2177711