Finite Group Databasebeta

← #14789918

Subgroups — #14789918

Order 2816.

1066 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
1C2×#10623001C211yesmaximal, characteristic
2C2×#10622442C212yesmaximal, complemented
3C4×#66977C211yesmaximal, characteristic
4#13069950C211yesmaximal, characteristic, complemented
5#13069945C211yesmaximal, characteristic, complemented
6#13069758C211yesmaximal, characteristic, complemented
7#13069758C211yesmaximal, characteristic, complemented
8#13069571C212yesmaximal, complemented
9#13068926C212yesmaximal
10#13068817C212yesmaximal
11C11×#1797C211yesmaximal, characteristic
12C22×C42×C11C2211yescharacteristic
13C2×#66979C2212yes—
14C2×#66977—21no—
15C2×#66977—21no—
16C2×#66977C2211yescharacteristic
17#10623001—28no—
18C2×#66923C2212yes—
19C2×#66922C2211yescharacteristic
20C2×#66922C2211yescharacteristic
21C2×C4×Dic22—21no—
22C2×C4×Dic22—21no—
23C2×C4×Dic22C2211yescharacteristic
24C2×C11×C4⋊C8C2212yes—
25C2×C11×C22⋊C8C2212yes—
26#10622825C2211yescharacteristic
27#10622823—21no—
28C4×C22⋊Dic11—21no—
29C2×#66882C2212yes—
30C2×#66881C2212yes—
31C2×#66880C2212yes—
32C2×#66880C2212yes—
33C4×C4⋊Dic11C2211yescharacteristic
34#10622546—21no—
35C4×Dic11⋊C4—21no—
36#10622542C2211yescharacteristic
37#10622541C2211yescharacteristic
38#10622442—28no—
39C11×#336C2211yescharacteristic
40C11×#336C2211yescharacteristic
41#10622189C2212yes—
42#10622189C2212yes—
43#10622189C2212yes—
44#10622189C2212yes—
45C23×C4×C11D411yescharacteristic
46C23×C4×C11D411yescharacteristic
47C23×C4×C11C2311yescharacteristic
48C22×Dic22—21no—
49C22×Dic22—21no—
50C22×Dic22C2311yescharacteristic
51C22×C8×C11C2312yes—
52C22×C8×C11C2312yes—
53C2×C42×C11—24no—
54C2×C42×C11D411yescharacteristic
55C2×C42×C11D411yescharacteristic
56C2×C42×C11C2311yescharacteristic
57C2×C42×C11C2311yescharacteristic
58C2×C22⋊Dic11—22no—
59C2×C22⋊Dic11—21no—
60C2×C22⋊Dic11—21no—
61#66979—24no—
62#66979—24no—
63C2×C4⋊Dic11—21no—
64C2×C4⋊Dic11—21no—
65C2×C4⋊Dic11C2311yescharacteristic
66C2×C4⋊Dic11C2311yescharacteristic
67C2×C4⋊Dic11C2312yes—
68C2×C4⋊Dic11C2312yes—
69C2×C4⋊Dic11C2311yescharacteristic
70#66977—28no—
71#66977—28no—
72#66977—22no—
73#66977Q814yescomplemented
74C2×Dic11⋊C4—22no—
75C2×Dic11⋊C4—22no—
76C2×Dic11⋊C4—21no—
77C2×Dic11⋊C4—21no—
78C2×Dic11⋊C4—21no—
79C2×C4×Dic11—22no—
80C2×C4×Dic11—21no—
81C2×C4×Dic11—21no—
82#66923—24no—
83#66923D418yes—
84#66922—24no—
85#66922—24no—
86C4×Dic22—28no—
87C4×Dic22—28no—
88C4×Dic22—24no—
89C11×C4⋊C8—24no—
90C11×C22⋊C8—24no—
91C11⋊#95—21no—
92C11⋊#95—21no—
93C11⋊#95—21no—
94C11⋊#95—21no—
95C11⋊#95—21no—
96#66882—24no—
97#66881—24no—
98#66880—24no—
99#66880—24no—
100#24306—111nomaximal, Sylow
101C24×C11C2×D411yescharacteristic
102C22×C4×C11—22no—
103C22×C4×C11—22no—
104C22×C4×C11—22no—
105C22×C4×C11—22no—
106C22×C4×C11—22no—
107C22×C4×C11—22no—
108C22×C4×C11—41no—
109C22×C4×C11—41no—
110C22×C4×C11C2×D411yescharacteristic
111C22×C4×C11C2×D411yescharacteristic
112C22×C4×C11C2×D411yescharacteristic
113C22×C4×C11C2×D411yescharacteristic
114C22×C4×C11C2×D411yescharacteristic
115C22×C4×C11C2×Q812yes—
116C22×C4×C11#4112yes—
117C22×C4×C11C2411yescharacteristic
118C22×Dic11—21no—
119C22×Dic11—21no—
120C22×Dic11—22no—
121C22×Dic11—22no—
122C22×Dic11—21no—
123C22×Dic11—21no—
124C2×Dic22—24no—
125C2×Dic22—24no—
126C2×Dic22—22no—
127C2×Dic22—21no—
128C2×Dic22—22no—
129C2×Dic22—42no—
130C2×Dic22—42no—
131C2×Dic22—42no—
132C2×Dic22—42no—
133C2×Dic22C2×Q812yes—
134C2×C8×C11—22no—
135C2×C8×C11—22no—
136C2×C8×C11—22no—
137C2×C8×C11—22no—
138C2×C8×C11—22no—
139C2×C8×C11—22no—
140C42×C11—22no—
141C42×C11—22no—
142C42×C11—22no—
143C42×C11C2×D414yes—
144C22⋊Dic11—28no—
145C22⋊Dic11—24no—
146C22⋊Dic11—42no—
147C4⋊Dic11—24no—
148C4⋊Dic11—24no—
149C4⋊Dic11—22no—
150C4⋊Dic11—22no—
151C4⋊Dic11—22no—
152C4⋊Dic11—21no—
153C4⋊Dic11—22no—
154C4⋊Dic11C2×D414yes—
155C4⋊Dic11C2×D414yes—
156C4⋊Dic11C2×Q812yes—
157Dic11⋊C4—28no—
158Dic11⋊C4—28no—
159Dic11⋊C4—24no—
160Dic11⋊C4—42no—
161Dic11⋊C4—42no—
162C4×Dic11—28no—
163C4×Dic11—42no—
164C4×Dic11—42no—
165C2×#533—111no—
166C2×#479—112no—
167C4×C22⋊Q8—111no—
168#2058—111no—
169#2057—111no—
170#2024—111no—
171#2024—111no—
172#1989—112no—
173#1873—112no—
174#1854—112no—
175#1797D1111yescharacteristic
176C23×C11—21no—
177C23×C11—21no—
178C23×C11—21no—
179C23×C11—21no—
180C23×C11—21no—
181C23×C11—21no—
182C23×C11C22⋊Q812yes—
183C23×C11C22×D411yescharacteristic
184C2×C4×C11—21no—
185C2×C4×C11—21no—
186C2×C4×C11—21no—
187C2×C4×C11—21no—
188C2×C4×C11—21no—
189C2×C4×C11—21no—
190C2×C4×C11—21no—
191C2×C4×C11—21no—
192C2×C4×C11—21no—
193C2×C4×C11—21no—
194C2×C4×C11—24no—
195C2×C4×C11—24no—
196C2×C4×C11—44no—
197C2×C4×C11—44no—
198C2×C4×C11#12012yes—
199C2×C4×C11#12012yes—
200C2×C4×C11C22⋊Q812yes—
201C2×C4×C11#13512yes—
202C2×C4×C11#13712yes—
203C2×C4×C11C22×D412yes—
204C2×C4×C11C22×Q811yescharacteristic
205C2×C4×C11C2×#4111yescharacteristic
206C2×C4×C112−1+412yes—
207C2×Dic11—24no—
208C2×Dic11—24no—
209C2×Dic11—22no—
210C2×Dic11—22no—
211C2×Dic11—41no—
212C2×Dic11—41no—
213C2×Dic11—41no—
214C2×Dic11—42no—
215C2×Dic11—42no—
216C2×Dic11—41no—
217C2×Dic11—42no—
218C2×Dic11—42no—
219C2×Dic11—41no—
220C2×Dic11—41no—
221C2×Dic11—41no—
222C2×Dic11—41no—
223C2×Dic11—41no—
224C2×Dic11—41no—
225Dic22—24no—
226Dic22—48no—
227Dic22—48no—
228Dic22—42no—
229C8×C11—42no—
230C8×C11—42no—
231#533—228no—
232C2×C4⋊Q8—111no—
233C2×C4⋊Q8—111no—
234C2×#125—112no—
235C2×C22⋊Q8—111no—
236C2×C22⋊Q8—221no—
237C2×C22⋊Q8—221no—
238C2×C4×Q8—111no—
239C2×C4×Q8—221no—
240C2×C4×Q8—221no—
241C2×#117—112no—
242C22×C42C2×D1111yescharacteristic
243#479—228no—
244C2×#107—112no—
245C2×#106—112no—
246C2×C4⋊C8C2×D1112yes—
247C2×#103—112no—
248C2×#103—112no—
249C2×C22⋊C8C2×D1112yes—
250#391—111no—
251#389—111no—
252#387—221no—
253#385—221no—
254#380—111no—
255C4×C4⋊C4—111no—
256C4×C4⋊C4—221no—
257C4×C22⋊C4—221no—
258#340—112no—
259#340—112no—
260#340—112no—
261#340—112no—
262#336C2×D1111yescharacteristic
263#336C2×D1111yescharacteristic
264C22×C11—22no—
265C22×C11—22no—
266C22×C11—22no—
267C22×C11—41no—
268C22×C11—41no—
269C22×C11—41no—
270C22×C11—41no—
271C22×C11C2×#12011yescharacteristic
272C22×C11C2×C22⋊Q811yescharacteristic
273C22×C11#53611yescharacteristic
274C22×C11C2×#13511yescharacteristic
275C22×C11C2×#13711yescharacteristic
276C22×C11#57511yescharacteristic
277C22×C11#57511yescharacteristic
278C4×C11—22no—
279C4×C11—22no—
280C4×C11—22no—
281C4×C11—22no—
282C4×C11—22no—
283C4×C11#54812yes—
284C4×C11D4×Q812yes—
285Dic11—42no—
286Dic11—42no—
287Dic11—44no—
288Dic11—44no—
289Dic11—81no—
290Dic11—81no—
291C22×Q8—111no—
292C22×Q8—221no—
293C22×Q8—221no—
294C23×C4D4411yescharacteristic
295C23×C4D4411yescharacteristic
296C23×C4C22×D1111yescharacteristic
297C22×C8C22×D1112yes—
298C22×C8C22×D1112yes—
299C4⋊Q8—224no—
300C4⋊Q8—224no—
301#125—118no—
302#125—224no—
303C22⋊Q8—114no—
304C22⋊Q8—228no—
305C22⋊Q8—228no—
306C22⋊Q8—222no—
307C4×Q8—228no—
308C4×Q8—228no—
309C4×Q8—224no—
310#117—224no—
311#117—224no—
312C2×C4⋊C4—112no—
313C2×C4⋊C4—112no—
314C2×C4⋊C4—111no—
315C2×C4⋊C4—111no—
316C2×C4⋊C4—111no—
317C2×C4⋊C4—222no—
318C2×C4⋊C4—222no—
319C2×C4⋊C4—221no—
320C2×C4⋊C4—221no—
321C2×C4⋊C4—221no—
322C2×C4⋊C4—221no—
323C2×C4⋊C4—221no—
324C2×C22⋊C4—222no—
325C2×C22⋊C4—221no—
326C2×C22⋊C4—221no—
327C2×C42—24no—
328C2×C42—222no—
329C2×C42—221no—
330C2×C42—221no—
331C2×C42D4411yescharacteristic
332C2×C42D4411yescharacteristic
333C2×C42C22×D1111yescharacteristic
334C2×C42C22×D1111yescharacteristic
335#107—224no—
336#106—224no—
337C4⋊C8—24no—
338#103—224no—
339#103—224no—
340C22⋊C8—24no—
341#95—221no—
342#95—221no—
343#95—221no—
344#95—221no—
345#95—221no—
346C2×C11—42no—
347C2×C11#240911yescharacteristic
348C2×C11#300211yescharacteristic
349C2×C11#300711yescharacteristic
350C2×C11#307611yescharacteristic
351C2×C11#307811yescharacteristic
352C2×C11#323111yescharacteristic
353C2×C11#324011yescharacteristic
354C24C2×D4411yescharacteristic
355C2×Q8—112no—
356C2×Q8—224no—
357C2×Q8—224no—
358C2×Q8—221no—
359C2×Q8—222no—
360C2×Q8—222no—
361C2×Q8—442no—
362C2×Q8—442no—
363C2×Q8—442no—
364C2×Q8—442no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—221no—
374C22×C4—221no—
375C22×C4—222no—
376C22×C4—222no—
377C22×C4—221no—
378C22×C4—221no—
379C22×C4C2×D4411yescharacteristic
380C22×C4C2×D4411yescharacteristic
381C22×C4C2×D4411yescharacteristic
382C22×C4C2×D4411yescharacteristic
383C22×C4C2×D4411yescharacteristic
384C22×C4#434112yes—
385C22×C4Q8×D1112yes—
386C22×C4C23×D1111yescharacteristic
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C4⋊C4—114no—
394C4⋊C4—114no—
395C4⋊C4—112no—
396C4⋊C4—224no—
397C4⋊C4—224no—
398C4⋊C4—228no—
399C4⋊C4—228no—
400C4⋊C4—224no—
401C4⋊C4—222no—
402C4⋊C4—222no—
403C4⋊C4—221no—
404C4⋊C4—222no—
405C4⋊C4—222no—
406C4⋊C4—442no—
407C4⋊C4—442no—
408C22⋊C4—228no—
409C22⋊C4—224no—
410C22⋊C4—442no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—228no—
415C42—442no—
416C42—442no—
417C42D4×D1114yes—
418C11#2430611yescharacteristic, complemented, Sylow
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23#6695012yes—
426C23C22×D4411yescharacteristic
427Q8—224no—
428Q8—448no—
429Q8—448no—
430Q8—442no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—224no—
446C2×C4—224no—
447C2×C4—222no—
448C2×C4—222no—
449C2×C4—441no—
450C2×C4—442no—
451C2×C4—441no—
452C2×C4—442no—
453C2×C4—441no—
454C2×C4—442no—
455C2×C4—442no—
456C2×C4—441no—
457C2×C4—441no—
458C2×C4—441no—
459C2×C4—441no—
460C2×C4—441no—
461C2×C4—441no—
462C2×C4—441no—
463C2×C4#6693512yes—
464C2×C4#6693512yes—
465C2×C4#6695012yes—
466C2×C4#6695712yes—
467C2×C4#6696212yes—
468C2×C4C2×D4×D1112yes—
469C2×C4C2×#434111yescharacteristic
470C2×C4C2×Q8×D1111yescharacteristic
471C2×C4#6704312yes—
472C8—42no—
473C8—42no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22C2×#6693511yescharacteristic
482C22C2×#6695011yescharacteristic
483C22#1062305711yescharacteristic
484C22C2×#6695711yescharacteristic
485C22C2×#6696211yescharacteristic
486C22#1062327911yescharacteristic
487C22#1062327911yescharacteristic
488C4—22no—
489C4—22no—
490C4—22no—
491C4—22no—
492C4—22no—
493C4—444no—
494C4—442no—
495C4—444no—
496C4—442no—
497C4—881no—
498C4—881no—
499C4#1062312912yes—
500C4#1062316312yes—
501C2—42no—
502C2#1307198211yescharacteristic
503C2#1307783811yescharacteristic
504C2#1307787811yescharacteristic
505C2#1307840611yescharacteristic
506C2#1307842311yescharacteristic
507C2#1307977711yescharacteristic
508C2#1307985811yescharacteristic
subgroupG/H
centerC23C22×D44
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC11×#1797C2
socleC23×C11C22×D4
prime radicalC24×C11C2×D4
hypercenter#1797D11
hypocenterC11#24306
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
2. C2×#10622442C2semidirect
4. #13069950C2semidirect
5. #13069945C2semidirect
6. #13069758C2semidirect
7. #13069758C2semidirect
8. #13069571C2semidirect
73. #66977Q8semidirect
418. C11#24306semidirect

Maximal subgroups

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

subgroupindex
C2×#106230012
C2×#106224422
C4×#669772
#130699502
#130699452
#130697582
#130695712
#130689262
#130688172
C11×#17972
#2430611