Finite Group Databasebeta

← #14671265

Subgroups — #14671265

Order 2816.

818 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622850C211yesmaximal, characteristic, complemented
2#13071931C211yesmaximal, characteristic
3C2×#10622109C212yesmaximal, complemented
4#13066733C212yesmaximal, complemented
5C11×#1782C211yesmaximal, characteristic, complemented
6C22×C11×#41C2211yescharacteristic
7C2×#66990C411yescharacteristic, complemented
8C2×#66990C2211yescharacteristic
9C22×(C2×D11)⋊C4C411yescharacteristic, complemented
10C22×#4318—21no—
11C2×C11×#99—21no—
12C2×C11×#99C2212yes—
13C11×#381—21no—
14#10622850—24no—
15#10622844—21no—
16C2×#66900—21no—
17#10622836—21no—
18#10622833C411yescharacteristic, complemented
19#10622833C2211yescharacteristic
20C2×#66897C2212yes—
21#10622826—21no—
22#10622826—21no—
23#10622824C411yescharacteristic, complemented
24#10622109—24no—
25C24×D11C2×C411yescharacteristic
26C2×C11×#41—22no—
27C2×C11×#41—22no—
28C2×C11×#41—41no—
29C2×C11×#41—41no—
30C2×C11×#41—42no—
31C22×Q8×C11D411yescharacteristic
32C22×D4×C11C2×C411yescharacteristic
33C22×D4×C11D411yescharacteristic
34C22×D4×C11C2311yescharacteristic
35C23×C4×C11C2×C411yescharacteristic
36C23×C4×C11D411yescharacteristic
37C23×C4×C11D411yescharacteristic
38C22×C22⋊D11—21no—
39C22×C22⋊D11—21no—
40C23×Dic11C2×C411yescharacteristic
41C2×C11×C22⋊C4—22no—
42C2×C22⋊Dic11—21no—
43C2×C22⋊Dic11—21no—
44C2×C22⋊Dic11C2×C411yescharacteristic
45C2×C22⋊Dic11C2×C411yescharacteristic
46#66990—22no—
47#66990—22no—
48#66990—42no—
49C2×(C2×D11)⋊C4—21no—
50C2×(C2×D11)⋊C4—21no—
51C2×(C2×D11)⋊C4—21no—
52C2×(C2×D11)⋊C4—21no—
53C2×(C2×D11)⋊C4—41no—
54C2×(C2×D11)⋊C4—41no—
55C2×#4318—24no—
56C2×#4318—22no—
57C2×#4318—42no—
58C2×#4318—41no—
59C22×C11⋊C8—41no—
60C11×#99—22no—
61C11×#99—22no—
62C11×#99—42no—
63C11×#95—41no—
64C11×#95—41no—
65#66900—24no—
66#66897—22no—
67#66897D414yescomplemented
68C11⋊#95—21no—
69C11⋊#95—21no—
70C11⋊#95—21no—
71C11⋊#95—21no—
72#13083—111nomaximal, Sylow
73C24×C11C22⋊C411yescharacteristic
74C24×C11C22×C411yescharacteristic
75C24×C11C2×D411yescharacteristic
76C23×D11—21no—
77C23×D11—22no—
78C23×D11—42no—
79C23×D11—41no—
80C23×D11—41no—
81C23×D11—81no—
82C11×#41—44no—
83C11×#41—44no—
84C11×#41—44no—
85C11×#41—44no—
86C2×Q8×C11—21no—
87C2×Q8×C11—21no—
88C2×Q8×C11—41no—
89C2×Q8×C11—41no—
90C2×D4×C11—22no—
91C2×D4×C11—22no—
92C2×D4×C11—22no—
93C2×D4×C11—22no—
94C2×D4×C11—41no—
95C2×D4×C11—41no—
96C2×D4×C11—41no—
97C2×D4×C11—41no—
98C2×D4×C11C2×D412yes—
99C2×D4×C11#4112yes—
100C22×C4×C11—22no—
101C22×C4×C11—21no—
102C22×C4×C11—21no—
103C22×C4×C11—21no—
104C22×C4×C11—21no—
105C22×C4×C11—22no—
106C22×C4×C11—22no—
107C22×C4×C11—21no—
108C22×C4×C11—21no—
109C22×C4×C11—21no—
110C22×C4×C11—21no—
111C22×C4×C11—41no—
112C22×C4×C11—41no—
113C22×C4×C11—42no—
114C22×C4×C11C22⋊C411yescharacteristic
115C22×C4×C11C22⋊C411yescharacteristic
116C22×C4×C11C22⋊C411yescharacteristic
117C22×C4×C11C2×D411yescharacteristic
118C2×C22⋊D11—42no—
119C2×C22⋊D11—42no—
120C2×C22⋊D11—42no—
121C2×C22⋊D11—42no—
122C2×C22⋊D11—81no—
123C2×C22⋊D11—81no—
124C22×Dic11—21no—
125C22×Dic11—21no—
126C22×Dic11—21no—
127C22×Dic11—41no—
128C22×Dic11—81no—
129C11×C22⋊C4—42no—
130C11×C22⋊C4—42no—
131C22⋊Dic11—22no—
132C22⋊Dic11—41no—
133C22⋊Dic11—81no—
134C22⋊Dic11—81no—
135(C2×D11)⋊C4—42no—
136(C2×D11)⋊C4—42no—
137(C2×D11)⋊C4—41no—
138(C2×D11)⋊C4—41no—
139(C2×D11)⋊C4—41no—
140(C2×D11)⋊C4—41no—
141#4318—24no—
142#4318—44no—
143#4318—42no—
144C2×C11⋊C8—44no—
145C2×C11⋊C8—42no—
146C2×#413—111no—
147#2404—111no—
148C2×AΣL(1,𝔽16)—112no—
149#1782D1111yescharacteristic, complemented
150#1390—112no—
151C23×C11—21no—
152C23×C11—21no—
153C23×C11—21no—
154C23×C11—21no—
155C23×C11—21no—
156C23×C11—21no—
157C23×C11—41no—
158C23×C11—41no—
159C23×C11—41no—
160C23×C11—41no—
161C23×C11—41no—
162C23×C11—41no—
163C23×C11#9912yes—
164C23×C11C2×C22⋊C411yescharacteristic
165C23×C11C4×D412yes—
166C23×C11#12112yes—
167C22×D11—22no—
168C22×D11—41no—
169C22×D11—41no—
170C22×D11—41no—
171C22×D11—42no—
172C22×D11—41no—
173C22×D11—41no—
174C22×D11—41no—
175C22×D11—42no—
176C22×D11—84no—
177C22×D11—84no—
178C22×D11—82no—
179C22×D11—82no—
180Q8×C11—42no—
181Q8×C11—42no—
182D4×C11—42no—
183D4×C11—42no—
184D4×C11—44no—
185D4×C11—44no—
186C2×C4×C11—22no—
187C2×C4×C11—22no—
188C2×C4×C11—22no—
189C2×C4×C11—22no—
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—42no—
201C2×C4×C11—42no—
202C2×C4×C11—42no—
203C2×C4×C11—42no—
204C2×C4×C11—82no—
205C2×C4×C11—82no—
206C2×C4×C11—82no—
207C2×C4×C11#9911yescharacteristic
208C2×C4×C11#9911yescharacteristic
209C2×C4×C11#12011yescharacteristic
210C2×C4×C11#12311yescharacteristic
211C22⋊D11—84no—
212C22⋊D11—84no—
213C2×Dic11—41no—
214C2×Dic11—41no—
215C2×Dic11—41no—
216C2×Dic11—41no—
217C2×Dic11—41no—
218C2×Dic11—41no—
219C2×Dic11—41no—
220C2×Dic11—41no—
221C2×Dic11—41no—
222C2×Dic11—82no—
223C11⋊C8—44no—
224C22×#41C2×D1111yescharacteristic
225C22×M16—221no—
226C2×#120—111no—
227C2×#120—111no—
228C22×C22⋊C4—111no—
229#413—224no—
230C2×#101—221no—
231C2×#100—112no—
232C2×#99—21no—
233C2×#99C2×D1112yes—
234#393—221no—
235#392—221no—
236#386—111no—
237#386—111no—
238#386—221no—
239#386—221no—
240#381—21no—
241#381—111no—
242AΣL(1,𝔽16)—224no—
243C22×C11—21no—
244C22×C11—21no—
245C22×C11—22no—
246C22×C11—22no—
247C22×C11—22no—
248C22×C11—42no—
249C22×C11—42no—
250C22×C11—42no—
251C22×C11—42no—
252C22×C11—41no—
253C22×C11—41no—
254C22×C11—41no—
255C22×C11—41no—
256C22×C11—41no—
257C22×C11—41no—
258C22×C11—81no—
259C22×C11—81no—
260C22×C11#38611yescharacteristic
261C22×C11C2×#9911yescharacteristic
262C22×C11C2×#9911yescharacteristic
263C2×D11—42no—
264C2×D11—42no—
265C2×D11—42no—
266C2×D11—81no—
267C2×D11—81no—
268C2×D11—84no—
269C2×D11—81no—
270C2×D11—81no—
271C2×D11—84no—
272C4×C11—21no—
273C4×C11—21no—
274C4×C11—21no—
275C4×C11—21no—
276C4×C11—41no—
277C4×C11—41no—
278C4×C11—162no—
279Dic11—81no—
280Dic11—81no—
281Dic11—81no—
282C25—111no—
283C2×#41—22no—
284C2×#41—22no—
285C2×#41—41no—
286C2×#41—41no—
287C2×#41—42no—
288C22×Q8D4411yescharacteristic
289C22×D4—221no—
290C22×D4—221no—
291C22×D4C4×D1111yescharacteristic
292C22×D4C22⋊D1111yescharacteristic
293C22×D4C22×D1111yescharacteristic
294C23×C4—111no—
295C23×C4C4×D1111yescharacteristic
296C23×C4D4411yescharacteristic
297C23×C4C22⋊D1111yescharacteristic
298C2×M16—224no—
299C2×M16—222no—
300C2×M16—442no—
301C2×M16—441no—
302C22×C8—441no—
303#120—222no—
304#120—222no—
305#120—442no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—111no—
308C2×C22⋊C4—111no—
309C2×C22⋊C4—221no—
310C2×C22⋊C4—221no—
311C2×C22⋊C4—221no—
312C2×C22⋊C4—221no—
313C2×C22⋊C4—221no—
314C2×C22⋊C4—221no—
315C2×C22⋊C4—441no—
316C2×C22⋊C4—441no—
317#101—224no—
318#100—114no—
319#100—222no—
320#99—22no—
321#99—22no—
322#99—42no—
323#95—41no—
324#95—41no—
325#95—221no—
326#95—221no—
327#95—221no—
328#95—221no—
329C2×C11—21no—
330C2×C11—21no—
331C2×C11—42no—
332C2×C11—42no—
333C2×C11—42no—
334C2×C11#189811yescharacteristic
335C2×C11#212011yescharacteristic
336C2×C11#212211yescharacteristic
337D11—82no—
338C24—221no—
339C24—222no—
340C24—441no—
341C24—441no—
342C24—442no—
343C24—881no—
344C24(C2×D11)⋊C411yescharacteristic
345C24C2×C4×D1111yescharacteristic
346C24C2×C22⋊D1111yescharacteristic
347#41—44no—
348#41—44no—
349#41—44no—
350#41—44no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—41no—
354C2×Q8—41no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—442no—
364C2×D4—442no—
365C2×D4—442no—
366C2×D4—442no—
367C2×D4—881no—
368C2×D4—881no—
369C2×D4D4×D1112yes—
370C2×D4#434112yes—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—221no—
386C22×C4—221no—
387C22×C4—221no—
388C22×C4—441no—
389C22×C4—881no—
390C22×C4(C2×D11)⋊C411yescharacteristic
391C22×C4(C2×D11)⋊C411yescharacteristic
392C22×C4(C2×D11)⋊C411yescharacteristic
393C22×C4C2×D4411yescharacteristic
394M16—224no—
395M16—444no—
396M16—442no—
397C2×C8—444no—
398C2×C8—442no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—222no—
402C22⋊C4—442no—
403C22⋊C4—442no—
404C22⋊C4—441no—
405C22⋊C4—441no—
406C22⋊C4—441no—
407C22⋊C4—441no—
408C22⋊C4—441no—
409C22⋊C4—881no—
410C22⋊C4—881no—
411C11#1308311yescharacteristic, complemented, Sylow
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—222no—
425C23—441no—
426C23—441no—
427C23—441no—
428C23—442no—
429C23—442no—
430C23—441no—
431C23—441no—
432C23—441no—
433C23—882no—
434C23—884no—
435C23—884no—
436C23—882no—
437C23#6687012yes—
438C23(C22⋊D11)⋊C412yes—
439C23C2×(C2×D11)⋊C411yescharacteristic
440C23#6699212yes—
441Q8—42no—
442Q8—42no—
443D4—42no—
444D4—42no—
445D4—44no—
446D4—44no—
447D4—884no—
448D4—884no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—441no—
471C2×C4—441no—
472C2×C4—441no—
473C2×C4—441no—
474C2×C4—441no—
475C2×C4—441no—
476C2×C4—441no—
477C2×C4—441no—
478C2×C4—441no—
479C2×C4—882no—
480C2×C4#6687011yescharacteristic
481C2×C4#6687011yescharacteristic
482C2×C4#6693511yescharacteristic
483C2×C4#6693911yescharacteristic
484C8—444no—
485C22—21no—
486C22—21no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—81no—
501C22—81no—
502C22—442no—
503C22—442no—
504C22—442no—
505C22—881no—
506C22—881no—
507C22—884no—
508C22—884no—
509C22—881no—
510C22—881no—
511C22C2×#6687011yescharacteristic
512C22C2×#6687011yescharacteristic
513C22#1062257111yescharacteristic
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—41no—
519C4—41no—
520C4—162no—
521C4—881no—
522C4—881no—
523C4—881no—
524C2—21no—
525C2—21no—
526C2—42no—
527C2—42no—
528C2—42no—
529C2—882no—
530C2#1306906711yescharacteristic
531C2#1307025511yescharacteristic
532C2#1307026511yescharacteristic
subgroupG/H
centerC22C2×#66870
commutatorC24×C11C22×C4
FrattiniC22×D4C22×D11
FittingC11×#1782C2
socleC22×C11C2×#99
prime radical#13071931C2
cosocleC22×D4×C11C23
hypercenter#1782D11
hypocenterC11#13083
LCSG ▷ C24×C11 ▷ C22×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #1782

Complements

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

NH ≅ G/Ntype
1. C2×#10622850C2semidirect
3. C2×#10622109C2semidirect
4. #13066733C2semidirect
5. C11×#1782C2semidirect
7. C2×#66990C4semidirect
9. C22×(C2×D11)⋊C4C4semidirect
18. #10622833C4semidirect
23. #10622824C4semidirect
67. #66897D4semidirect
149. #1782D11semidirect
411. C11#13083semidirect

Maximal subgroups

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

subgroupindex
C2×#106228502
#130719312
C2×#106221092
#130667332
C11×#17822
#1308311