Finite Group Databasebeta

← #14786410

Subgroups — #14786410

Order 2816.

1500 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622656C212yesmaximal, complemented
2C2×#10622825C211yesmaximal, characteristic
3#13069809C214yesmaximal, complemented
4#13068924C212yesmaximal, complemented
5#13068809C212yesmaximal, complemented
6#13068551C212yesmaximal, complemented
7#13068526C211yesmaximal, characteristic
8C22×C11×C22⋊Q8C211yesmaximal, characteristic
9C23×Q8×C11C2211yescharacteristic
10C24×C4×C11C2211yescharacteristic
11C2×C11×C22⋊Q8—21no—
12C2×C11×C22⋊Q8—21no—
13C2×C11×C22⋊Q8—28no—
14C2×C11×C22⋊Q8C2212yes—
15C2×C11×C22⋊Q8C2212yes—
16C22×C11×C4⋊C4—21no—
17C22×C11×C4⋊C4C2211yescharacteristic
18C22×C11×C22⋊C4—21no—
19C22×C22⋊Dic11—21no—
20C22×C4⋊Dic11C2211yescharacteristic
21C22×C4⋊Dic11C2211yescharacteristic
22C2×#66899C2214yescomplemented
23C2×#66899C2214yescomplemented
24#10622825—21no—
25#10622825—21no—
26#10622825C2212yes—
27#10622825C2212yes—
28C2×C11⋊#95—21no—
29C2×C22⋊(C11⋊C8)C2212yes—
30C2×C22⋊(C11⋊C8)C2212yes—
31C2×C22⋊(C11⋊C8)C2212yes—
32#10622656—28no—
33C2×#66872C2214yescomplemented
34#10622167C2214yescomplemented
35C25×C11D411yescharacteristic
36C22×Q8×C11—24no—
37C22×Q8×C11—21no—
38C22×Q8×C11—24no—
39C22×Q8×C11—42no—
40C22×Q8×C11C2312yes—
41C23×C4×C11—21no—
42C23×C4×C11—21no—
43C23×C4×C11—24no—
44C23×C4×C11—24no—
45C23×C4×C11—42no—
46C23×C4×C11D411yescharacteristic
47C23×C4×C11D411yescharacteristic
48C23×C4×C11D411yescharacteristic
49C23×C4×C11C2311yescharacteristic
50C23×C4×C11C2311yescharacteristic
51C23×C4×C11C2311yescharacteristic
52C23×Dic11—21no—
53C23×Dic11—21no—
54C11×C22⋊Q8—28no—
55C11×C22⋊Q8—28no—
56C11×C22⋊Q8—24no—
57C11×C22⋊Q8—24no—
58C11×C22⋊Q8—24no—
59C11×C22⋊Q8D418yes—
60C2×C11×C4⋊C4—28no—
61C2×C11×C4⋊C4—22no—
62C2×C11×C4⋊C4—22no—
63C2×C11×C4⋊C4—21no—
64C2×C11×C4⋊C4—24no—
65C2×C11×C4⋊C4C2312yes—
66C2×C11×C22⋊C4—28no—
67C2×C11×C22⋊C4—22no—
68C2×C11×C22⋊C4—22no—
69C2×C22⋊Dic11—22no—
70C2×C22⋊Dic11—22no—
71C2×C22⋊Dic11—44no—
72C2×C4⋊Dic11—24no—
73C2×C4⋊Dic11—21no—
74C2×C4⋊Dic11—21no—
75C2×C4⋊Dic11—42no—
76C2×C4⋊Dic11C2312yes—
77C2×C4⋊Dic11C2312yes—
78C22×C11⋊C8C2314yes—
79#66899—28no—
80#66899—44no—
81C11⋊#95—22no—
82C11⋊#95—22no—
83C22⋊(C11⋊C8)—24no—
84C22⋊(C11⋊C8)—24no—
85C22⋊(C11⋊C8)—42no—
86#66872—28no—
87#24003—111nomaximal, Sylow
88C24×C11—22no—
89C24×C11—22no—
90C24×C11—22no—
91C24×C11—22no—
92C24×C11—41no—
93C24×C11—41no—
94C24×C11SD1614yes—
95C24×C11C2×D411yescharacteristic
96C24×C11C2×D411yescharacteristic
97C24×C11C2×D411yescharacteristic
98C2×Q8×C11—24no—
99C2×Q8×C11—22no—
100C2×Q8×C11—24no—
101C2×Q8×C11—24no—
102C2×Q8×C11—416no—
103C2×Q8×C11—42no—
104C2×Q8×C11—42no—
105C2×Q8×C11C2×D414yes—
106C22×C4×C11—22no—
107C22×C4×C11—22no—
108C22×C4×C11—22no—
109C22×C4×C11—22no—
110C22×C4×C11—24no—
111C22×C4×C11—24no—
112C22×C4×C11—22no—
113C22×C4×C11—22no—
114C22×C4×C11—22no—
115C22×C4×C11—22no—
116C22×C4×C11—22no—
117C22×C4×C11—22no—
118C22×C4×C11—22no—
119C22×C4×C11—22no—
120C22×C4×C11—22no—
121C22×C4×C11—22no—
122C22×C4×C11—22no—
123C22×C4×C11—41no—
124C22×C4×C11—41no—
125C22×C4×C11—41no—
126C22×C4×C11—416no—
127C22×C4×C11—42no—
128C22×C4×C11—42no—
129C22×C4×C11—42no—
130C22×C4×C11—42no—
131C22×C4×C11C2×D411yescharacteristic
132C22×C4×C11C2×D411yescharacteristic
133C22×C4×C11C2×D414yes—
134C22×C4×C11C2×D411yescharacteristic
135C22×C4×C11#4114yes—
136C22×C4×C11C2411yescharacteristic
137C22×Dic11—22no—
138C22×Dic11—22no—
139C22×Dic11—42no—
140C22×Dic11—42no—
141C22×Dic11—42no—
142C22×Dic11—42no—
143C22×Dic11—42no—
144C22×Dic11—42no—
145C11×C4⋊C4—24no—
146C11×C4⋊C4—28no—
147C11×C4⋊C4—28no—
148C11×C4⋊C4—22no—
149C11×C4⋊C4C2×D414yes—
150C11×C22⋊C4—28no—
151C11×C22⋊C4—28no—
152C22⋊Dic11—44no—
153C22⋊Dic11—44no—
154C4⋊Dic11—24no—
155C4⋊Dic11—24no—
156C4⋊Dic11—42no—
157C4⋊Dic11—42no—
158C2×C11⋊C8—24no—
159C2×C11⋊C8—24no—
160C2×C11⋊C8—24no—
161C22×C22⋊Q8D1111yescharacteristic
162C2×#483—112no—
163C2×#380—111no—
164#2035—114no—
165#1872—112no—
166#1853—112no—
167#1787—112no—
168#1779—111no—
169C23×C11—21no—
170C23×C11—21no—
171C23×C11—21no—
172C23×C11—21no—
173C23×C11—21no—
174C23×C11—21no—
175C23×C11—21no—
176C23×C11—21no—
177C23×C11—21no—
178C23×C11—21no—
179C23×C11—21no—
180C23×C11—21no—
181C23×C11—21no—
182C23×C11—21no—
183C23×C11—28no—
184C23×C11—28no—
185C23×C11—44no—
186C23×C11—44no—
187C23×C11—44no—
188C23×C11—44no—
189C23×C11—44no—
190C23×C11#12012yes—
191C23×C11#12312yes—
192C23×C11C2×SD1612yes—
193C23×C11C2×SD1612yes—
194C23×C11C2×SD1612yes—
195C23×C11#13712yes—
196C23×C11#13712yes—
197C23×C11C22×D411yescharacteristic
198Q8×C11—28no—
199Q8×C11—48no—
200Q8×C11—44no—
201C2×C4×C11—21no—
202C2×C4×C11—21no—
203C2×C4×C11—21no—
204C2×C4×C11—21no—
205C2×C4×C11—28no—
206C2×C4×C11—28no—
207C2×C4×C11—24no—
208C2×C4×C11—24no—
209C2×C4×C11—21no—
210C2×C4×C11—21no—
211C2×C4×C11—21no—
212C2×C4×C11—44no—
213C2×C4×C11—44no—
214C2×C4×C11—44no—
215C2×C4×C11—48no—
216C2×C4×C11—48no—
217C2×C4×C11—42no—
218C2×C4×C11—42no—
219C2×C4×C11—42no—
220C2×C4×C11—42no—
221C2×C4×C11—44no—
222C2×C4×C11#12012yes—
223C2×C4×C11#12312yes—
224C2×C4×C11C22×D412yes—
225C2×C4×C11C2×#4112yes—
226C2×C4×C11H(5,𝔽2)12yes—
227C2×Dic11—42no—
228C2×Dic11—42no—
229C2×Dic11—42no—
230C2×Dic11—42no—
231C2×Dic11—42no—
232C2×Dic11—42no—
233C2×Dic11—84no—
234C2×Dic11—84no—
235C11⋊C8—44no—
236C23×Q8C2×D1111yescharacteristic
237C24×C4C2×D1111yescharacteristic
238C2×C22⋊Q8—21no—
239C2×C22⋊Q8—21no—
240C2×C22⋊Q8—28no—
241C2×C22⋊Q8C2×D1112yes—
242C2×C22⋊Q8C2×D1112yes—
243C22×C4⋊C4—21no—
244C22×C4⋊C4—111no—
245C22×C4⋊C4—111no—
246C22×C4⋊C4C2×D1111yescharacteristic
247C22×C22⋊C4—21no—
248C22×C22⋊C4—221no—
249#483—228no—
250C2×#106—114no—
251C2×#103—114no—
252C2×#103—114no—
253C2×C22⋊C8—112no—
254C2×C22⋊C8—112no—
255C2×C22⋊C8—112no—
256#380—112no—
257#380—112no—
258#380—221no—
259#380—221no—
260C2×#95—221no—
261#340—114no—
262C22×C11—22no—
263C22×C11—22no—
264C22×C11—22no—
265C22×C11—22no—
266C22×C11—22no—
267C22×C11—22no—
268C22×C11—22no—
269C22×C11—22no—
270C22×C11—22no—
271C22×C11—22no—
272C22×C11—22no—
273C22×C11—22no—
274C22×C11—22no—
275C22×C11—22no—
276C22×C11—22no—
277C22×C11—416no—
278C22×C11—41no—
279C22×C11—41no—
280C22×C11—41no—
281C22×C11—41no—
282C22×C11#44814yes—
283C22×C11#48314yescomplemented
284C22×C11C2×#12011yescharacteristic
285C22×C11C2×#12311yescharacteristic
286C22×C11#53411yescharacteristic
287C22×C11C22×SD1611yescharacteristic
288C22×C11C2×#13711yescharacteristic
289C22×C11C2×#13711yescharacteristic
290C22×C11#57511yescharacteristic
291C4×C11—22no—
292C4×C11—22no—
293C4×C11—41no—
294C4×C11—44no—
295C4×C11—44no—
296C4×C11#54614yes—
297Dic11—82no—
298Dic11—82no—
299C25C22⋊D1111yescharacteristic
300C22×Q8—24no—
301C22×Q8—21no—
302C22×Q8—24no—
303C22×Q8—42no—
304C22×Q8C22×D1112yes—
305C23×C4—21no—
306C23×C4—21no—
307C23×C4—24no—
308C23×C4—24no—
309C23×C4—42no—
310C23×C4—221no—
311C23×C4—221no—
312C23×C4C22⋊D1111yescharacteristic
313C23×C4C22⋊D1111yescharacteristic
314C23×C4C22⋊D1111yescharacteristic
315C23×C4C22×D1111yescharacteristic
316C23×C4C22×D1111yescharacteristic
317C23×C4C22×D1111yescharacteristic
318C22×C8—114no—
319C22⋊Q8—28no—
320C22⋊Q8—28no—
321C22⋊Q8—24no—
322C22⋊Q8—24no—
323C22⋊Q8—24no—
324C22⋊Q8C22⋊D1118yes—
325C2×C4⋊C4—28no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—112no—
331C2×C4⋊C4—112no—
332C2×C4⋊C4—224no—
333C2×C4⋊C4—221no—
334C2×C4⋊C4—221no—
335C2×C4⋊C4—442no—
336C2×C4⋊C4C22×D1112yes—
337C2×C22⋊C4—28no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—222no—
341C2×C22⋊C4—222no—
342C2×C22⋊C4—444no—
343#106—228no—
344#103—228no—
345#103—444no—
346C22⋊C8—224no—
347C22⋊C8—224no—
348C22⋊C8—442no—
349#95—222no—
350#95—222no—
351C2×C11—24no—
352C2×C11—24no—
353C2×C11—42no—
354C2×C11#240511yescharacteristic
355C2×C11C2×#44811yescharacteristic
356C2×C11#300711yescharacteristic
357C2×C11C2×#48311yescharacteristic
358C2×C11#309511yescharacteristic
359C2×C11#318711yescharacteristic
360C2×C11#319511yescharacteristic
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—41no—
366C24—41no—
367C24#432514yes—
368C24C2×C22⋊D1111yescharacteristic
369C24C2×C22⋊D1111yescharacteristic
370C24C2×C22⋊D1111yescharacteristic
371C2×Q8—24no—
372C2×Q8—22no—
373C2×Q8—24no—
374C2×Q8—24no—
375C2×Q8—416no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8C2×C22⋊D1114yes—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—416no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—222no—
405C22×C4—222no—
406C22×C4—442no—
407C22×C4—442no—
408C22×C4—442no—
409C22×C4—442no—
410C22×C4—442no—
411C22×C4—442no—
412C22×C4#434114yes—
413C22×C4C2×C22⋊D1111yescharacteristic
414C22×C4C2×C22⋊D1111yescharacteristic
415C22×C4C2×C22⋊D1111yescharacteristic
416C22×C4C2×C22⋊D1114yes—
417C22×C4C23×D1111yescharacteristic
418C2×C8—224no—
419C2×C8—224no—
420C2×C8—224no—
421C4⋊C4—24no—
422C4⋊C4—28no—
423C4⋊C4—28no—
424C4⋊C4—22no—
425C4⋊C4—224no—
426C4⋊C4—224no—
427C4⋊C4—442no—
428C4⋊C4—442no—
429C4⋊C4C2×C22⋊D1114yes—
430C22⋊C4—28no—
431C22⋊C4—28no—
432C22⋊C4—444no—
433C22⋊C4—444no—
434C11#2400311yescharacteristic, complemented, Sylow
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—28no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—28no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23#6698812yes—
457C23C2×#432512yes—
458C23C2×#432512yes—
459C23C2×#432512yes—
460C23#6699512yes—
461C23#6700412yes—
462C23#6700612yes—
463C23C22×C22⋊D1111yescharacteristic
464Q8—28no—
465Q8—48no—
466Q8—44no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—28no—
472C2×C4—28no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—21no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—48no—
482C2×C4—48no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—442no—
489C2×C4—442no—
490C2×C4—442no—
491C2×C4—442no—
492C2×C4—442no—
493C2×C4—442no—
494C2×C4—884no—
495C2×C4—884no—
496C2×C4#6698812yes—
497C2×C4#6700612yes—
498C2×C4C2×#434112yes—
499C2×C4#6703712yes—
500C2×C4C22×C22⋊D1112yes—
501C8—444no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—416no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22#1062265614yescomplemented
523C22#1062284014yes—
524C22C2×#6698811yescharacteristic
525C22#1062333511yescharacteristic
526C22C22×#432511yescharacteristic
527C22C2×#6699511yescharacteristic
528C22#1062335311yescharacteristic
529C22C2×#6700411yescharacteristic
530C22C2×#6700611yescharacteristic
531C4—22no—
532C4—22no—
533C4—41no—
534C4—44no—
535C4—44no—
536C4—882no—
537C4—882no—
538C4#1062333714yes—
539C2—24no—
540C2—24no—
541C2—42no—
542C2#1307193911yescharacteristic
543C2C2×#1062284011yescharacteristic
544C2#1307787611yescharacteristic
545C2C2×#1062265611yescharacteristic
546C2#1307856011yescharacteristic
547C2#1307936911yescharacteristic
548C2#1307943711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D11
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC22×C11×C22⋊Q8C2
socleC23×C11C22×D4
prime radicalC25×C11D4
hypercenterC22×C22⋊Q8D11
hypocenterC11#24003
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#10622656C2semidirect
3. #13069809C2semidirect
4. #13068924C2semidirect
5. #13068809C2semidirect
6. #13068551C2semidirect
22. C2×#66899C22semidirect
23. C2×#66899C22semidirect
33. C2×#66872C22semidirect
34. #10622167C22semidirect
283. C22×C11#483semidirect
434. C11#24003semidirect
522. C22#10622656semidirect

Maximal subgroups

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

subgroupindex
C2×#106226562
C2×#106228252
#130698092
#130689242
#130688092
#130685512
#130685262
C22×C11×C22⋊Q82
#2400311