Finite Group Databasebeta

← #14684800

Subgroups — #14684800

Order 2816.

2222 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66991C211yesmaximal, characteristic
2C2×#10622277C211yesmaximal, characteristic, complemented
3C2×#10622280C211yesmaximal, characteristic, complemented
4#13066885C211yesmaximal, characteristic, complemented
5#13066883C211yesmaximal, characteristic, complemented
6#13066859C214yesmaximal, complemented
7#13066857C211yesmaximal, characteristic, complemented
8#13066855C214yesmaximal, complemented
9C11×#1438C211yesmaximal, characteristic, complemented
10C23×D4×C11C2211yescharacteristic
11C23×C22⋊D11C2212yes—
12C23×C4×D11C2211yescharacteristic
13C22×C11×C22⋊C4C2214yes—
14C22×C22⋊Dic11C2212yes—
15C22×C22⋊Dic11C2214yes—
16C2×#66991—24no—
17C2×#66991—24no—
18C2×#66991C418yescomplemented
19C22×(C2×D11)⋊C4C2212yes—
20C22×(C2×D11)⋊C4C2212yes—
21C22×C4⋊Dic11C2211yescharacteristic
22C22×Dic11⋊C4C2212yes—
23C2×C11×#95C2211yescharacteristic
24C2×C11×#95C2211yescharacteristic
25C2×C11⋊#95C2211yescharacteristic
26C2×C11⋊#95C2211yescharacteristic
27C2×C11⋊#95C2212yes—
28C2×C11⋊#95C2214yes—
29C2×C11⋊#95C2214yes—
30#10622280—24no—
31#10622277—24no—
32C25×C11C2312yes—
33C24×D11C2311yescharacteristic
34C22×D4×C11—22no—
35C22×D4×C11—22no—
36C22×D4×C11—44no—
37C22×D4×C11C2×C414yes—
38C23×C4×C11C2312yes—
39C23×C4×C11C2312yes—
40C23×C4×C11C2311yescharacteristic
41C22×C22⋊D11—24no—
42C22×C22⋊D11—24no—
43C22×C22⋊D11—44no—
44C22×C22⋊D11—44no—
45C22×C22⋊D11C2×C418yes—
46C23×Dic11C2312yes—
47C23×Dic11C2312yes—
48C23×Dic11C2312yes—
49C23×Dic11C2311yescharacteristic
50C22×C4×D11—44no—
51C22×C4×D11C2×C414yes—
52C22×C4×D11D414yes—
53C22×C4×D11D414yes—
54C2×C11×C22⋊C4—44no—
55C2×C11×C22⋊C4—44no—
56C2×C11×C22⋊C4—44no—
57C2×C22⋊Dic11—24no—
58C2×C22⋊Dic11—24no—
59C2×C22⋊Dic11—44no—
60C2×C22⋊Dic11—44no—
61C2×C22⋊Dic11—44no—
62C2×C22⋊Dic11C2×C418yes—
63#66991—232no—
64C2×(C2×D11)⋊C4—24no—
65C2×(C2×D11)⋊C4—24no—
66C2×(C2×D11)⋊C4—44no—
67C2×(C2×D11)⋊C4—44no—
68C2×C4⋊Dic11—22no—
69C2×C4⋊Dic11—22no—
70C2×C4⋊Dic11C2×C414yes—
71C2×Dic11⋊C4—24no—
72C2×Dic11⋊C4—44no—
73C11×#95—22no—
74C11×#95—22no—
75C11⋊#95—22no—
76C11⋊#95—22no—
77C11⋊#95—24no—
78C11⋊#95—44no—
79C11⋊#95—44no—
80#14659—111nomaximal, Sylow
81C24×C11—22no—
82C24×C11—22no—
83C24×C11—24no—
84C24×C11—24no—
85C24×C11—24no—
86C24×C11—24no—
87C24×C11—24no—
88C24×C11—24no—
89C24×C11C22×C414yes—
90C24×C11C2411yescharacteristic
91C23×D11—22no—
92C23×D11—24no—
93C23×D11—22no—
94C23×D11—22no—
95C23×D11—22no—
96C23×D11C22×C412yes—
97C23×D11C2×D412yes—
98C23×D11C2×D412yes—
99C2×D4×C11—28no—
100C2×D4×C11—48no—
101C2×D4×C11—48no—
102C2×D4×C11—48no—
103C22×C4×C11—24no—
104C22×C4×C11—22no—
105C22×C4×C11—22no—
106C22×C4×C11—22no—
107C22×C4×C11—24no—
108C22×C4×C11—24no—
109C22×C4×C11—24no—
110C22×C4×C11—24no—
111C22×C4×C11—22no—
112C22×C4×C11—22no—
113C22×C4×C11—22no—
114C22×C4×C11C22×C412yes—
115C22×C4×C11C2×D412yes—
116C22×C4×C11C2×D412yes—
117C2×C22⋊D11—216no—
118C2×C22⋊D11—416no—
119C2×C22⋊D11—48no—
120C2×C22⋊D11—416no—
121C2×C22⋊D11—48no—
122C22×Dic11—22no—
123C22×Dic11—22no—
124C22×Dic11—22no—
125C22×Dic11—22no—
126C22×Dic11—24no—
127C22×Dic11—24no—
128C22×Dic11—24no—
129C22×Dic11—22no—
130C22×Dic11—24no—
131C22×Dic11—24no—
132C22×Dic11—24no—
133C22×Dic11—24no—
134C22×Dic11—22no—
135C22×Dic11—22no—
136C22×Dic11—22no—
137C22×Dic11C22×C414yes—
138C22×Dic11C22×C412yes—
139C22×Dic11C2×D412yes—
140C22×Dic11C2×D412yes—
141C2×C4×D11—48no—
142C2×C4×D11—48no—
143C2×C4×D11—48no—
144C2×C4×D11C22⋊C4116yescomplemented
145C11×C22⋊C4—416no—
146C22⋊Dic11—216no—
147C22⋊Dic11—416no—
148(C2×D11)⋊C4—48no—
149(C2×D11)⋊C4—48no—
150C4⋊Dic11—28no—
151Dic11⋊C4—48no—
152C22×#121—111no—
153C2×#381—111no—
154C2×#380—111no—
155#1445—111no—
156#1445—111no—
157#1439—114no—
158#1438—111no—
159#1438—114no—
160#1438D1111yescharacteristic, complemented
161C23×C11—24no—
162C23×C11—28no—
163C23×C11—24no—
164C23×C11—24no—
165C23×C11—28no—
166C23×C11—44no—
167C23×C11—44no—
168C23×C11—42no—
169C23×C11—44no—
170C23×C11—42no—
171C23×C11—42no—
172C23×C11—42no—
173C23×C11—44no—
174C23×C11—44no—
175C23×C11—44no—
176C23×C11—44no—
177C23×C11—44no—
178C23×C11—44no—
179C23×C11—44no—
180C23×C11—44no—
181C23×C11—44no—
182C23×C11C23×C411yescharacteristic
183C23×C11C22×D411yescharacteristic
184C23×C11C22×D411yescharacteristic
185C23×C11H(5,𝔽2)14yes—
186C23×C11H(5,𝔽2)12yes—
187C23×C11H(5,𝔽2)12yes—
188C23×C11H(5,𝔽2)12yes—
189C23×C112−1+412yes—
190C22×D11—24no—
191C22×D11—24no—
192C22×D11—24no—
193C22×D11—24no—
194C22×D11—24no—
195C22×D11—22no—
196C22×D11—22no—
197C22×D11—24no—
198C22×D11—22no—
199C22×D11—22no—
200C22×D11—24no—
201C22×D11—44no—
202C22×D11—44no—
203C22×D11—44no—
204C22×D11—44no—
205C22×D11C2×C22⋊C414yes—
206D4×C11—416no—
207C2×C4×C11—24no—
208C2×C4×C11—24no—
209C2×C4×C11—24no—
210C2×C4×C11—42no—
211C2×C4×C11—42no—
212C2×C4×C11—42no—
213C2×C4×C11—42no—
214C2×C4×C11—42no—
215C2×C4×C11—42no—
216C2×C4×C11—44no—
217C2×C4×C11—44no—
218C2×C4×C11—44no—
219C2×C4×C11—44no—
220C2×C4×C11C2×C22⋊C414yes—
221C22⋊D11—432no—
222C2×Dic11—24no—
223C2×Dic11—24no—
224C2×Dic11—24no—
225C2×Dic11—24no—
226C2×Dic11—28no—
227C2×Dic11—42no—
228C2×Dic11—44no—
229C2×Dic11—42no—
230C2×Dic11—42no—
231C2×Dic11—42no—
232C2×Dic11—42no—
233C2×Dic11—42no—
234C2×Dic11—42no—
235C2×Dic11—42no—
236C2×Dic11—44no—
237C2×Dic11—44no—
238C2×Dic11—44no—
239C2×Dic11—44no—
240C2×Dic11C2×C22⋊C414yes—
241C4×D11—416no—
242C23×D4—112no—
243C23×D4C2×D1111yescharacteristic
244C24×C4—111no—
245C2×#121—118no—
246C2×#121—224no—
247C2×#121—224no—
248C22×C4⋊C4—111no—
249C22×C4⋊C4—112no—
250C22×C22⋊C4—112no—
251C22×C22⋊C4—112no—
252C22×C22⋊C4—112no—
253C22×C22⋊C4—114no—
254C22×C22⋊C4C2×D1114yes—
255#381—224no—
256#380—224no—
257C2×#95—111no—
258C2×#95—111no—
259C2×#95—112no—
260C2×#95—114no—
261C2×#95—114no—
262C2×#95C2×D1111yescharacteristic
263C2×#95C2×D1111yescharacteristic
264C22×C11—44no—
265C22×C11—44no—
266C22×C11—44no—
267C22×C11—48no—
268C22×C11—48no—
269C22×C11—84no—
270C22×C11—84no—
271C22×C11—84no—
272C22×C11—84no—
273C22×C11C22×C22⋊C411yescharacteristic
274C22×C11#51812yes—
275C22×C11#51812yes—
276C22×C11#52012yes—
277C22×C11#53412yes—
278C22×C11#53412yes—
279C22×C11#53511yescharacteristic
280C22×C11#53511yescharacteristic
281C22×C11#53512yes—
282C22×C11#53511yescharacteristic
283C22×C11#53611yescharacteristic
284C22×C11#53712yes—
285C22×C11#56014yes—
286C22×C11#56014yes—
287C22×C11#56214yes—
288C22×C11#56214yes—
289C2×D11—24no—
290C2×D11—24no—
291C2×D11—24no—
292C2×D11—44no—
293C2×D11—48no—
294C2×D11—44no—
295C2×D11—44no—
296C2×D11—44no—
297C4×C11—24no—
298C4×C11—82no—
299C4×C11—82no—
300Dic11—24no—
301Dic11—44no—
302Dic11—82no—
303Dic11—82no—
304C25—111no—
305C25C22×D1112yes—
306C22×D4—22no—
307C22×D4—22no—
308C22×D4—44no—
309C22×D4—118no—
310C22×D4—224no—
311C22×D4—224no—
312C22×D4—444no—
313C22×D4—444no—
314C22×D4C4×D1114yes—
315C23×C4—111no—
316C23×C4—114no—
317C23×C4—114no—
318C23×C4—114no—
319C23×C4—112no—
320C23×C4—112no—
321C23×C4—112no—
322C23×C4—444no—
323C23×C4C22×D1112yes—
324C23×C4C22×D1112yes—
325C23×C4C22×D1111yescharacteristic
326#121—2232no—
327C2×C4⋊C4—114no—
328C2×C4⋊C4—222no—
329C2×C4⋊C4—222no—
330C2×C4⋊C4—224no—
331C2×C4⋊C4—444no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—118no—
336C2×C22⋊C4—224no—
337C2×C22⋊C4—224no—
338C2×C22⋊C4—224no—
339C2×C22⋊C4—224no—
340C2×C22⋊C4—444no—
341C2×C22⋊C4—444no—
342C2×C22⋊C4—444no—
343C2×C22⋊C4—444no—
344C2×C22⋊C4—444no—
345#95—22no—
346#95—22no—
347#95—224no—
348#95—222no—
349#95—222no—
350#95—444no—
351#95—444no—
352C2×C11—84no—
353C2×C11#230811yescharacteristic
354C2×C11#230911yescharacteristic
355C2×C11#231111yescharacteristic
356C2×C11#237714yes—
357C2×C11#281312yes—
358C2×C11#281312yes—
359C2×C11#281512yes—
360C2×C11#281612yes—
361D11—44no—
362C24—22no—
363C24—22no—
364C24—24no—
365C24—24no—
366C24—24no—
367C24—24no—
368C24—24no—
369C24—24no—
370C24—112no—
371C24—112no—
372C24—112no—
373C24—222no—
374C24—224no—
375C24—222no—
376C24—222no—
377C24—222no—
378C24C2×C4×D1114yes—
379C24C23×D1111yescharacteristic
380C2×D4—28no—
381C2×D4—48no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—2216no—
385C2×D4—4416no—
386C2×D4—448no—
387C2×D4—4416no—
388C2×D4—448no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—112no—
401C22×C4—112no—
402C22×C4—112no—
403C22×C4—1116no—
404C22×C4—114no—
405C22×C4—224no—
406C22×C4—222no—
407C22×C4—222no—
408C22×C4—224no—
409C22×C4—224no—
410C22×C4—222no—
411C22×C4—222no—
412C22×C4—222no—
413C22×C4—224no—
414C22×C4—224no—
415C22×C4—222no—
416C22×C4—224no—
417C22×C4—224no—
418C22×C4—222no—
419C22×C4—222no—
420C22×C4—448no—
421C22×C4—448no—
422C22×C4—448no—
423C22×C4C2×C4×D1112yes—
424C22×C4D4×D1112yes—
425C22×C4D4×D1112yes—
426C4⋊C4—228no—
427C4⋊C4—448no—
428C22⋊C4—416no—
429C22⋊C4—2216no—
430C22⋊C4—448no—
431C22⋊C4—448no—
432C22⋊C4—4416no—
433C11#1465911yescharacteristic, complemented, Sylow
434C23—24no—
435C23—24no—
436C23—24no—
437C23—28no—
438C23—28no—
439C23—42no—
440C23—42no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—42no—
454C23—42no—
455C23—114no—
456C23—222no—
457C23—222no—
458C23—224no—
459C23—224no—
460C23—224no—
461C23—224no—
462C23—224no—
463C23—224no—
464C23—224no—
465C23—222no—
466C23—222no—
467C23—444no—
468C23—444no—
469C23—444no—
470C23—444no—
471C23C22×C4×D1111yescharacteristic
472C23C2×D4×D1111yescharacteristic
473C23C2×D4×D1111yescharacteristic
474C23#6703712yes—
475C23#6703712yes—
476C23#6703714yes—
477C23#6704212yes—
478C23#6704312yes—
479D4—416no—
480D4—4432no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—114no—
495C2×C4—224no—
496C2×C4—224no—
497C2×C4—224no—
498C2×C4—224no—
499C2×C4—228no—
500C2×C4—4416no—
501C2×C4—442no—
502C2×C4—444no—
503C2×C4—442no—
504C2×C4—442no—
505C2×C4—442no—
506C2×C4—442no—
507C2×C4—444no—
508C2×C4—442no—
509C2×C4—442no—
510C2×C4—444no—
511C2×C4—444no—
512C2×C4—444no—
513C2×C4—442no—
514C2×C4C22⋊C4×D1114yes—
515C22—44no—
516C22—44no—
517C22—44no—
518C22—48no—
519C22—48no—
520C22—84no—
521C22—84no—
522C22—84no—
523C22—84no—
524C22—224no—
525C22—224no—
526C22—224no—
527C22—444no—
528C22—444no—
529C22—448no—
530C22—444no—
531C22—444no—
532C22C2×C22⋊C4×D1111yescharacteristic
533C22#1062301712yes—
534C22#1062307612yes—
535C22#1062307712yes—
536C22#1062312012yes—
537C22#1062312012yes—
538C22#1062312614yes—
539C22#1062313812yes—
540C22#1062313912yes—
541C22#1062315114yes—
542C22#1062315214yes—
543C22#1062319814yes—
544C22#1062320511yescharacteristic
545C22#1062320611yescharacteristic
546C22#1062324611yescharacteristic
547C22#1062324611yescharacteristic
548C4—24no—
549C4—82no—
550C4—82no—
551C4—224no—
552C4—444no—
553C4—882no—
554C4—882no—
555C2—84no—
556C2—444no—
557C2#1307119211yescharacteristic
558C2#1307119511yescharacteristic
559C2#1307120611yescharacteristic
560C2#1307176814yes—
561C2#1307661512yes—
562C2#1307661712yes—
563C2#1307663712yes—
564C2#1307664712yes—
subgroupG/H
center, FrattiniC24C23×D11
commutatorC23×C11C23×C4
FittingC11×#1438C2
socle, cosocleC24×C11C24
prime radicalC22×#66991C2
hypercenter#1438D11
hypocenterC11#14659
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ #1438

Complements

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

NH ≅ G/Ntype
2. C2×#10622277C2semidirect
3. C2×#10622280C2semidirect
4. #13066885C2semidirect
5. #13066883C2semidirect
6. #13066859C2semidirect
7. #13066857C2semidirect
8. #13066855C2semidirect
9. C11×#1438C2semidirect
18. C2×#66991C4semidirect
144. C2×C4×D11C22⋊C4semidirect
160. #1438D11semidirect
433. C11#14659semidirect

Maximal subgroups

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

subgroupindex
C22×#669912
C2×#106222772
C2×#106222802
#130668852
#130668832
#130668592
#130668572
#130668552
C11×#14382
#1465911