Finite Group Databasebeta

← #14809948

Subgroups — #14809948

Order 2816.

1338 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66997C211yesmaximal, characteristic
2C2×#10622405C212yesmaximal, complemented
3#13076966C211yesmaximal, characteristic
4#13069861C212yesmaximal, complemented
5#13069860C212yesmaximal, complemented
6#13069416C212yesmaximal, complemented
7#13068552C211yesmaximal, characteristic
8#13068551C211yesmaximal, characteristic
9#13068550C211yesmaximal, characteristic
10C22⋊C8×Dic11C211yesmaximal, characteristic
11C11×#1787C211yesmaximal, characteristic
12C23×Q8×C11C2211yescharacteristic
13C23×Dic22C2211yescharacteristic
14C2×#66997—28no—
15C2×#66997—21no—
16C2×#66997—21no—
17C2×#66997C2212yes—
18C2×#66997C2212yes—
19C22×Dic11⋊C4—21no—
20C22×Dic11⋊C4—21no—
21C22×C4×Dic11C2211yescharacteristic
22C2×#66941C2211yescharacteristic
23C2×#66931—21no—
24C2×C11×#103C2212yes—
25C2×C11×#103C2212yes—
26C2×C11×C22⋊C8C2211yescharacteristic
27C11×#380C2211yescharacteristic
28C2×#66899C2212yes—
29C2×#66899C2212yes—
30#10622825C2211yescharacteristic
31C2×C22⋊(C11⋊C8)C2211yescharacteristic
32C2×#66880C2212yes—
33C2×#66880C2212yes—
34C2×C8×Dic11C2212yes—
35#10622598C2211yescharacteristic
36#10622589C2211yescharacteristic
37#10622586C2211yescharacteristic
38C2×#66874C2212yes—
39C2×#66874C2212yes—
40#10622555—21no—
41#10622405—28no—
42#10622260—21no—
43#10622259—21no—
44#10622188C2212yes—
45C22×Q8×C11—24no—
46C22×Q8×C11—21no—
47C22×Q8×C11—22no—
48C22×Q8×C11—22no—
49C22×Q8×C11—42no—
50C22×Q8×C11C2312yes—
51C23×C4×C11—21no—
52C23×C4×C11C2311yescharacteristic
53C23×Dic11—21no—
54C23×Dic11D411yescharacteristic
55C23×Dic11D411yescharacteristic
56C22×Dic22—21no—
57C22×Dic22—24no—
58C22×Dic22—22no—
59C22×Dic22—22no—
60C22×Dic22—21no—
61C22×Dic22—42no—
62C22×Dic22C2312yes—
63C22×C8×C11C2312yes—
64C2×C11×C4⋊C4C2311yescharacteristic
65C2×C11×C4⋊C4C2311yescharacteristic
66C2×C11×C22⋊C4—21no—
67C2×C22⋊Dic11—21no—
68#66997—28no—
69#66997—28no—
70#66997—24no—
71#66997—24no—
72#66997—24no—
73#66997D418yes—
74C2×C4⋊Dic11—21no—
75C2×C4⋊Dic11C2311yescharacteristic
76C2×C4⋊Dic11C2311yescharacteristic
77C2×Dic11⋊C4—28no—
78C2×Dic11⋊C4—28no—
79C2×Dic11⋊C4—22no—
80C2×Dic11⋊C4—22no—
81C2×Dic11⋊C4—22no—
82C2×Dic11⋊C4—22no—
83C2×Dic11⋊C4—21no—
84C2×Dic11⋊C4—21no—
85C2×Dic11⋊C4—21no—
86C2×Dic11⋊C4—21no—
87C2×C4×Dic11—24no—
88C2×C4×Dic11D411yescharacteristic
89C2×C4×Dic11D411yescharacteristic
90C2×C4×Dic11C2311yescharacteristic
91C2×C4×Dic11C2311yescharacteristic
92C22×C11⋊C8C2312yes—
93#66941—24no—
94#66931—44no—
95C11×#103—24no—
96C11×#103—42no—
97C11×C22⋊C8—22no—
98C11×#95—21no—
99#66899—24no—
100#66899—42no—
101C11⋊#95—21no—
102C11⋊#95—21no—
103C11⋊#95—21no—
104C22⋊(C11⋊C8)—22no—
105#66880—24no—
106#66880—42no—
107C8×Dic11—24no—
108#66874—24no—
109#66874—42no—
110#25856—111nomaximal, Sylow
111C24×C11C2×D411yescharacteristic
112C2×Q8×C11—24no—
113C2×Q8×C11—22no—
114C2×Q8×C11—22no—
115C2×Q8×C11—22no—
116C2×Q8×C11—22no—
117C2×Q8×C11—22no—
118C2×Q8×C11—48no—
119C2×Q8×C11—48no—
120C2×Q8×C11—42no—
121C2×Q8×C11—42no—
122C2×Q8×C11C2×D414yes—
123C22×C4×C11—22no—
124C22×C4×C11—22no—
125C22×C4×C11—24no—
126C22×C4×C11—22no—
127C22×C4×C11—21no—
128C22×C4×C11—21no—
129C22×C4×C11—42no—
130C22×C4×C11—42no—
131C22×C4×C11C2×D411yescharacteristic
132C22×C4×C11#4112yes—
133C22×C4×C11#4112yes—
134C22×C4×C11C2411yescharacteristic
135C22×Dic11—22no—
136C22×Dic11—22no—
137C22×Dic11—22no—
138C22×Dic11—22no—
139C22×Dic11—24no—
140C22×Dic11—22no—
141C22×Dic11—21no—
142C22×Dic11—21no—
143C22×Dic11—42no—
144C22×Dic11—42no—
145C22×Dic11SD1614yes—
146C22×Dic11Q1614yes—
147C22×Dic11C2×D411yescharacteristic
148C22×Dic11C2×D411yescharacteristic
149C22×Dic11C2×D411yescharacteristic
150C22×Dic11C2×D411yescharacteristic
151C2×Dic22—24no—
152C2×Dic22—22no—
153C2×Dic22—22no—
154C2×Dic22—22no—
155C2×Dic22—22no—
156C2×Dic22—22no—
157C2×Dic22—48no—
158C2×Dic22—48no—
159C2×Dic22—42no—
160C2×Dic22—42no—
161C2×Dic22—42no—
162C2×Dic22—42no—
163C2×Dic22—42no—
164C2×Dic22C2×D414yes—
165C2×C8×C11—22no—
166C2×C8×C11—22no—
167C2×C8×C11—22no—
168C11×C4⋊C4—22no—
169C11×C4⋊C4—22no—
170C11×C22⋊C4—42no—
171C22⋊Dic11—42no—
172C4⋊Dic11—22no—
173C4⋊Dic11—22no—
174C4⋊Dic11—42no—
175Dic11⋊C4—28no—
176Dic11⋊C4—28no—
177Dic11⋊C4—28no—
178Dic11⋊C4—28no—
179Dic11⋊C4—42no—
180Dic11⋊C4—42no—
181Dic11⋊C4—42no—
182Dic11⋊C4—42no—
183C4×Dic11—22no—
184C4×Dic11—22no—
185C4×Dic11—22no—
186C4×Dic11C2×D414yes—
187C2×C11⋊C8—22no—
188C2×C11⋊C8—22no—
189C2×C11⋊C8—22no—
190C22×C4⋊Q8—111no—
191C2×#487—112no—
192#2857—111no—
193#2043—112no—
194#2043—112no—
195#1956—112no—
196#1787—111no—
197#1787—111no—
198#1787—111no—
199#1787D1111yescharacteristic
200C4×C22⋊C8—111no—
201C23×C11—21no—
202C23×C11—21no—
203C23×C11—21no—
204C23×C11—21no—
205C23×C11#12412yes—
206C23×C11C2×SD1612yes—
207C23×C11C2×Q1612yes—
208C23×C11C22×D411yescharacteristic
209Q8×C11—24no—
210Q8×C11—24no—
211Q8×C11—48no—
212Q8×C11—42no—
213Q8×C11—42no—
214C2×C4×C11—21no—
215C2×C4×C11—21no—
216C2×C4×C11—24no—
217C2×C4×C11—24no—
218C2×C4×C11—24no—
219C2×C4×C11—48no—
220C2×C4×C11—42no—
221C2×C4×C11—42no—
222C2×C4×C11—41no—
223C2×C4×C11—41no—
224C2×C4×C11—41no—
225C2×C4×C11—41no—
226C2×C4×C11—41no—
227C2×C4×C11—41no—
228C2×C4×C11#12412yes—
229C2×C4×C11C22×D412yes—
230C2×C4×C11C2×#4111yescharacteristic
231C2×C4×C11C2×#4111yescharacteristic
232C2×C4×C11H(5,𝔽2)12yes—
233C2×Dic11—21no—
234C2×Dic11—21no—
235C2×Dic11—21no—
236C2×Dic11—21no—
237C2×Dic11—28no—
238C2×Dic11—28no—
239C2×Dic11—24no—
240C2×Dic11—48no—
241C2×Dic11—42no—
242C2×Dic11—42no—
243C2×Dic11—41no—
244C2×Dic11—41no—
245C2×Dic11—41no—
246C2×Dic11—41no—
247C2×Dic11—41no—
248C2×Dic11—41no—
249C2×Dic11#12012yes—
250C2×Dic11#12012yes—
251C2×Dic11C2×SD1612yes—
252C2×Dic11C2×SD1612yes—
253C2×Dic11C2×Q1612yes—
254C2×Dic11C2×Q1612yes—
255C2×Dic11#13712yes—
256C2×Dic11#13712yes—
257Dic22—24no—
258Dic22—24no—
259Dic22—48no—
260Dic22—42no—
261Dic22—42no—
262Dic22—84no—
263C8×C11—42no—
264C11⋊C8—42no—
265C23×Q8—111no—
266C23×Q8C2×D1111yescharacteristic
267C2×C4⋊Q8—111no—
268C2×C4⋊Q8—112no—
269C2×C4⋊Q8—112no—
270C2×C4⋊Q8—221no—
271C2×C4⋊Q8—221no—
272C2×C4⋊Q8—228no—
273C2×C22⋊Q8—221no—
274C22×C4⋊C4—221no—
275C22×C4⋊C4—221no—
276C22×C42—111no—
277#487—228no—
278C2×#103—112no—
279C2×#103—112no—
280C2×#103—112no—
281C2×#103—112no—
282C2×#103—112no—
283C2×#103—112no—
284C2×#103C2×D1112yes—
285C2×#103C2×D1112yes—
286C2×C22⋊C8—111no—
287C2×C22⋊C8C2×D1111yescharacteristic
288C2×C4×C8—112no—
289#391—221no—
290#389—111no—
291#389—111no—
292#389—221no—
293#384—111no—
294#380—111no—
295#380—221no—
296#380C2×D1111yescharacteristic
297#336—112no—
298C22×C11—22no—
299C22×C11—22no—
300C22×C11—22no—
301C22×C11—22no—
302C22×C11—22no—
303C22×C11—22no—
304C22×C11#48714yescomplemented
305C22×C11C2×#12011yescharacteristic
306C22×C11C2×#12411yescharacteristic
307C22×C11#53511yescharacteristic
308C22×C11C22×SD1611yescharacteristic
309C22×C11C22×Q1611yescharacteristic
310C22×C11C2×#13711yescharacteristic
311C22×C11C2×#13711yescharacteristic
312C4×C11—22no—
313C4×C11—44no—
314C4×C11—81no—
315C4×C11—81no—
316C4×C11#54612yes—
317C4×C11#54612yes—
318Dic11—22no—
319Dic11—22no—
320Dic11—44no—
321Dic11—81no—
322Dic11—81no—
323Dic11#44814yes—
324Dic11#45114yes—
325C22×Q8—24no—
326C22×Q8—21no—
327C22×Q8—22no—
328C22×Q8—22no—
329C22×Q8—42no—
330C22×Q8—112no—
331C22×Q8—221no—
332C22×Q8—224no—
333C22×Q8—221no—
334C22×Q8—222no—
335C22×Q8—222no—
336C22×Q8—442no—
337C22×Q8C22×D1112yes—
338C23×C4—21no—
339C23×C4—111no—
340C23×C4—111no—
341C23×C4—221no—
342C23×C4C22×D1111yescharacteristic
343C22×C8—112no—
344C22×C8C22×D1112yes—
345C4⋊Q8—118no—
346C4⋊Q8—224no—
347C4⋊Q8—224no—
348C4⋊Q8—224no—
349C4⋊Q8—228no—
350C4⋊Q8—228no—
351C4⋊Q8—224no—
352C22⋊Q8—444no—
353C2×C4⋊C4—111no—
354C2×C4⋊C4—111no—
355C2×C4⋊C4—222no—
356C2×C4⋊C4—222no—
357C2×C4⋊C4—222no—
358C2×C4⋊C4—222no—
359C2×C4⋊C4—228no—
360C2×C4⋊C4—228no—
361C2×C4⋊C4—221no—
362C2×C4⋊C4—221no—
363C2×C4⋊C4—221no—
364C2×C4⋊C4—221no—
365C2×C4⋊C4—221no—
366C2×C4⋊C4C22×D1111yescharacteristic
367C2×C4⋊C4C22×D1111yescharacteristic
368C2×C22⋊C4—21no—
369C2×C22⋊C4—221no—
370C2×C42—111no—
371C2×C42—111no—
372C2×C42—111no—
373C2×C42—111no—
374C2×C42—224no—
375#103—24no—
376#103—42no—
377#103—224no—
378#103—224no—
379#103—224no—
380#103—442no—
381#103—442no—
382#103—442no—
383C22⋊C8—22no—
384C22⋊C8—222no—
385C4×C8—224no—
386#95—21no—
387#95—221no—
388#95—221no—
389#95—221no—
390C2×C11—24no—
391C2×C11#243511yescharacteristic
392C2×C11C2×#44811yescharacteristic
393C2×C11C2×#45111yescharacteristic
394C2×C11C2×#48711yescharacteristic
395C2×C11#313911yescharacteristic
396C2×C11#320611yescharacteristic
397C2×C11#320911yescharacteristic
398C24D4×D1111yescharacteristic
399C2×Q8—24no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—48no—
406C2×Q8—48no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—114no—
410C2×Q8—224no—
411C2×Q8—222no—
412C2×Q8—222no—
413C2×Q8—222no—
414C2×Q8—222no—
415C2×Q8—222no—
416C2×Q8—448no—
417C2×Q8—448no—
418C2×Q8—442no—
419C2×Q8—442no—
420C2×Q8—442no—
421C2×Q8—442no—
422C2×Q8—442no—
423C2×Q8D4×D1114yes—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—111no—
433C22×C4—111no—
434C22×C4—114no—
435C22×C4—114no—
436C22×C4—111no—
437C22×C4—111no—
438C22×C4—222no—
439C22×C4—222no—
440C22×C4—222no—
441C22×C4—222no—
442C22×C4—224no—
443C22×C4—222no—
444C22×C4—221no—
445C22×C4—221no—
446C22×C4—442no—
447C22×C4—442no—
448C22×C4#433912yes—
449C22×C4D4×D1111yescharacteristic
450C22×C4#434112yes—
451C22×C4C23×D1111yescharacteristic
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—222no—
456C2×C8—222no—
457C2×C8—222no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—228no—
461C4⋊C4—228no—
462C4⋊C4—228no—
463C4⋊C4—228no—
464C4⋊C4—222no—
465C4⋊C4—222no—
466C4⋊C4—442no—
467C4⋊C4—442no—
468C4⋊C4—442no—
469C4⋊C4—442no—
470C4⋊C4—442no—
471C22⋊C4—42no—
472C22⋊C4—442no—
473C42—114no—
474C42—222no—
475C42—222no—
476C42—222no—
477C11#2585611yescharacteristic, complemented, Sylow
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23#6693812yes—
483C23SD16×D1112yes—
484C23Q16×D1112yes—
485C23C2×D4×D1111yescharacteristic
486Q8—24no—
487Q8—24no—
488Q8—48no—
489Q8—42no—
490Q8—42no—
491Q8—224no—
492Q8—224no—
493Q8—448no—
494Q8—442no—
495Q8—442no—
496Q8—884no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—48no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—112no—
512C2×C4—112no—
513C2×C4—112no—
514C2×C4—112no—
515C2×C4—112no—
516C2×C4—112no—
517C2×C4—112no—
518C2×C4—112no—
519C2×C4—228no—
520C2×C4—228no—
521C2×C4—221no—
522C2×C4—221no—
523C2×C4—221no—
524C2×C4—221no—
525C2×C4—224no—
526C2×C4—448no—
527C2×C4—442no—
528C2×C4—442no—
529C2×C4—441no—
530C2×C4—441no—
531C2×C4—441no—
532C2×C4—441no—
533C2×C4—441no—
534C2×C4—441no—
535C2×C4#6693812yes—
536C2×C4C2×#433911yescharacteristic
537C2×C4C2×D4×D1112yes—
538C2×C4C2×#434111yescharacteristic
539C2×C4#6703712yes—
540C8—42no—
541C8—442no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22#1062240514yescomplemented
549C22C2×#6693811yescharacteristic
550C22D11×#12011yescharacteristic
551C22#1062312311yescharacteristic
552C22C2×SD16×D1111yescharacteristic
553C22C2×Q16×D1111yescharacteristic
554C22D11×#13711yescharacteristic
555C22D11×#13711yescharacteristic
556C4—22no—
557C4—44no—
558C4—81no—
559C4—81no—
560C4—114no—
561C4—114no—
562C4—222no—
563C4—222no—
564C4—444no—
565C4—881no—
566C4—881no—
567C4#1062302112yes—
568C4#1062312212yes—
569C2—24no—
570C2#1307226511yescharacteristic
571C2D11×#44811yescharacteristic
572C2D11×#45111yescharacteristic
573C2C2×#1062240511yescharacteristic
574C2#1307899411yescharacteristic
575C2#1307953911yescharacteristic
576C2#1307956211yescharacteristic
subgroupG/H
centerC23C2×D4×D11
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC11×#1787C2
socleC23×C11C22×D4
prime radicalC24×C11C2×D4
hypercenter#1787D11
hypocenterC11#25856
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
2. C2×#10622405C2semidirect
4. #13069861C2semidirect
5. #13069860C2semidirect
6. #13069416C2semidirect
304. C22×C11#487semidirect
477. C11#25856semidirect
548. C22#10622405semidirect

Maximal subgroups

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

subgroupindex
C22×#669972
C2×#106224052
#130769662
#130698612
#130698602
#130694162
#130685522
#130685512
#130685502
C22⋊C8×Dic112
C11×#17872
#2585611