Finite Group Databasebeta

← #14759275

Subgroups — #14759275

Order 2816.

1598 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#4313C211yesmaximal, characteristic, complemented
2C2×#10622724C212yesmaximal, complemented
3C4×#66990C211yesmaximal, characteristic
4#13068910C212yesmaximal, complemented
5#13068909C211yesmaximal, characteristic, complemented
6#13068797C212yesmaximal, complemented
7#13068793C211yesmaximal, characteristic, complemented
8#13068525C211yesmaximal, characteristic, complemented
9#13068399C212yesmaximal, complemented
10#13068392C211yesmaximal, characteristic, complemented
11C2×C8×D4×C11C211yesmaximal, characteristic, complemented
12C23×C4×D11C2211yescharacteristic
13C23×C8×C11C2211yescharacteristic
14C23×C8×C11C2211yescharacteristic, complemented
15C2×C4×D4×C11C2211yescharacteristic
16C2×#66990C411yescharacteristic
17C2×C4×C22⋊D11C2212yes—
18C23×C11⋊C8C2211yescharacteristic
19C22×#4313—24no—
20C22×#4313—28no—
21C22×#4313C2212yes—
22C22×#4313C2212yes—
23C8×D4×C11—24no—
24C2×C11×C4⋊C8C2211yescharacteristic, complemented
25C2×C11×C22⋊C8C2211yescharacteristic
26C2×C11×C22⋊C8C2211yescharacteristic, complemented
27C2×C4×C8×C11C2211yescharacteristic, complemented
28#10622833C411yescharacteristic
29C4×C22⋊Dic11C2211yescharacteristic
30C2×C22⋊(C11⋊C8)C2211yescharacteristic
31C2×C22⋊(C11⋊C8)C2212yescomplemented
32#10622724—28no—
33C2×(C2×D11)⋊C8C2212yes—
34C2×(C2×D11)⋊C8C2212yes—
35C2×#66879C2212yescomplemented
36C2×Dic11⋊C8C2212yescomplemented
37#10622602C412yes—
38#10622569C412yes—
39#10622568C411yescharacteristic
40#10622559C411yescharacteristic
41C4×(C2×D11)⋊C4C2212yes—
42#10622188C2212yescomplemented
43#10622188C2212yescomplemented
44#10622162C2212yescomplemented
45C24×D11C2×C411yescharacteristic
46C22×D4×C11C2×C411yescharacteristic
47C23×C4×C11C2311yescharacteristic
48C23×C4×C11C2311yescharacteristic
49C22×C22⋊D11C2×C412yes—
50C23×Dic11C2×C411yescharacteristic
51C22×C4×D11—22no—
52C22×C4×D11—22no—
53C22×C4×D11—24no—
54C22×C4×D11—24no—
55C22×C4×D11C2×C412yes—
56C22×C4×D11C2312yes—
57C22×C8×C11—24no—
58C22×C8×C11—24no—
59C22×C8×C11—22no—
60C22×C8×C11D414yescomplemented
61C22×C8×C11C2311yescharacteristic
62C22×C8×C11C2311yescharacteristic
63C22×C8×C11C2311yescharacteristic
64C22×C8×C11C2311yescharacteristic
65C4×D4×C11—22no—
66C4×D4×C11—22no—
67C2×C11×C4⋊C4C2×C411yescharacteristic
68C2×C11×C22⋊C4C2×C411yescharacteristic
69C2×C11×C22⋊C4C2×C411yescharacteristic
70C2×C42×C11C2311yescharacteristic
71C2×C22⋊Dic11C2×C412yes—
72C2×C22⋊Dic11C2×C411yescharacteristic
73C2×C22⋊Dic11C2×C411yescharacteristic
74#66990—24no—
75C4×C22⋊D11—24no—
76C4×C22⋊D11—24no—
77C2×(C2×D11)⋊C4C2×C412yes—
78C2×(C2×D11)⋊C4C2×C412yes—
79C2×(C2×D11)⋊C4C2×C412yes—
80C2×Dic11⋊C4C2×C412yes—
81C2×C4×Dic11C2312yes—
82C22×C11⋊C8—22no—
83C22×C11⋊C8—24no—
84C22×C11⋊C8C2312yes—
85C22×C11⋊C8C2312yes—
86C2×#4313—232no—
87C2×#4313—24no—
88C2×#4313—24no—
89C2×#4313—24no—
90C2×#4313—24no—
91C2×#4313—24no—
92C2×#4313D418yescomplemented
93C11×C4⋊C8—22no—
94C11×C22⋊C8—22no—
95C11×C22⋊C8—22no—
96C4×C8×C11—22no—
97C11⋊#95C2×C412yes—
98C11⋊#95C2×C412yes—
99C11⋊#95C2×C412yes—
100C22⋊(C11⋊C8)—22no—
101C22⋊(C11⋊C8)—24no—
102(C2×D11)⋊C8—24no—
103(C2×D11)⋊C8—24no—
104#66879—24no—
105Dic11⋊C8—24no—
106#21439—111nomaximal, Sylow
107C24×C11C22×C411yescharacteristic
108C24×C11C22×C411yescharacteristic
109C23×D11—22no—
110C23×D11—24no—
111C23×D11—22no—
112C23×D11—22no—
113C23×D11—42no—
114C23×D11C22×C412yes—
115C2×D4×C11—22no—
116C2×D4×C11—22no—
117C2×D4×C11M1614yescomplemented
118C22×C4×C11—22no—
119C22×C4×C11—22no—
120C22×C4×C11—22no—
121C22×C4×C11—21no—
122C22×C4×C11—22no—
123C22×C4×C11—21no—
124C22×C4×C11C22×C411yescharacteristic
125C22×C4×C11C22×C411yescharacteristic
126C22×C4×C11C22×C411yescharacteristic
127C22×C4×C11C22×C411yescharacteristic
128C22×C4×C11C2×D412yes—
129C22×C4×C11#4112yes—
130C22×C4×C11C2411yescharacteristic
131C2×C22⋊D11—24no—
132C2×C22⋊D11—24no—
133C2×C22⋊D11—42no—
134C22×Dic11—24no—
135C22×Dic11—22no—
136C22×Dic11C22×C412yes—
137C22×Dic11C22×C412yes—
138C22×Dic11C22×C412yes—
139C2×C4×D11—28no—
140C2×C4×D11—28no—
141C2×C4×D11—216no—
142C2×C4×D11—24no—
143C2×C4×D11—24no—
144C2×C4×D11—24no—
145C2×C4×D11—22no—
146C2×C4×D11—22no—
147C2×C4×D11—22no—
148C2×C4×D11—22no—
149C2×C4×D11C2×D414yes—
150C2×C4×D11#4114yes—
151C2×C8×C11—28no—
152C2×C8×C11—22no—
153C2×C8×C11—22no—
154C2×C8×C11—21no—
155C2×C8×C11—21no—
156C2×C8×C11—22no—
157C2×C8×C11—44no—
158C2×C8×C11C2×D414yes—
159C2×C8×C11C2×D412yes—
160C2×C8×C11C2×D412yes—
161C11×C4⋊C4—22no—
162C11×C22⋊C4—22no—
163C11×C22⋊C4—22no—
164C42×C11—21no—
165C42×C11—21no—
166C22⋊Dic11—24no—
167C22⋊Dic11—22no—
168C22⋊Dic11—22no—
169(C2×D11)⋊C4—24no—
170(C2×D11)⋊C4—24no—
171(C2×D11)⋊C4—42no—
172Dic11⋊C4—24no—
173C4×Dic11—22no—
174C4×Dic11—22no—
175C2×C11⋊C8—28no—
176C2×C11⋊C8—22no—
177C2×C11⋊C8—22no—
178C2×C11⋊C8—22no—
179C2×C11⋊C8—22no—
180C2×C11⋊C8—22no—
181C2×C11⋊C8C2×D414yes—
182#4313—216no—
183#4313—216no—
184C23×M16—111no—
185C2×#435—112no—
186C2×C8×D4D1111yescharacteristic, complemented
187C4×#120—111no—
188#1870—112no—
189#1869—111no—
190#1851—112no—
191#1850—111no—
192#1779—111no—
193#1753—112no—
194#1752—111no—
195C23×C11—22no—
196C23×C11—22no—
197C23×C11—22no—
198C23×C11—21no—
199C23×C11—21no—
200C23×C11—22no—
201C23×C11C4×D412yes—
202C23×C11C2×M1612yes—
203C23×C11C2×M1612yescomplemented
204C23×C11#13112yes—
205C23×C11C23×C411yescharacteristic
206C22×D11—28no—
207C22×D11—22no—
208C22×D11—22no—
209C22×D11—22no—
210C22×D11—22no—
211C22×D11—22no—
212C22×D11—22no—
213C22×D11—48no—
214C22×D11—48no—
215C22×D11—44no—
216C22×D11—44no—
217C22×D11C4×D414yes—
218D4×C11—28no—
219C2×C4×C11—21no—
220C2×C4×C11—22no—
221C2×C4×C11—21no—
222C2×C4×C11—22no—
223C2×C4×C11—22no—
224C2×C4×C11—24no—
225C2×C4×C11—22no—
226C2×C4×C11—21no—
227C2×C4×C11—21no—
228C2×C4×C11—22no—
229C2×C4×C11—41no—
230C2×C4×C11—42no—
231C2×C4×C11—41no—
232C2×C4×C11C4×D412yes—
233C2×C4×C11C2×M1612yescomplemented
234C2×C4×C11C22×D412yes—
235C2×C4×C11C22×D411yescharacteristic
236C2×C4×C11C2×#4111yescharacteristic
237C2×C4×C11C2×#4112yes—
238C22⋊D11—48no—
239C2×Dic11—28no—
240C2×Dic11—22no—
241C2×Dic11—22no—
242C2×Dic11—22no—
243C2×Dic11—22no—
244C2×Dic11—22no—
245C2×Dic11—22no—
246C2×Dic11—22no—
247C2×Dic11—22no—
248C2×Dic11C4×D414yes—
249C4×D11—216no—
250C4×D11—24no—
251C4×D11—24no—
252C4×D11—24no—
253C4×D11—24no—
254C8×C11—22no—
255C8×C11—41no—
256C8×C11—41no—
257C8×C11#12014yescomplemented
258C11⋊C8—24no—
259C11⋊C8—42no—
260C24×C4—111no—
261C22×M16—112no—
262C22×M16—112no—
263C22×M16—228no—
264C22×M16—224no—
265C23×C8—111no—
266C23×C8C2×D1111yescharacteristic
267C23×C8C2×D1111yescharacteristic, complemented
268C2×#120—111no—
269C2×C4×D4—112no—
270C2×C4×D4C2×D1111yescharacteristic
271#435—228no—
272C8×D4—24no—
273C2×C4⋊C8—112no—
274C2×C4⋊C8C2×D1111yescharacteristic, complemented
275C2×C22⋊C8—112no—
276C2×C22⋊C8—111no—
277C2×C22⋊C8—112no—
278C2×C22⋊C8—112no—
279C2×C22⋊C8C2×D1111yescharacteristic
280C2×C22⋊C8C2×D1111yescharacteristic, complemented
281C2×#97—112no—
282C2×C4×C8C2×D1111yescharacteristic, complemented
283#386—111no—
284#386—112no—
285#385—112no—
286#385—111no—
287#379—111no—
288C4×C22⋊C4—111no—
289C4×C22⋊C4—112no—
290#336—112no—
291#336—112no—
292#336—112no—
293C22×C11—22no—
294C22×C11—24no—
295C22×C11—24no—
296C22×C11—22no—
297C22×C11—22no—
298C22×C11—24no—
299C22×C11—41no—
300C22×C11—42no—
301C22×C11—41no—
302C22×C11#43514yescomplemented
303C22×C11C2×C4×D411yescharacteristic
304C22×C11C2×C4×D412yes—
305C22×C11C22×M1611yescharacteristic
306C22×C11C2×#13111yescharacteristic
307C22×C11#56812yes—
308C2×D11—24no—
309C2×D11—24no—
310C2×D11—24no—
311C2×D11—416no—
312C2×D11—42no—
313C2×D11—42no—
314C2×D11—42no—
315C2×D11—42no—
316C4×C11—22no—
317C4×C11—22no—
318C4×C11—41no—
319C4×C11—41no—
320C4×C11C2×#12011yescharacteristic
321C4×C11#52512yes—
322C4×C11#52511yescharacteristic
323Dic11—24no—
324Dic11—42no—
325Dic11—42no—
326C25—111no—
327C22×D4—112no—
328C22×D4C4×D1111yescharacteristic
329C23×C4—112no—
330C23×C4—111no—
331C23×C4—112no—
332C23×C4—224no—
333C23×C4—224no—
334C23×C4—222no—
335C23×C4—222no—
336C23×C4C22×D1111yescharacteristic
337C23×C4C22×D1111yescharacteristic
338C2×M16—118no—
339C2×M16—224no—
340C2×M16—224no—
341C2×M16—2232no—
342C2×M16—224no—
343C2×M16—224no—
344C2×M16—224no—
345C22×C8—24no—
346C22×C8—24no—
347C22×C8—22no—
348C22×C8—112no—
349C22×C8—112no—
350C22×C8—224no—
351C22×C8—222no—
352C22×C8C22⋊D1114yescomplemented
353C22×C8C22×D1111yescharacteristic
354C22×C8C22×D1111yescharacteristic
355C22×C8C22×D1111yescharacteristic
356C22×C8C22×D1111yescharacteristic
357#120—224no—
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—224no—
361C4×D4—224no—
362C2×C4⋊C4—112no—
363C2×C4⋊C4C4×D1111yescharacteristic
364C2×C22⋊C4—112no—
365C2×C22⋊C4—111no—
366C2×C22⋊C4—111no—
367C2×C22⋊C4—112no—
368C2×C22⋊C4—112no—
369C2×C22⋊C4—112no—
370C2×C22⋊C4C4×D1111yescharacteristic
371C2×C22⋊C4C4×D1111yescharacteristic
372C2×C42—112no—
373C2×C42C22×D1111yescharacteristic
374C4⋊C8—22no—
375C4⋊C8—224no—
376C22⋊C8—22no—
377C22⋊C8—22no—
378C22⋊C8—224no—
379C22⋊C8—224no—
380C22⋊C8—224no—
381C22⋊C8—222no—
382#97—224no—
383C4×C8—22no—
384#95—112no—
385#95—112no—
386#95—112no—
387C2×C11—24no—
388C2×C11—42no—
389C2×C11C4×#12011yescharacteristic
390C2×C11C2×#43511yescharacteristic
391C2×C11#293011yescharacteristic
392C2×C11#293312yes—
393C2×C11D4×M1612yes—
394D11—44no—
395C24—112no—
396C24—224no—
397C24—222no—
398C24—222no—
399C24—222no—
400C24—442no—
401C24C2×C4×D1111yescharacteristic
402C24C2×C4×D1111yescharacteristic
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—224no—
406C2×D4—224no—
407C2×D4—442no—
408C2×D4#431314yescomplemented
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—112no—
416C22×C4—114no—
417C22×C4—114no—
418C22×C4—112no—
419C22×C4—112no—
420C22×C4—224no—
421C22×C4—224no—
422C22×C4—228no—
423C22×C4—2216no—
424C22×C4—228no—
425C22×C4—224no—
426C22×C4—222no—
427C22×C4—224no—
428C22×C4—222no—
429C22×C4—222no—
430C22×C4—222no—
431C22×C4—222no—
432C22×C4C2×C4×D1111yescharacteristic
433C22×C4C2×C4×D1111yescharacteristic
434C22×C4C2×C4×D1111yescharacteristic
435C22×C4C2×C4×D1111yescharacteristic
436C22×C4#433912yes—
437C22×C4C2×C22⋊D1112yes—
438C22×C4C23×D1111yescharacteristic
439M16—2216no—
440M16—2216no—
441C2×C8—22no—
442C2×C8—28no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—44no—
448C2×C8—114no—
449C2×C8—228no—
450C2×C8—222no—
451C2×C8—222no—
452C2×C8—222no—
453C2×C8—222no—
454C2×C8—222no—
455C2×C8D4×D1114yes—
456C2×C8C2×C22⋊D1112yes—
457C2×C8C2×C22⋊D1112yes—
458C4⋊C4—22no—
459C4⋊C4—224no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—224no—
463C22⋊C4—224no—
464C22⋊C4—222no—
465C22⋊C4—222no—
466C22⋊C4—224no—
467C22⋊C4—442no—
468C42—21no—
469C42—21no—
470C42—222no—
471C42—222no—
472C11#2143911yescharacteristic, complemented, Sylow
473C23—22no—
474C23—22no—
475C23—21no—
476C23—22no—
477C23—21no—
478C23—22no—
479C23—114no—
480C23—228no—
481C23—222no—
482C23—222no—
483C23—222no—
484C23—222no—
485C23—222no—
486C23—222no—
487C23—448no—
488C23—448no—
489C23—444no—
490C23—444no—
491C23C2×#431312yes—
492C23C2×#431312yescomplemented
493C23#6695412yes—
494C23C4×C22⋊D1112yes—
495C23C22×C4×D1111yescharacteristic
496D4—28no—
497D4—448no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—114no—
512C2×C4—2216no—
513C2×C4—228no—
514C2×C4—222no—
515C2×C4—222no—
516C2×C4—224no—
517C2×C4—224no—
518C2×C4—224no—
519C2×C4—224no—
520C2×C4—222no—
521C2×C4—222no—
522C2×C4—222no—
523C2×C4—222no—
524C2×C4—222no—
525C2×C4—222no—
526C2×C4C2×#431312yescomplemented
527C2×C4C4×C22⋊D1112yes—
528C2×C4C2×#433911yescharacteristic
529C2×C4C2×D4×D1112yes—
530C2×C4#41×D1112yes—
531C2×C4C22×C22⋊D1111yescharacteristic
532C8—22no—
533C8—41no—
534C8—41no—
535C8—224no—
536C8—442no—
537C8#6699014yescomplemented
538C22—22no—
539C22—22no—
540C22—24no—
541C22—24no—
542C22—22no—
543C22—24no—
544C22—41no—
545C22—41no—
546C22—42no—
547C22—224no—
548C22—224no—
549C22—224no—
550C22—4416no—
551C22—442no—
552C22—442no—
553C22—442no—
554C22—442no—
555C22#1062272414yescomplemented
556C22C4×D4×D1112yes—
557C22C22×#431311yescharacteristic
558C22C2×#6695411yescharacteristic
559C22#1062328112yes—
560C22C2×C4×C22⋊D1111yescharacteristic
561C4—22no—
562C4—22no—
563C4—41no—
564C4—41no—
565C4—224no—
566C4—442no—
567C4—442no—
568C4#1062307812yes—
569C4C2×#6699011yescharacteristic
570C4#1062335911yescharacteristic
571C2—24no—
572C2—42no—
573C2—444no—
574C2C4×#6699011yescharacteristic
575C2C2×#1062272411yescharacteristic
576C2#1307731711yescharacteristic
577C2#1307733612yes—
578C2D4×#431312yes—
subgroupG/H
center, FrattiniC22×C4C23×D11
commutator, socleC23×C11C23×C4
FittingC2×C8×D4×C11C2
prime radicalC2×#66990C4
cosocleC22×C4×C11C24
hypercenterC2×C8×D4D11
hypocenterC11#21439
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C22×C4 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
1. C23×#4313C2semidirect
2. C2×#10622724C2semidirect
4. #13068910C2semidirect
5. #13068909C2semidirect
6. #13068797C2semidirect
7. #13068793C2semidirect
8. #13068525C2semidirect
9. #13068399C2semidirect
10. #13068392C2semidirect
11. C2×C8×D4×C11C2semidirect
14. C23×C8×C11C22semidirect
24. C2×C11×C4⋊C8C22semidirect
26. C2×C11×C22⋊C8C22semidirect
27. C2×C4×C8×C11C22semidirect
31. C2×C22⋊(C11⋊C8)C22semidirect
35. C2×#66879C22semidirect
36. C2×Dic11⋊C8C22semidirect
42. #10622188C22semidirect
43. #10622188C22semidirect
44. #10622162C22semidirect
60. C22×C8×C11D4semidirect
92. C2×#4313D4semidirect
117. C2×D4×C11M16semidirect
186. C2×C8×D4D11semidirect
203. C23×C11C2×M16semidirect
233. C2×C4×C11C2×M16semidirect
257. C8×C11#120semidirect
267. C23×C8C2×D11semidirect
274. C2×C4⋊C8C2×D11semidirect
280. C2×C22⋊C8C2×D11semidirect
282. C2×C4×C8C2×D11semidirect
302. C22×C11#435semidirect
352. C22×C8C22⋊D11semidirect
408. C2×D4#4313semidirect
472. C11#21439semidirect
492. C23C2×#4313semidirect
526. C2×C4C2×#4313semidirect
537. C8#66990semidirect
555. C22#10622724semidirect

Maximal subgroups

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

subgroupindex
C23×#43132
C2×#106227242
C4×#669902
#130689102
#130689092
#130687972
#130687932
#130685252
#130683992
#130683922
C2×C8×D4×C112
#2143911