Finite Group Databasebeta

← #14769070

Subgroups — #14769070

Order 2816.

1908 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D44C211yesmaximal, characteristic
2C2×#10622301C212yesmaximal, complemented
3C2×#10622339C212yesmaximal, complemented
4#13069410C212yesmaximal, complemented
5#13068912C214yesmaximal, complemented
6#13068596C211yesmaximal, characteristic, complemented
7#13068525C212yesmaximal, complemented
8C4×C11×C22⋊C8C211yesmaximal, characteristic, complemented
9C23×D44C411yescharacteristic
10C23×C4×D11C2212yes—
11C22×C42×C11C2211yescharacteristic
12C22×(C2×D11)⋊C4C412yes—
13C22×C4⋊Dic11C411yescharacteristic
14C22×#4313C2214yescomplemented
15C2×C4×D44—28no—
16C2×C4×D44C411yescharacteristic
17C2×C4×D44C412yes—
18C2×C4×D44C411yescharacteristic
19C2×C4×D44C2211yescharacteristic
20C2×C4×D44C2212yes—
21C2×C4×D44C2211yescharacteristic
22C2×C11×C22⋊C8C2211yescharacteristic
23C2×C11×C22⋊C8C2211yescharacteristic
24C2×C4×C8×C11C2212yescomplemented
25C2×C22⋊(C11⋊C8)C2212yes—
26C2×(C2×D11)⋊C8C2214yescomplemented
27C2×(C2×D11)⋊C8C2214yescomplemented
28C2×(C2×D11)⋊C8C2214yescomplemented
29C2×C4⋊(C11⋊C8)C2212yescomplemented
30#10622339—28no—
31#10622301—28no—
32C11×#336C2212yescomplemented
33#10622162C2212yescomplemented
34C24×D11C2×C412yes—
35C23×C4×C11C2×C411yescharacteristic
36C23×C4×C11C2×C411yescharacteristic
37C23×C4×C11C2311yescharacteristic
38C23×Dic11C2×C412yes—
39C22×D44—24no—
40C22×D44—24no—
41C22×D44—24no—
42C22×D44C2×C412yes—
43C22×D44C2×C411yescharacteristic
44C22×D44C2×C411yescharacteristic
45C22×C4×D11—24no—
46C22×C4×D11—24no—
47C22×C4×D11—28no—
48C22×C4×D11—44no—
49C22×C4×D11C2×C412yes—
50C22×C4×D11C2×C412yes—
51C22×C4×D11C2312yes—
52C22×C4×D11C2312yes—
53C22×C8×C11C2312yes—
54C22×C8×C11C2312yes—
55C2×C42×C11—24no—
56C2×C42×C11C2×C411yescharacteristic
57C2×C42×C11C2×C411yescharacteristic
58C2×C42×C11C2311yescharacteristic
59C2×C42×C11C2311yescharacteristic
60C2×(C2×D11)⋊C4—28no—
61C2×(C2×D11)⋊C4C2×C412yes—
62C2×(C2×D11)⋊C4C2×C412yes—
63C2×(C2×D11)⋊C4C2×C412yes—
64C2×(C2×D11)⋊C4C2×C412yes—
65C2×C4⋊Dic11—24no—
66C2×C4⋊Dic11C2×C412yes—
67C2×C4⋊Dic11C2×C411yescharacteristic
68C2×C4⋊Dic11C2×C411yescharacteristic
69C22×C11⋊C8C2314yes—
70C2×#4313—28no—
71C2×#4313—28no—
72C2×#4313—44no—
73C4×D44—24no—
74C4×D44—28no—
75C4×D44—24no—
76C4×D44—28no—
77C4×D44—24no—
78C4×D44D418yes—
79C11×C22⋊C8—22no—
80C11×C22⋊C8D414yescomplemented
81C4×C8×C11—24no—
82C22⋊(C11⋊C8)—24no—
83(C2×D11)⋊C8—28no—
84(C2×D11)⋊C8—28no—
85(C2×D11)⋊C8—44no—
86C4⋊(C11⋊C8)—24no—
87#22401—111nomaximal, Sylow
88C24×C11C22×C411yescharacteristic
89C23×D11—24no—
90C23×D11—24no—
91C23×D11—24no—
92C23×D11—24no—
93C23×D11—24no—
94C23×D11—42no—
95C23×D11—42no—
96C23×D11C22×C412yes—
97C23×D11C22×C412yes—
98C22×C4×C11—22no—
99C22×C4×C11—22no—
100C22×C4×C11—22no—
101C22×C4×C11—22no—
102C22×C4×C11—22no—
103C22×C4×C11—22no—
104C22×C4×C11—41no—
105C22×C4×C11M1614yescomplemented
106C22×C4×C11C22×C411yescharacteristic
107C22×C4×C11C22×C411yescharacteristic
108C22×C4×C11C22×C411yescharacteristic
109C22×C4×C11C22×C411yescharacteristic
110C22×C4×C11C22×C411yescharacteristic
111C22×C4×C11C2×D412yes—
112C22×C4×C11#4112yes—
113C22×C4×C11C2411yescharacteristic
114C22×Dic11—24no—
115C22×Dic11—24no—
116C22×Dic11—42no—
117C22×Dic11C22×C412yes—
118C22×Dic11C22×C412yes—
119C2×D44—232no—
120C2×D44—24no—
121C2×D44—24no—
122C2×D44—22no—
123C2×D44—24no—
124C2×D44—22no—
125C2×D44—22no—
126C2×D44—22no—
127C2×D44C22⋊C414yes—
128C2×D44M1614yescomplemented
129C2×C4×D11—24no—
130C2×C4×D11—24no—
131C2×C4×D11—24no—
132C2×C4×D11—24no—
133C2×C4×D11—22no—
134C2×C4×D11—22no—
135C2×C4×D11—24no—
136C2×C4×D11—22no—
137C2×C4×D11—22no—
138C2×C4×D11—24no—
139C2×C4×D11—24no—
140C2×C4×D11—42no—
141C2×C4×D11—42no—
142C2×C4×D11—416no—
143C2×C4×D11—416no—
144C2×C4×D11C2×D418yes—
145C2×C8×C11—22no—
146C2×C8×C11—22no—
147C2×C8×C11—22no—
148C2×C8×C11—22no—
149C2×C8×C11—22no—
150C2×C8×C11C2×D414yescomplemented
151C42×C11—22no—
152C42×C11—22no—
153C42×C11—22no—
154C42×C11C2×D414yes—
155(C2×D11)⋊C4—24no—
156(C2×D11)⋊C4—24no—
157(C2×D11)⋊C4—24no—
158(C2×D11)⋊C4C22⋊C418yes—
159C4⋊Dic11—24no—
160C4⋊Dic11—22no—
161C4⋊Dic11C22⋊C414yes—
162C2×C11⋊C8—24no—
163C2×C11⋊C8—24no—
164C2×C11⋊C8—24no—
165#4313—416no—
166C22×C4×D4—111no—
167C2×#436—112no—
168C2×#407—112no—
169#1954—112no—
170#1870—114no—
171#1797—111no—
172#1779—112no—
173C4×C22⋊C8D1111yescharacteristic, complemented
174C23×C11—21no—
175C23×C11—21no—
176C23×C11—21no—
177C23×C11—21no—
178C23×C11C4×D412yes—
179C23×C11C2×M1612yes—
180C23×C11#13112yes—
181C23×C11C23×C411yescharacteristic
182C22×D11—216no—
183C22×D11—22no—
184C22×D11—22no—
185C22×D11—22no—
186C22×D11—22no—
187C22×D11—22no—
188C22×D11—22no—
189C22×D11—22no—
190C22×D11—22no—
191C22×D11—22no—
192C22×D11—22no—
193C22×D11—22no—
194C22×D11—22no—
195C22×D11—48no—
196C22×D11—48no—
197C22×D11—48no—
198C22×D11—48no—
199C22×D11—48no—
200C22×D11—48no—
201C22×D11C2×C22⋊C414yes—
202C22×D11C2×M1614yes—
203C2×C4×C11—21no—
204C2×C4×C11—21no—
205C2×C4×C11—21no—
206C2×C4×C11—21no—
207C2×C4×C11—21no—
208C2×C4×C11—21no—
209C2×C4×C11—24no—
210C2×C4×C11—24no—
211C2×C4×C11—28no—
212C2×C4×C11—21no—
213C2×C4×C11—21no—
214C2×C4×C11—21no—
215C2×C4×C11—44no—
216C2×C4×C11C2×C22⋊C412yes—
217C2×C4×C11C2×C22⋊C412yes—
218C2×C4×C11C4×D412yes—
219C2×C4×C11C2×M1612yes—
220C2×C4×C11C2×M1612yes—
221C2×C4×C11C2×M1612yes—
222C2×C4×C11C22×D411yescharacteristic
223C2×C4×C11C22×D412yes—
224C2×C4×C11C2×#4111yescharacteristic
225C2×C4×C11H(5,𝔽2)12yes—
226C2×Dic11—22no—
227C2×Dic11—22no—
228C2×Dic11—22no—
229C2×Dic11—22no—
230C2×Dic11—22no—
231C2×Dic11—48no—
232C2×Dic11C2×C22⋊C414yes—
233D44—216no—
234D44—28no—
235D44—28no—
236C4×D11—28no—
237C4×D11—28no—
238C4×D11—48no—
239C4×D11—48no—
240C4×D11—44no—
241C4×D11—44no—
242C8×C11—24no—
243C8×C11—42no—
244C11⋊C8—44no—
245C23×D4—111no—
246C24×C4—112no—
247C22×M16—114no—
248C2×C4×D4—111no—
249C2×C4×D4—112no—
250C2×C4×D4—111no—
251C2×C4×D4—112no—
252C2×C4×D4—111no—
253C2×C4×D4—111no—
254C2×C4×D4—228no—
255C22×C4⋊C4—111no—
256C22×C22⋊C4—112no—
257C22×C42C2×D1111yescharacteristic
258#436—228no—
259C2×C4⋊C8—112no—
260#407—228no—
261C2×C22⋊C8—112no—
262C2×C22⋊C8—114no—
263C2×C22⋊C8—114no—
264C2×C22⋊C8—114no—
265C2×C22⋊C8C2×D1111yescharacteristic
266C2×C22⋊C8C2×D1111yescharacteristic
267C2×C4×C8C2×D1112yescomplemented
268#336—112no—
269#336C2×D1112yescomplemented
270C22×C11—22no—
271C22×C11—22no—
272C22×C11—22no—
273C22×C11—22no—
274C22×C11—22no—
275C22×C11—22no—
276C22×C11#43614yescomplemented
277C22×C11C22×C22⋊C411yescharacteristic
278C22×C11C2×C4×D411yescharacteristic
279C22×C11#51811yescharacteristic
280C22×C11C22×M1611yescharacteristic
281C22×C11C22×M1611yescharacteristic
282C22×C11C2×#13111yescharacteristic
283C22×C11#56811yescharacteristic
284C2×D11—24no—
285C2×D11—24no—
286C2×D11—24no—
287C2×D11—24no—
288C2×D11—24no—
289C2×D11—24no—
290C2×D11—432no—
291C2×D11—42no—
292C2×D11—42no—
293C2×D11—42no—
294C2×D11—42no—
295C2×D11—42no—
296C2×D11—42no—
297C2×D11—42no—
298C2×D11—42no—
299C4×C11—22no—
300C4×C11—22no—
301C4×C11—22no—
302C4×C11—41no—
303C4×C11#40714yescomplemented
304C4×C11D4212yes—
305C4×C11#54612yes—
306Dic11—24no—
307Dic11—42no—
308C25—112no—
309C22×D4—111no—
310C22×D4—111no—
311C22×D4—112no—
312C22×D4—224no—
313C22×D4—224no—
314C22×D4—224no—
315C23×C4—112no—
316C23×C4—112no—
317C23×C4—112no—
318C23×C4—112no—
319C23×C4—112no—
320C23×C4—228no—
321C23×C4—224no—
322C23×C4—224no—
323C23×C4—444no—
324C23×C4C4×D1111yescharacteristic
325C23×C4C4×D1111yescharacteristic
326C23×C4C22×D1111yescharacteristic
327C2×M16—228no—
328C2×M16—228no—
329C2×M16—444no—
330C22×C8—114no—
331C22×C8C22×D1112yes—
332C22×C8C22×D1112yes—
333C4×D4—118no—
334C4×D4—228no—
335C4×D4—224no—
336C4×D4—228no—
337C4×D4—224no—
338C4×D4—224no—
339C2×C4⋊C4—111no—
340C2×C4⋊C4—112no—
341C2×C4⋊C4—111no—
342C2×C4⋊C4—224no—
343C2×C22⋊C4—112no—
344C2×C22⋊C4—112no—
345C2×C22⋊C4—112no—
346C2×C22⋊C4—112no—
347C2×C22⋊C4—228no—
348C2×C42—24no—
349C2×C42C4×D1111yescharacteristic
350C2×C42C4×D1111yescharacteristic
351C2×C42C22×D1111yescharacteristic
352C2×C42C22×D1111yescharacteristic
353C4⋊C8—224no—
354C22⋊C8—22no—
355C22⋊C8—228no—
356C22⋊C8—224no—
357C22⋊C8—228no—
358C22⋊C8—444no—
359C22⋊C8D4414yescomplemented
360C4×C8—24no—
361C2×C11—24no—
362C2×C11D4×C22⋊C411yescharacteristic
363C2×C11C2×#40711yescharacteristic
364C2×C11#288011yescharacteristic
365C2×C11C2×#43611yescharacteristic
366C2×C11D4×M1611yescharacteristic
367C2×C11#297411yescharacteristic
368C2×C11#297511yescharacteristic
369D11—44no—
370D11—44no—
371C24—112no—
372C24—112no—
373C24—224no—
374C24—224no—
375C24—224no—
376C24—224no—
377C24—224no—
378C24—442no—
379C24—442no—
380C24C2×C4×D1111yescharacteristic
381C2×D4—114no—
382C2×D4—114no—
383C2×D4—224no—
384C2×D4—224no—
385C2×D4—2232no—
386C2×D4—222no—
387C2×D4—224no—
388C2×D4—222no—
389C2×D4—222no—
390C2×D4—222no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—41no—
398C22×C4—112no—
399C22×C4—118no—
400C22×C4—112no—
401C22×C4—224no—
402C22×C4—224no—
403C22×C4—224no—
404C22×C4—222no—
405C22×C4—222no—
406C22×C4—224no—
407C22×C4—224no—
408C22×C4—224no—
409C22×C4—224no—
410C22×C4—222no—
411C22×C4—222no—
412C22×C4—224no—
413C22×C4—224no—
414C22×C4—442no—
415C22×C4—442no—
416C22×C4—4416no—
417C22×C4—4416no—
418C22×C4—442no—
419C22×C4#431314yescomplemented
420C22×C4C2×C4×D1111yescharacteristic
421C22×C4C2×C4×D1111yescharacteristic
422C22×C4C2×C4×D1111yescharacteristic
423C22×C4C2×C4×D1111yescharacteristic
424C22×C4C2×C4×D1111yescharacteristic
425C22×C4C2×D4412yes—
426C22×C4#433912yes—
427C22×C4C23×D1111yescharacteristic
428M16—4416no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—224no—
435C2×C8—224no—
436C2×C8—224no—
437C2×C8C2×D4414yescomplemented
438C4⋊C4—114no—
439C4⋊C4—224no—
440C4⋊C4—222no—
441C22⋊C4—118no—
442C22⋊C4—224no—
443C22⋊C4—224no—
444C22⋊C4—224no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42D4×D1114yes—
449C11#2240111yescharacteristic, complemented, Sylow
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—114no—
455C23—114no—
456C23—2216no—
457C23—222no—
458C23—222no—
459C23—222no—
460C23—222no—
461C23—222no—
462C23—222no—
463C23—222no—
464C23—222no—
465C23—222no—
466C23—222no—
467C23—222no—
468C23—222no—
469C23—448no—
470C23—448no—
471C23—448no—
472C23—448no—
473C23—448no—
474C23—448no—
475C23C4×D4412yes—
476C23C2×#431312yes—
477C23#6695412yes—
478C23C22×C4×D1111yescharacteristic
479D4—2216no—
480D4—228no—
481D4—228no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—44no—
495C2×C4—114no—
496C2×C4—222no—
497C2×C4—222no—
498C2×C4—228no—
499C2×C4—228no—
500C2×C4—222no—
501C2×C4—222no—
502C2×C4—222no—
503C2×C4—448no—
504C2×C4—448no—
505C2×C4—448no—
506C2×C4—444no—
507C2×C4—444no—
508C2×C4C4×D4412yes—
509C2×C4C22⋊C4×D1112yes—
510C2×C4C22⋊C4×D1112yes—
511C2×C4C2×#431312yes—
512C2×C4C2×#431312yes—
513C2×C4M16×D1112yes—
514C2×C4C22×D4411yescharacteristic
515C2×C4C2×#433911yescharacteristic
516C2×C4C2×D4×D1112yes—
517C2×C4#6704212yes—
518C8—24no—
519C8—42no—
520C8—444no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—224no—
528C22—224no—
529C22—224no—
530C22—224no—
531C22—224no—
532C22—224no—
533C22—4432no—
534C22—442no—
535C22—442no—
536C22—442no—
537C22—442no—
538C22—442no—
539C22—442no—
540C22—442no—
541C22—442no—
542C22#1062230114yescomplemented
543C22C2×C4×D4411yescharacteristic
544C22C2×C22⋊C4×D1111yescharacteristic
545C22#1062305211yescharacteristic
546C22C22×#431311yescharacteristic
547C22C2×#6695411yescharacteristic
548C22C2×M16×D1111yescharacteristic
549C22#1062327611yescharacteristic
550C4—22no—
551C4—22no—
552C4—22no—
553C4—41no—
554C4—224no—
555C4—442no—
556C4#1062233914yescomplemented
557C4D4×D4412yes—
558C4#1062308112yes—
559C2—24no—
560C2—444no—
561C2—444no—
562C2C22⋊C4×D4411yescharacteristic
563C2C2×#1062233911yescharacteristic
564C2#1307706211yescharacteristic
565C2C2×#1062230111yescharacteristic
566C2M16×D4411yescharacteristic
567C2#1307765011yescharacteristic
568C2#1307765611yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D11
commutator, socleC23×C11C23×C4
FittingC4×C11×C22⋊C8C2
prime radicalC23×D44C4
cosocleC22×C4×C11C24
hypercenterC4×C22⋊C8D11
hypocenterC11#22401
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#10622301C2semidirect
3. C2×#10622339C2semidirect
4. #13069410C2semidirect
5. #13068912C2semidirect
6. #13068596C2semidirect
7. #13068525C2semidirect
8. C4×C11×C22⋊C8C2semidirect
14. C22×#4313C22semidirect
24. C2×C4×C8×C11C22semidirect
26. C2×(C2×D11)⋊C8C22semidirect
27. C2×(C2×D11)⋊C8C22semidirect
28. C2×(C2×D11)⋊C8C22semidirect
29. C2×C4⋊(C11⋊C8)C22semidirect
32. C11×#336C22semidirect
33. #10622162C22semidirect
80. C11×C22⋊C8D4semidirect
105. C22×C4×C11M16semidirect
128. C2×D44M16semidirect
150. C2×C8×C11C2×D4semidirect
173. C4×C22⋊C8D11semidirect
267. C2×C4×C8C2×D11semidirect
269. #336C2×D11semidirect
276. C22×C11#436semidirect
303. C4×C11#407semidirect
359. C22⋊C8D44semidirect
419. C22×C4#4313semidirect
437. C2×C8C2×D44semidirect
449. C11#22401semidirect
542. C22#10622301semidirect
556. C4#10622339semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D442
C2×#106223012
C2×#106223392
#130694102
#130689122
#130685962
#130685252
C4×C11×C22⋊C82
#2240111