Finite Group Databasebeta

← #14809049

Subgroups — #14809049

Order 2816.

2090 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622857C214yesmaximal, complemented
2#13072254C212yesmaximal, complemented
3#13071972C211yesmaximal, characteristic, complemented
4#13069740C214yesmaximal, complemented
5#13069453C212yesmaximal, complemented
6C4×#66898C211yesmaximal, characteristic, complemented
7C11×#3496C211yesmaximal, characteristic, complemented
8C2×C11×H(5,𝔽2)C2212yes—
9C23×D44C2211yescharacteristic, complemented
10C11×#547—22no—
11C11×#546—24no—
12D42×C11—22no—
13C11×#537—21no—
14C11×#534—21no—
15C11×#534C2212yes—
16C11×#532—21no—
17C2×C11×#121—21no—
18C2×C11×#121C2212yes—
19C2×C4×D4×C11C2211yescharacteristic
20C2×#66993—22no—
21C2×#66990C2212yes—
22C2×#66990C2218yescomplemented
23C2×#66983—21no—
24#10622857—216no—
25C2×#66898C2212yes—
26C2×#66898C2218yescomplemented
27#10622825C2211yescharacteristic, complemented
28C4×C22⋊Dic11C2212yescomplemented
29#10622604—21no—
30#10622601—22no—
31#10622573—21no—
32#10622573—24no—
33#10622568C2212yescomplemented
34#10622552—21no—
35#10622164C2212yescomplemented
36C24×D11C2312yes—
37C11×H(5,𝔽2)—28no—
38C11×H(5,𝔽2)—44no—
39C2×C11×#41—22no—
40C2×C11×#41—24no—
41C22×D4×C11—22no—
42C22×D4×C11—24no—
43C22×D4×C11—21no—
44C22×D4×C11D412yescomplemented
45C22×D4×C11C2314yes—
46C22×D4×C11C2311yescharacteristic
47C23×C4×C11D411yescharacteristic, complemented
48C23×C4×C11C2311yescharacteristic
49C22×C22⋊D11—22no—
50C22×C22⋊D11—28no—
51C22×D44—21no—
52C22×D44—22no—
53C22×D44—22no—
54C22×D44—21no—
55C22×D44—44no—
56C11×C4⋊Q8—21no—
57C11×#127—21no—
58C11×#124—22no—
59C11×#123—24no—
60C11×#123—24no—
61C11×C22⋊Q8—24no—
62C11×#121—22no—
63C11×#121—22no—
64C11×#121—24no—
65C11×#121—22no—
66C11×#121—22no—
67C11×#121—22no—
68C11×#121—22no—
69C11×#121—22no—
70C11×#121—22no—
71C11×#120—24no—
72C11×#120—24no—
73C4×D4×C11—22no—
74C4×D4×C11—22no—
75C4×D4×C11—22no—
76C4×D4×C11—22no—
77C2×C11×C4⋊C4—21no—
78C2×C11×C4⋊C4D412yescomplemented
79C2×C11×C4⋊C4D411yescharacteristic, complemented
80C2×C11×C22⋊C4—22no—
81C2×C11×C22⋊C4D411yescharacteristic, complemented
82C2×C11×C22⋊C4D414yescomplemented
83C2×C11×C22⋊C4C2311yescharacteristic
84C2×C42×C11D411yescharacteristic, complemented
85C2×C22⋊Dic11C2312yes—
86C2×C22⋊Dic11C2314yes—
87#66993—48no—
88#66990—28no—
89#66990—216no—
90#66990—48no—
91#66983—44no—
92C2×(C2×D11)⋊C4—22no—
93C2×(C2×D11)⋊C4—22no—
94C2×(C2×D11)⋊C4—28no—
95C2×(C2×D11)⋊C4—22no—
96C2×C4⋊Dic11—21no—
97C2×C4×Dic11—22no—
98#66898—216no—
99#66898D418yescomplemented
100C11⋊#95—22no—
101C11⋊#95—21no—
102#25771—111nomaximal, Sylow
103C24×C11—22no—
104C24×C11C2×D414yes—
105C24×C11C2×D411yescharacteristic
106C24×C11C2411yescharacteristic
107C23×D11—22no—
108C23×D11—24no—
109C23×D11—24no—
110C23×D11—24no—
111C23×D11—44no—
112C23×D11—82no—
113C11×#41—416no—
114C11×#41—44no—
115C11×#41—44no—
116C2×Q8×C11—22no—
117C2×D4×C11—22no—
118C2×D4×C11—22no—
119C2×D4×C11—24no—
120C2×D4×C11—22no—
121C2×D4×C11—28no—
122C2×D4×C11—24no—
123C2×D4×C11—24no—
124C2×D4×C11—28no—
125C2×D4×C11—22no—
126C2×D4×C11—22no—
127C2×D4×C11—22no—
128C2×D4×C11—22no—
129C2×D4×C11—44no—
130C2×D4×C11—416no—
131C2×D4×C11—44no—
132C2×D4×C11—44no—
133C2×D4×C11—82no—
134C2×D4×C11C2×D414yes—
135C22×C4×C11—22no—
136C22×C4×C11—24no—
137C22×C4×C11—21no—
138C22×C4×C11—21no—
139C22×C4×C11—42no—
140C22×C4×C11—42no—
141C22×C4×C11C2×D414yes—
142C22×C4×C11C2×D412yes—
143C22×C4×C11C2×D411yescharacteristic
144C22×C4×C11C2×D411yescharacteristic
145C22×C4×C11C2×D411yescharacteristic
146C22×C4×C11C2×D411yescharacteristic
147C22×C4×C11C2×D412yes—
148C22×C4×C11C2×D411yescharacteristic
149C2×C22⋊D11—44no—
150C2×C22⋊D11—416no—
151C2×C22⋊D11—416no—
152C2×C22⋊D11—44no—
153C2×C22⋊D11—82no—
154C2×C22⋊D11—88no—
155C22×Dic11—24no—
156C22×Dic11—22no—
157C2×D44—24no—
158C2×D44—28no—
159C2×D44—44no—
160C2×D44—42no—
161C2×D44—44no—
162C2×D44—48no—
163C2×D44—41no—
164C2×D44—42no—
165C2×D44—41no—
166C11×C4⋊C4—22no—
167C11×C4⋊C4—22no—
168C11×C4⋊C4—22no—
169C11×C4⋊C4—22no—
170C11×C4⋊C4—22no—
171C11×C22⋊C4—24no—
172C11×C22⋊C4—24no—
173C11×C22⋊C4—22no—
174C11×C22⋊C4—24no—
175C11×C22⋊C4—22no—
176C11×C22⋊C4—24no—
177C42×C11—21no—
178C42×C11—21no—
179C22⋊Dic11—24no—
180C22⋊Dic11—24no—
181C22⋊Dic11C2×D418yescomplemented
182(C2×D11)⋊C4—416no—
183(C2×D11)⋊C4—44no—
184(C2×D11)⋊C4—44no—
185(C2×D11)⋊C4—82no—
186C4⋊Dic11—42no—
187C4×Dic11—44no—
188#3496D1111yescharacteristic, complemented
189C2×B(4,𝔽2)—114no—
190#2433—112no—
191#2408—111no—
192#2021—114no—
193#1964—112no—
194C4×#99—111no—
195C23×C11—21no—
196C23×C11—21no—
197C23×C11—24no—
198C23×C11—28no—
199C23×C11—41no—
200C23×C11—41no—
201C23×C11—41no—
202C23×C11—41no—
203C23×C11—44no—
204C23×C11—44no—
205C23×C11—44no—
206C23×C11—82no—
207C23×C11—82no—
208C23×C11#12712yescomplemented
209C23×C11C22×D412yes—
210C23×C11C22×D411yescharacteristic
211C23×C11C22×D412yes—
212C23×C11H(5,𝔽2)12yes—
213C22×D11—24no—
214C22×D11—28no—
215C22×D11—42no—
216C22×D11—42no—
217C22×D11—44no—
218C22×D11—42no—
219C22×D11—48no—
220C22×D11—42no—
221C22×D11—42no—
222C22×D11—42no—
223C22×D11—48no—
224C22×D11—88no—
225C22×D11—88no—
226Q8×C11—44no—
227D4×C11—28no—
228D4×C11—44no—
229D4×C11—48no—
230D4×C11—44no—
231D4×C11—44no—
232D4×C11—48no—
233D4×C11—44no—
234D4×C11—82no—
235D4×C11—82no—
236D4×C11—82no—
237C2×C4×C11—21no—
238C2×C4×C11—21no—
239C2×C4×C11—24no—
240C2×C4×C11—22no—
241C2×C4×C11—21no—
242C2×C4×C11—21no—
243C2×C4×C11—24no—
244C2×C4×C11—28no—
245C2×C4×C11—42no—
246C2×C4×C11—41no—
247C2×C4×C11—48no—
248C2×C4×C11—44no—
249C2×C4×C11—42no—
250C2×C4×C11—41no—
251C2×C4×C11—41no—
252C2×C4×C11—81no—
253C2×C4×C11#12011yescharacteristic
254C2×C4×C11#12012yes—
255C2×C4×C11#12011yescharacteristic
256C2×C4×C11#12712yescomplemented
257C22⋊D11—832no—
258C22⋊D11—88no—
259C2×Dic11—28no—
260C2×Dic11—44no—
261C2×Dic11—44no—
262C2×Dic11—42no—
263C2×Dic11—42no—
264C2×Dic11—42no—
265D44—44no—
266D44—48no—
267D44—44no—
268C2×H(5,𝔽2)C2×D1112yes—
269C23×D4—111no—
270#547—22no—
271#546—24no—
272D42—22no—
273#537—21no—
274#534—21no—
275#534C2×D1112yes—
276#532—21no—
277C2×#127—222no—
278C2×#121—21no—
279C2×#121—221no—
280C2×#121C2×D1112yes—
281C2×#120—112no—
282C2×#120—118no—
283C2×C4×D4C2×D1111yescharacteristic
284B(4,𝔽2)—2216no—
285C2×#99—112no—
286C2×#99—118no—
287#394—221no—
288#392—221no—
289#392—224no—
290#390—222no—
291#390—221no—
292#380—111no—
293#379—112no—
294C4×C22⋊C4—112no—
295#342—112no—
296C22×C11—21no—
297C22×C11—22no—
298C22×C11—21no—
299C22×C11—22no—
300C22×C11—22no—
301C22×C11—42no—
302C22×C11—42no—
303C22×C11—42no—
304C22×C11—41no—
305C22×C11—41no—
306C22×C11—44no—
307C22×C11—44no—
308C22×C11—44no—
309C22×C11—81no—
310C22×C11—88no—
311C22×C11—164no—
312C22×C11C2×#12011yescharacteristic
313C22×C11C2×#12711yescharacteristic
314C22×C11#53511yescharacteristic
315C22×C11D4214yes—
316C2×D11—44no—
317C2×D11—416no—
318C2×D11—44no—
319C2×D11—44no—
320C2×D11—88no—
321C2×D11—82no—
322C2×D11—82no—
323C2×D11—82no—
324C2×D11—82no—
325C4×C11—22no—
326C4×C11—21no—
327C4×C11—22no—
328C4×C11—44no—
329C4×C11—41no—
330C4×C11—41no—
331C4×C11B(4,𝔽2)12yescomplemented
332Dic11—48no—
333Dic11—82no—
334C25—112no—
335H(5,𝔽2)—28no—
336H(5,𝔽2)—44no—
337C2×#41—22no—
338C2×#41—24no—
339C22×D4—22no—
340C22×D4—24no—
341C22×D4—21no—
342C22×D4—222no—
343C22×D4—228no—
344C22×D4—221no—
345C22×D4—222no—
346C22×D4—222no—
347C22×D4—221no—
348C22×D4—444no—
349C22×D4C22⋊D1112yescomplemented
350C22×D4C22×D1114yes—
351C22×D4C22×D1111yescharacteristic
352C23×C4C22⋊D1111yescharacteristic, complemented
353C23×C4C22×D1111yescharacteristic
354C4⋊Q8—21no—
355#127—21no—
356#127—448no—
357#124—22no—
358#123—24no—
359#123—24no—
360C22⋊Q8—24no—
361#121—22no—
362#121—22no—
363#121—24no—
364#121—22no—
365#121—22no—
366#121—22no—
367#121—22no—
368#121—22no—
369#121—22no—
370#121—444no—
371#120—24no—
372#120—24no—
373#120—2216no—
374#120—228no—
375#120—448no—
376C4×D4—22no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4—22no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—221no—
382C2×C4⋊C4C22⋊D1111yescharacteristic, complemented
383C2×C4⋊C4C22⋊D1112yescomplemented
384C2×C22⋊C4—22no—
385C2×C22⋊C4—114no—
386C2×C22⋊C4—112no—
387C2×C22⋊C4—228no—
388C2×C22⋊C4—222no—
389C2×C22⋊C4—222no—
390C2×C22⋊C4—222no—
391C2×C22⋊C4C22⋊D1111yescharacteristic, complemented
392C2×C22⋊C4C22⋊D1114yescomplemented
393C2×C22⋊C4C22×D1111yescharacteristic
394C2×C42—222no—
395C2×C42C22⋊D1111yescharacteristic, complemented
396#99—118no—
397#99—2216no—
398#95—222no—
399#95—221no—
400C2×C11—24no—
401C2×C11—21no—
402C2×C11—21no—
403C2×C11—42no—
404C2×C11—84no—
405C2×C11#243311yescharacteristic
406C2×C11C2×B(4,𝔽2)11yescharacteristic
407C2×C11#302711yescharacteristic
408D11—84no—
409C24—22no—
410C24—222no—
411C24—224no—
412C24—224no—
413C24—224no—
414C24—444no—
415C24—882no—
416C24C2×C22⋊D1114yes—
417C24C2×C22⋊D1111yescharacteristic
418C24C23×D1111yescharacteristic
419#41—416no—
420#41—44no—
421#41—44no—
422C2×Q8—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—24no—
426C2×D4—22no—
427C2×D4—28no—
428C2×D4—28no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—44no—
436C2×D4—416no—
437C2×D4—44no—
438C2×D4—44no—
439C2×D4—82no—
440C2×D4—224no—
441C2×D4—228no—
442C2×D4—444no—
443C2×D4—442no—
444C2×D4—444no—
445C2×D4—448no—
446C2×D4—4416no—
447C2×D4—444no—
448C2×D4—4416no—
449C2×D4—444no—
450C2×D4—441no—
451C2×D4—442no—
452C2×D4—441no—
453C2×D4—888no—
454C2×D4—882no—
455C2×D4D4×D1114yes—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—224no—
463C22×C4—222no—
464C22×C4C2×C22⋊D1114yes—
465C22×C4C2×C22⋊D1111yescharacteristic
466C22×C4C2×C22⋊D1112yes—
467C22×C4C2×C22⋊D1111yescharacteristic
468C22×C4C2×C22⋊D1111yescharacteristic
469C22×C4C2×C22⋊D1111yescharacteristic
470C22×C4C2×C22⋊D1112yes—
471C22×C4C2×C22⋊D1111yescharacteristic
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—442no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—22no—
481C22⋊C4—24no—
482C22⋊C4—22no—
483C22⋊C4—24no—
484C22⋊C4—118no—
485C22⋊C4—224no—
486C22⋊C4—224no—
487C22⋊C4—4416no—
488C22⋊C4—444no—
489C22⋊C4—444no—
490C22⋊C4—882no—
491C42—21no—
492C42—21no—
493C42—444no—
494C11#2577111yescharacteristic, complemented, Sylow
495C23—21no—
496C23—21no—
497C23—24no—
498C23—28no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—82no—
507C23—82no—
508C23—224no—
509C23—228no—
510C23—442no—
511C23—442no—
512C23—444no—
513C23—442no—
514C23—448no—
515C23—448no—
516C23—442no—
517C23—442no—
518C23—442no—
519C23—888no—
520C23—888no—
521C23#6699312yescomplemented
522C23C2×D4×D1112yes—
523C23#6703712yes—
524C23C22×C22⋊D1112yes—
525C23C22×C22⋊D1111yescharacteristic
526Q8—44no—
527D4—28no—
528D4—44no—
529D4—48no—
530D4—48no—
531D4—44no—
532D4—44no—
533D4—44no—
534D4—82no—
535D4—82no—
536D4—82no—
537D4—444no—
538D4—448no—
539D4—444no—
540D4—8832no—
541D4—888no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—28no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—24no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—48no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—81no—
558C2×C4—228no—
559C2×C4—444no—
560C2×C4—444no—
561C2×C4—442no—
562C2×C4—442no—
563C2×C4—442no—
564C2×C4#6699312yescomplemented
565C2×C4#6700611yescharacteristic
566C2×C4#6700612yes—
567C2×C4#6700611yescharacteristic
568C22—21no—
569C22—22no—
570C22—21no—
571C22—22no—
572C22—22no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—41no—
577C22—41no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—81no—
582C22—88no—
583C22—164no—
584C22—4416no—
585C22—444no—
586C22—444no—
587C22—444no—
588C22—888no—
589C22—882no—
590C22—882no—
591C22—882no—
592C22—882no—
593C22D4×C22⋊D1114yes—
594C22C2×#6699311yescharacteristic
595C22#1062333611yescharacteristic
596C22C2×#6700611yescharacteristic
597C4—22no—
598C4—21no—
599C4—22no—
600C4—41no—
601C4—44no—
602C4—41no—
603C4—448no—
604C4—882no—
605C4#1062285712yescomplemented
606C2—24no—
607C2—21no—
608C2—21no—
609C2—42no—
610C2—84no—
611C2—884no—
612C2#1307224911yescharacteristic
613C2C2×#1062285711yescharacteristic
614C2#1307801811yescharacteristic
subgroupG/H
centerC22C2×#67006
commutator, cosocleC24×C11C24
FrattiniC24C23×D11
FittingC11×#3496C2
socleC22×C11C2×#120
hypercenter#3496D11
hypocenterC11#25771
LCSG ▷ C24×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3496

Complements

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

NH ≅ G/Ntype
1. C2×#10622857C2semidirect
2. #13072254C2semidirect
3. #13071972C2semidirect
4. #13069740C2semidirect
5. #13069453C2semidirect
6. C4×#66898C2semidirect
7. C11×#3496C2semidirect
9. C23×D44C22semidirect
22. C2×#66990C22semidirect
26. C2×#66898C22semidirect
27. #10622825C22semidirect
28. C4×C22⋊Dic11C22semidirect
33. #10622568C22semidirect
35. #10622164C22semidirect
44. C22×D4×C11D4semidirect
47. C23×C4×C11D4semidirect
78. C2×C11×C4⋊C4D4semidirect
79. C2×C11×C4⋊C4D4semidirect
81. C2×C11×C22⋊C4D4semidirect
82. C2×C11×C22⋊C4D4semidirect
84. C2×C42×C11D4semidirect
99. #66898D4semidirect
181. C22⋊Dic11C2×D4semidirect
188. #3496D11semidirect
208. C23×C11#127semidirect
256. C2×C4×C11#127semidirect
331. C4×C11B(4,𝔽2)semidirect
349. C22×D4C22⋊D11semidirect
352. C23×C4C22⋊D11semidirect
382. C2×C4⋊C4C22⋊D11semidirect
383. C2×C4⋊C4C22⋊D11semidirect
391. C2×C22⋊C4C22⋊D11semidirect
392. C2×C22⋊C4C22⋊D11semidirect
395. C2×C42C22⋊D11semidirect
494. C11#25771semidirect
521. C23#66993semidirect
564. C2×C4#66993semidirect
605. C4#10622857semidirect

Maximal subgroups

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

subgroupindex
C2×#106228572
#130722542
#130719722
#130697402
#130694532
C4×#668982
C11×#34962
#2577111