Finite Group Databasebeta

← #14755299

Subgroups — #14755299

Order 2816.

1702 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13081819C211yesmaximal, characteristic
2C2×#10622849C211yesmaximal, characteristic
3C2×#10622838C211yesmaximal, characteristic, complemented
4#13071202C211yesmaximal, characteristic, complemented
5#13068987C218yesmaximal, complemented
6#13068354C211yesmaximal, characteristic, complemented
7#13068327C211yesmaximal, characteristic, complemented
8C22×C11×#137C211yesmaximal, characteristic
9C22×C11×#41C2211yescharacteristic
10C23×Q8×C11C2211yescharacteristic
11C2×C11×#137—24no—
12C2×C11×#137—22no—
13C2×C11×#137C418yescomplemented
14C2×C11×#137C2214yes—
15C22×C11×Q16—21no—
16C22×C11×SD16—21no—
17C22×C11×M16C2211yescharacteristic
18#10623358—22no—
19#10623358C2214yes—
20#41×Dic11—22no—
21#41×Dic11—22no—
22#10623343—21no—
23#10623329—21no—
24#10623329—22no—
25C2×#66979—22no—
26C2×#66979C2211yescharacteristic
27C2×#66979C2211yescharacteristic
28C2×#66979C2211yescharacteristic
29C22×#4318C2211yescharacteristic
30#10622849—24no—
31C2×#66901C2214yes—
32#10622843—22no—
33#10622843C2214yescomplemented
34#10622838—22no—
35#10622838C2214yescomplemented
36C2×#66899—22no—
37#10622824—22no—
38#10622193C2214yescomplemented
39#10622192C2214yescomplemented
40C2×C11×#41—21no—
41C2×C11×#41—21no—
42C2×C11×#41—22no—
43C2×C11×#41—22no—
44C2×C11×#41C2×C414yes—
45C2×C11×#41D414yes—
46C2×C11×#41D412yes—
47C2×C11×#41C2312yes—
48C22×Q8×C11—21no—
49C22×Q8×C11—22no—
50C22×Q8×C11—21no—
51C22×Q8×C11—21no—
52C22×Q8×C11—21no—
53C22×Q8×C11—21no—
54C22×Q8×C11—22no—
55C22×Q8×C11—22no—
56C22×Q8×C11C2×C414yescomplemented
57C22×Q8×C11D411yescharacteristic
58C22×Q8×C11C2312yes—
59C22×D4×C11—21no—
60C22×D4×C11D411yescharacteristic
61C23×C4×C11—21no—
62C23×C4×C11—21no—
63C23×C4×C11C2311yescharacteristic
64C11×#137—216no—
65C11×#137—28no—
66C11×#137C2×C4116yescomplemented
67C2×C11×Q16—28no—
68C2×C11×Q16—24no—
69C2×C11×SD16—28no—
70C2×C11×SD16—24no—
71C2×C11×M16—22no—
72C2×C11×M16—21no—
73C2×C11×M16C2×C414yescomplemented
74C2×C11×M16C2312yes—
75C22×C8×C11—21no—
76C2×C22⋊Dic11—24no—
77C2×C22⋊Dic11—21no—
78C2×C22⋊Dic11—21no—
79C2×C22⋊Dic11D412yes—
80Q8×Dic11—24no—
81Q8×Dic11—24no—
82D4×Dic11—28no—
83D4×Dic11—24no—
84D4×Dic11—24no—
85D4×Dic11—28no—
86#66979—24no—
87#66979—24no—
88#66979—21no—
89#66979—21no—
90#66979—21no—
91#66979—21no—
92#66979—21no—
93#66979—21no—
94#66979—24no—
95#66979—22no—
96#66979—22no—
97#66979D412yes—
98#66979C2312yes—
99C2×C4⋊Dic11—24no—
100C2×C4⋊Dic11—21no—
101C2×C4⋊Dic11D412yes—
102C2×C4⋊Dic11C2312yes—
103C2×C4×Dic11—24no—
104C2×C4×Dic11—21no—
105C2×C4×Dic11D412yes—
106C2×C4×Dic11C2312yes—
107C2×#4318—21no—
108C2×#4318—22no—
109C2×#4318—22no—
110C2×#4318C2312yes—
111C22×C11⋊C8—21no—
112#66901—28no—
113#66899—28no—
114C11⋊#95—28no—
115#20944—111nomaximal, Sylow
116C24×C11—21no—
117C24×C11C2×D411yescharacteristic
118C11×#41—24no—
119C11×#41—24no—
120C11×#41—24no—
121C11×#41—24no—
122C11×#41—24no—
123C11×#41—22no—
124C11×#41—22no—
125C11×#41C22⋊C418yes—
126C11×#41C22×C414yes—
127C11×#41C2×D414yes—
128C2×Q8×C11—24no—
129C2×Q8×C11—21no—
130C2×Q8×C11—22no—
131C2×Q8×C11—22no—
132C2×Q8×C11—24no—
133C2×Q8×C11—24no—
134C2×Q8×C11—24no—
135C2×Q8×C11—24no—
136C2×Q8×C11—22no—
137C2×Q8×C11—22no—
138C2×Q8×C11—22no—
139C2×Q8×C11—22no—
140C2×Q8×C11—41no—
141C2×Q8×C11—41no—
142C2×Q8×C11—42no—
143C2×Q8×C11—42no—
144C2×Q8×C11—42no—
145C2×Q8×C11—42no—
146C2×Q8×C11—42no—
147C2×Q8×C11C22⋊C414yes—
148C2×Q8×C11C22×C414yes—
149C2×Q8×C11C2×D412yes—
150C2×D4×C11—24no—
151C2×D4×C11—22no—
152C2×D4×C11—24no—
153C2×D4×C11—22no—
154C2×D4×C11—21no—
155C2×D4×C11—22no—
156C2×D4×C11C22⋊C414yes—
157C2×D4×C11C2×D412yes—
158C22×C4×C11—21no—
159C22×C4×C11—24no—
160C22×C4×C11—22no—
161C22×C4×C11—24no—
162C22×C4×C11—22no—
163C22×C4×C11—24no—
164C22×C4×C11—22no—
165C22×C4×C11—41no—
166C22×C4×C11—41no—
167C22×C4×C11—42no—
168C22×C4×C11C22×C412yes—
169C22×C4×C11C2×D411yescharacteristic
170C22×C4×C11C2×D411yescharacteristic
171C22×C4×C11C2×D412yes—
172C22×C4×C11C2×D411yescharacteristic
173C22×C4×C11C2×D411yescharacteristic
174C22×C4×C11C2×D411yescharacteristic
175C22×C4×C11#4112yes—
176C22×C4×C11#4112yes—
177C22×C4×C11C2411yescharacteristic
178C22×Dic11—28no—
179C22×Dic11—22no—
180C22×Dic11—22no—
181C22×Dic11C2×D414yes—
182C11×Q16—216no—
183C11×SD16—216no—
184C11×M16—24no—
185C11×M16—22no—
186C11×M16C22×C414yes—
187C2×C8×C11—24no—
188C2×C8×C11—22no—
189C22⋊Dic11—28no—
190C22⋊Dic11—22no—
191C22⋊Dic11—22no—
192C22⋊Dic11—22no—
193C22⋊Dic11—22no—
194C22⋊Dic11—42no—
195C4⋊Dic11—28no—
196C4⋊Dic11—22no—
197C4⋊Dic11—22no—
198C4⋊Dic11—22no—
199C4⋊Dic11—42no—
200C4⋊Dic11C2×D414yes—
201C4×Dic11—28no—
202C4×Dic11—22no—
203C4×Dic11—22no—
204C4×Dic11—22no—
205C4×Dic11—22no—
206C4×Dic11—22no—
207C4×Dic11—22no—
208#4318—22no—
209#4318—22no—
210#4318—42no—
211C2×C11⋊C8—22no—
212C2×C11⋊C8—42no—
213C22×#137D1111yescharacteristic
214#3428—111no—
215C2×#421—111no—
216C2×#419—111no—
217#2310—111no—
218#1882—118no—
219#1742—111no—
220#1733—111no—
221C23×C11—22no—
222C23×C11—24no—
223C23×C11—42no—
224C23×C11—82no—
225C23×C11C4×D412yes—
226C23×C11#12011yescharacteristic
227C23×C11#12011yescharacteristic
228C23×C11#12112yes—
229C23×C11C22×D411yescharacteristic
230Q8×C11—22no—
231Q8×C11—24no—
232Q8×C11—24no—
233Q8×C11—24no—
234Q8×C11—42no—
235Q8×C11—42no—
236Q8×C11—44no—
237Q8×C11—44no—
238Q8×C11C2×C22⋊C414yes—
239D4×C11—24no—
240D4×C11—24no—
241D4×C11—28no—
242D4×C11—22no—
243D4×C11—24no—
244D4×C11C2×C22⋊C414yes—
245C2×C4×C11—22no—
246C2×C4×C11—24no—
247C2×C4×C11—24no—
248C2×C4×C11—24no—
249C2×C4×C11—28no—
250C2×C4×C11—24no—
251C2×C4×C11—42no—
252C2×C4×C11—42no—
253C2×C4×C11—44no—
254C2×C4×C11—44no—
255C2×C4×C11C2×C22⋊C412yes—
256C2×C4×C11C2×C22⋊C412yes—
257C2×C4×C11C4×D412yes—
258C2×C4×C11C4×D412yes—
259C2×C4×C11C4×D412yes—
260C2×C4×C11#12011yescharacteristic
261C2×C4×C11#12011yescharacteristic
262C2×C4×C11#12112yes—
263C2×C4×C11#12112yes—
264C2×C4×C11#12112yes—
265C2×C4×C11#12312yes—
266C2×C4×C11#12312yes—
267C2×C4×C11C23×C411yescharacteristic
268C2×C4×C11C22×D411yescharacteristic
269C2×C4×C11C22×D411yescharacteristic
270C2×C4×C11C22×D411yescharacteristic
271C2×C4×C11C2×#4111yescharacteristic
272C2×C4×C11C2×#4111yescharacteristic
273C2×Dic11—24no—
274C2×Dic11—24no—
275C2×Dic11—44no—
276C2×Dic11—44no—
277C2×Dic11—42no—
278C2×Dic11—82no—
279C2×Dic11—88no—
280C8×C11—24no—
281C11⋊C8—42no—
282C22×#41C2×D1111yescharacteristic
283C23×Q8C2×D1111yescharacteristic
284C2×#137—24no—
285C2×#137—22no—
286C2×#137Dic1118yescomplemented
287C2×#137C2×D1114yes—
288C22×Q16—21no—
289C22×SD16—21no—
290C22×M16—111no—
291C22×M16C2×D1111yescharacteristic
292#520—114no—
293#520—222no—
294#519—221no—
295#518—222no—
296#518—221no—
297C4×#41—222no—
298C4×#41—222no—
299C2×#117—111no—
300C2×#117—111no—
301C2×#117—111no—
302C2×#117—222no—
303#421—224no—
304C2×#104—114no—
305#419—114no—
306#419—222no—
307C2×#103—222no—
308#391—114no—
309#391—222no—
310#381—222no—
311#343—114no—
312#339—114no—
313C22×C11—22no—
314C22×C11—22no—
315C22×C11—22no—
316C22×C11—24no—
317C22×C11—42no—
318C22×C11—42no—
319C22×C11—44no—
320C22×C11—84no—
321C22×C11#38612yes—
322C22×C11#38612yes—
323C22×C11C2×C4×D411yescharacteristic
324C22×C11C2×#12011yescharacteristic
325C22×C11C2×#12111yescharacteristic
326C4×C11—24no—
327C4×C11—44no—
328C4×C11#38612yes—
329C4×C11#38612yes—
330C4×C11C22×C22⋊C411yescharacteristic
331C4×C11C2×C4×D411yescharacteristic
332C4×C11C2×#12111yescharacteristic
333C4×C11C2×#12311yescharacteristic
334Dic11—44no—
335Dic11—44no—
336Dic11—82no—
337C2×#41—21no—
338C2×#41—21no—
339C2×#41—22no—
340C2×#41—22no—
341C2×#41C2×Dic1114yes—
342C2×#41C22⋊D1114yes—
343C2×#41C22⋊D1112yes—
344C2×#41C22×D1112yes—
345C22×Q8—21no—
346C22×Q8—22no—
347C22×Q8—21no—
348C22×Q8—21no—
349C22×Q8—22no—
350C22×Q8—21no—
351C22×Q8—21no—
352C22×Q8—22no—
353C22×Q8C2×Dic1114yescomplemented
354C22×Q8C22⋊D1111yescharacteristic
355C22×Q8C22×D1112yes—
356C22×D4—21no—
357C22×D4C22⋊D1111yescharacteristic
358C23×C4—21no—
359C23×C4—21no—
360C23×C4C22×D1111yescharacteristic
361#137—216no—
362#137—28no—
363#137C2×Dic11116yescomplemented
364C2×Q16—28no—
365C2×Q16—24no—
366C2×SD16—28no—
367C2×SD16—24no—
368C2×M16—22no—
369C2×M16—21no—
370C2×M16—112no—
371C2×M16—221no—
372C2×M16—222no—
373C2×M16—222no—
374C2×M16C2×Dic1114yescomplemented
375C2×M16C22×D1112yes—
376C22×C8—21no—
377C22×C8—221no—
378C4×Q8—224no—
379C4×Q8—224no—
380C4×D4—228no—
381C4×D4—228no—
382C4×D4—224no—
383C4×D4—224no—
384#117—112no—
385#117—112no—
386#117—224no—
387#117—224no—
388#117—224no—
389#117—221no—
390#117—221no—
391#117—221no—
392#117—221no—
393#117—221no—
394#117—221no—
395#117—222no—
396#117—222no—
397C2×C4⋊C4—112no—
398C2×C4⋊C4—112no—
399C2×C4⋊C4—224no—
400C2×C4⋊C4—221no—
401C2×C22⋊C4—112no—
402C2×C22⋊C4—224no—
403C2×C22⋊C4—221no—
404C2×C22⋊C4—221no—
405C2×C42—112no—
406C2×C42—112no—
407C2×C42—224no—
408C2×C42—221no—
409#104—228no—
410#103—228no—
411#95—228no—
412C2×C11—21no—
413C2×C11—22no—
414C2×C11—21no—
415C2×C11—22no—
416C2×C11—44no—
417C2×C11C2×#38611yescharacteristic
418C2×C11#302212yes—
419C24—21no—
420C24D4×D1111yescharacteristic
421#41—24no—
422#41—24no—
423#41—24no—
424#41—24no—
425#41—24no—
426#41—22no—
427#41—22no—
428#41C22⋊Dic1118yes—
429#41C22×Dic1114yes—
430#41C2×C22⋊D1114yes—
431C2×Q8—24no—
432C2×Q8—22no—
433C2×Q8—21no—
434C2×Q8—22no—
435C2×Q8—24no—
436C2×Q8—24no—
437C2×Q8—24no—
438C2×Q8—24no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—41no—
446C2×Q8—41no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8C22⋊Dic1114yes—
451C2×Q8C22×Dic1114yes—
452C2×Q8C2×C22⋊D1112yes—
453C2×D4—24no—
454C2×D4—22no—
455C2×D4—24no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—21no—
459C2×D4C22⋊Dic1114yes—
460C2×D4C2×C22⋊D1112yes—
461C22×C4—21no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—42no—
471C22×C4—114no—
472C22×C4—228no—
473C22×C4—222no—
474C22×C4—222no—
475C22×C4D4×D1111yescharacteristic
476C22×C4D4×D1111yescharacteristic
477C22×C4D4×D1111yescharacteristic
478C22×C4#434112yes—
479C22×C4#434112yes—
480C22×C4C22×Dic1112yes—
481C22×C4C2×C22⋊D1111yescharacteristic
482C22×C4C2×C22⋊D1111yescharacteristic
483C22×C4C2×C22⋊D1112yes—
484C22×C4C23×D1111yescharacteristic
485Q16—216no—
486SD16—216no—
487M16—24no—
488M16—22no—
489M16—222no—
490M16—222no—
491M16—442no—
492M16C22×Dic1114yes—
493C2×C8—24no—
494C2×C8—22no—
495C2×C8—222no—
496C2×C8—442no—
497C4⋊C4—114no—
498C4⋊C4—228no—
499C4⋊C4—222no—
500C4⋊C4—222no—
501C4⋊C4—222no—
502C4⋊C4—442no—
503C22⋊C4—228no—
504C22⋊C4—222no—
505C22⋊C4—222no—
506C22⋊C4—222no—
507C22⋊C4—222no—
508C22⋊C4—442no—
509C42—228no—
510C42—222no—
511C42—222no—
512C42—222no—
513C42—222no—
514C42—222no—
515C42—222no—
516C11#2094411yescharacteristic, complemented, Sylow
517C23—22no—
518C23—24no—
519C23—42no—
520C23—82no—
521C23D4×Dic1112yes—
522C23#6699011yescharacteristic
523C23#6699011yescharacteristic
524C23#6699212yes—
525C23C2×D4×D1111yescharacteristic
526Q8—22no—
527Q8—24no—
528Q8—24no—
529Q8—24no—
530Q8—42no—
531Q8—42no—
532Q8—44no—
533Q8—44no—
534Q8C2×C22⋊Dic1114yes—
535D4—24no—
536D4—24no—
537D4—28no—
538D4—22no—
539D4—24no—
540D4C2×C22⋊Dic1114yes—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—224no—
552C2×C4—224no—
553C2×C4—444no—
554C2×C4—444no—
555C2×C4—442no—
556C2×C4—882no—
557C2×C4—888no—
558C2×C4D4×Dic1112yes—
559C2×C4D4×Dic1112yes—
560C2×C4D4×Dic1112yes—
561C2×C4#6698812yes—
562C2×C4#6698812yes—
563C2×C4#6699011yescharacteristic
564C2×C4#6699011yescharacteristic
565C2×C4#6699212yes—
566C2×C4#6699212yes—
567C2×C4#6699212yes—
568C2×C4C2×C22⋊Dic1112yes—
569C2×C4C2×C22⋊Dic1112yes—
570C2×C4C2×D4×D1111yescharacteristic
571C2×C4C2×#434111yescharacteristic
572C2×C4C2×#434111yescharacteristic
573C2×C4C23×Dic1111yescharacteristic
574C2×C4C22×C22⋊D1111yescharacteristic
575C2×C4C22×C22⋊D1111yescharacteristic
576C8—24no—
577C8—442no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—24no—
582C22—42no—
583C22—42no—
584C22—44no—
585C22—84no—
586C22#1062283312yes—
587C22#1062283312yes—
588C22C2×D4×Dic1111yescharacteristic
589C22C2×#6699011yescharacteristic
590C22C2×#6699211yescharacteristic
591C4—24no—
592C4—44no—
593C4—444no—
594C4—444no—
595C4—882no—
596C4#1062283312yes—
597C4#1062283312yes—
598C4C2×D4×Dic1111yescharacteristic
599C4C2×#6698811yescharacteristic
600C4C2×#6699211yescharacteristic
601C4C22×C22⋊Dic1111yescharacteristic
602C2—21no—
603C2—22no—
604C2—21no—
605C2—22no—
606C2—44no—
607C2C2×#1062283311yescharacteristic
608C2#1307799212yes—
subgroupG/H
centerC22C2×#66990
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC22×C11×#137C2
socleC22×C11C2×#120
prime radicalC22×C11×#41C22
cosocleC22×C4×C11C24
hypercenterC22×#137D11
hypocenterC11#20944
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#137

Complements

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

NH ≅ G/Ntype
3. C2×#10622838C2semidirect
4. #13071202C2semidirect
5. #13068987C2semidirect
6. #13068354C2semidirect
7. #13068327C2semidirect
13. C2×C11×#137C4semidirect
33. #10622843C22semidirect
35. #10622838C22semidirect
38. #10622193C22semidirect
39. #10622192C22semidirect
56. C22×Q8×C11C2×C4semidirect
66. C11×#137C2×C4semidirect
73. C2×C11×M16C2×C4semidirect
286. C2×#137Dic11semidirect
353. C22×Q8C2×Dic11semidirect
363. #137C2×Dic11semidirect
374. C2×M16C2×Dic11semidirect
516. C11#20944semidirect

Maximal subgroups

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

subgroupindex
#130818192
C2×#106228492
C2×#106228382
#130712022
#130689872
#130683542
#130683272
C22×C11×#1372
#2094411