Finite Group Databasebeta

← #14755179

Subgroups — #14755179

Order 2816.

1738 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66962C211yesmaximal, characteristic
2C2×#10623047C211yesmaximal, characteristic, complemented
3C2×#10622615C211yesmaximal, characteristic
4C22×#66874C211yesmaximal, characteristic, complemented
5#13071209C211yesmaximal, characteristic
6#13068972C214yesmaximal, complemented
7#13068958C214yesmaximal, complemented
8#13068314C211yesmaximal, characteristic, complemented
9C11×#1736C211yesmaximal, characteristic, complemented
10C22×#4339C2211yescharacteristic
11C23×Dic22C2211yescharacteristic
12C22×C11×M16C2211yescharacteristic
13C2×C11×#117C2211yescharacteristic
14C22×C11×C4⋊C4C2211yescharacteristic
15C2×#66979C2211yescharacteristic
16C23×C11⋊C8C2211yescharacteristic
17C2×#66962—24no—
18C2×#66962—24no—
19C2×#66962C418yescomplemented
20C22×Dic44C2212yes—
21C22×#4314C2212yes—
22#10623047—24no—
23#10623047—24no—
24C2×#66940C2214yes—
25C2×#66930—22no—
26C2×C4×Dic22—22no—
27#10622826—22no—
28#10622615—24no—
29#10622601C2212yes—
30#10622600—22no—
31#10622587C2212yes—
32C2×#66874—24no—
33C2×#66874C2214yes—
34C2×#66874C2212yes—
35C2×#66873C2212yes—
36C11×#340C2214yes—
37#10622188C2214yes—
38C23×C4×C11—21no—
39C23×C4×C11C2311yescharacteristic
40C22×C22⋊D11—21no—
41C23×Dic11—21no—
42C2×#4339—22no—
43C2×#4339—22no—
44C2×#4339—22no—
45C2×#4339—22no—
46C2×#4339—22no—
47C2×#4339C2×C414yes—
48C22×D44C2311yescharacteristic
49C22×C4×D11—21no—
50C22×Dic22—21no—
51C22×Dic22—22no—
52C22×Dic22—22no—
53C22×Dic22—22no—
54C22×Dic22C2×C414yes—
55C22×Dic22D414yes—
56C22×Dic22D414yes—
57C22×Dic22C2311yescharacteristic
58C22×Dic22C2312yes—
59C2×C11×M16—22no—
60C2×C11×M16—22no—
61C2×C11×M16C2×C414yes—
62C22×C8×C11C2312yes—
63C11×#117—22no—
64C11×#117D414yes—
65C2×C11×C4⋊C4—22no—
66C2×C11×C4⋊C4—22no—
67C2×C11×C4⋊C4—21no—
68C2×C11×C4⋊C4C2312yes—
69C2×C11×C4⋊C4C2312yes—
70C2×C11×C22⋊C4—21no—
71C2×C42×C11—21no—
72C2×C22⋊Dic11D412yes—
73C2×(C2×D11)⋊C4—24no—
74#66979—22no—
75#66979—22no—
76C2×C4⋊Dic11D412yes—
77C2×Dic11⋊C4—24no—
78C2×C4×Dic11—24no—
79C2×C4×Dic11C2312yes—
80C22×C11⋊C8—22no—
81C22×C11⋊C8—22no—
82C22×C11⋊C8—22no—
83C22×C11⋊C8C2312yes—
84#66962—232no—
85C2×Dic44—24no—
86C2×Dic44—24no—
87C2×Dic44C2×C418yescomplemented
88C2×#4314—24no—
89C2×#4314—24no—
90C2×#4314C2×C418yescomplemented
91#66940—28no—
92#66940—28no—
93#66930—216no—
94C4×Dic22—216no—
95C11⋊#95—24no—
96#66874—28no—
97#66874—24no—
98#66873—24no—
99#20930—111nomaximal, Sylow
100C24×C11C2×D411yescharacteristic
101C23×D11—21no—
102C22×C4×C11—22no—
103C22×C4×C11—22no—
104C22×C4×C11—42no—
105C22×C4×C11—82no—
106C22×C4×C11C22×C412yes—
107C22×C4×C11C2×D412yes—
108C22×C4×C11C2×D412yes—
109C22×C4×C11C2×D411yescharacteristic
110C22×C4×C11C2×D412yes—
111C22×C4×C11#4112yes—
112C22×C4×C11#4112yes—
113C22×C4×C11C2411yescharacteristic
114C2×C22⋊D11—24no—
115C2×C22⋊D11—42no—
116C2×C22⋊D11—42no—
117C22×Dic11—24no—
118C22×Dic11—24no—
119C22×Dic11—24no—
120C22×Dic11—22no—
121C22×Dic11—21no—
122C22×Dic11C2×D414yes—
123#4339—28no—
124#4339—28no—
125#4339—44no—
126#4339—44no—
127C2×D44—21no—
128C2×D44—21no—
129C2×D44—21no—
130C2×D44—21no—
131C2×D44—21no—
132C2×D44C22×C412yes—
133C2×C4×D11—24no—
134C2×C4×D11—42no—
135C2×C4×D11—42no—
136C2×Dic22—28no—
137C2×Dic22—24no—
138C2×Dic22—24no—
139C2×Dic22—24no—
140C2×Dic22—28no—
141C2×Dic22—28no—
142C2×Dic22—22no—
143C2×Dic22—21no—
144C2×Dic22—21no—
145C2×Dic22—21no—
146C2×Dic22—22no—
147C2×Dic22—22no—
148C2×Dic22—21no—
149C2×Dic22—21no—
150C2×Dic22C22⋊C4116yescomplemented
151C2×Dic22C22×C412yes—
152C2×Dic22C22×C414yes—
153C2×Dic22C2×D414yes—
154C2×Dic22C2×D414yes—
155C11×M16—28no—
156C2×C8×C11—22no—
157C2×C8×C11—22no—
158C2×C8×C11C22×C414yes—
159C11×C4⋊C4—22no—
160C11×C4⋊C4—22no—
161C11×C4⋊C4—22no—
162C11×C4⋊C4—42no—
163C11×C4⋊C4C2×D414yes—
164C11×C22⋊C4—24no—
165C42×C11—24no—
166C22⋊Dic11—24no—
167(C2×D11)⋊C4—84no—
168C4⋊Dic11—24no—
169Dic11⋊C4—216no—
170C4×Dic11—216no—
171C4×Dic11—22no—
172C4×Dic11—22no—
173C2×C11⋊C8—22no—
174C2×C11⋊C8—22no—
175C2×C11⋊C8—22no—
176C2×C11⋊C8—44no—
177Dic44—216no—
178#4314—216no—
179C22×#137—111no—
180C2×#519—111no—
181C2×#416—111no—
182C22×#103—111no—
183#2311—111no—
184#1880—114no—
185#1878—114no—
186#1736D1111yescharacteristic, complemented
187#1729—111no—
188C23×C11—22no—
189C23×C11—22no—
190C23×C11C4×D412yes—
191C23×C11#12012yes—
192C23×C11#12112yes—
193C23×C11C22×D411yescharacteristic
194C22×D11—22no—
195C22×D11—41no—
196C22×D11—41no—
197C22×D11—42no—
198C22×D11—42no—
199C2×C4×C11—24no—
200C2×C4×C11—42no—
201C2×C4×C11—42no—
202C2×C4×C11—42no—
203C2×C4×C11—44no—
204C2×C4×C11—84no—
205C2×C4×C11C2×C22⋊C414yes—
206C2×C4×C11C4×D414yes—
207C2×C4×C11C4×D412yes—
208C2×C4×C11#12012yes—
209C2×C4×C11#12112yes—
210C2×C4×C11#12114yes—
211C2×C4×C11#12314yes—
212C2×C4×C11C23×C411yescharacteristic
213C2×C4×C11C22×D411yescharacteristic
214C2×C4×C11C22×D411yescharacteristic
215C2×C4×C11C22×D411yescharacteristic
216C2×C4×C11C2×#4111yescharacteristic
217C2×C4×C11C2×#4111yescharacteristic
218C22⋊D11—48no—
219C2×Dic11—24no—
220C2×Dic11—24no—
221C2×Dic11—24no—
222C2×Dic11—216no—
223C2×Dic11—22no—
224C2×Dic11—24no—
225C2×Dic11—24no—
226C2×Dic11—24no—
227C2×Dic11—41no—
228C2×Dic11—41no—
229D44—22no—
230D44—22no—
231D44—41no—
232D44—41no—
233C4×D11—48no—
234Dic22—24no—
235Dic22—24no—
236Dic22—24no—
237Dic22—216no—
238Dic22—22no—
239Dic22—22no—
240Dic22—41no—
241Dic22—41no—
242Dic22C2×C22⋊C418yes—
243C8×C11—24no—
244C11⋊C8—42no—
245C22×#41—111no—
246C23×Q8—111no—
247C2×#137—118no—
248C2×#137—224no—
249C2×#137—224no—
250C22×Q16—112no—
251C22×SD16—112no—
252C22×M16C2×D1111yescharacteristic
253C23×C8—111no—
254#519—224no—
255#519—224no—
256C2×C4×Q8—114no—
257C2×C4×Q8—222no—
258C2×#117—111no—
259C2×#117—222no—
260C2×#117C2×D1111yescharacteristic
261C22×C4⋊C4C2×D1111yescharacteristic
262#416—224no—
263C2×#103—114no—
264C2×#103—112no—
265C2×#103—224no—
266C2×#102—112no—
267#391—112no—
268#390—112no—
269#386—222no—
270#380—222no—
271#340C2×D1114yes—
272#336—114no—
273C22×C11—24no—
274C22×C11—24no—
275C22×C11—22no—
276C22×C11—22no—
277C22×C11#38614yes—
278C22×C11C2×C4×D411yescharacteristic
279C22×C11C2×#12011yescharacteristic
280C22×C11C2×#12111yescharacteristic
281C22×C11#57712yes—
282C22×C11#57812yes—
283C2×D11—44no—
284C2×D11—42no—
285C2×D11—82no—
286C2×D11—81no—
287C2×D11—81no—
288C4×C11—44no—
289C4×C11—82no—
290C4×C11#38614yes—
291C4×C11C22×C22⋊C411yescharacteristic
292C4×C11C2×C4×D411yescharacteristic
293C4×C11C2×#12111yescharacteristic
294C4×C11C2×#12311yescharacteristic
295Dic11—28no—
296Dic11—44no—
297Dic11—42no—
298C2×#41—114no—
299C2×#41—222no—
300C2×#41—222no—
301C2×#41—222no—
302C2×#41—222no—
303C2×#41—222no—
304C22×Q8—114no—
305C22×Q8—114no—
306C22×Q8—114no—
307C22×Q8—112no—
308C22×Q8—111no—
309C22×Q8—222no—
310C22×Q8—222no—
311C22×Q8—221no—
312C22×Q8—222no—
313C22×D4—111no—
314C22×D4—221no—
315C23×C4—21no—
316C23×C4—221no—
317C23×C4—221no—
318C23×C4C22×D1111yescharacteristic
319#137—2232no—
320C2×Q16—118no—
321C2×Q16—224no—
322C2×Q16—224no—
323C2×SD16—118no—
324C2×SD16—224no—
325C2×SD16—224no—
326C2×M16—22no—
327C2×M16—22no—
328C2×M16C4×D1114yes—
329C22×C8—112no—
330C22×C8—222no—
331C22×C8—222no—
332C22×C8—222no—
333C22×C8C22×D1112yes—
334C4×Q8—2216no—
335C4×Q8—228no—
336C4×Q8—228no—
337#117—22no—
338#117—2216no—
339#117—222no—
340#117—222no—
341#117D4414yes—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—112no—
346C2×C4⋊C4—224no—
347C2×C4⋊C4C22×D1112yes—
348C2×C4⋊C4C22×D1112yes—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—112no—
351C2×C22⋊C4—224no—
352C2×C42—21no—
353C2×C42—112no—
354C2×C42—224no—
355#103—228no—
356#103—224no—
357#102—224no—
358#95—224no—
359C2×C11—24no—
360C2×C11C2×#38611yescharacteristic
361C2×C11#294712yes—
362C2×C11#301312yes—
363C2×C11#304311yescharacteristic
364C2×C11#304411yescharacteristic
365D11—82no—
366C24—221no—
367C24D4×D1111yescharacteristic
368#41—228no—
369#41—228no—
370#41—444no—
371#41—444no—
372C2×Q8—1116no—
373C2×Q8—114no—
374C2×Q8—112no—
375C2×Q8—114no—
376C2×Q8—114no—
377C2×Q8—228no—
378C2×Q8—224no—
379C2×Q8—224no—
380C2×Q8—224no—
381C2×Q8—228no—
382C2×Q8—228no—
383C2×Q8—222no—
384C2×Q8—222no—
385C2×Q8—222no—
386C2×Q8—221no—
387C2×Q8—221no—
388C2×Q8—221no—
389C2×Q8—221no—
390C2×Q8—221no—
391C2×D4—112no—
392C2×D4—224no—
393C2×D4—221no—
394C2×D4—221no—
395C2×D4—221no—
396C2×D4—221no—
397C2×D4—221no—
398C2×D4—442no—
399C2×D4—442no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—42no—
403C22×C4—82no—
404C22×C4—114no—
405C22×C4—224no—
406C22×C4—224no—
407C22×C4—224no—
408C22×C4—222no—
409C22×C4—221no—
410C22×C4—224no—
411C22×C4—442no—
412C22×C4—442no—
413C22×C4C2×C4×D1112yes—
414C22×C4C2×D4412yes—
415C22×C4#433912yes—
416C22×C4D4×D1112yes—
417C22×C4D4×D1112yes—
418C22×C4D4×D1111yescharacteristic
419C22×C4#434312yes—
420C22×C4C23×D1111yescharacteristic
421Q16—2216no—
422SD16—2216no—
423M16—28no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—222no—
427C2×C8—222no—
428C2×C8—222no—
429C2×C8—444no—
430C2×C8C2×C4×D1114yes—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—42no—
435C4⋊C4—2216no—
436C4⋊C4—224no—
437C4⋊C4C2×D4414yes—
438C22⋊C4—24no—
439C22⋊C4—224no—
440C22⋊C4—884no—
441C42—24no—
442C42—2216no—
443C42—222no—
444C42—222no—
445C11#2093011yescharacteristic, complemented, Sylow
446C23—22no—
447C23—22no—
448C23—222no—
449C23—441no—
450C23—441no—
451C23—442no—
452C23—442no—
453C23#6693512yes—
454C23#6693712yes—
455C23#6694612yes—
456C23C2×D4×D1111yescharacteristic
457Q8—118no—
458Q8—224no—
459Q8—224no—
460Q8—224no—
461Q8—2216no—
462Q8—222no—
463Q8—222no—
464Q8—441no—
465Q8—441no—
466D4—222no—
467D4—222no—
468D4—448no—
469D4—441no—
470D4—441no—
471C2×C4—24no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—84no—
477C2×C4—224no—
478C2×C4—224no—
479C2×C4—224no—
480C2×C4—2216no—
481C2×C4—224no—
482C2×C4—222no—
483C2×C4—224no—
484C2×C4—224no—
485C2×C4—441no—
486C2×C4—441no—
487C2×C4—448no—
488C2×C4C4×D4414yes—
489C2×C4C22⋊C4×D1114yes—
490C2×C4#6693512yes—
491C2×C4#6693712yes—
492C2×C4#6694612yes—
493C2×C4#6694714yes—
494C2×C4#6694814yes—
495C2×C4C22×C4×D1111yescharacteristic
496C2×C4C22×D4411yescharacteristic
497C2×C4C2×#433911yescharacteristic
498C2×C4C2×D4×D1111yescharacteristic
499C2×C4C2×D4×D1111yescharacteristic
500C2×C4C2×#434311yescharacteristic
501C8—24no—
502C8—442no—
503C22—22no—
504C22—22no—
505C22—24no—
506C22—24no—
507C22—442no—
508C22—444no—
509C22—881no—
510C22—881no—
511C22—882no—
512C22#1062227814yes—
513C22C2×#6693511yescharacteristic
514C22C2×#6693711yescharacteristic
515C22C2×#6694611yescharacteristic
516C22#1062330612yes—
517C22#1062331012yes—
518C4—44no—
519C4—82no—
520C4—228no—
521C4—444no—
522C4—442no—
523C4#1062227814yes—
524C4C2×C4×D4411yescharacteristic
525C4C2×C22⋊C4×D1111yescharacteristic
526C4C2×#6694711yescharacteristic
527C4C2×#6694811yescharacteristic
528C2—24no—
529C2—882no—
530C2C2×#1062227811yescharacteristic
531C2#1307745712yes—
532C2#1307793312yes—
533C2#1307813611yescharacteristic
534C2#1307814511yescharacteristic
subgroupG/H
centerC23C2×D4×D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1736C2
socleC23×C11C22×D4
prime radicalC22×#4339C22
cosocleC22×C4×C11C24
hypercenter#1736D11
hypocenterC11#20930
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
2. C2×#10623047C2semidirect
4. C22×#66874C2semidirect
6. #13068972C2semidirect
7. #13068958C2semidirect
8. #13068314C2semidirect
9. C11×#1736C2semidirect
19. C2×#66962C4semidirect
87. C2×Dic44C2×C4semidirect
90. C2×#4314C2×C4semidirect
150. C2×Dic22C22⋊C4semidirect
186. #1736D11semidirect
445. C11#20930semidirect

Maximal subgroups

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

subgroupindex
C22×#669622
C2×#106230472
C2×#106226152
C22×#668742
#130712092
#130689722
#130689582
#130683142
C11×#17362
#2093011