Finite Group Databasebeta

← #14804272

Subgroups — #14804272

Order 2816.

1086 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622689C211yesmaximal, characteristic
2C2×#10622775C212yesmaximal, complemented
3#13073601C211yesmaximal, characteristic, complemented
4#13071681C211yesmaximal, characteristic
5#13069932C212yesmaximal, complemented
6#13069697C212yesmaximal, complemented
7#13069538C211yesmaximal, characteristic, complemented
8#13069332C211yesmaximal, characteristic, complemented
9#13068948C212yesmaximal, complemented
10C4×#66899C211yesmaximal, characteristic, complemented
11C2×C11×#460C211yesmaximal, characteristic
12C22×C11×SD16C2212yes—
13C2×C11×#127C2211yescharacteristic
14C2×C4×Q8×C11C2211yescharacteristic
15C2×#66997C2212yes—
16C2×D4×Dic11—21no—
17C2×D4×Dic11C2212yes—
18C22×#4324C2212yes—
19C22×#4324C2212yes—
20C2×#66922C2211yescharacteristic
21C11×#460—24no—
22C2×C11×C4⋊C8C2211yescharacteristic
23C2×C11×#102C2212yes—
24C2×#66899C2211yescharacteristic
25C2×#66899C2211yescharacteristic
26#10622834—21no—
27#10622834C2212yes—
28#10622833—22no—
29#10622775—28no—
30C2×#66880C2212yes—
31C2×#66880C2212yes—
32C2×Dic11⋊C8C2212yes—
33#10622689—24no—
34#10622596—22no—
35#10622587C2212yes—
36C4×C4⋊Dic11C2211yescharacteristic
37C4×C4⋊Dic11C2211yescharacteristic
38#10622541—21no—
39#10622541—21no—
40C2×C4×C11⋊C8C2211yescharacteristic
41#10622188C2212yes—
42#10622167C2211yescharacteristic
43#10622167C2211yescharacteristic
44C22×Q8×C11C2311yescharacteristic
45C22×D4×C11—21no—
46C22×D4×C11—21no—
47C22×D4×C11C2312yes—
48C23×Dic11—22no—
49C22×Dic22C2312yes—
50C2×C11×SD16—24no—
51C2×C11×SD16—24no—
52C2×C11×SD16—42no—
53C22×C8×C11C2312yes—
54C11×#127—22no—
55C11×#127D414yes—
56C4×Q8×C11—22no—
57C4×Q8×C11—22no—
58C2×C11×C4⋊C4—21no—
59C2×C11×C4⋊C4C2311yescharacteristic
60C2×C42×C11—21no—
61C2×C42×C11C2311yescharacteristic
62C2×C22⋊Dic11—22no—
63C2×C22⋊Dic11—22no—
64C2×C22⋊Dic11—22no—
65#66997—24no—
66#66997—42no—
67D4×Dic11—24no—
68D4×Dic11—42no—
69D4×Dic11—44no—
70C2×#4324—24no—
71C2×#4324—24no—
72C2×#4324—24no—
73C2×#4324—24no—
74C2×#4324—42no—
75C2×#4324—42no—
76C2×C4⋊Dic11—21no—
77C2×C4⋊Dic11—21no—
78C2×C4⋊Dic11D411yescharacteristic
79C2×C4⋊Dic11D411yescharacteristic
80C2×C4⋊Dic11C2311yescharacteristic
81C2×C4⋊Dic11C2311yescharacteristic
82C2×Dic11⋊C4—22no—
83C2×Dic11⋊C4—22no—
84C2×C4×Dic11—21no—
85C2×C4×Dic11C2312yes—
86C22×C11⋊C8C2311yescharacteristic
87C22×C11⋊C8C2311yescharacteristic
88#66922—22no—
89#66922—22no—
90C11×C4⋊C8—22no—
91C11×#102—24no—
92#66899—22no—
93#66899D414yescomplemented
94C11⋊#95—21no—
95C11⋊#95—22no—
96C11⋊#95—22no—
97C11⋊#95D412yes—
98#66880—24no—
99#66880—42no—
100Dic11⋊C8—24no—
101C4×C11⋊C8—22no—
102#25368—111nomaximal, Sylow
103C24×C11—22no—
104C2×Q8×C11—21no—
105C2×Q8×C11—21no—
106C2×Q8×C11—21no—
107C2×Q8×C11—21no—
108C2×Q8×C11—21no—
109C2×Q8×C11C2×D412yes—
110C2×D4×C11—24no—
111C2×D4×C11—24no—
112C2×D4×C11—22no—
113C2×D4×C11—22no—
114C2×D4×C11—22no—
115C2×D4×C11—22no—
116C2×D4×C11—22no—
117C2×D4×C11—42no—
118C2×D4×C11—42no—
119C2×D4×C11—81no—
120C2×D4×C11—81no—
121C2×D4×C11C2×D414yes—
122C22×C4×C11—21no—
123C22×C4×C11—21no—
124C22×C4×C11C2×D411yescharacteristic
125C22×C4×C11C2×D411yescharacteristic
126C22×C4×C11C2411yescharacteristic
127C22×Dic11—21no—
128C22×Dic11—22no—
129C22×Dic11—21no—
130C22×Dic11—44no—
131C22×Dic11—44no—
132C22×Dic11—82no—
133C22×Dic11C2×D412yes—
134C22×Dic11C2×D412yes—
135C2×Dic22—22no—
136C2×Dic22—22no—
137C2×Dic22—22no—
138C2×Dic22—22no—
139C2×Dic22—22no—
140C2×Dic22—22no—
141C11×SD16—48no—
142C2×C8×C11—22no—
143C2×C8×C11—22no—
144C2×C8×C11—22no—
145C11×C4⋊C4—21no—
146C11×C4⋊C4—21no—
147C11×C4⋊C4—42no—
148C42×C11—42no—
149C42×C11C2×D412yes—
150C42×C11#4112yes—
151C22⋊Dic11—44no—
152C22⋊Dic11—44no—
153C22⋊Dic11—82no—
154#4324—48no—
155#4324—48no—
156C4⋊Dic11—22no—
157C4⋊Dic11—21no—
158C4⋊Dic11—21no—
159C4⋊Dic11—21no—
160C4⋊Dic11—42no—
161C4⋊Dic11—42no—
162C4⋊Dic11SD1614yescomplemented
163C4⋊Dic11C2×D412yes—
164Dic11⋊C4—44no—
165Dic11⋊C4—44no—
166C4×Dic11—22no—
167C4×Dic11—22no—
168C4×Dic11—42no—
169C2×C11⋊C8—21no—
170C2×C11⋊C8—21no—
171C2×C11⋊C8—21no—
172C2×C11⋊C8—21no—
173C2×C11⋊C8C2×D412yes—
174C2×C11⋊C8#4112yes—
175C2×#492—111no—
176C2×#460D1111yescharacteristic
177C2×#461—112no—
178#2546—111no—
179#2366—111no—
180#2055—112no—
181#2013—112no—
182#1981—111no—
183#1939—111no—
184#1877—112no—
185C4×#103—111no—
186C23×C11—24no—
187C23×C11—42no—
188C23×C11—42no—
189C23×C11—42no—
190C23×C11—42no—
191C23×C11—42no—
192C23×C11—42no—
193C23×C11C22×D411yescharacteristic
194Q8×C11—22no—
195Q8×C11—22no—
196Q8×C11—41no—
197Q8×C11—41no—
198D4×C11—24no—
199D4×C11—24no—
200D4×C11—42no—
201D4×C11—42no—
202D4×C11—48no—
203D4×C11—84no—
204C2×C4×C11—21no—
205C2×C4×C11—21no—
206C2×C4×C11—22no—
207C2×C4×C11—41no—
208C2×C4×C11—41no—
209C2×C4×C11—41no—
210C2×C4×C11—41no—
211C2×C4×C11—41no—
212C2×C4×C11#12112yes—
213C2×C4×C11#12112yes—
214C2×C4×C11C2×SD1612yes—
215C2×C4×C11#13712yes—
216C2×C4×C11C22×D411yescharacteristic
217C2×C4×C11C22×D411yescharacteristic
218C2×C4×C11C2×#4111yescharacteristic
219C2×C4×C11C2×#4111yescharacteristic
220C2×C4×C11H(5,𝔽2)11yescharacteristic
221C2×C4×C11H(5,𝔽2)11yescharacteristic
222C2×Dic11—22no—
223C2×Dic11—22no—
224C2×Dic11—22no—
225C2×Dic11—22no—
226C2×Dic11—22no—
227C2×Dic11—22no—
228C2×Dic11—41no—
229C2×Dic11—41no—
230C2×Dic11—42no—
231C2×Dic11—42no—
232C2×Dic11—41no—
233C2×Dic11—42no—
234C2×Dic11—41no—
235C2×Dic11—41no—
236C2×Dic11—88no—
237C2×Dic11C2×SD1614yes—
238Dic22—42no—
239Dic22—42no—
240Dic22—42no—
241Dic22—42no—
242C8×C11—42no—
243C11⋊C8—22no—
244C11⋊C8—22no—
245C22×SD16—112no—
246C22×SD16—112no—
247C22×SD16C2×D1112yes—
248C2×C4⋊Q8—112no—
249C2×C4⋊Q8—111no—
250C2×#127C2×D1111yescharacteristic
251C2×C4×Q8C2×D1111yescharacteristic
252C2×C4×D4—112no—
253C2×C4×D4—221no—
254#492—224no—
255#461—228no—
256#460—24no—
257C2×C4⋊C8—112no—
258C2×C4⋊C8C2×D1111yescharacteristic
259C2×#103—111no—
260C2×#103—111no—
261C2×#103—112no—
262C2×#103—112no—
263C2×#102C2×D1112yes—
264C2×C4×C8—111no—
265#398—222no—
266#391—112no—
267#390—112no—
268#390—221no—
269#389—221no—
270#389—221no—
271#386—222no—
272C4×C4⋊C4—111no—
273C4×C4⋊C4—111no—
274#340—111no—
275#340—111no—
276#336—112no—
277C22×C11—44no—
278C22×C11—44no—
279C22×C11—44no—
280C22×C11—82no—
281C22×C11—82no—
282C22×C11—82no—
283C22×C11—82no—
284C22×C11C2×#12111yescharacteristic
285C22×C11D4211yescharacteristic
286C22×C11#55011yescharacteristic
287C22×C11C22×SD1611yescharacteristic
288C22×C11C2×#13711yescharacteristic
289C22×C11#57711yescharacteristic
290C22×C11#57711yescharacteristic
291C4×C11—22no—
292C4×C11—42no—
293C4×C11—81no—
294C4×C11#46114yescomplemented
295C4×C11#53211yescharacteristic
296C4×C11#54011yescharacteristic
297C4×C11D4211yescharacteristic
298C4×C11#54611yescharacteristic
299Dic11—24no—
300Dic11—42no—
301Dic11—42no—
302Dic11—81no—
303Dic11—82no—
304C22×Q8—112no—
305C22×Q8C22×D1111yescharacteristic
306C22×D4—21no—
307C22×D4—21no—
308C22×D4C22×D1112yes—
309C23×C4—222no—
310C2×SD16—24no—
311C2×SD16—24no—
312C2×SD16—42no—
313C2×SD16—224no—
314C2×SD16—224no—
315C2×SD16—224no—
316C2×SD16—224no—
317C2×SD16—442no—
318C2×SD16—442no—
319C22×C8—111no—
320C22×C8—111no—
321C22×C8C22×D1112yes—
322C4⋊Q8—224no—
323C4⋊Q8—222no—
324C4⋊Q8—222no—
325C4⋊Q8—442no—
326#127—22no—
327#127C22⋊D1114yes—
328C4×Q8—22no—
329C4×Q8—22no—
330C4×D4—224no—
331C4×D4—442no—
332C4×D4—444no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—111no—
335C2×C4⋊C4—111no—
336C2×C4⋊C4—111no—
337C2×C4⋊C4—111no—
338C2×C4⋊C4—222no—
339C2×C4⋊C4—222no—
340C2×C4⋊C4—221no—
341C2×C4⋊C4—221no—
342C2×C4⋊C4C22×D1111yescharacteristic
343C2×C22⋊C4—222no—
344C2×C22⋊C4—222no—
345C2×C22⋊C4—222no—
346C2×C42—21no—
347C2×C42—112no—
348C2×C42—221no—
349C2×C42C22×D1111yescharacteristic
350C4⋊C8—22no—
351C4⋊C8—224no—
352#103—114no—
353#103—224no—
354#103—222no—
355#103—442no—
356#102—24no—
357C4×C8—222no—
358#95—112no—
359#95—222no—
360#95—222no—
361#95—221no—
362C2×C11—84no—
363C2×C11#242811yescharacteristic
364C2×C11C2×#46111yescharacteristic
365C2×C11#304211yescharacteristic
366C2×C11#327811yescharacteristic
367C2×C11D4×SD1611yescharacteristic
368C2×C11#339211yescharacteristic
369C2×C11#340011yescharacteristic
370C24—22no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—222no—
377C2×Q8—222no—
378C2×Q8—222no—
379C2×Q8—222no—
380C2×Q8—222no—
381C2×Q8—222no—
382C2×Q8D4×D1112yes—
383C2×D4—24no—
384C2×D4—24no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—81no—
393C2×D4—81no—
394C2×D4C2×C22⋊D1114yes—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—112no—
398C22×C4—112no—
399C22×C4—221no—
400C22×C4—222no—
401C22×C4—221no—
402C22×C4—444no—
403C22×C4—444no—
404C22×C4—882no—
405C22×C4D4×D1111yescharacteristic
406C22×C4D4×D1111yescharacteristic
407C22×C4C23×D1111yescharacteristic
408SD16—48no—
409SD16—448no—
410SD16—448no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—112no—
415C2×C8—112no—
416C2×C8—221no—
417C2×C8—221no—
418C2×C8—221no—
419C2×C8—221no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—42no—
423C4⋊C4—114no—
424C4⋊C4—112no—
425C4⋊C4—222no—
426C4⋊C4—221no—
427C4⋊C4—221no—
428C4⋊C4—221no—
429C4⋊C4—444no—
430C4⋊C4—444no—
431C4⋊C4—442no—
432C4⋊C4—442no—
433C22⋊C4—444no—
434C22⋊C4—444no—
435C22⋊C4—882no—
436C42—42no—
437C42—222no—
438C42—222no—
439C42—442no—
440C42#434112yes—
441C42C2×C22⋊D1112yes—
442C11#2536811yescharacteristic, complemented, Sylow
443C23—24no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23C2×D4×D1111yescharacteristic
451Q8—22no—
452Q8—22no—
453Q8—41no—
454Q8—41no—
455Q8—442no—
456Q8—442no—
457Q8—442no—
458Q8—442no—
459D4—24no—
460D4—24no—
461D4—48no—
462D4—42no—
463D4—42no—
464D4—84no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—114no—
474C2×C4—222no—
475C2×C4—222no—
476C2×C4—222no—
477C2×C4—222no—
478C2×C4—222no—
479C2×C4—222no—
480C2×C4—441no—
481C2×C4—441no—
482C2×C4—441no—
483C2×C4—441no—
484C2×C4—442no—
485C2×C4—441no—
486C2×C4—442no—
487C2×C4—442no—
488C2×C4—888no—
489C2×C4SD16×D1112yes—
490C2×C4#6696812yes—
491C2×C4#6699212yes—
492C2×C4#6699212yes—
493C2×C4C2×D4×D1111yescharacteristic
494C2×C4C2×#434111yescharacteristic
495C2×C4#6703711yescharacteristic
496C2×C4#41×D1111yescharacteristic
497C2×C4#6704211yescharacteristic
498C2×C4C22×C22⋊D1111yescharacteristic
499C8—42no—
500C8—222no—
501C8—222no—
502C22—44no—
503C22—44no—
504C22—44no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22#1062313011yescharacteristic
510C22#1062314011yescharacteristic
511C22C2×SD16×D1111yescharacteristic
512C22C2×#6696811yescharacteristic
513C22#1062330611yescharacteristic
514C22#1062330611yescharacteristic
515C22C2×#6699211yescharacteristic
516C4—22no—
517C4—42no—
518C4—81no—
519C4—224no—
520C4—442no—
521C4—442no—
522C4—882no—
523C4—881no—
524C4#1062277514yescomplemented
525C4#1062312711yescharacteristic
526C4#1062315011yescharacteristic
527C4D4×C22⋊D1111yescharacteristic
528C4#1062336011yescharacteristic
529C2—84no—
530C2#1307220611yescharacteristic
531C2C2×#1062277511yescharacteristic
532C2#1307813011yescharacteristic
533C2#1308028511yescharacteristic
534C2#1308030711yescharacteristic
535C2#1308159611yescharacteristic
536C2#1308166811yescharacteristic
subgroupG/H
centerC23C2×D4×D11
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC2×C11×#460C2
socleC23×C11C22×D4
prime radicalC2×C11×#127C22
hypercenterC2×#460D11
hypocenterC11#25368
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
2. C2×#10622775C2semidirect
3. #13073601C2semidirect
5. #13069932C2semidirect
6. #13069697C2semidirect
7. #13069538C2semidirect
8. #13069332C2semidirect
9. #13068948C2semidirect
10. C4×#66899C2semidirect
93. #66899D4semidirect
162. C4⋊Dic11SD16semidirect
294. C4×C11#461semidirect
442. C11#25368semidirect
524. C4#10622775semidirect

Maximal subgroups

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

subgroupindex
C2×#106226892
C2×#106227752
#130736012
#130716812
#130699322
#130696972
#130695382
#130693322
#130689482
C4×#668992
C2×C11×#4602
#2536811