Finite Group Databasebeta

← #14788815

Subgroups — #14788815

Order 2816.

1514 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66922C211yesmaximal, characteristic, complemented
2C2×#10622632C212yesmaximal, complemented
3C2×#10622653C212yesmaximal
4#13069862C212yesmaximal, complemented
5#13069698C212yesmaximal, complemented
6#13069539C212yesmaximal, complemented
7#13068596C211yesmaximal, characteristic, complemented
8#13068550C212yesmaximal, complemented
9C4×C11×#121C211yesmaximal, characteristic
10C23×Dic22C2212yes—
11C2×C11×#121C2211yescharacteristic
12C2×C4×D4×C11—21no—
13C2×C4×D4×C11C2211yescharacteristic
14C22×C42×C11C2211yescharacteristic
15C22×#4324C2214yes—
16C22×C4⋊Dic11—21no—
17C22×C4⋊Dic11—21no—
18C2×#66922—21no—
19C2×#66922—21no—
20C2×#66922—28no—
21C2×#66922C2212yes—
22C2×#66922C2212yes—
23C11×#390C2211yescharacteristic
24C11×#389C2211yescharacteristic
25C11×#388—21no—
26C11×#386—21no—
27C11×#380C2211yescharacteristic
28C4×C11×C4⋊C4C2211yescharacteristic
29C4×C11×C22⋊C4—21no—
30C2×C22⋊(C11⋊C8)C2212yes—
31#10622653—28no—
32#10622632—28no—
33C2×#66874C2214yes—
34C2×#66874C2214yes—
35C2×#66874C2214yes—
36C2×C4⋊(C11⋊C8)C2212yes—
37#10622162C2212yes—
38C22×D4×C11—21no—
39C22×D4×C11C2311yescharacteristic
40C23×C4×C11—21no—
41C23×C4×C11D411yescharacteristic
42C23×C4×C11D411yescharacteristic
43C23×C4×C11C2311yescharacteristic
44C23×Dic11—22no—
45C22×Dic22—22no—
46C22×Dic22—28no—
47C22×Dic22—24no—
48C22×Dic22—24no—
49C22×Dic22—44no—
50C22×Dic22C2314yes—
51C11×#121—22no—
52C11×#121D414yes—
53C4×D4×C11—24no—
54C4×D4×C11—44no—
55C2×C11×C4⋊C4—21no—
56C2×C11×C4⋊C4C2311yescharacteristic
57C2×C11×C4⋊C4C2311yescharacteristic
58C2×C11×C4⋊C4C2311yescharacteristic
59C2×C11×C22⋊C4—21no—
60C2×C11×C22⋊C4—21no—
61C2×C11×C22⋊C4—21no—
62C2×C11×C22⋊C4—21no—
63C2×C11×C22⋊C4—21no—
64C2×C42×C11—24no—
65C2×C42×C11—21no—
66C2×C42×C11D411yescharacteristic
67C2×C42×C11D411yescharacteristic
68C2×C42×C11C2311yescharacteristic
69C2×C42×C11C2311yescharacteristic
70C2×#4324—28no—
71C2×#4324—28no—
72C2×#4324—44no—
73C2×C4⋊Dic11—28no—
74C2×C4⋊Dic11—28no—
75C2×C4⋊Dic11—22no—
76C2×C4⋊Dic11—22no—
77C2×C4⋊Dic11—22no—
78C2×C4⋊Dic11—22no—
79C22×C11⋊C8C2314yes—
80#66922—28no—
81#66922—28no—
82#66922—28no—
83#66922—24no—
84#66922D418yescomplemented
85C11×#95—21no—
86C11×#95—21no—
87C11×#95—21no—
88C22⋊(C11⋊C8)—24no—
89#66874—28no—
90#66874—28no—
91#66874—44no—
92C4⋊(C11⋊C8)—24no—
93#24209—111nomaximal, Sylow
94C24×C11—21no—
95C24×C11C2×D411yescharacteristic
96C2×D4×C11—24no—
97C2×D4×C11—22no—
98C2×D4×C11—21no—
99C2×D4×C11—22no—
100C2×D4×C11—42no—
101C2×D4×C11—42no—
102C2×D4×C11C2×D412yes—
103C22×C4×C11—22no—
104C22×C4×C11—22no—
105C22×C4×C11—22no—
106C22×C4×C11—22no—
107C22×C4×C11—22no—
108C22×C4×C11—22no—
109C22×C4×C11—22no—
110C22×C4×C11—21no—
111C22×C4×C11—21no—
112C22×C4×C11—21no—
113C22×C4×C11—42no—
114C22×C4×C11—42no—
115C22×C4×C11—42no—
116C22×C4×C11SD1614yescomplemented
117C22×C4×C11C2×D411yescharacteristic
118C22×C4×C11C2×D411yescharacteristic
119C22×C4×C11C2×D412yes—
120C22×C4×C11C2×D411yescharacteristic
121C22×C4×C11C2×D411yescharacteristic
122C22×C4×C11C2×D411yescharacteristic
123C22×C4×C11#4112yes—
124C22×C4×C11C2411yescharacteristic
125C22×Dic11—28no—
126C22×Dic11—24no—
127C22×Dic11—44no—
128C22×Dic11—44no—
129C2×Dic22—28no—
130C2×Dic22—24no—
131C2×Dic22—24no—
132C2×Dic22—24no—
133C2×Dic22—24no—
134C2×Dic22—24no—
135C2×Dic22—416no—
136C2×Dic22—416no—
137C2×Dic22—44no—
138C2×Dic22—44no—
139C2×Dic22C2×D418yes—
140C11×C4⋊C4—21no—
141C11×C4⋊C4—22no—
142C11×C4⋊C4—22no—
143C11×C4⋊C4—42no—
144C11×C4⋊C4C2×D412yes—
145C11×C22⋊C4—24no—
146C11×C22⋊C4—42no—
147C11×C22⋊C4—42no—
148C11×C22⋊C4—42no—
149C11×C22⋊C4—42no—
150C42×C11—22no—
151C42×C11—22no—
152C42×C11—22no—
153C42×C11—42no—
154C42×C11C2×D414yes—
155#4324—416no—
156C4⋊Dic11—28no—
157C4⋊Dic11—28no—
158C4⋊Dic11—28no—
159C4⋊Dic11—28no—
160C2×C11⋊C8—24no—
161C2×C11⋊C8—24no—
162C2×C11⋊C8—24no—
163C22×C4⋊Q8—111no—
164C2×#461—112no—
165C2×#448—112no—
166C4×#121D1111yescharacteristic
167#2043—112no—
168#2013—112no—
169#1981—112no—
170#1797—111no—
171#1787—112no—
172C23×C11—21no—
173C23×C11—21no—
174C23×C11—21no—
175C23×C11—21no—
176C23×C11—22no—
177C23×C11—41no—
178C23×C11—41no—
179C23×C11—41no—
180C23×C11—41no—
181C23×C11—41no—
182C23×C11—41no—
183C23×C11#12112yes—
184C23×C11C2×SD1612yes—
185C23×C11#13712yes—
186C23×C11C22×D411yescharacteristic
187D4×C11—24no—
188D4×C11—48no—
189D4×C11—42no—
190C2×C4×C11—28no—
191C2×C4×C11—21no—
192C2×C4×C11—21no—
193C2×C4×C11—21no—
194C2×C4×C11—21no—
195C2×C4×C11—21no—
196C2×C4×C11—21no—
197C2×C4×C11—21no—
198C2×C4×C11—21no—
199C2×C4×C11—24no—
200C2×C4×C11—24no—
201C2×C4×C11—28no—
202C2×C4×C11—22no—
203C2×C4×C11—41no—
204C2×C4×C11—41no—
205C2×C4×C11—41no—
206C2×C4×C11—41no—
207C2×C4×C11—41no—
208C2×C4×C11—41no—
209C2×C4×C11—41no—
210C2×C4×C11—41no—
211C2×C4×C11—84no—
212C2×C4×C11#12012yes—
213C2×C4×C11#12012yes—
214C2×C4×C11#12112yes—
215C2×C4×C11C2×SD1612yes—
216C2×C4×C11C2×SD1612yes—
217C2×C4×C11#13712yes—
218C2×C4×C11#13712yes—
219C2×C4×C11C22×D411yescharacteristic
220C2×C4×C11C22×D412yes—
221C2×C4×C11C2×#4111yescharacteristic
222C2×C4×C11H(5,𝔽2)12yes—
223C2×Dic11—28no—
224C2×Dic11—416no—
225C2×Dic11—44no—
226C2×Dic11—44no—
227Dic22—28no—
228Dic22—28no—
229Dic22—416no—
230Dic22—44no—
231Dic22—44no—
232C11⋊C8—44no—
233C23×Q8—112no—
234C22×SD16—114no—
235C2×C4⋊Q8—112no—
236C2×C4⋊Q8—112no—
237C2×C4⋊Q8—228no—
238C2×C4⋊Q8—221no—
239C2×C4⋊Q8—221no—
240C2×#121C2×D1111yescharacteristic
241C2×C4×D4—21no—
242C2×C4×D4C2×D1111yescharacteristic
243C22×C4⋊C4—221no—
244C22×C4⋊C4—221no—
245C22×C42C2×D1111yescharacteristic
246#461—228no—
247#448—228no—
248C2×C4⋊C8—112no—
249C2×#103—114no—
250C2×#103—114no—
251C2×#103—114no—
252C2×C22⋊C8—112no—
253#390C2×D1111yescharacteristic
254#389C2×D1111yescharacteristic
255#388—21no—
256#386—21no—
257#380C2×D1111yescharacteristic
258C4×C4⋊C4C2×D1111yescharacteristic
259C4×C22⋊C4—21no—
260#336—112no—
261C22×C11—22no—
262C22×C11—22no—
263C22×C11—22no—
264C22×C11—22no—
265C22×C11—22no—
266C22×C11—22no—
267C22×C11—42no—
268C22×C11—42no—
269C22×C11—42no—
270C22×C11—81no—
271C22×C11—81no—
272C22×C11—81no—
273C22×C11—81no—
274C22×C11#46114yescomplemented
275C22×C11C2×#12011yescharacteristic
276C22×C11C2×#12111yescharacteristic
277C22×C11#53511yescharacteristic
278C22×C11C22×SD1611yescharacteristic
279C22×C11C2×#13711yescharacteristic
280C22×C11C2×#13711yescharacteristic
281C22×C11C2×#13711yescharacteristic
282C4×C11—22no—
283C4×C11—22no—
284C4×C11—22no—
285C4×C11—22no—
286C4×C11—42no—
287C4×C11—81no—
288C4×C11—81no—
289C4×C11#44814yescomplemented
290C4×C11D4212yes—
291C4×C11#54612yes—
292Dic11—48no—
293C22×Q8—114no—
294C22×Q8—228no—
295C22×Q8—222no—
296C22×Q8—224no—
297C22×Q8—224no—
298C22×Q8—444no—
299C22×D4—21no—
300C22×D4C22×D1111yescharacteristic
301C23×C4—21no—
302C23×C4—222no—
303C23×C4C22⋊D1111yescharacteristic
304C23×C4C22⋊D1111yescharacteristic
305C23×C4C22×D1111yescharacteristic
306C2×SD16—228no—
307C2×SD16—228no—
308C2×SD16—444no—
309C22×C8—114no—
310C4⋊Q8—118no—
311C4⋊Q8—228no—
312C4⋊Q8—228no—
313C4⋊Q8—228no—
314C4⋊Q8—224no—
315#121—22no—
316#121D4414yes—
317C4×D4—24no—
318C4×D4—44no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—228no—
321C2×C4⋊C4—228no—
322C2×C4⋊C4—222no—
323C2×C4⋊C4—222no—
324C2×C4⋊C4—222no—
325C2×C4⋊C4—222no—
326C2×C4⋊C4C22×D1111yescharacteristic
327C2×C4⋊C4C22×D1111yescharacteristic
328C2×C4⋊C4C22×D1111yescharacteristic
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C42—24no—
335C2×C42—21no—
336C2×C42C22⋊D1111yescharacteristic
337C2×C42C22⋊D1111yescharacteristic
338C2×C42C22×D1111yescharacteristic
339C2×C42C22×D1111yescharacteristic
340C4⋊C8—224no—
341#103—228no—
342#103—228no—
343#103—444no—
344C22⋊C8—224no—
345#95—21no—
346#95—21no—
347#95—21no—
348C2×C11—24no—
349C2×C11—82no—
350C2×C11#240811yescharacteristic
351C2×C11C2×#44811yescharacteristic
352C2×C11#300411yescharacteristic
353C2×C11C2×#46111yescharacteristic
354C2×C11#303911yescharacteristic
355C2×C11#320611yescharacteristic
356C2×C11#321811yescharacteristic
357C24—21no—
358C24C2×C22⋊D1111yescharacteristic
359C2×Q8—118no—
360C2×Q8—228no—
361C2×Q8—224no—
362C2×Q8—224no—
363C2×Q8—224no—
364C2×Q8—224no—
365C2×Q8—224no—
366C2×Q8—4416no—
367C2×Q8—4416no—
368C2×Q8—444no—
369C2×Q8—444no—
370C2×D4—24no—
371C2×D4—21no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4C2×D4412yes—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—228no—
391C22×C4—224no—
392C22×C4—444no—
393C22×C4—444no—
394C22×C4#432414yescomplemented
395C22×C4C2×D4412yes—
396C22×C4#433912yes—
397C22×C4C2×C22⋊D1111yescharacteristic
398C22×C4C2×C22⋊D1111yescharacteristic
399C22×C4C2×C22⋊D1111yescharacteristic
400C22×C4C2×C22⋊D1111yescharacteristic
401C22×C4C2×C22⋊D1111yescharacteristic
402C22×C4C23×D1111yescharacteristic
403SD16—4416no—
404C2×C8—224no—
405C2×C8—224no—
406C2×C8—224no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—42no—
411C4⋊C4—228no—
412C4⋊C4—228no—
413C4⋊C4—228no—
414C4⋊C4—228no—
415C4⋊C4C2×D4412yes—
416C22⋊C4—24no—
417C22⋊C4—42no—
418C22⋊C4—42no—
419C22⋊C4—42no—
420C22⋊C4—42no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—42no—
425C42D4×D1114yes—
426C11#2420911yescharacteristic, complemented, Sylow
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—22no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23#6698312yes—
439C23C2×#432412yes—
440C23#6700412yes—
441C23C22×C22⋊D1111yescharacteristic
442Q8—228no—
443Q8—228no—
444Q8—4416no—
445Q8—444no—
446Q8—444no—
447D4—24no—
448D4—48no—
449D4—42no—
450C2×C4—28no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—28no—
462C2×C4—22no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—84no—
472C2×C4—228no—
473C2×C4—4416no—
474C2×C4—444no—
475C2×C4—444no—
476C2×C4#6698312yes—
477C2×C4C2×#432412yes—
478C2×C4C2×#432412yes—
479C2×C4#6699012yes—
480C2×C4#6699012yes—
481C2×C4#6700412yes—
482C2×C4#6700412yes—
483C2×C4C22×D4411yescharacteristic
484C2×C4C2×#433911yescharacteristic
485C2×C4C2×D4×D1112yes—
486C2×C4#6704212yes—
487C8—444no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22#1062263214yescomplemented
502C22C2×#6698311yescharacteristic
503C22C22×#432411yescharacteristic
504C22C2×#6699011yescharacteristic
505C22C2×#6700411yescharacteristic
506C22C2×#6700411yescharacteristic
507C22C2×#6700411yescharacteristic
508C22#1062336111yescharacteristic
509C4—22no—
510C4—22no—
511C4—22no—
512C4—22no—
513C4—42no—
514C4—81no—
515C4—81no—
516C4—448no—
517C4#1062265314yescomplemented
518C4D4×D4412yes—
519C4#1062308112yes—
520C2—24no—
521C2—82no—
522C2#1307197211yescharacteristic
523C2C2×#1062265311yescharacteristic
524C2#1307784911yescharacteristic
525C2C2×#1062263211yescharacteristic
526C2#1307810311yescharacteristic
527C2#1307953811yescharacteristic
528C2#1307964011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D11
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC4×C11×#121C2
socleC23×C11C22×D4
prime radicalC2×C11×#121C22
hypercenterC4×#121D11
hypocenterC11#24209
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. C22×#66922C2semidirect
2. C2×#10622632C2semidirect
4. #13069862C2semidirect
5. #13069698C2semidirect
6. #13069539C2semidirect
7. #13068596C2semidirect
8. #13068550C2semidirect
84. #66922D4semidirect
116. C22×C4×C11SD16semidirect
274. C22×C11#461semidirect
289. C4×C11#448semidirect
394. C22×C4#4324semidirect
426. C11#24209semidirect
501. C22#10622632semidirect
517. C4#10622653semidirect

Maximal subgroups

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

subgroupindex
C22×#669222
C2×#106226322
C2×#106226532
#130698622
#130696982
#130695392
#130685962
#130685502
C4×C11×#1212
#2420911