Finite Group Databasebeta

← #14804279

Subgroups — #14804279

Order 2816.

1014 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
1C2×#10622306C211yesmaximal, characteristic
2C2×#10622458C212yesmaximal, complemented
3C2×#10622462C211yesmaximal, characteristic, complemented
4#13073618C211yesmaximal, characteristic
5#13071672C211yesmaximal, characteristic, complemented
6#13069934C212yesmaximal, complemented
7#13069713C212yesmaximal, complemented
8#13069542C211yesmaximal, characteristic
9#13068970C212yesmaximal, complemented
10C4×#66880C211yesmaximal, characteristic, complemented
11C11×#1939C211yesmaximal, characteristic, complemented
12C22×Dic44C2212yes—
13C22×#4314C2212yes—
14C22×#4314C2212yes—
15C2×#66928C2212yes—
16C2×#66927C2211yescharacteristic
17C2×#66922C2211yescharacteristic
18C2×C4×Dic22C2211yescharacteristic
19C2×C4×Dic22C2212yes—
20C2×C11×C4⋊C8C2211yescharacteristic
21C2×C11×C4⋊C8C2212yes—
22C2×C4×C8×C11C2211yescharacteristic
23C4×C11×C4⋊C4C2211yescharacteristic
24C2×#66885C2212yes—
25C2×#66885C2212yes—
26C2×#66880C2212yes—
27C2×#66880C2211yescharacteristic
28C2×#66880C2211yescharacteristic
29#10622601—21no—
30#10622601—21no—
31C4⋊C4×Dic11—21no—
32#10622586—21no—
33#10622552C2212yes—
34C4×C4⋊Dic11C2211yescharacteristic
35#10622542C2212yes—
36#10622462—24no—
37#10622458—28no—
38#10622306—24no—
39#10622285—22no—
40#10622267—22no—
41C11×#336C2212yes—
42#10622189C2211yescharacteristic
43#10622189C2211yescharacteristic
44C22×D44—21no—
45C22×D44—21no—
46C22×D44C2312yes—
47C22×Dic22C2312yes—
48C22×Dic22C2311yescharacteristic
49C22×C8×C11C2312yes—
50C22×C8×C11C2311yescharacteristic
51C22×C8×C11C2311yescharacteristic
52C2×C11×C4⋊C4D411yescharacteristic
53C2×C11×C4⋊C4D411yescharacteristic
54C2×C42×C11C2312yes—
55C2×C42×C11C2311yescharacteristic
56C2×(C2×D11)⋊C4—22no—
57C2×(C2×D11)⋊C4—22no—
58C2×(C2×D11)⋊C4—22no—
59C2×(C2×D11)⋊C4—22no—
60C2×C4⋊Dic11—21no—
61C2×C4⋊Dic11—21no—
62C2×C4⋊Dic11C2311yescharacteristic
63C2×C4⋊Dic11C2311yescharacteristic
64C2×C4⋊Dic11C2311yescharacteristic
65C2×Dic11⋊C4—22no—
66C2×Dic11⋊C4—21no—
67C2×C4×Dic11—22no—
68C2×C4×Dic11—21no—
69C2×C4×Dic11—21no—
70C2×Dic44—24no—
71C2×Dic44—24no—
72C2×Dic44—42no—
73C2×#4314—24no—
74C2×#4314—24no—
75C2×#4314—24no—
76C2×#4314—24no—
77C2×#4314—42no—
78C2×#4314—42no—
79#66928—24no—
80#66928—42no—
81#66927—22no—
82#66927—22no—
83#66922—22no—
84#66922D414yes—
85C4×Dic22—22no—
86C4×Dic22—22no—
87C4×Dic22—24no—
88C4×Dic22—42no—
89C11×C4⋊C8—24no—
90C11×C4⋊C8—22no—
91C4×C8×C11—22no—
92C11×#95D412yes—
93C11⋊#95—21no—
94C11⋊#95—22no—
95C11⋊#95—22no—
96#66885—24no—
97#66885—42no—
98#66880—24no—
99#66880—22no—
100#66880D414yescomplemented
101#25369—111nomaximal, Sylow
102C23×D11—22no—
103C22×C4×C11C2×D412yes—
104C22×C4×C11C2×D412yes—
105C22×C4×C11C2×D411yescharacteristic
106C22×C4×C11C2×D411yescharacteristic
107C22×C4×C11C2411yescharacteristic
108C22×Dic11—22no—
109C22×Dic11—21no—
110C22×Dic11—21no—
111C22×Dic11—21no—
112C22×Dic11—21no—
113C2×D44—22no—
114C2×D44—22no—
115C2×D44—22no—
116C2×D44—22no—
117C2×D44—22no—
118C2×D44—22no—
119C2×D44—42no—
120C2×D44—42no—
121C2×D44—42no—
122C2×D44—42no—
123C2×D44—81no—
124C2×D44—81no—
125C2×Dic22—22no—
126C2×Dic22—21no—
127C2×Dic22—21no—
128C2×Dic22—21no—
129C2×Dic22—22no—
130C2×Dic22—22no—
131C2×Dic22—22no—
132C2×Dic22—22no—
133C2×Dic22—21no—
134C2×Dic22—21no—
135C2×Dic22C2×D414yes—
136C2×Dic22C2×D412yes—
137C2×C8×C11—21no—
138C2×C8×C11—21no—
139C2×C8×C11—22no—
140C2×C8×C11—22no—
141C2×C8×C11—21no—
142C2×C8×C11—21no—
143C2×C8×C11—22no—
144C2×C8×C11C2×D412yes—
145C2×C8×C11#4112yes—
146C11×C4⋊C4—22no—
147C11×C4⋊C4—22no—
148C42×C11—22no—
149C42×C11—22no—
150C42×C11C2×D412yes—
151C42×C11#4112yes—
152(C2×D11)⋊C4—44no—
153(C2×D11)⋊C4—44no—
154(C2×D11)⋊C4—82no—
155(C2×D11)⋊C4—82no—
156C4⋊Dic11—24no—
157C4⋊Dic11—24no—
158C4⋊Dic11—21no—
159C4⋊Dic11—21no—
160C4⋊Dic11—21no—
161C4⋊Dic11—21no—
162C4⋊Dic11—21no—
163C4⋊Dic11C2×D412yes—
164Dic11⋊C4—44no—
165Dic11⋊C4—42no—
166C4×Dic11—44no—
167C4×Dic11—42no—
168C4×Dic11—42no—
169Dic44—48no—
170#4314—48no—
171#4314—48no—
172C2×#492—111no—
173C2×#464—112no—
174C2×#462—111no—
175#2548—111no—
176#2365—111no—
177#2055—112no—
178#2015—112no—
179#1982—111no—
180#1939D1111yescharacteristic, complemented
181#1880—112no—
182C4×#103—111no—
183C23×C11C22×D411yescharacteristic
184C22×D11—42no—
185C22×D11—42no—
186C22×D11—42no—
187C22×D11—42no—
188C22×D11—42no—
189C22×D11—42no—
190C22×D11—42no—
191C2×C4×C11—21no—
192C2×C4×C11—21no—
193C2×C4×C11—22no—
194C2×C4×C11—22no—
195C2×C4×C11—22no—
196C2×C4×C11—22no—
197C2×C4×C11—22no—
198C2×C4×C11—22no—
199C2×C4×C11#12112yes—
200C2×C4×C11#12112yes—
201C2×C4×C11#13512yes—
202C2×C4×C11#13712yes—
203C2×C4×C11C22×D411yescharacteristic
204C2×C4×C11C22×D411yescharacteristic
205C2×C4×C11C2×#4111yescharacteristic
206C2×C4×C11C2×#4111yescharacteristic
207C2×C4×C11H(5,𝔽2)11yescharacteristic
208C2×C4×C11H(5,𝔽2)11yescharacteristic
209C2×Dic11—24no—
210C2×Dic11—22no—
211C2×Dic11—22no—
212C2×Dic11—41no—
213C2×Dic11—42no—
214C2×Dic11—41no—
215C2×Dic11—42no—
216C2×Dic11—41no—
217C2×Dic11—41no—
218C2×Dic11—41no—
219C2×Dic11—41no—
220C2×Dic11—41no—
221C2×Dic11—41no—
222C2×Dic11—41no—
223C2×Dic11—41no—
224D44—42no—
225D44—42no—
226D44—42no—
227D44—42no—
228D44—84no—
229D44—84no—
230Dic22—22no—
231Dic22—24no—
232Dic22—22no—
233Dic22—24no—
234Dic22—42no—
235Dic22—41no—
236Dic22—42no—
237Dic22—41no—
238C8×C11—22no—
239C8×C11—22no—
240C8×C11—42no—
241C22×Q16—112no—
242C22×SD16—112no—
243C22×SD16—112no—
244C2×C4⋊Q8—111no—
245C2×#127—111no—
246C2×#124—112no—
247C2×C4×Q8—111no—
248C2×C4×Q8—112no—
249#492—224no—
250#464—228no—
251#462—224no—
252C2×C4⋊C8C2×D1112yes—
253C2×C4⋊C8C2×D1111yescharacteristic
254C2×#103—111no—
255C2×#103—112no—
256C2×#103—111no—
257C2×#102—112no—
258C2×#102—112no—
259C2×C4×C8C2×D1111yescharacteristic
260#394—222no—
261#391—112no—
262#390—112no—
263#390—221no—
264#390—221no—
265#389—221no—
266#387—222no—
267C4×C4⋊C4—111no—
268C4×C4⋊C4—221no—
269C4×C4⋊C4C2×D1111yescharacteristic
270#340—111no—
271#340—111no—
272#336C2×D1112yes—
273C22×C11C2×#12111yescharacteristic
274C22×C11D4211yescharacteristic
275C22×C11#55011yescharacteristic
276C22×C11C2×#13511yescharacteristic
277C22×C11C2×#13711yescharacteristic
278C22×C11#57711yescharacteristic
279C22×C11#57711yescharacteristic
280C2×D11—82no—
281C2×D11—82no—
282C2×D11—82no—
283C2×D11—82no—
284C2×D11—82no—
285C2×D11—82no—
286C2×D11—82no—
287C4×C11—22no—
288C4×C11—42no—
289C4×C11—42no—
290C4×C11#46414yescomplemented
291C4×C11#53211yescharacteristic
292C4×C11#54011yescharacteristic
293C4×C11D4211yescharacteristic
294C4×C11#54611yescharacteristic
295Dic11—42no—
296Dic11—44no—
297Dic11—42no—
298Dic11—81no—
299Dic11—81no—
300C22×Q8—111no—
301C22×Q8—112no—
302C22×D4—112no—
303C22×D4—221no—
304C22×D4—221no—
305C2×Q16—224no—
306C2×Q16—224no—
307C2×Q16—442no—
308C2×SD16—224no—
309C2×SD16—224no—
310C2×SD16—224no—
311C2×SD16—224no—
312C2×SD16—442no—
313C2×SD16—442no—
314C22×C8C22×D1112yes—
315C22×C8C22×D1111yescharacteristic
316C22×C8C22×D1111yescharacteristic
317C4⋊Q8—114no—
318C4⋊Q8—222no—
319#127—222no—
320#127—222no—
321#124—224no—
322#124—442no—
323C4×Q8—224no—
324C4×Q8—222no—
325C4×Q8—222no—
326C4×Q8—442no—
327C2×C4⋊C4—111no—
328C2×C4⋊C4—111no—
329C2×C4⋊C4—111no—
330C2×C4⋊C4—221no—
331C2×C4⋊C4—221no—
332C2×C4⋊C4—222no—
333C2×C4⋊C4—221no—
334C2×C4⋊C4D4411yescharacteristic
335C2×C4⋊C4D4411yescharacteristic
336C2×C22⋊C4—222no—
337C2×C22⋊C4—222no—
338C2×C22⋊C4—222no—
339C2×C22⋊C4—222no—
340C2×C42—222no—
341C2×C42—221no—
342C2×C42—221no—
343C2×C42C22×D1112yes—
344C2×C42C22×D1111yescharacteristic
345C4⋊C8—24no—
346C4⋊C8—22no—
347#103—114no—
348#103—224no—
349#103—222no—
350#102—224no—
351#102—442no—
352C4×C8—22no—
353#95—221no—
354#95—222no—
355#95—222no—
356#95D4412yes—
357C2×C11#242811yescharacteristic
358C2×C11C2×#46411yescharacteristic
359C2×C11#304311yescharacteristic
360C2×C11#327811yescharacteristic
361C2×C11#328411yescharacteristic
362C2×C11#339411yescharacteristic
363C2×C11#340011yescharacteristic
364D11—162no—
365C24—222no—
366C2×Q8—112no—
367C2×Q8—114no—
368C2×Q8—222no—
369C2×Q8—221no—
370C2×Q8—222no—
371C2×Q8—221no—
372C2×Q8—221no—
373C2×Q8—221no—
374C2×Q8—222no—
375C2×Q8—222no—
376C2×Q8—222no—
377C2×Q8—221no—
378C2×D4—222no—
379C2×D4—222no—
380C2×D4—222no—
381C2×D4—222no—
382C2×D4—222no—
383C2×D4—222no—
384C2×D4—442no—
385C2×D4—442no—
386C2×D4—442no—
387C2×D4—442no—
388C2×D4—881no—
389C2×D4—881no—
390C22×C4—222no—
391C22×C4—221no—
392C22×C4—221no—
393C22×C4—221no—
394C22×C4—221no—
395C22×C4C2×D4412yes—
396C22×C4C2×D4412yes—
397C22×C4C2×D4411yescharacteristic
398C22×C4C2×D4411yescharacteristic
399C22×C4C23×D1111yescharacteristic
400Q16—448no—
401SD16—448no—
402SD16—448no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8D4×D1112yes—
411C2×C8#434312yes—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—112no—
415C4⋊C4—224no—
416C4⋊C4—224no—
417C4⋊C4—221no—
418C4⋊C4—221no—
419C4⋊C4—221no—
420C4⋊C4—221no—
421C4⋊C4—221no—
422C4⋊C4—444no—
423C4⋊C4—442no—
424C22⋊C4—444no—
425C22⋊C4—444no—
426C22⋊C4—882no—
427C22⋊C4—882no—
428C42—22no—
429C42—22no—
430C42—444no—
431C42—442no—
432C42—442no—
433C42D4×D1112yes—
434C42#434312yes—
435C11#2536911yescharacteristic, complemented, Sylow
436C23—442no—
437C23—442no—
438C23—442no—
439C23—442no—
440C23—442no—
441C23—442no—
442C23—442no—
443C23C22×D4411yescharacteristic
444Q8—222no—
445Q8—224no—
446Q8—222no—
447Q8—224no—
448Q8—442no—
449Q8—441no—
450Q8—442no—
451Q8—441no—
452D4—442no—
453D4—442no—
454D4—442no—
455D4—442no—
456D4—884no—
457D4—884no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—224no—
467C2×C4—222no—
468C2×C4—222no—
469C2×C4—441no—
470C2×C4—441no—
471C2×C4—441no—
472C2×C4—442no—
473C2×C4—442no—
474C2×C4—441no—
475C2×C4—441no—
476C2×C4—441no—
477C2×C4—441no—
478C2×C4—441no—
479C2×C4—441no—
480C2×C4—441no—
481C2×C4#6694812yes—
482C2×C4#6694812yes—
483C2×C4#6695712yes—
484C2×C4#6696212yes—
485C2×C4C2×D4×D1111yescharacteristic
486C2×C4C2×D4×D1111yescharacteristic
487C2×C4C2×#434311yescharacteristic
488C2×C4C2×#434311yescharacteristic
489C2×C4#6704211yescharacteristic
490C2×C4#6704211yescharacteristic
491C8—22no—
492C8—22no—
493C8—42no—
494C22—882no—
495C22—882no—
496C22—882no—
497C22—882no—
498C22—882no—
499C22—882no—
500C22—882no—
501C22C2×#6694811yescharacteristic
502C22D4×D4411yescharacteristic
503C22#1062310811yescharacteristic
504C22C2×#6695711yescharacteristic
505C22C2×#6696211yescharacteristic
506C22#1062328211yescharacteristic
507C22#1062328211yescharacteristic
508C4—22no—
509C4—42no—
510C4—42no—
511C4—442no—
512C4—444no—
513C4—442no—
514C4—881no—
515C4—881no—
516C4#1062245814yescomplemented
517C4#1062314011yescharacteristic
518C4#1062316111yescharacteristic
519C4#1062322911yescharacteristic
520C4#1062325611yescharacteristic
521C2—1762no—
522C2#1307220211yescharacteristic
523C2C2×#1062245811yescharacteristic
524C2#1307813811yescharacteristic
525C2#1308028411yescharacteristic
526C2#1308034811yescharacteristic
527C2#1308161411yescharacteristic
528C2#1308167111yescharacteristic
subgroupG/H
centerC23C22×D44
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC11×#1939C2
socleC23×C11C22×D4
prime radicalC2×#66927C22
hypercenter#1939D11
hypocenterC11#25369
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1939

Complements

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

NH ≅ G/Ntype
2. C2×#10622458C2semidirect
3. C2×#10622462C2semidirect
5. #13071672C2semidirect
6. #13069934C2semidirect
7. #13069713C2semidirect
9. #13068970C2semidirect
10. C4×#66880C2semidirect
11. C11×#1939C2semidirect
100. #66880D4semidirect
180. #1939D11semidirect
290. C4×C11#464semidirect
435. C11#25369semidirect
516. C4#10622458semidirect

Maximal subgroups

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

subgroupindex
C2×#106223062
C2×#106224582
C2×#106224622
#130736182
#130716722
#130699342
#130697132
#130695422
#130689702
C4×#668802
C11×#19392
#2536911