Finite Group Databasebeta

← #14960627

Subgroups — #14960627

Order 2816.

2422 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623031C211yesmaximal, characteristic, complemented
2#13076903C212yesmaximal, complemented
3#13075132C218yesmaximal, complemented
4#13074897C218yesmaximal, complemented
5#13074421C212yesmaximal, complemented
6#13073996C212yesmaximal, complemented
7#13072890C212yesmaximal, complemented
8#13072890C212yesmaximal, complemented
9#13072263C212yesmaximal
10#13071219C211yesmaximal, characteristic, complemented
11C22×C11×#123C211yesmaximal, characteristic
12C23×D4×C11C2211yescharacteristic
13C24×C4×C11C2211yescharacteristic
14C2×C11×#123—28no—
15C2×C11×#123C2218yes—
16C22×C11×C4⋊C4C2211yescharacteristic
17C22×C11×C4⋊C4C2211yescharacteristic
18C22×C11×C22⋊C4C2211yescharacteristic
19C22×C11×C22⋊C4C2211yescharacteristic
20C22×C11×C22⋊C4C2211yescharacteristic
21C2×#66989C2214yes—
22C2×#66979C2212yes—
23C2×#66977C2212yes—
24C2×#66977C2212yes—
25C2×#66977C2214yescomplemented
26#10623031—28no—
27C2×#66943C2214yescomplemented
28C2×#66941C2214yescomplemented
29#10622860C2214yes—
30#10622860C2214yes—
31#10622835C2218yes—
32#10622834C2214yes—
33#10622825C2212yes—
34#10622825C2214yescomplemented
35#10622824C2212yes—
36#10622823C2214yes—
37#10622823C2214yes—
38#10622823C2218yescomplemented
39C4×C22⋊Dic11C2212yes—
40#10622596C2214yescomplemented
41#10622594C2218yescomplemented
42#10622592C2214yescomplemented
43#10622585C2218yescomplemented
44#10622566C2214yescomplemented
45#10622566C2214yescomplemented
46#10622562C2214yescomplemented
47#10622562C2214yescomplemented
48#10622562C2214yescomplemented
49#10622561C2218yescomplemented
50#10622560C2214yescomplemented
51#10622557C2218yescomplemented
52#10622557C2218yescomplemented
53C25×C11C2311yescharacteristic
54C25×C11C2311yescharacteristic
55C22×D4×C11—24no—
56C22×D4×C11—44no—
57C22×D4×C11C2314yes—
58C23×C4×C11—24no—
59C23×C4×C11—44no—
60C23×C4×C11C2314yes—
61C23×C4×C11C2311yescharacteristic
62C23×C4×C11C2311yescharacteristic
63C23×C4×C11C2311yescharacteristic
64C23×C4×C11C2311yescharacteristic
65C23×C4×C11C2311yescharacteristic
66C22×Dic22C2312yes—
67C22×Dic22C2312yes—
68C11×#123—232no—
69C2×C11×C4⋊C4—24no—
70C2×C11×C4⋊C4D418yes—
71C2×C11×C4⋊C4C2314yes—
72C2×C11×C4⋊C4C2314yes—
73C2×C11×C22⋊C4—24no—
74C2×C11×C22⋊C4—24no—
75C2×C11×C22⋊C4—24no—
76C2×C11×C22⋊C4C2314yes—
77C2×C11×C22⋊C4C2314yes—
78C2×C11×C22⋊C4C2314yes—
79C2×C22⋊Dic11C2314yes—
80C2×C22⋊Dic11C2318yes—
81C2×C22⋊Dic11C2314yes—
82C2×C22⋊Dic11C2318yes—
83C2×C22⋊Dic11C2318yes—
84C2×C22⋊Dic11C2318yes—
85#66989—48no—
86#66979—28no—
87C2×C4⋊Dic11C2312yes—
88C2×C4⋊Dic11C2312yes—
89C2×C4⋊Dic11C2312yes—
90C2×C4⋊Dic11C2312yes—
91C2×C4⋊Dic11C2314yes—
92#66977—28no—
93#66977—28no—
94#66977—48no—
95C2×Dic11⋊C4C2314yes—
96C2×Dic11⋊C4C2314yes—
97C2×Dic11⋊C4C2318yes—
98C2×Dic11⋊C4C2318yes—
99C2×C4×Dic11C2314yes—
100C2×C4×Dic11C2314yes—
101#66943—216no—
102#66941—216no—
103C11⋊#95C2314yes—
104C11⋊#95C2314yes—
105C11⋊#95C2318yes—
106C11⋊#95C2318yes—
107C11⋊#95C2314yes—
108C11⋊#95C2314yes—
109#36711—111nomaximal, Sylow
110C24×C11—22no—
111C24×C11—24no—
112C24×C11—22no—
113C24×C11—24no—
114C24×C11—24no—
115C24×C11—42no—
116C24×C11—42no—
117C24×C11—42no—
118C24×C11C2412yes—
119C24×C11C2412yes—
120C24×C11C2411yescharacteristic
121C2×D4×C11—28no—
122C2×D4×C11—28no—
123C2×D4×C11—44no—
124C2×D4×C11—416no—
125C22×C4×C11—24no—
126C22×C4×C11—22no—
127C22×C4×C11—22no—
128C22×C4×C11—28no—
129C22×C4×C11—28no—
130C22×C4×C11—42no—
131C22×C4×C11—44no—
132C22×C4×C11—42no—
133C22×C4×C11—42no—
134C22×C4×C11—42no—
135C22×C4×C11—416no—
136C22×C4×C11C2×D414yes—
137C22×C4×C11C2×D414yes—
138C22×C4×C11C2×D414yes—
139C22×C4×C11C2412yes—
140C22×C4×C11C2412yes—
141C22×C4×C11C2412yes—
142C22×C4×C11C2412yes—
143C22×C4×C11C2412yes—
144C22×Dic11C2418yes—
145C22×Dic11C2414yes—
146C22×Dic11C2414yes—
147C2×Dic22—24no—
148C2×Dic22—24no—
149C2×Dic22—44no—
150C2×Dic22—44no—
151C11×C4⋊C4—28no—
152C11×C4⋊C4C2×D4116yes—
153C11×C22⋊C4—28no—
154C11×C22⋊C4—28no—
155C11×C22⋊C4—28no—
156C22⋊Dic11—28no—
157C22⋊Dic11—216no—
158C22⋊Dic11—28no—
159C22⋊Dic11—48no—
160C22⋊Dic11—48no—
161C22⋊Dic11—48no—
162C4⋊Dic11—24no—
163C4⋊Dic11—24no—
164C4⋊Dic11—24no—
165C4⋊Dic11—24no—
166C4⋊Dic11—44no—
167Dic11⋊C4—28no—
168Dic11⋊C4—28no—
169Dic11⋊C4—48no—
170Dic11⋊C4—48no—
171C4×Dic11—28no—
172C4×Dic11—44no—
173C2×#544—111no—
174C22×#123D1111yescharacteristic
175#2848—112no—
176#2694—118no—
177#2674—118no—
178#2626—112no—
179#2583—112no—
180#2482—112no—
181#2482—112no—
182#2435—112no—
183#2313—111no—
184C23×C11—22no—
185C23×C11—22no—
186C23×C11—28no—
187C23×C11—22no—
188C23×C11—22no—
189C23×C11—24no—
190C23×C11—24no—
191C23×C11—22no—
192C23×C11—24no—
193C23×C11—24no—
194C23×C11—24no—
195C23×C11—416no—
196C23×C11—44no—
197C23×C11—44no—
198C23×C11—44no—
199C23×C11—44no—
200C23×C11—44no—
201C23×C11—88no—
202C23×C11C22×D412yes—
203C23×C11H(5,𝔽2)14yes—
204C23×C112−1+414yes—
205C23×C11C2511yescharacteristic
206D4×C11—416no—
207C2×C4×C11—28no—
208C2×C4×C11—24no—
209C2×C4×C11—24no—
210C2×C4×C11—24no—
211C2×C4×C11—22no—
212C2×C4×C11—22no—
213C2×C4×C11—24no—
214C2×C4×C11—24no—
215C2×C4×C11—44no—
216C2×C4×C11—44no—
217C2×C4×C11—416no—
218C2×C4×C11—44no—
219C2×C4×C11—44no—
220C2×C4×C11C22×D414yes—
221C2×C4×C11C22×D414yes—
222C2×C4×C11C22×D414yes—
223C2×Dic11—24no—
224C2×Dic11—24no—
225C2×Dic11—28no—
226C2×Dic11—28no—
227C2×Dic11—28no—
228C2×Dic11—28no—
229C2×Dic11—28no—
230Dic22—84no—
231Dic22—84no—
232C23×D4C2×D1111yescharacteristic
233C24×C4C2×D1111yescharacteristic
234#544—228no—
235C2×C4⋊Q8—114no—
236C2×#125—114no—
237C2×#124—114no—
238C2×#123—28no—
239C2×#123C2×D1118yes—
240C2×C22⋊Q8—112no—
241C2×C22⋊Q8—112no—
242C2×C22⋊Q8—114no—
243C2×#117—112no—
244C22×C4⋊C4C2×D1111yescharacteristic
245C22×C4⋊C4C2×D1111yescharacteristic
246C22×C22⋊C4C2×D1111yescharacteristic
247C22×C22⋊C4C2×D1111yescharacteristic
248C22×C22⋊C4C2×D1111yescharacteristic
249#399—114no—
250#399—114no—
251#399—114no—
252#398—114no—
253#397—114no—
254#397—114no—
255#396—118no—
256#395—114no—
257#394—118no—
258#393—114no—
259#390—114no—
260#389—118no—
261#388—118no—
262#388—118no—
263#387—118no—
264#385—114no—
265#385—114no—
266#385—118no—
267#381—112no—
268#380—112no—
269#380—114no—
270#379—114no—
271#379—114no—
272C4×C22⋊C4—112no—
273C22×C11—22no—
274C22×C11—22no—
275C22×C11—28no—
276C22×C11—24no—
277C22×C11—28no—
278C22×C11—416no—
279C22×C11—42no—
280C22×C11—42no—
281C22×C11—44no—
282C22×C11—42no—
283C22×C11—42no—
284C22×C11—42no—
285C22×C11—48no—
286C22×C11—88no—
287C22×C11#53614yes—
288C22×C11C23×D411yescharacteristic
289C22×C11C2×H(5,𝔽2)12yes—
290C22×C11C2×2−1+412yes—
291C22×C11#58512yes—
292C4×C11—24no—
293C4×C11—24no—
294C4×C11—24no—
295C4×C11—42no—
296C4×C11—42no—
297Dic11—44no—
298Dic11—44no—
299Dic11—48no—
300C25C22×D1111yescharacteristic
301C25C22×D1111yescharacteristic
302C22×Q8—112no—
303C22×Q8—112no—
304C22×D4—24no—
305C22×D4—44no—
306C22×D4C22×D1114yes—
307C23×C4—24no—
308C23×C4—44no—
309C23×C4C22×D1114yes—
310C23×C4C22×D1111yescharacteristic
311C23×C4C22×D1111yescharacteristic
312C23×C4C22×D1111yescharacteristic
313C23×C4C22×D1111yescharacteristic
314C23×C4C22×D1111yescharacteristic
315C4⋊Q8—2216no—
316#125—2216no—
317#124—448no—
318#123—232no—
319C22⋊Q8—228no—
320C22⋊Q8—228no—
321C22⋊Q8—448no—
322#117—228no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—112no—
325C2×C4⋊C4—112no—
326C2×C4⋊C4—112no—
327C2×C4⋊C4—112no—
328C2×C4⋊C4—114no—
329C2×C4⋊C4—114no—
330C2×C4⋊C4—114no—
331C2×C4⋊C4—118no—
332C2×C4⋊C4—118no—
333C2×C4⋊C4C22⋊D1118yes—
334C2×C4⋊C4C22×D1114yes—
335C2×C4⋊C4C22×D1114yes—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—114no—
340C2×C22⋊C4—118no—
341C2×C22⋊C4—114no—
342C2×C22⋊C4—118no—
343C2×C22⋊C4—118no—
344C2×C22⋊C4—118no—
345C2×C22⋊C4C22×D1114yes—
346C2×C22⋊C4C22×D1114yes—
347C2×C22⋊C4C22×D1114yes—
348C2×C42—114no—
349C2×C42—114no—
350#95—114no—
351#95—114no—
352#95—118no—
353#95—114no—
354#95—114no—
355#95—118no—
356C2×C11—24no—
357C2×C11—44no—
358C2×C11—82no—
359C2×C11C2×#53611yescharacteristic
360C2×C11#348411yescharacteristic
361C2×C11#348711yescharacteristic
362C2×C11#353514yes—
363C24—24no—
364C24—22no—
365C24—24no—
366C24—22no—
367C24—24no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24C23×D1112yes—
372C24C23×D1112yes—
373C24C23×D1111yescharacteristic
374C2×Q8—224no—
375C2×Q8—224no—
376C2×Q8—444no—
377C2×Q8—444no—
378C2×D4—28no—
379C2×D4—28no—
380C2×D4—44no—
381C2×D4—416no—
382C22×C4—24no—
383C22×C4—28no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—28no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—416no—
393C22×C4—118no—
394C22×C4—114no—
395C22×C4—114no—
396C22×C4C2×C22⋊D1114yes—
397C22×C4C2×C22⋊D1114yes—
398C22×C4C2×C22⋊D1114yes—
399C22×C4C23×D1112yes—
400C22×C4C23×D1112yes—
401C22×C4C23×D1112yes—
402C22×C4C23×D1112yes—
403C22×C4C23×D1112yes—
404C4⋊C4—28no—
405C4⋊C4—224no—
406C4⋊C4—224no—
407C4⋊C4—224no—
408C4⋊C4—224no—
409C4⋊C4—228no—
410C4⋊C4—228no—
411C4⋊C4—448no—
412C4⋊C4—444no—
413C4⋊C4—448no—
414C4⋊C4C2×C22⋊D11116yes—
415C22⋊C4—28no—
416C22⋊C4—28no—
417C22⋊C4—28no—
418C22⋊C4—2216no—
419C22⋊C4—228no—
420C22⋊C4—228no—
421C22⋊C4—448no—
422C22⋊C4—448no—
423C22⋊C4—448no—
424C42—228no—
425C42—444no—
426C11#3671111yescharacteristic, complemented, Sylow
427C23—22no—
428C23—22no—
429C23—28no—
430C23—22no—
431C23—24no—
432C23—24no—
433C23—22no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—22no—
438C23—416no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—88no—
445C23#6703714yes—
446C23#6704314yes—
447C23C22×C22⋊D1112yes—
448C23C24×D1111yescharacteristic
449Q8—884no—
450Q8—884no—
451D4—416no—
452C2×C4—28no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—416no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—228no—
466C2×C4—228no—
467C2×C4—228no—
468C2×C4—228no—
469C2×C4—228no—
470C2×C4—224no—
471C2×C4—224no—
472C2×C4C22×C22⋊D1114yes—
473C2×C4C22×C22⋊D1114yes—
474C2×C4C22×C22⋊D1114yes—
475C22—22no—
476C22—22no—
477C22—28no—
478C22—24no—
479C22—28no—
480C22—416no—
481C22—42no—
482C22—42no—
483C22—44no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—48no—
488C22—88no—
489C22#1062335614yes—
490C22C2×#6703712yes—
491C22C2×#6704312yes—
492C22#1062345312yes—
493C22C23×C22⋊D1111yescharacteristic
494C4—24no—
495C4—24no—
496C4—24no—
497C4—42no—
498C4—42no—
499C4—444no—
500C4—444no—
501C4—448no—
502C2—24no—
503C2—44no—
504C2—82no—
505C2C2×#1062335611yescharacteristic
506C2#1308225111yescharacteristic
507C2#1308227411yescharacteristic
508C2#1308282914yes—
subgroupG/H
center, FrattiniC23C24×D11
commutator, socle, cosocleC23×C11C25
FittingC22×C11×#123C2
prime radicalC23×D4×C11C22
hypercenterC22×#123D11
hypocenterC11#36711
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
1. C2×#10623031C2semidirect
2. #13076903C2semidirect
3. #13075132C2semidirect
4. #13074897C2semidirect
5. #13074421C2semidirect
6. #13073996C2semidirect
7. #13072890C2semidirect
8. #13072890C2semidirect
10. #13071219C2semidirect
25. C2×#66977C22semidirect
27. C2×#66943C22semidirect
28. C2×#66941C22semidirect
34. #10622825C22semidirect
38. #10622823C22semidirect
40. #10622596C22semidirect
41. #10622594C22semidirect
42. #10622592C22semidirect
43. #10622585C22semidirect
44. #10622566C22semidirect
45. #10622566C22semidirect
46. #10622562C22semidirect
47. #10622562C22semidirect
48. #10622562C22semidirect
49. #10622561C22semidirect
50. #10622560C22semidirect
51. #10622557C22semidirect
52. #10622557C22semidirect
426. C11#36711semidirect

Maximal subgroups

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

subgroupindex
C2×#106230312
#130769032
#130751322
#130748972
#130744212
#130739962
#130728902
#130722632
#130712192
C22×C11×#1232
#3671111