Finite Group Databasebeta

← #14913924

Subgroups — #14913924

Order 2816.

2342 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623053C211yesmaximal, characteristic
2#13071585C214yesmaximal, complemented
3#13071577C214yesmaximal, complemented
4#13071550C218yesmaximal, complemented
5#13071221C214yesmaximal, complemented
6#13071209C211yesmaximal, characteristic
7#13071204C211yesmaximal, characteristic
8#13071019C211yesmaximal, characteristic, complemented
9#13071012C214yesmaximal, complemented
10C2×#10622590C211yesmaximal, characteristic, complemented
11#13070972C211yesmaximal, characteristic, complemented
12C11×#2274C211yesmaximal, characteristic, complemented
13C22×#4339C2211yescharacteristic
14C2×C11×#117C2211yescharacteristic
15C2×C11×#117C2211yescharacteristic
16C22×C11×C4⋊C4C2211yescharacteristic
17C2×#66979C2211yescharacteristic
18C22×C4⋊Dic11C2211yescharacteristic
19C22×C4×Dic11C2211yescharacteristic
20#10623053C4116yescomplemented
21C2×#66946C2212yes—
22C2×#66940C2212yes—
23C2×(C22⋊D11)⋊C4C2214yes—
24C2×#66925C2212yes—
25C2×C42×D11C2212yes—
26C4×C11×C4⋊C4C2214yes—
27C4×C11×C4⋊C4C2214yes—
28C4×C11×C22⋊C4C2214yes—
29#10622826C2214yes—
30C2×C11⋊#95C2214yes—
31#10622601C2212yes—
32#10622600C2214yes—
33#10622600C2214yes—
34#10622590C2212yes—
35#10622590C2212yes—
36#10622590C2214yes—
37#10622590C2212yes—
38#10622589C2212yes—
39C4⋊C4×Dic11C2214yes—
40#10622587C2212yes—
41#10622571C2214yes—
42#10622555C2214yes—
43C22⋊C4×Dic11C2214yes—
44#10622553C2218yes—
45#10622552C2212yes—
46C4×(C2×D11)⋊C4C2218yes—
47C4×C4⋊Dic11C2214yes—
48#10622546C2218yes—
49#10622545C2218yes—
50C4×Dic11⋊C4C2214yes—
51#10622542C2212yes—
52#10622282C2218yes—
53#10622277C2218yes—
54#10622265C2218yes—
55#10622264C2218yes—
56C23×C4×C11C2312yes—
57C23×C4×C11C2311yescharacteristic
58C22×C22⋊D11C2312yes—
59C23×Dic11C2312yes—
60C23×Dic11C2312yes—
61C2×#4339—24no—
62C2×#4339—24no—
63C2×#4339C2×C418yes—
64C22×D44C2311yescharacteristic
65C22×C4×D11C2312yes—
66C22×C4×D11C2314yes—
67C22×Dic22C2311yescharacteristic
68C11×#117—24no—
69C11×#117C2×C418yes—
70C2×C11×C4⋊C4—24no—
71C2×C11×C4⋊C4C2312yes—
72C2×C11×C4⋊C4C2312yes—
73C2×C11×C4⋊C4C2312yes—
74C2×C11×C4⋊C4C2312yes—
75C2×C11×C22⋊C4C2312yes—
76C2×C11×C22⋊C4C2312yes—
77C2×C42×C11C2312yes—
78C2×C42×C11C2318yes—
79C2×C42×C11C2312yes—
80C2×C22⋊Dic11C2312yes—
81C2×(C2×D11)⋊C4C2318yes—
82C2×(C2×D11)⋊C4C2318yes—
83C2×(C2×D11)⋊C4C2314yes—
84#66979—24no—
85C2×C4⋊Dic11—24no—
86C2×C4⋊Dic11C2312yes—
87C2×C4⋊Dic11C2312yes—
88C2×C4⋊Dic11C2312yes—
89C2×Dic11⋊C4C2314yes—
90C2×Dic11⋊C4C2314yes—
91C2×Dic11⋊C4C2314yes—
92C2×C4×Dic11C2×C418yes—
93C2×C4×Dic11C2312yes—
94C2×C4×Dic11C2318yes—
95C2×C4×Dic11C2311yescharacteristic
96C2×C4×Dic11C2314yes—
97C2×C4×Dic11C2311yescharacteristic
98C2×C4×Dic11C2312yes—
99#66946C2×C4116yescomplemented
100#66940C2×C4116yescomplemented
101(C22⋊D11)⋊C4C2×C4132yes—
102#66925C2×C4116yescomplemented
103C42×D11C2×C4116yescomplemented
104C11×#95C2318yes—
105C11⋊#95C2314yes—
106C11⋊#95C2318yes—
107C11⋊#95C2318yes—
108C11⋊#95C2318yes—
109C11⋊#95C2318yes—
110C11⋊#95C2318yes—
111C11⋊#95C2314yes—
112#34189—111nomaximal, Sylow
113C24×C11C2411yescharacteristic
114C23×D11C2412yes—
115C22×C4×C11—24no—
116C22×C4×C11—44no—
117C22×C4×C11C22×C414yes—
118C22×C4×C11#4114yes—
119C22×C4×C11#4114yes—
120C22×C4×C11C2414yes—
121C22×C4×C11C2414yes—
122C22×C4×C11C2414yes—
123C22×C4×C11C2411yescharacteristic
124C22×C4×C11C2411yescharacteristic
125C2×C22⋊D11—24no—
126C2×C22⋊D11—24no—
127C2×C22⋊D11C22×C418yes—
128C22×Dic11—24no—
129C22×Dic11—42no—
130C22×Dic11—42no—
131C22×Dic11—44no—
132C22×Dic11C22×C418yes—
133C22×Dic11C2414yes—
134C22×Dic11C2414yes—
135C22×Dic11C2412yes—
136C22×Dic11C2412yes—
137C22×Dic11C2412yes—
138#4339—232no—
139C2×D44—22no—
140C2×D44—22no—
141C2×D44C22×C414yes—
142C2×C4×D11—24no—
143C2×C4×D11—24no—
144C2×C4×D11—44no—
145C2×C4×D11—44no—
146C2×C4×D11C22×C418yes—
147C2×C4×D11C22×C4116yes—
148C2×Dic22—22no—
149C2×Dic22—22no—
150C2×Dic22C22×C414yes—
151C11×C4⋊C4—24no—
152C11×C4⋊C4—24no—
153C11×C4⋊C4—24no—
154C11×C4⋊C4C22×C418yes—
155C11×C22⋊C4—24no—
156C11×C22⋊C4C22×C418yes—
157C42×C11—24no—
158C42×C11—48no—
159C42×C11C22×C418yes—
160C22⋊Dic11—24no—
161(C2×D11)⋊C4—48no—
162(C2×D11)⋊C4—48no—
163(C2×D11)⋊C4C22×C4116yes—
164C4⋊Dic11—24no—
165C4⋊Dic11—24no—
166C4⋊Dic11—24no—
167Dic11⋊C4—44no—
168Dic11⋊C4—44no—
169Dic11⋊C4C22×C4116yes—
170C4×Dic11—24no—
171C4×Dic11—48no—
172C4×Dic11C22×C414yes—
173C4×Dic11C22×C4116yes—
174C4×Dic11C22×C414yes—
175C4×Dic11C22×C418yes—
176C2×C4×#41—111no—
177#2357—114no—
178#2356—114no—
179#2353—118no—
180#2314—114no—
181#2311—111no—
182#2311—111no—
183#2285—111no—
184#2283—114no—
185C2×#382—111no—
186#2274—111no—
187#2274D1111yescharacteristic, complemented
188C23×C11—22no—
189C23×C11—22no—
190C23×C11C23×C412yes—
191C23×C11C2×#4112yes—
192C23×C11C2×#4112yes—
193C23×C11C2511yescharacteristic
194C22×D11—22no—
195C22×D11—22no—
196C22×D11—24no—
197C22×D11—24no—
198C22×D11C23×C414yes—
199C2×C4×C11—24no—
200C2×C4×C11—24no—
201C2×C4×C11—24no—
202C2×C4×C11—24no—
203C2×C4×C11—24no—
204C2×C4×C11—24no—
205C2×C4×C11—28no—
206C2×C4×C11—84no—
207C2×C4×C11#117116yes—
208C2×C4×C11C23×C418yes—
209C2×C4×C11C23×C412yes—
210C2×C4×C11C23×C412yes—
211C2×C4×C11C2×#4112yes—
212C2×C4×C11C2×#4112yes—
213C2×C4×C11C2×#4112yes—
214C2×C4×C11C2×#4112yes—
215C22⋊D11—216no—
216C2×Dic11—22no—
217C2×Dic11—22no—
218C2×Dic11—22no—
219C2×Dic11—22no—
220C2×Dic11—22no—
221C2×Dic11—22no—
222C2×Dic11—24no—
223C2×Dic11—24no—
224C2×Dic11—24no—
225C2×Dic11—28no—
226C2×Dic11—24no—
227C2×Dic11—48no—
228C2×Dic11—84no—
229C2×Dic11C23×C414yes—
230C2×Dic11C23×C414yes—
231C2×Dic11C23×C414yes—
232D44—28no—
233C4×D11—216no—
234C4×D11—416no—
235Dic22—28no—
236C22×#41—111no—
237C4×#41—1116no—
238C2×C4×Q8—112no—
239C2×C4×D4—112no—
240C2×C4×D4—114no—
241C2×#117—111no—
242C2×#117—112no—
243C2×#117C2×D1111yescharacteristic
244C2×#117C2×D1111yescharacteristic
245C22×C4⋊C4—111no—
246C22×C4⋊C4C2×D1111yescharacteristic
247C22×C42—111no—
248C22×C42—112no—
249#391—112no—
250#391—112no—
251#390—112no—
252#390—112no—
253#388—118no—
254#388—118no—
255#387—118no—
256#387—118no—
257#386—114no—
258#386—114no—
259#384—112no—
260#383—118no—
261#382—112no—
262#382—114no—
263#382—112no—
264#382—112no—
265#381—118no—
266#380—114no—
267#380—114no—
268#380—114no—
269C4×C4⋊C4—114no—
270C4×C4⋊C4—114no—
271C4×C4⋊C4—114no—
272C4×C4⋊C4C2×D1114yes—
273C4×C4⋊C4C2×D1114yes—
274C4×C22⋊C4—114no—
275C4×C22⋊C4—118no—
276C4×C22⋊C4C2×D1114yes—
277#376—118no—
278C2×#95—114no—
279C22×C11—24no—
280C22×C11—22no—
281C22×C11—22no—
282C22×C11—24no—
283C22×C11C2×#11714yes—
284C22×C11C24×C411yescharacteristic
285C22×C11C22×#4111yescharacteristic
286C22×C11C22×#4111yescharacteristic
287C22×C11#58512yes—
288C22×C11#58512yes—
289C2×D11—24no—
290C2×D11—28no—
291C2×D11—42no—
292C2×D11—42no—
293C2×D11—44no—
294C4×C11—28no—
295C4×C11—44no—
296C4×C11—44no—
297C4×C11C2×#11714yes—
298C4×C11C2×#11714yes—
299Dic11—24no—
300Dic11—24no—
301Dic11—24no—
302Dic11—44no—
303Dic11—44no—
304C2×#41—118no—
305C2×#41—224no—
306C2×#41—224no—
307C22×Q8—111no—
308C22×D4—112no—
309C22×D4—111no—
310C23×C4—112no—
311C23×C4—112no—
312C23×C4—114no—
313C23×C4—112no—
314C23×C4C22×D1112yes—
315C23×C4C22×D1111yescharacteristic
316C4×Q8—1116no—
317C4×D4—1116no—
318C4×D4—1132no—
319#117—24no—
320#117—1116no—
321#117—224no—
322#117C4×D1118yes—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—114no—
325C2×C4⋊C4—112no—
326C2×C4⋊C4—112no—
327C2×C4⋊C4—112no—
328C2×C4⋊C4—114no—
329C2×C4⋊C4—114no—
330C2×C4⋊C4—224no—
331C2×C4⋊C4C22×D1112yes—
332C2×C4⋊C4C22×D1112yes—
333C2×C4⋊C4C22×D1112yes—
334C2×C4⋊C4C22×D1112yes—
335C2×C22⋊C4—114no—
336C2×C22⋊C4—112no—
337C2×C22⋊C4—118no—
338C2×C22⋊C4—118no—
339C2×C22⋊C4C22×D1112yes—
340C2×C22⋊C4C22×D1112yes—
341C2×C42—118no—
342C2×C42—111no—
343C2×C42—114no—
344C2×C42—112no—
345C2×C42—111no—
346C2×C42—1116no—
347C2×C42—112no—
348C2×C42—118no—
349C2×C42C22×D1112yes—
350C2×C42C22×D1118yes—
351C2×C42C22×D1112yes—
352#95—118no—
353#95—118no—
354#95—114no—
355#95—118no—
356#95—118no—
357#95—114no—
358#95—118no—
359#95C22×D1118yes—
360C2×C11—24no—
361C2×C11C22×#11711yescharacteristic
362C2×C11#342812yes—
363C2×C11#351011yescharacteristic
364C2×C11#351011yescharacteristic
365C2×C11#351212yes—
366D11—44no—
367C24—112no—
368C24C23×D1111yescharacteristic
369#41—2232no—
370C2×Q8—114no—
371C2×Q8—222no—
372C2×Q8—222no—
373C2×D4—118no—
374C2×D4—114no—
375C2×D4—224no—
376C2×D4—224no—
377C2×D4—222no—
378C2×D4—222no—
379C22×C4—24no—
380C22×C4—44no—
381C22×C4—112no—
382C22×C4—112no—
383C22×C4—112no—
384C22×C4—118no—
385C22×C4—118no—
386C22×C4—1116no—
387C22×C4—114no—
388C22×C4—114no—
389C22×C4—224no—
390C22×C4—224no—
391C22×C4—224no—
392C22×C4—444no—
393C22×C4—444no—
394C22×C4—442no—
395C22×C4—442no—
396C22×C4—444no—
397C22×C4C2×C4×D1114yes—
398C22×C4#433914yes—
399C22×C4#433914yes—
400C22×C4C23×D1114yes—
401C22×C4C23×D1114yes—
402C22×C4C23×D1114yes—
403C22×C4C23×D1111yescharacteristic
404C22×C4C23×D1111yescharacteristic
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—1116no—
409C4⋊C4—224no—
410C4⋊C4—224no—
411C4⋊C4—224no—
412C4⋊C4—444no—
413C4⋊C4—444no—
414C4⋊C4C2×C4×D1118yes—
415C22⋊C4—24no—
416C22⋊C4—1116no—
417C22⋊C4—224no—
418C22⋊C4—448no—
419C22⋊C4—448no—
420C22⋊C4C2×C4×D1118yes—
421C42—24no—
422C42—48no—
423C42—114no—
424C42—114no—
425C42—118no—
426C42—1116no—
427C42—224no—
428C42—448no—
429C42C2×C4×D1118yes—
430C11#3418911yescharacteristic, complemented, Sylow
431C23—22no—
432C23—22no—
433C23—114no—
434C23—222no—
435C23—222no—
436C23—224no—
437C23—224no—
438C23C22×C4×D1112yes—
439C23C2×#433912yes—
440C23C2×#433912yes—
441C23C24×D1111yescharacteristic
442Q8—228no—
443D4—2216no—
444D4—228no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—28no—
451C2×C4—24no—
452C2×C4—84no—
453C2×C4—114no—
454C2×C4—114no—
455C2×C4—114no—
456C2×C4—222no—
457C2×C4—222no—
458C2×C4—222no—
459C2×C4—222no—
460C2×C4—222no—
461C2×C4—222no—
462C2×C4—2216no—
463C2×C4—224no—
464C2×C4—224no—
465C2×C4—224no—
466C2×C4—224no—
467C2×C4—228no—
468C2×C4—4416no—
469C2×C4—448no—
470C2×C4—884no—
471C2×C4#66925116yes—
472C2×C4C22×C4×D1118yes—
473C2×C4C22×C4×D1112yes—
474C2×C4C22×C4×D1112yes—
475C2×C4C2×#433912yes—
476C2×C4C2×#433912yes—
477C2×C4C2×#433912yes—
478C2×C4C2×#433912yes—
479C22—22no—
480C22—22no—
481C22—24no—
482C22—24no—
483C22—224no—
484C22—228no—
485C22—442no—
486C22—442no—
487C22—444no—
488C22C2×#6692514yes—
489C22C23×C4×D1111yescharacteristic
490C22C22×#433911yescharacteristic
491C22C22×#433911yescharacteristic
492C22#1062345512yes—
493C22#1062345712yes—
494C4—28no—
495C4—44no—
496C4—44no—
497C4—224no—
498C4—224no—
499C4—224no—
500C4—444no—
501C4—444no—
502C4C2×#6692514yes—
503C4C2×#6692514yes—
504C2—24no—
505C2—444no—
506C2C22×#6692511yescharacteristic
507C2#1308181612yes—
508C2#1308250711yescharacteristic
509C2#1308251311yescharacteristic
510C2#1308253312yes—
subgroupG/H
center, FrattiniC23C24×D11
commutatorC22×C11C24×C4
FittingC11×#2274C2
socle, cosocleC23×C11C25
prime radicalC22×#4339C22
hypercenter#2274D11
hypocenterC11#34189
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C23 ◁ #2274

Complements

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

NH ≅ G/Ntype
2. #13071585C2semidirect
3. #13071577C2semidirect
4. #13071550C2semidirect
5. #13071221C2semidirect
8. #13071019C2semidirect
9. #13071012C2semidirect
10. C2×#10622590C2semidirect
11. #13070972C2semidirect
12. C11×#2274C2semidirect
20. #10623053C4semidirect
99. #66946C2×C4semidirect
100. #66940C2×C4semidirect
102. #66925C2×C4semidirect
103. C42×D11C2×C4semidirect
187. #2274D11semidirect
430. C11#34189semidirect

Maximal subgroups

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

subgroupindex
C2×#106230532
#130715852
#130715772
#130715502
#130712212
#130712092
#130712042
#130710192
#130710122
C2×#106225902
#130709722
C11×#22742
#3418911