Finite Group Databasebeta

← #14676471

Subgroups — #14676471

Order 2816.

1862 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622549C211yesmaximal, characteristic, complemented
2C2×#10622545C211yesmaximal, characteristic, complemented
3#13066869C212yesmaximal, complemented
4#13066861C212yesmaximal
5#13066853C212yesmaximal
6#13066848C212yesmaximal, complemented
7#13066843C212yesmaximal, complemented
8#13066839C212yesmaximal
9C22×C11×#117C211yesmaximal, characteristic
10C24×C4×C11C2211yescharacteristic
11C2×C11×#117—24no—
12C2×C11×#117—24no—
13C2×C11×#117C418yescomplemented
14C22×C11×C4⋊C4C2211yescharacteristic
15C22×C11×C4⋊C4C2211yescharacteristic
16C22×C11×C22⋊C4C2211yescharacteristic
17C22×C11×C22⋊C4C2211yescharacteristic
18C22×C42×C11C2211yescharacteristic
19C22×C42×C11C2211yescharacteristic
20C22×C22⋊Dic11C2214yes—
21C22×C4⋊Dic11C2212yes—
22C22×C4×Dic11C2212yes—
23C2×C11⋊#95C2212yes—
24C2×C11⋊#95C2212yes—
25C2×C11⋊#95C2214yes—
26C2×C11⋊#95C2214yes—
27C2×C11⋊#95C2212yes—
28C2×C11⋊#95C2212yes—
29C2×C11⋊#95C2214yes—
30#10622549—24no—
31#10622545—24no—
32C25×C11C2311yescharacteristic
33C23×C4×C11—24no—
34C23×C4×C11—22no—
35C23×C4×C11—22no—
36C23×C4×C11—24no—
37C23×C4×C11C2×C414yes—
38C23×C4×C11C2311yescharacteristic
39C23×C4×C11C2311yescharacteristic
40C23×C4×C11C2311yescharacteristic
41C23×C4×C11C2311yescharacteristic
42C23×C4×C11C2311yescharacteristic
43C23×C4×C11C2311yescharacteristic
44C23×Dic11C2314yes—
45C23×Dic11C2314yes—
46C11×#117—232no—
47C2×C11×C4⋊C4—22no—
48C2×C11×C4⋊C4—22no—
49C2×C11×C4⋊C4—22no—
50C2×C11×C4⋊C4—22no—
51C2×C11×C4⋊C4C2×C414yes—
52C2×C11×C4⋊C4C2×C414yes—
53C2×C11×C22⋊C4—22no—
54C2×C11×C22⋊C4—22no—
55C2×C11×C22⋊C4—22no—
56C2×C11×C22⋊C4—22no—
57C2×C11×C22⋊C4C2×C414yes—
58C2×C11×C22⋊C4C2×C414yes—
59C2×C42×C11—22no—
60C2×C42×C11—22no—
61C2×C42×C11C2×C414yes—
62C2×C42×C11C2×C414yes—
63C2×C42×C11D414yes—
64C2×C42×C11D414yes—
65C2×C22⋊Dic11—28no—
66C2×C22⋊Dic11—28no—
67C2×C22⋊Dic11—44no—
68C2×C4⋊Dic11—24no—
69C2×C4⋊Dic11—44no—
70C2×C4×Dic11—24no—
71C2×C4×Dic11—28no—
72C11⋊#95—24no—
73C11⋊#95—24no—
74C11⋊#95—24no—
75C11⋊#95—24no—
76C11⋊#95—28no—
77C11⋊#95—28no—
78C11⋊#95—44no—
79#13957—111nomaximal, Sylow
80C24×C11—21no—
81C24×C11—21no—
82C24×C11—24no—
83C24×C11—22no—
84C24×C11—22no—
85C24×C11—22no—
86C24×C11C22×C412yes—
87C24×C11#4114yes—
88C24×C11C2411yescharacteristic
89C22×C4×C11—216no—
90C22×C4×C11—216no—
91C22×C4×C11—28no—
92C22×C4×C11—21no—
93C22×C4×C11—21no—
94C22×C4×C11—24no—
95C22×C4×C11—22no—
96C22×C4×C11—22no—
97C22×C4×C11—22no—
98C22×C4×C11—22no—
99C22×C4×C11—22no—
100C22×C4×C11—28no—
101C22×C4×C11—28no—
102C22×C4×C11—21no—
103C22×C4×C11—21no—
104C22×C4×C11—21no—
105C22×C4×C11—21no—
106C22×C4×C11—24no—
107C22×C4×C11—22no—
108C22×C4×C11—22no—
109C22×C4×C11C22×C412yes—
110C22×C4×C11C22×C412yes—
111C22×C4×C11C22×C412yes—
112C22×C4×C11C22×C412yes—
113C22×C4×C11C22×C412yes—
114C22×C4×C11C22×C412yes—
115C22×C4×C11C2×D412yes—
116C22×C4×C11C2×D412yes—
117C22×C4×C11C2×D412yes—
118C22×C4×C11C2×D412yes—
119C22×C4×C11C2×D412yes—
120C22×C4×C11C2×D412yes—
121C22×C4×C11#4114yes—
122C22×Dic11—24no—
123C22×Dic11—24no—
124C22×Dic11—24no—
125C22×Dic11—24no—
126C22×Dic11—24no—
127C22×Dic11—24no—
128C22×Dic11—24no—
129C22×Dic11—24no—
130C22×Dic11—24no—
131C22×Dic11—24no—
132C22×Dic11—24no—
133C22×Dic11—24no—
134C22×Dic11—24no—
135C22×Dic11#4118yes—
136C11×C4⋊C4—28no—
137C11×C4⋊C4—28no—
138C11×C22⋊C4—28no—
139C11×C22⋊C4—28no—
140C42×C11—28no—
141C42×C11C22⋊C4116yescomplemented
142C22⋊Dic11—416no—
143C4⋊Dic11—48no—
144C4×Dic11—216no—
145C22×#117D1111yescharacteristic
146C2×#382—111no—
147C2×#376—111no—
148#1441—112no—
149#1439—112no—
150#1437—112no—
151#1436—112no—
152#1434—112no—
153C22⋊#95—112no—
154C23×C11—22no—
155C23×C11—22no—
156C23×C11—24no—
157C23×C11—24no—
158C23×C11—24no—
159C23×C11—24no—
160C23×C11—28no—
161C23×C11—28no—
162C23×C11—42no—
163C23×C11—42no—
164C23×C11—41no—
165C23×C11—41no—
166C23×C11—42no—
167C23×C11—44no—
168C23×C11—44no—
169C23×C11#11714yes—
170C23×C11C23×C411yescharacteristic
171C23×C11C22×D411yescharacteristic
172C23×C11C22×D411yescharacteristic
173C23×C11C2×#4112yes—
174C23×C11C2×#4112yes—
175C23×C11C2×#4114yes—
176C23×C11H(5,𝔽2)12yes—
177C23×C112−1+412yes—
178C2×C4×C11—232no—
179C2×C4×C11—22no—
180C2×C4×C11—22no—
181C2×C4×C11—22no—
182C2×C4×C11—22no—
183C2×C4×C11—24no—
184C2×C4×C11—24no—
185C2×C4×C11—24no—
186C2×C4×C11—24no—
187C2×C4×C11—24no—
188C2×C4×C11—24no—
189C2×C4×C11—24no—
190C2×C4×C11—24no—
191C2×C4×C11—22no—
192C2×C4×C11—22no—
193C2×C4×C11—24no—
194C2×C4×C11—22no—
195C2×C4×C11—22no—
196C2×C4×C11—24no—
197C2×C4×C11—24no—
198C2×C4×C11—42no—
199C2×C4×C11—42no—
200C2×C4×C11—41no—
201C2×C4×C11—41no—
202C2×C4×C11—42no—
203C2×C4×C11C2×C22⋊C414yes—
204C2×C4×C11C2×C22⋊C414yes—
205C2×C4×C11C2×C22⋊C414yes—
206C2×C4×C11#11714yes—
207C2×Dic11—28no—
208C2×Dic11—28no—
209C2×Dic11—28no—
210C2×Dic11—44no—
211C2×Dic11—44no—
212C2×Dic11—44no—
213C2×Dic11—44no—
214C2×Dic11—44no—
215C2×Dic11—44no—
216C2×Dic11—44no—
217C2×Dic11—44no—
218C2×Dic11—44no—
219C2×Dic11—44no—
220C2×Dic11—44no—
221C24×C4C2×D1111yescharacteristic
222C2×#117—24no—
223C2×#117—24no—
224C2×#117Dic1118yescomplemented
225C22×C4⋊C4—112no—
226C22×C4⋊C4C2×D1111yescharacteristic
227C22×C4⋊C4C2×D1111yescharacteristic
228C22×C22⋊C4—114no—
229C22×C22⋊C4C2×D1111yescharacteristic
230C22×C22⋊C4C2×D1111yescharacteristic
231C22×C42—112no—
232C22×C42C2×D1111yescharacteristic
233C22×C42C2×D1111yescharacteristic
234#382—224no—
235#376—224no—
236C2×#95—112no—
237C2×#95—112no—
238C2×#95—114no—
239C2×#95—114no—
240C2×#95—112no—
241C2×#95—112no—
242C2×#95—114no—
243C22×C11—28no—
244C22×C11—24no—
245C22×C11—42no—
246C22×C11—44no—
247C22×C11—48no—
248C22×C11—44no—
249C22×C11—44no—
250C22×C11—42no—
251C22×C11C22×C22⋊C411yescharacteristic
252C22×C11C2×#11711yescharacteristic
253C22×C11C2×#11711yescharacteristic
254C22×C11C4×#4112yes—
255C22×C11#52012yes—
256C22×C11#52512yes—
257C22×C11#52512yes—
258C22×C11#52512yes—
259C22×C11#53212yes—
260C22×C11#53511yescharacteristic
261C22×C11#53611yescharacteristic
262C22×C11#53712yes—
263C22×C11#55114yes—
264C22×C11#55214yes—
265C22×C11#55214yes—
266C22×C11#55514yes—
267C4×C11—24no—
268C4×C11—24no—
269C4×C11—24no—
270C4×C11—24no—
271C4×C11—42no—
272C4×C11—42no—
273Dic11—48no—
274Dic11—84no—
275C25C22×D1111yescharacteristic
276C23×C4—24no—
277C23×C4—24no—
278C23×C4—22no—
279C23×C4—22no—
280C23×C4—114no—
281C23×C4—114no—
282C23×C4C2×Dic1114yes—
283C23×C4C22×D1111yescharacteristic
284C23×C4C22×D1111yescharacteristic
285C23×C4C22×D1111yescharacteristic
286C23×C4C22×D1111yescharacteristic
287C23×C4C22×D1111yescharacteristic
288C23×C4C22×D1111yescharacteristic
289#117—232no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—224no—
295C2×C4⋊C4—444no—
296C2×C4⋊C4C2×Dic1114yes—
297C2×C4⋊C4C2×Dic1114yes—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—228no—
303C2×C22⋊C4—228no—
304C2×C22⋊C4—444no—
305C2×C22⋊C4C2×Dic1114yes—
306C2×C22⋊C4C2×Dic1114yes—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—228no—
310C2×C42—224no—
311C2×C42C2×Dic1114yes—
312C2×C42C2×Dic1114yes—
313C2×C42C22⋊D1114yes—
314C2×C42C22⋊D1114yes—
315#95—228no—
316#95—228no—
317#95—224no—
318#95—224no—
319#95—224no—
320#95—224no—
321#95—444no—
322C2×C11—44no—
323C2×C11#228011yescharacteristic
324C2×C11#229711yescharacteristic
325C2×C11#231111yescharacteristic
326C2×C11#234312yes—
327C2×C11#234612yes—
328C2×C11#260914yes—
329C2×C11#261012yes—
330C2×C11#261112yes—
331C24—22no—
332C24—24no—
333C24—22no—
334C24—22no—
335C24—21no—
336C24—21no—
337C24#433914yes—
338C24C22×Dic1112yes—
339C24C23×D1111yescharacteristic
340C22×C4—216no—
341C22×C4—216no—
342C22×C4—28no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—118no—
361C22×C4—224no—
362C22×C4—224no—
363C22×C4—224no—
364C22×C4—224no—
365C22×C4—224no—
366C22×C4—224no—
367C22×C4—224no—
368C22×C4—224no—
369C22×C4—224no—
370C22×C4—224no—
371C22×C4—224no—
372C22×C4—224no—
373C22×C4—224no—
374C22×C4#433914yes—
375C22×C4C22×Dic1112yes—
376C22×C4C22×Dic1112yes—
377C22×C4C22×Dic1112yes—
378C22×C4C22×Dic1112yes—
379C22×C4C22×Dic1112yes—
380C22×C4C22×Dic1112yes—
381C22×C4C2×C22⋊D1112yes—
382C22×C4C2×C22⋊D1112yes—
383C22×C4C2×C22⋊D1112yes—
384C22×C4C2×C22⋊D1112yes—
385C22×C4C2×C22⋊D1112yes—
386C22×C4C2×C22⋊D1112yes—
387C4⋊C4—28no—
388C4⋊C4—28no—
389C4⋊C4—448no—
390C22⋊C4—28no—
391C22⋊C4—28no—
392C22⋊C4—4416no—
393C42—28no—
394C42—2216no—
395C42C22⋊Dic11116yescomplemented
396C11#1395711yescharacteristic, complemented, Sylow
397C23—28no—
398C23—22no—
399C23—28no—
400C23—24no—
401C23—24no—
402C23—24no—
403C23—24no—
404C23—22no—
405C23—44no—
406C23—44no—
407C23—41no—
408C23—42no—
409C23—41no—
410C23—42no—
411C23—42no—
412C23#6697914yes—
413C23C2×#433912yes—
414C23C2×#433912yes—
415C23#41×D1114yes—
416C23#6704212yes—
417C23#6704312yes—
418C23C23×Dic1111yescharacteristic
419C23C22×C22⋊D1111yescharacteristic
420C23C22×C22⋊D1111yescharacteristic
421C2×C4—232no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—228no—
447C2×C4—228no—
448C2×C4—228no—
449C2×C4—444no—
450C2×C4—444no—
451C2×C4—444no—
452C2×C4—444no—
453C2×C4—444no—
454C2×C4—444no—
455C2×C4—444no—
456C2×C4—444no—
457C2×C4—444no—
458C2×C4—444no—
459C2×C4—444no—
460C2×C4#6697914yes—
461C2×C4C2×C22⋊Dic1114yes—
462C2×C4C2×C22⋊Dic1114yes—
463C2×C4C2×C22⋊Dic1114yes—
464C22—28no—
465C22—24no—
466C22—42no—
467C22—42no—
468C22—48no—
469C22—44no—
470C22—44no—
471C22—44no—
472C22#1062305814yes—
473C22#1062306014yes—
474C22#1062306214yes—
475C22#1062306314yes—
476C22C2×#6697911yescharacteristic
477C22C2×#6697911yescharacteristic
478C22#1062332212yes—
479C22#1062332212yes—
480C22#1062335611yescharacteristic
481C22#41×Dic1112yes—
482C22#1062335812yes—
483C22#1062335912yes—
484C22#1062336111yescharacteristic
485C22#1062336312yes—
486C22#1062336412yes—
487C22C22×C22⋊Dic1111yescharacteristic
488C4—24no—
489C4—24no—
490C4—24no—
491C4—24no—
492C4—42no—
493C4—42no—
494C4—448no—
495C4—884no—
496C2—44no—
497C2#1307099911yescharacteristic
498C2#1307110711yescharacteristic
499C2#1307120811yescharacteristic
500C2#1307147612yes—
501C2#1307150112yes—
502C2#1307428814yes—
503C2#1307430312yes—
504C2#1307431412yes—
subgroupG/H
center, FrattiniC24C23×D11
commutatorC23×C11C23×C4
FittingC22×C11×#117C2
socle, cosocleC24×C11C24
prime radicalC25×C11C23
hypercenterC22×#117D11
hypocenterC11#13957
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
1. C2×#10622549C2semidirect
2. C2×#10622545C2semidirect
3. #13066869C2semidirect
6. #13066848C2semidirect
7. #13066843C2semidirect
13. C2×C11×#117C4semidirect
141. C42×C11C22⋊C4semidirect
224. C2×#117Dic11semidirect
395. C42C22⋊Dic11semidirect
396. C11#13957semidirect

Maximal subgroups

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

subgroupindex
C2×#106225492
C2×#106225452
#130668692
#130668612
#130668532
#130668482
#130668432
#130668392
C22×C11×#1172
#1395711