Finite Group Databasebeta

← #14684809

Subgroups — #14684809

Order 2816.

1702 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66950C211yesmaximal, characteristic
2C2×#10622277C211yesmaximal, characteristic
3C2×#10622280C211yesmaximal, characteristic
4#13066886C211yesmaximal, characteristic, complemented
5#13066886C211yesmaximal, characteristic, complemented
6#13066884C211yesmaximal, characteristic, complemented
7#13066868C214yesmaximal, complemented
8#13066864C214yesmaximal
9C11×#1445C211yesmaximal, characteristic, complemented
10C23×C4×D11C2211yescharacteristic
11C23×Dic22C2211yescharacteristic
12C22×C11×C4⋊C4C2211yescharacteristic
13C22×(C2×D11)⋊C4C2212yes—
14C22×(C2×D11)⋊C4C2212yes—
15C22×(C2×D11)⋊C4C2212yes—
16C22×C4⋊Dic11C2212yes—
17C22×Dic11⋊C4C2212yes—
18C2×#66950—24no—
19C2×#66950—24no—
20C2×#66950C418yescomplemented
21C2×C11×#95C2211yescharacteristic
22C2×C11×#95C2211yescharacteristic
23C2×C11×#95C2214yes—
24C2×C11⋊#95C2212yes—
25C2×C11⋊#95C2211yescharacteristic
26C2×C11⋊#95C2211yescharacteristic
27C2×C11⋊#95C2214yes—
28C2×C11⋊#95C2214yes—
29C2×C11⋊#95C2214yes—
30#10622280—24no—
31#10622277—24no—
32C24×D11C2311yescharacteristic
33C23×C4×C11C2312yes—
34C23×C4×C11C2312yes—
35C23×C4×C11C2312yes—
36C23×C4×C11C2311yescharacteristic
37C23×Dic11C2312yes—
38C23×Dic11C2312yes—
39C23×Dic11C2312yes—
40C23×Dic11C2311yescharacteristic
41C22×C4×D11—44no—
42C22×C4×D11C2×C414yes—
43C22×C4×D11D414yes—
44C22×C4×D11D414yes—
45C22×Dic22—22no—
46C22×Dic22—22no—
47C22×Dic22—44no—
48C22×Dic22C2×C414yes—
49C2×C11×C4⋊C4—22no—
50C2×C11×C4⋊C4—22no—
51C2×C11×C4⋊C4C2×C414yes—
52C2×(C2×D11)⋊C4—24no—
53C2×(C2×D11)⋊C4—24no—
54C2×(C2×D11)⋊C4—24no—
55C2×(C2×D11)⋊C4—24no—
56C2×(C2×D11)⋊C4—44no—
57C2×(C2×D11)⋊C4—44no—
58C2×(C2×D11)⋊C4C2×C418yes—
59C2×C4⋊Dic11—24no—
60C2×C4⋊Dic11—24no—
61C2×C4⋊Dic11C2×C418yes—
62C2×Dic11⋊C4—24no—
63C2×Dic11⋊C4—44no—
64#66950—232no—
65C11×#95—22no—
66C11×#95—22no—
67C11×#95—44no—
68C11⋊#95—22no—
69C11⋊#95—22no—
70C11⋊#95—24no—
71C11⋊#95—44no—
72C11⋊#95—44no—
73C11⋊#95—44no—
74#14660—111nomaximal, Sylow
75C24×C11C2411yescharacteristic
76C23×D11—22no—
77C23×D11—24no—
78C23×D11—22no—
79C23×D11—22no—
80C23×D11—22no—
81C23×D11C22×C412yes—
82C23×D11C2×D412yes—
83C23×D11C2×D412yes—
84C22×C4×C11—24no—
85C22×C4×C11—22no—
86C22×C4×C11—22no—
87C22×C4×C11—24no—
88C22×C4×C11—24no—
89C22×C4×C11—22no—
90C22×C4×C11—22no—
91C22×C4×C11—24no—
92C22×C4×C11—24no—
93C22×C4×C11—22no—
94C22×C4×C11—24no—
95C22×C4×C11—22no—
96C22×C4×C11—22no—
97C22×C4×C11—24no—
98C22×C4×C11—22no—
99C22×C4×C11C22×C414yes—
100C22×C4×C11C22×C412yes—
101C22×C4×C11C2×D412yes—
102C22×C4×C11C2×D412yes—
103C22×Dic11—22no—
104C22×Dic11—22no—
105C22×Dic11—22no—
106C22×Dic11—22no—
107C22×Dic11—24no—
108C22×Dic11—24no—
109C22×Dic11—24no—
110C22×Dic11—22no—
111C22×Dic11—24no—
112C22×Dic11—24no—
113C22×Dic11—24no—
114C22×Dic11—22no—
115C22×Dic11—24no—
116C22×Dic11—22no—
117C22×Dic11—22no—
118C22×Dic11C22×C414yes—
119C22×Dic11C22×C412yes—
120C22×Dic11C2×D412yes—
121C22×Dic11C2×D412yes—
122C2×C4×D11—48no—
123C2×C4×D11—48no—
124C2×C4×D11—48no—
125C2×C4×D11C22⋊C4116yes—
126C2×Dic22—28no—
127C2×Dic22—48no—
128C2×Dic22—48no—
129C2×Dic22—48no—
130C11×C4⋊C4—28no—
131(C2×D11)⋊C4—216no—
132(C2×D11)⋊C4—48no—
133(C2×D11)⋊C4—48no—
134C4⋊Dic11—216no—
135Dic11⋊C4—48no—
136C22×C22⋊Q8—111no—
137C2×#381—111no—
138C2×#380—111no—
139#1446—111no—
140#1446—111no—
141#1445—111no—
142#1445D1111yescharacteristic, complemented
143#1441—114no—
144#1440—114no—
145C23×C11C23×C411yescharacteristic
146C23×C11C22×D411yescharacteristic
147C23×C11C22×D411yescharacteristic
148C23×C11H(5,𝔽2)12yes—
149C23×C11H(5,𝔽2)14yes—
150C23×C112−1+412yes—
151C23×C112−1+412yes—
152C23×C112−1+412yes—
153C22×D11—24no—
154C22×D11—24no—
155C22×D11—24no—
156C22×D11—24no—
157C22×D11—24no—
158C22×D11—22no—
159C22×D11—22no—
160C22×D11—24no—
161C22×D11—22no—
162C22×D11—22no—
163C22×D11—24no—
164C22×D11—44no—
165C22×D11—44no—
166C22×D11—44no—
167C22×D11—44no—
168C22×D11C2×C22⋊C414yes—
169C2×C4×C11—24no—
170C2×C4×C11—24no—
171C2×C4×C11—24no—
172C2×C4×C11—24no—
173C2×C4×C11—28no—
174C2×C4×C11—42no—
175C2×C4×C11—42no—
176C2×C4×C11—44no—
177C2×C4×C11—42no—
178C2×C4×C11—42no—
179C2×C4×C11—42no—
180C2×C4×C11—42no—
181C2×C4×C11—42no—
182C2×C4×C11—42no—
183C2×C4×C11—44no—
184C2×C4×C11—44no—
185C2×C4×C11—44no—
186C2×C4×C11—44no—
187C2×C4×C11C2×C22⋊C414yes—
188C2×Dic11—24no—
189C2×Dic11—24no—
190C2×Dic11—24no—
191C2×Dic11—24no—
192C2×Dic11—24no—
193C2×Dic11—24no—
194C2×Dic11—44no—
195C2×Dic11—44no—
196C2×Dic11—42no—
197C2×Dic11—42no—
198C2×Dic11—42no—
199C2×Dic11—42no—
200C2×Dic11—42no—
201C2×Dic11—42no—
202C2×Dic11—44no—
203C2×Dic11—44no—
204C2×Dic11—44no—
205C2×Dic11—44no—
206C2×Dic11C2×C22⋊C414yes—
207C4×D11—416no—
208Dic22—416no—
209C23×Q8—111no—
210C24×C4—111no—
211C2×C22⋊Q8—118no—
212C2×C22⋊Q8—224no—
213C2×C22⋊Q8—224no—
214C22×C4⋊C4—112no—
215C22×C4⋊C4—112no—
216C22×C4⋊C4C2×D1111yescharacteristic
217C22×C22⋊C4—112no—
218C22×C22⋊C4—112no—
219C22×C22⋊C4—112no—
220#381—224no—
221#380—224no—
222C2×#95—112no—
223C2×#95—111no—
224C2×#95—111no—
225C2×#95—114no—
226C2×#95—114no—
227C2×#95—114no—
228C2×#95C2×D1111yescharacteristic
229C2×#95C2×D1111yescharacteristic
230C2×#95C2×D1114yes—
231C22×C11C22×C22⋊C411yescharacteristic
232C22×C11#51812yes—
233C22×C11#51912yes—
234C22×C11#52012yes—
235C22×C11#53412yes—
236C22×C11#53412yes—
237C22×C11#53511yescharacteristic
238C22×C11#53611yescharacteristic
239C22×C11#53612yes—
240C22×C11#53611yescharacteristic
241C22×C11#53611yescharacteristic
242C22×C11#53712yes—
243C22×C11#56214yes—
244C22×C11#56214yes—
245C22×C11#56314yes—
246C22×C11#56314yes—
247C2×D11—24no—
248C2×D11—24no—
249C2×D11—24no—
250C2×D11—44no—
251C2×D11—44no—
252C2×D11—48no—
253C2×D11—44no—
254C2×D11—44no—
255C4×C11—24no—
256C4×C11—44no—
257C4×C11—82no—
258C4×C11—82no—
259Dic11—24no—
260Dic11—44no—
261Dic11—82no—
262Dic11—82no—
263C25—111no—
264C22×Q8—114no—
265C22×Q8—222no—
266C22×Q8—222no—
267C22×Q8—444no—
268C23×C4—111no—
269C23×C4—114no—
270C23×C4—114no—
271C23×C4—114no—
272C23×C4—112no—
273C23×C4—112no—
274C23×C4—112no—
275C23×C4—444no—
276C23×C4C22×D1112yes—
277C23×C4C22×D1112yes—
278C23×C4C22×D1112yes—
279C23×C4C22×D1111yescharacteristic
280C22⋊Q8—2232no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—118no—
284C2×C4⋊C4—224no—
285C2×C4⋊C4—224no—
286C2×C4⋊C4—224no—
287C2×C4⋊C4—444no—
288C2×C4⋊C4C4×D1114yes—
289C2×C22⋊C4—118no—
290C2×C22⋊C4—224no—
291C2×C22⋊C4—224no—
292C2×C22⋊C4—224no—
293C2×C22⋊C4—224no—
294C2×C22⋊C4—444no—
295C2×C22⋊C4—444no—
296#95—22no—
297#95—22no—
298#95—44no—
299#95—224no—
300#95—222no—
301#95—222no—
302#95—444no—
303#95—444no—
304#95—444no—
305C2×C11#230811yescharacteristic
306C2×C11#231011yescharacteristic
307C2×C11#231111yescharacteristic
308C2×C11#237814yes—
309C2×C11#281412yes—
310C2×C11#281412yes—
311C2×C11#281612yes—
312C2×C11#281712yes—
313D11—44no—
314C24—112no—
315C24—112no—
316C24—112no—
317C24—222no—
318C24—222no—
319C24—224no—
320C24—222no—
321C24—222no—
322C24C23×D1111yescharacteristic
323C2×Q8—228no—
324C2×Q8—448no—
325C2×Q8—448no—
326C2×Q8—448no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—112no—
343C22×C4—112no—
344C22×C4—112no—
345C22×C4—1116no—
346C22×C4—114no—
347C22×C4—224no—
348C22×C4—222no—
349C22×C4—222no—
350C22×C4—224no—
351C22×C4—224no—
352C22×C4—222no—
353C22×C4—222no—
354C22×C4—222no—
355C22×C4—224no—
356C22×C4—224no—
357C22×C4—222no—
358C22×C4—224no—
359C22×C4—224no—
360C22×C4—222no—
361C22×C4—222no—
362C22×C4—448no—
363C22×C4—448no—
364C22×C4—448no—
365C22×C4C2×C4×D1114yes—
366C22×C4C2×C4×D1112yes—
367C22×C4D4×D1112yes—
368C22×C4D4×D1112yes—
369C4⋊C4—28no—
370C4⋊C4—2216no—
371C4⋊C4—448no—
372C22⋊C4—2216no—
373C22⋊C4—448no—
374C22⋊C4—448no—
375C11#1466011yescharacteristic, complemented, Sylow
376C23—114no—
377C23—222no—
378C23—222no—
379C23—224no—
380C23—224no—
381C23—224no—
382C23—224no—
383C23—224no—
384C23—224no—
385C23—224no—
386C23—222no—
387C23—222no—
388C23—444no—
389C23—444no—
390C23—444no—
391C23—444no—
392C23C22×C4×D1111yescharacteristic
393C23C2×D4×D1111yescharacteristic
394C23C2×D4×D1111yescharacteristic
395C23#6703712yes—
396C23#6704012yes—
397C23#6704214yes—
398C23#6704312yes—
399C23#6704312yes—
400Q8—4416no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—28no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—114no—
420C2×C4—224no—
421C2×C4—224no—
422C2×C4—224no—
423C2×C4—224no—
424C2×C4—224no—
425C2×C4—224no—
426C2×C4—4416no—
427C2×C4—444no—
428C2×C4—444no—
429C2×C4—442no—
430C2×C4—442no—
431C2×C4—442no—
432C2×C4—442no—
433C2×C4—444no—
434C2×C4—444no—
435C2×C4—442no—
436C2×C4—442no—
437C2×C4—444no—
438C2×C4—444no—
439C2×C4C22⋊C4×D1114yes—
440C22—224no—
441C22—224no—
442C22—224no—
443C22—444no—
444C22—444no—
445C22—448no—
446C22—444no—
447C22—444no—
448C22C2×C22⋊C4×D1111yescharacteristic
449C22#1062305212yes—
450C22#1062307712yes—
451C22#1062310312yes—
452C22#1062313912yes—
453C22#1062315912yes—
454C22#1062317814yes—
455C22#1062318412yes—
456C22#1062318412yes—
457C22#1062319814yes—
458C22#1062320611yescharacteristic
459C22#1062320611yescharacteristic
460C22#1062323014yes—
461C22#1062324314yes—
462C22#1062324611yescharacteristic
463C22#1062325511yescharacteristic
464C4—24no—
465C4—44no—
466C4—82no—
467C4—82no—
468C4—224no—
469C4—444no—
470C4—882no—
471C4—882no—
472C2—444no—
473C2#1307119311yescharacteristic
474C2#1307119911yescharacteristic
475C2#1307120611yescharacteristic
476C2#1307178014yes—
477C2#1307663012yes—
478C2#1307663212yes—
479C2#1307664512yes—
480C2#1307665212yes—
subgroupG/H
center, FrattiniC24C23×D11
commutatorC23×C11C23×C4
FittingC11×#1445C2
socle, cosocleC24×C11C24
prime radicalC24×D11C23
hypercenter#1445D11
hypocenterC11#14660
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ #1445

Complements

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

NH ≅ G/Ntype
4. #13066886C2semidirect
5. #13066886C2semidirect
6. #13066884C2semidirect
7. #13066868C2semidirect
9. C11×#1445C2semidirect
20. C2×#66950C4semidirect
142. #1445D11semidirect
375. C11#14660semidirect

Maximal subgroups

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

subgroupindex
C22×#669502
C2×#106222772
C2×#106222802
#130668862
#130668842
#130668682
#130668642
C11×#14452
#1466011