Finite Group Databasebeta

← #14763521

Subgroups — #14763521

Order 2816.

1118 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622539C211yesmaximal, characteristic, complemented
2#13071023C211yesmaximal, characteristic, complemented
3#13069586C212yesmaximal, complemented
4#13069433C212yesmaximal, complemented
5#13068939C212yesmaximal
6#13068929C212yesmaximal, complemented
7#13068631C212yesmaximal, complemented
8#13068626C212yesmaximal, complemented
9C2×C11×#529C211yesmaximal, characteristic
10C11×#529—24no—
11C11×#529C418yescomplemented
12C2×C11×#126—21no—
13C2×C11×#125C2211yescharacteristic
14C2×C11×#124C2211yescharacteristic
15C2×C11×#123—21no—
16C2×C11×C22⋊Q8C2211yescharacteristic
17C2×C11×#121C2211yescharacteristic
18C2×C4×Q8×C11C2211yescharacteristic
19C2×C4×D4×C11—21no—
20C2×C4×D4×C11C2211yescharacteristic
21C2×C11×#117—21no—
22C22×C42×C11C2211yescharacteristic
23C2×#66979C2212yes—
24C2×#66899C2214yes—
25C2×#66896C2214yes—
26#10622825—21no—
27#10622825—21no—
28C2×C22⋊(C11⋊C8)C2212yes—
29#10622549C2212yes—
30#10622548C2212yes—
31C4×C4⋊Dic11—21no—
32C4×C4⋊Dic11—21no—
33#10622539—24no—
34C2×C4⋊(C11⋊C8)C2212yes—
35C2×#66867C2212yes—
36#10622167C2214yes—
37#10622167C2214yes—
38C22×Q8×C11C2311yescharacteristic
39C22×D4×C11—21no—
40C22×D4×C11C2311yescharacteristic
41C23×C4×C11—21no—
42C23×C4×C11D411yescharacteristic
43C23×C4×C11D411yescharacteristic
44C23×C4×C11C2311yescharacteristic
45C11×#126—28no—
46C11×#125—22no—
47C11×#125C2×C414yes—
48C11×#124—22no—
49C11×#124C2×C414yes—
50C11×#123—28no—
51C11×C22⋊Q8—22no—
52C11×C22⋊Q8C2×C414yes—
53C11×#121—22no—
54C11×#121C2×C414yes—
55C4×Q8×C11—22no—
56C4×Q8×C11C2×C414yes—
57C4×D4×C11—28no—
58C4×D4×C11—22no—
59C4×D4×C11C2×C414yes—
60C11×#117—28no—
61C2×C11×C4⋊C4—21no—
62C2×C11×C4⋊C4—21no—
63C2×C11×C4⋊C4—21no—
64C2×C11×C4⋊C4—21no—
65C2×C11×C4⋊C4C2311yescharacteristic
66C2×C11×C4⋊C4C2311yescharacteristic
67C2×C11×C22⋊C4—21no—
68C2×C11×C22⋊C4—21no—
69C2×C11×C22⋊C4—21no—
70C2×C11×C22⋊C4—21no—
71C2×C11×C22⋊C4—21no—
72C2×C42×C11—21no—
73C2×C42×C11C2×C414yes—
74C2×C42×C11D411yescharacteristic
75C2×C42×C11D411yescharacteristic
76C2×C42×C11D414yes—
77C2×C42×C11C2311yescharacteristic
78C2×C42×C11C2311yescharacteristic
79C2×C22⋊Dic11—22no—
80#66979—44no—
81C2×C4⋊Dic11—22no—
82C2×C4⋊Dic11—22no—
83C2×C4⋊Dic11C2314yes—
84C2×C4×Dic11—22no—
85C22×C11⋊C8C2314yes—
86#66899—44no—
87#66896—44no—
88C11⋊#95—22no—
89C11⋊#95—22no—
90C22⋊(C11⋊C8)—24no—
91C4⋊(C11⋊C8)—24no—
92#66867—24no—
93#21868—111nomaximal, Sylow
94C24×C11—21no—
95C24×C11C2×D411yescharacteristic
96C2×Q8×C11—21no—
97C2×Q8×C11—22no—
98C2×Q8×C11—22no—
99C2×Q8×C11C22×C412yes—
100C2×D4×C11—24no—
101C2×D4×C11—21no—
102C2×D4×C11—22no—
103C2×D4×C11—22no—
104C2×D4×C11—44no—
105C2×D4×C11C22×C412yes—
106C22×C4×C11—22no—
107C22×C4×C11—22no—
108C22×C4×C11—21no—
109C22×C4×C11—21no—
110C22×C4×C11—21no—
111C22×C4×C11—24no—
112C22×C4×C11—42no—
113C22×C4×C11—42no—
114C22×C4×C11—44no—
115C22×C4×C11C22⋊C414yes—
116C22×C4×C11C22⋊C414yes—
117C22×C4×C11C22×C412yes—
118C22×C4×C11C2×D411yescharacteristic
119C22×C4×C11C2×D411yescharacteristic
120C22×C4×C11C2×D411yescharacteristic
121C22×C4×C11C2×D411yescharacteristic
122C22×C4×C11C2×D411yescharacteristic
123C22×C4×C11C2×D412yes—
124C22×C4×C11C2411yescharacteristic
125C22×Dic11—24no—
126C11×C4⋊C4—24no—
127C11×C4⋊C4—24no—
128C11×C4⋊C4—24no—
129C11×C4⋊C4—24no—
130C11×C4⋊C4—21no—
131C11×C4⋊C4—21no—
132C11×C4⋊C4C22×C412yes—
133C11×C4⋊C4C22×C412yes—
134C11×C22⋊C4—24no—
135C11×C22⋊C4—24no—
136C11×C22⋊C4—24no—
137C11×C22⋊C4—24no—
138C11×C22⋊C4—24no—
139C42×C11—24no—
140C42×C11C22⋊C414yes—
141C42×C11C22⋊C414yes—
142C42×C11C22×C412yes—
143C42×C11C22×C412yes—
144C42×C11C2×D412yes—
145C42×C11C2×D412yes—
146C22⋊Dic11—44no—
147C4⋊Dic11—24no—
148C4⋊Dic11—24no—
149C4⋊Dic11—44no—
150C4⋊Dic11—44no—
151C4×Dic11—44no—
152C2×C11⋊C8—24no—
153C2×C11⋊C8—24no—
154C2×C11⋊C8—24no—
155C2×#529D1111yescharacteristic
156C2×#432—111no—
157#2285—111no—
158#1992—112no—
159#1959—112no—
160#1875—112no—
161#1873—112no—
162#1805—112no—
163#1804—112no—
164C23×C11—21no—
165C23×C11—21no—
166C23×C11—22no—
167C23×C11—22no—
168C23×C11—22no—
169C23×C11—41no—
170C23×C11—41no—
171C23×C11—42no—
172C23×C11—42no—
173C23×C11C2×C22⋊C412yes—
174C23×C11C22×D411yescharacteristic
175Q8×C11—24no—
176Q8×C11—42no—
177D4×C11—24no—
178D4×C11—48no—
179D4×C11—42no—
180C2×C4×C11—21no—
181C2×C4×C11—21no—
182C2×C4×C11—22no—
183C2×C4×C11—22no—
184C2×C4×C11—22no—
185C2×C4×C11—22no—
186C2×C4×C11—24no—
187C2×C4×C11—24no—
188C2×C4×C11—22no—
189C2×C4×C11—22no—
190C2×C4×C11—22no—
191C2×C4×C11—44no—
192C2×C4×C11—44no—
193C2×C4×C11—48no—
194C2×C4×C11—41no—
195C2×C4×C11—41no—
196C2×C4×C11—42no—
197C2×C4×C11—42no—
198C2×C4×C11C2×C22⋊C412yes—
199C2×C4×C11C2×C22⋊C412yes—
200C2×C4×C11C2×C22⋊C412yes—
201C2×C4×C11C2×C22⋊C412yes—
202C2×C4×C11C2×C22⋊C412yes—
203C2×C4×C11C23×C411yescharacteristic
204C2×C4×C11C22×D411yescharacteristic
205C2×C4×C11H(5,𝔽2)12yes—
206C2×C4×C112−1+412yes—
207C2×Dic11—44no—
208C2×Dic11—44no—
209C2×Dic11—44no—
210C11⋊C8—44no—
211#529—24no—
212#529Dic1118yescomplemented
213C2×#126—21no—
214C2×#125C2×D1111yescharacteristic
215C2×#124C2×D1111yescharacteristic
216C2×#123—21no—
217C2×C22⋊Q8C2×D1111yescharacteristic
218C2×#121C2×D1111yescharacteristic
219C2×C4×Q8C2×D1111yescharacteristic
220C2×C4×D4—21no—
221C2×C4×D4C2×D1111yescharacteristic
222C2×#117—21no—
223C2×#117—112no—
224C22×C42C2×D1111yescharacteristic
225#432—224no—
226C2×C4⋊C8—112no—
227C2×#103—114no—
228C2×#102—114no—
229C2×C22⋊C8—112no—
230C2×#97—112no—
231#384—112no—
232#382—112no—
233#380—221no—
234#380—221no—
235C4×C4⋊C4—221no—
236C4×C4⋊C4—221no—
237#340—114no—
238#340—114no—
239C22×C11—22no—
240C22×C11—22no—
241C22×C11—22no—
242C22×C11—42no—
243C22×C11—42no—
244C22×C11—42no—
245C22×C11—42no—
246C22×C11—42no—
247C22×C11—82no—
248C22×C11—82no—
249C22×C11C22×C22⋊C411yescharacteristic
250C22×C11#53511yescharacteristic
251C22×C11#53611yescharacteristic
252C22×C11#57512yes—
253C22×C11#57512yes—
254C4×C11—22no—
255C4×C11—22no—
256C4×C11—22no—
257C4×C11—22no—
258C4×C11—22no—
259C4×C11—42no—
260C4×C11—42no—
261C4×C11—42no—
262C4×C11#52012yes—
263C4×C11#53712yes—
264Dic11—84no—
265C22×Q8C22×D1111yescharacteristic
266C22×D4—21no—
267C22×D4C22×D1111yescharacteristic
268C23×C4—21no—
269C23×C4C22⋊D1111yescharacteristic
270C23×C4C22⋊D1111yescharacteristic
271C23×C4C22×D1111yescharacteristic
272C22×C8—114no—
273#126—28no—
274#125—22no—
275#125C2×Dic1114yes—
276#124—22no—
277#124C2×Dic1114yes—
278#123—28no—
279C22⋊Q8—22no—
280C22⋊Q8C2×Dic1114yes—
281#121—22no—
282#121C2×Dic1114yes—
283C4×Q8—22no—
284C4×Q8C2×Dic1114yes—
285C4×D4—28no—
286C4×D4—22no—
287C4×D4C2×Dic1114yes—
288#117—28no—
289#117—444no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—114no—
295C2×C4⋊C4—222no—
296C2×C4⋊C4—222no—
297C2×C4⋊C4C22×D1111yescharacteristic
298C2×C4⋊C4C22×D1111yescharacteristic
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—222no—
305C2×C42—21no—
306C2×C42—222no—
307C2×C42C2×Dic1114yes—
308C2×C42C22⋊D1111yescharacteristic
309C2×C42C22⋊D1111yescharacteristic
310C2×C42C22⋊D1114yes—
311C2×C42C22×D1111yescharacteristic
312C2×C42C22×D1111yescharacteristic
313C4⋊C8—224no—
314#103—444no—
315#102—444no—
316C22⋊C8—224no—
317#97—224no—
318#95—222no—
319#95—222no—
320C2×C11—42no—
321C2×C11—82no—
322C2×C11#231111yescharacteristic
323C2×C11#289611yescharacteristic
324C2×C11#289611yescharacteristic
325C2×C11#321912yes—
326C2×C11#324012yes—
327C24—21no—
328C24C2×C22⋊D1111yescharacteristic
329C2×Q8—21no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8C22×Dic1112yes—
333C2×D4—24no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—21no—
337C2×D4—44no—
338C2×D4C22×Dic1112yes—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—24no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—44no—
348C22×C4—224no—
349C22×C4C22⋊Dic1114yes—
350C22×C4C22⋊Dic1114yes—
351C22×C4C22×Dic1112yes—
352C22×C4C2×C22⋊D1111yescharacteristic
353C22×C4C2×C22⋊D1111yescharacteristic
354C22×C4C2×C22⋊D1111yescharacteristic
355C22×C4C2×C22⋊D1111yescharacteristic
356C22×C4C2×C22⋊D1111yescharacteristic
357C22×C4C2×C22⋊D1112yes—
358C22×C4C23×D1111yescharacteristic
359C2×C8—224no—
360C2×C8—224no—
361C2×C8—224no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—224no—
369C4⋊C4—224no—
370C4⋊C4—444no—
371C4⋊C4—444no—
372C4⋊C4C22×Dic1112yes—
373C4⋊C4C22×Dic1112yes—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—24no—
378C22⋊C4—24no—
379C22⋊C4—444no—
380C42—24no—
381C42—444no—
382C42C22⋊Dic1114yes—
383C42C22⋊Dic1114yes—
384C42C22×Dic1112yes—
385C42C22×Dic1112yes—
386C42C2×C22⋊D1112yes—
387C42C2×C22⋊D1112yes—
388C11#2186811yescharacteristic, complemented, Sylow
389C23—22no—
390C23—22no—
391C23—21no—
392C23—21no—
393C23—22no—
394C23—42no—
395C23—42no—
396C23—41no—
397C23—41no—
398C23C2×C22⋊Dic1112yes—
399C23C22×C22⋊D1111yescharacteristic
400Q8—24no—
401Q8—42no—
402D4—24no—
403D4—48no—
404D4—42no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—48no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—444no—
424C2×C4—444no—
425C2×C4—444no—
426C2×C4C2×C22⋊Dic1112yes—
427C2×C4C2×C22⋊Dic1112yes—
428C2×C4C2×C22⋊Dic1112yes—
429C2×C4C2×C22⋊Dic1112yes—
430C2×C4C2×C22⋊Dic1112yes—
431C2×C4#6704212yes—
432C2×C4#6704312yes—
433C2×C4C23×Dic1111yescharacteristic
434C2×C4C22×C22⋊D1111yescharacteristic
435C8—444no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—82no—
445C22—82no—
446C22#1062335312yes—
447C22#1062335312yes—
448C22#1062335611yescharacteristic
449C22#1062336111yescharacteristic
450C22C22×C22⋊Dic1111yescharacteristic
451C4—22no—
452C4—22no—
453C4—22no—
454C4—22no—
455C4—22no—
456C4—42no—
457C4—42no—
458C4—42no—
459C4—884no—
460C4#1062335812yes—
461C4#1062336412yes—
462C2—42no—
463C2—82no—
464C2#1307120811yescharacteristic
465C2#1307713511yescharacteristic
466C2#1307713511yescharacteristic
467C2#1307964912yes—
468C2#1307985212yes—
subgroupG/H
centerC23C22×C22⋊D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC2×C11×#529C2
socleC23×C11C22×D4
prime radicalC2×C11×#121C22
cosocleC22×C4×C11C24
hypercenterC2×#529D11
hypocenterC11#21868
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#529

Complements

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

NH ≅ G/Ntype
1. C2×#10622539C2semidirect
2. #13071023C2semidirect
3. #13069586C2semidirect
4. #13069433C2semidirect
6. #13068929C2semidirect
7. #13068631C2semidirect
8. #13068626C2semidirect
11. C11×#529C4semidirect
212. #529Dic11semidirect
388. C11#21868semidirect

Maximal subgroups

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

subgroupindex
C2×#106225392
#130710232
#130695862
#130694332
#130689392
#130689292
#130686312
#130686262
C2×C11×#5292
#2186811